Split (1)

train · 21.9k rows

text stringlengths 270 6.81k
cells of 1 skeleton then gives a covering space X projecting homeomorphically onto cells of X. Restricting p to the e e X 1→X 1 over the 1 skeleton of X. Show: e (a) Two such covering spaces X1→X and 2→X 1 are isomorphic. X 1 e 1→X 1 and X 1 X→X is a normal covering space iﬀ e (b) e e e X 1→X 1 is normal. X2→X are isomorphic iﬀ the restrictions (c) The groups of deck transformations of the coverings e isomorphic, via the restriction map. e X→X and X 1→X 1 are e e 33. In Example 1.44 let d be the greatest common divisor of m and n, and let m′ = m/d and n′ = n/d. Show that the graph Tm,n/K consists of m′ vertices labeled a, n′ vertices labeled b, together with d edges joining each a vertex to each b vertex. Deduce that the subgroup K ⊂ Gm,n is free on dm′n′ − m′ − n′ + 1 generators. Graphs and Free Groups Section 1.A 83 Since all groups can be realized as fundamental groups of spaces, this opens the way for using topology to study algebraic properties of groups. The topics in this section and the next give some illustrations of this principle, mainly using covering space theory. We remind the reader that the Additional Topics which form the remainder of this chapter are not to be regarded as an essential part of the basic core of the book. Readers who are eager to move on to new topics should feel free to skip ahead. By deﬁnition, a graph is a 1 dimensional CW complex, in other words, a space X obtained from a discrete set X 0 by attaching a collection of 1 cells eα. Thus X is obtained from the disjoint union of X 0 with closed intervals Iα by identifying the two endpoints of each Iα with points of X 0. The points of X 0 are the vertices and the 1 cells the edges of X. Note that with this deﬁnition an edge does not include its Since X has the quotient topology from the disjoint union X 0 endpoints, so an edge is an open subset of X. The two endpoints of an edge can be the same vertex
, so the closure eα of an edge eα is homeomorphic either to I or S 1. α Iα, a subset of X is open (or closed) iﬀ it intersects the closure eα of each edge eα in an open (or closed) set in eα. One says that X has the weak topology with respect to the subspaces eα. In this topology a sequence of points in the interiors of distinct edges forms a closed ` subset, hence never converges. This is true in particular if the edges containing the sequence all have a common vertex and one tries to choose the sequence so that it gets ‘closer and closer’ to the vertex. Thus if there is a vertex that is the endpoint of inﬁnitely many edges, then the weak topology cannot be a metric topology. An exercise at the end of this section is to show the converse, that the weak topology is a metric topology if each vertex is an endpoint of only ﬁnitely many edges. A basis for the topology of X consists of the open intervals in the edges together with the path-connected neighborhoods of the vertices. A neighborhood of the latter sort about a vertex v is the union of connected open neighborhoods Uα of v in eα for all eα containing v. In particular, we see that X is locally path-connected. Hence a graph is connected iﬀ it is path-connected. If X has only ﬁnitely many vertices and edges, then X is compact, being the continuous image of the compact space X 0 α Iα. The converse is also true, and more generally, a compact subset C of a graph X can meet only ﬁnitely many vertices and ` edges of X. To see this, let the subspace D ⊂ C consist of the vertices in C together with one point in each edge that C meets. Then D is a closed subset of X since it 84 Chapter 1 The Fundamental Group meets each eα in a closed set. For the same reason, any subset of D is closed, so D has the discrete topology. But D is compact, being a closed subset of the compact space C, so D must be ﬁnite. By the deﬁnition of D this means that C can meet only ﬁnitely many vertices and edges. A subgraph of a
graph X is a subspace Y ⊂ X that is a union of vertices and edges of X, such that eα ⊂ Y implies eα ⊂ Y. The latter condition just says that Y is a closed subspace of X. A tree is a contractible graph. By a tree in a graph X we mean a subgraph that is a tree. We call a tree in X maximal if it contains all the vertices of X. This is equivalent to the more obvious meaning of maximality, as we will see below. Proposition 1A.1. Every connected graph contains a maximal tree, and in fact any tree in the graph is contained in a maximal tree. Proof: Let X be a connected graph. We will describe a construction that embeds an arbitrary subgraph X0 ⊂ X as a deformation retract of a subgraph Y ⊂ X that contains all the vertices of X. By choosing X0 to be any subtree of X, for example a single vertex, this will prove the proposition. As a preliminary step, we construct a sequence of subgraphs X0 ⊂ X1 ⊂ X2 ⊂ ···, letting Xi+1 be obtained from Xi by adjoining the closures eα of all edges eα ⊂ X −Xi i Xi is open in X since a neighborhood having at least one endpoint in Xi. The union i Xi is closed since it is a union of a point in Xi is contained in Xi+1. Furthermore, of closed edges and X has the weak topology. So X = i Xi since X is connected. S S S Now to construct Y we begin by setting Y0 = X0. Then inductively, assuming that Yi ⊂ Xi has been constructed so as to contain all the vertices of Xi, let Yi+1 be obtained from Yi by adjoining one edge connecting each vertex of Xi+1 −Xi to Yi, and let Y = i Yi. It is evident that Yi+1 deformation retracts to Yi, and we may obtain a deformation retraction of Y to Y0 = X0 by performing the deformation retraction of Yi+1 to Yi during the time interval [1/2i+1, 1/2i]. Thus a point x ∈ Yi+1 − Yi is stationary until this interval, when it moves into Yi and thereafter continues moving until it reaches Y0. The resulting homotopy ht :
Y →Y is continuous since it is ⊔⊓ continuous on the closure of each edge and Y has the weak topology. S Given a maximal tree T ⊂ X and a base vertex x0 ∈ T, then each edge eα of X − T determines a loop fα in X that goes ﬁrst from x0 to one endpoint of eα by a path in T, then across eα, then back to x0 by a path in T. Strictly speaking, we should ﬁrst orient the edge eα in order to specify which direction to cross it. Note that the homotopy class of fα is independent of the choice of the paths in T since T is simply-connected. Proposition 1A.2. For a connected graph X with maximal tree T, π1(X) is a free group with basis the classes [fα] corresponding to the edges eα of X − T. Graphs and Free Groups Section 1.A 85 In particular this implies that a maximal tree is maximal in the sense of not being contained in any larger tree, since adjoining any edge to a maximal tree produces a graph with nontrivial fundamental group. Another consequence is that a graph is a tree iﬀ it is simply-connected. Proof: The quotient map X→X/T is a homotopy equivalence by Proposition 0.17. The quotient X/T is a graph with only one vertex, hence is a wedge sum of circles, whose fundamental group we showed in Example 1.21 to be free with basis the loops given by the edges of X/T, which are the images of the loops fα in X. ⊔⊓ Here is a very useful fact about graphs: Lemma 1A.3. Every covering space of a graph is also a graph, with vertices and edges the lifts of the vertices and edges in the base graph. e X→X be the covering space. For the vertices of X 0 = p−1(X 0). Writing X as a quotient space of X 0 Proof: Let p : X we take the discrete α Iα as in the deﬁnition set of a graph and applying the path lifting property to the resulting maps Iα→X, we get a unique lift Iα→ X passing through each point in p−1(x), for x ∈ e
α. These X is the lifts deﬁne the edges of a graph structure on e X. The resulting topology on same as its original topology since both topologies have the same basic open sets, the ` e e covering projection X→X being a local homeomorphism. e e ⊔⊓ We can now apply what we have proved about graphs and their fundamental e groups to prove a basic fact of group theory: Theorem 1A.4. Every subgroup of a free group is free. Proof: Given a free group F, choose a graph X with π1(X) ≈ F, for example a wedge of circles corresponding to a basis for F. For each subgroup G of F there is by Proposition 1.36 a covering space p : since p∗ is injective by Proposition 1.31. Since the group G ≈ π1( X) is free by Proposition 1A.2. e e X→X with p∗ π1( e X) = G, hence π1( X) ≈ G X is a graph by the preceding lemma, e ⊔⊓ e The structure of trees can be elucidated by looking more closely at the constructions in the proof of Proposition 1A.1. If X is a tree and v0 is any vertex of X, then the construction of a maximal tree Y ⊂ X starting with Y0 = {v0} yields an increasing sequence of subtrees Yn ⊂ X whose union is all of X since a tree has only one maximal subtree, namely itself. We can think of the vertices in Yn − Yn−1 as being at ‘height’ n, with the edges of Yn − Yn−1 connecting these vertices to vertices of height n − 1. In this way we get a ‘height function’ h : X→R assigning to each vertex its height, and monotone on edges. 86 Chapter 1 The Fundamental Group For each vertex v of X there is exactly one edge leading downward from v, so by following these downward edges we obtain a path from v to the base vertex v0. This is an example of an edgepath, which is a composition of ﬁnitely many paths each consisting of a single edge traversed monotonically. For any edgepath joining v to v0 other than the downward edgepath, the height function
would not be monotone and hence would have local maxima, occurring when the edgepath backtracked, retracing some edge it had just crossed. Thus in a tree there is a unique nonbacktracking edgepath joining any two points. All the vertices and edges along this edgepath are distinct. A tree can contain no subgraph homeomorphic to a circle, since two vertices in such a subgraph could be joined by more than one nonbacktracking edgepath. Conversely, if a connected graph X contains no circle subgraph, then it must be a tree. For if T is a maximal tree in X that is not equal to X, then the union of an edge of X − T with the nonbacktracking edgepath in T joining the endpoints of this edge is a circle subgraph of X. So if there are no circle subgraphs of X, we must have X = T, a tree. For an arbitrary connected graph X and a pair of vertices v0 and v1 in X there is a unique nonbacktracking edgepath in each homotopy class of paths from v0 to v1. X, which is a tree since it is simplyThis can be seen by lifting to the universal cover v0 of v0, a homotopy class of paths from v0 to v1 lifts to v1 of v1. Then v1 projects to the desired e v0 and ending at a unique lift v0 to the unique nonbacktracking edgepath in a homotopy class of paths starting at connected. Choosing a lift X from e e e nonbacktracking edgepath in X. e e e Exercises 1. Let X be a graph in which each vertex is an endpoint of only ﬁnitely many edges. Show that the weak topology on X is a metric topology. 2. Show that a connected graph retracts onto any connected subgraph. 3. For a ﬁnite graph X deﬁne the Euler characteristic χ (X) to be the number of vertices minus the number of edges. Show that χ (X) = 1 if X is a tree, and that the rank (number of elements in a basis) of π1(X) is 1 − χ (X) if X is connected. 4. If X is a ﬁnite graph and Y is a subgraph homeomorphic to S 1 and containing the basepoint x0
, show that π1(X, x0) has a basis in which one element is represented by the loop Y. 5. Construct a connected graph X and maps f, g : X→X such that f g = 11 but f and g do not induce isomorphisms on π1. [Note that f∗g∗ = 11 implies that f∗ is surjective and g∗ is injective.] 6. Let F be the free group on two generators and let F ′ be its commutator subgroup. Find a set of free generators for F ′ by considering the covering space of the graph S 1 ∨ S 1 corresponding to F ′. K(G,1) Spaces and Graphs of Groups Section 1.B 87 7. If F is a ﬁnitely generated free group and N is a nontrivial normal subgroup of inﬁnite index, show, using covering spaces, that N is not ﬁnitely generated. 8. Show that a ﬁnitely generated group has only a ﬁnite number of subgroups of a given ﬁnite index. [First do the case of free groups, using covering spaces of graphs. The general case then follows since every group is a quotient group of a free group.] 9. Using covering spaces, show that an index n subgroup H of a group G has at most n conjugate subgroups gHg−1 in G. Apply this to show that there exists a normal subgroup K ⊂ G of ﬁnite index with K ⊂ H. [For the latter statement, consider the intersection of all the conjugate subgroups gHg−1. This is the maximal normal subgroup of G contained in H.] 10. Let X be the wedge sum of n circles, with its natural graph structure, and let X→X be a covering space with Y ⊂ X a ﬁnite connected subgraph. Show there is a ﬁnite graph Z ⊃ Y having the same vertices as Y, such that the projection Y →X e extends to a covering space Z→X. 11. Apply the two preceding problems to show that if F is a ﬁnitely generated free e group and x ∈ F is not the identity element, then there is a normal subgroup H ⊂ F of
ﬁnite index such that x ∉ H. Hence x has nontrivial image in a ﬁnite quotient group of F. In this situation one says F is residually ﬁnite. 12. Let F be a ﬁnitely generated free group, H ⊂ F a ﬁnitely generated subgroup, and x ∈ F − H. Show there is a subgroup K of ﬁnite index in F such that K ⊃ H and x ∉ K. [Apply Exercise 10.] 13. Let x be a nontrivial element of a ﬁnitely generated free group F. Show there is a ﬁnite-index subgroup H ⊂ F in which x is one element of a basis. [Exercises 4 and 10 may be helpful.] 14. Show that the existence of maximal trees is equivalent to the Axiom of Choice. In this section we introduce a class of spaces whose homotopy type depends only on their fundamental group. These spaces arise many places in topology, especially in its interactions with group theory. A path-connected space whose fundamental group is isomorphic to a given group G and which has a contractible universal covering space is called a K ( G, 1) space. The ‘1’ here refers to π1. More general K(G, n) spaces are studied in §4.2. All these spaces are called Eilenberg–MacLane spaces, though in the case n = 1 they were studied by 88 Chapter 1 The Fundamental Group Hurewicz before Eilenberg and MacLane took up the general case. Here are some examples: Example 1B.1. S 1 is a K(Z, 1). More generally, a connected graph is a K(G, 1) with G a free group, since by the results of §1.A its universal cover is a tree, hence con- tractible. Example 1B.2. Closed surfaces with inﬁnite π1, in other words, closed surfaces other than S 2 and RP2, are K(G, 1) ’s. This will be shown in Example 1B.14 below. It also follows from the theorem in surface theory that the only simply-connected surfaces without boundary are S 2 and R2, so the universal cover of a closed surface with inﬁn
ite fundamental group must be R2 since it is noncompact. Nonclosed surfaces deformation retract onto graphs, so such surfaces are K(G, 1) ’s with G free. Example 1B.3. The inﬁnite-dimensional projective space RP∞ is a K(Z2, 1) since its universal cover is S ∞, which is contractible. To show the latter fact, a homotopy from the identity map of S ∞ to a constant map can be constructed in two stages as follows. First, deﬁne ft : R∞→R∞ by ft(x1, x2, ···) = (1 − t)(x1, x2, ···) + t(0, x1, x2, ···). This takes nonzero vectors to nonzero vectors for all t ∈ [0, 1], so ft/\|ft\| gives a homotopy from the identity map of S ∞ to the map (x1, x2, ···)֏(0, x1, x2, ···). Then a homotopy from this map to a constant map is given by gt/\|gt\| where gt(x1, x2, ···) = (1 − t)(0, x1, x2, ···) + t(1, 0, 0, ···). Example 1B.4. Generalizing the preceding example, we can construct a K(Zm, 1) as an inﬁnite-dimensional lens space S ∞/Zm, where Zm acts on S ∞, regarded as the unit sphere in C∞, by scalar multiplication by m th roots of unity, a generator of this action being the map (z1, z2, ···) ֏ e2π i/m(z1, z2, ···). It is not hard to check that this is a covering space action. Example 1B.5. A product K(G, 1)× K(H, 1) is a K(G× H, 1) since its universal cover is the product of the universal covers of K(G, 1) and K(H, 1). By taking products of circles and inﬁnite-dimensional lens spaces we therefore get K(G, 1) �
�s for arbitrary ﬁnitely generated abelian groups G. For example the n dimensional torus T n, the product of n circles, is a K(Zn, 1). Example 1B.6. For a closed connected subspace K of S 3 that is nonempty, the complement S 3 −K is a K(G, 1). This is a theorem in 3 manifold theory, but in the special case that K is a torus knot the result follows from our study of torus knot complements in Examples 1.24 and 1.35. Namely, we showed that for K the torus knot Km,n there is a deformation retraction of S 3 − K onto a certain 2 dimensional complex Xm,n having contractible universal cover. The homotopy lifting property then implies that the universal cover of S 3 − K is homotopy equivalent to the universal cover of Xm,n, hence is also contractible. K(G,1) Spaces and Graphs of Groups Section 1.B 89 ∆ complex deﬁned in §2.1. Let EG be the It is not hard to construct a K(G, 1) for an arbitrary group G, us- Example 1B.7. complex whose ing the notion of a n simplices are the ordered (n + 1) tuples [g0, ···, gn] of elements of G. Such an gi, ···, gn] in the obvious way, n simplex attaches to the (n − 1) simplices [g0, ···, gi means that this just as a standard simplex attaches to its faces. vertex is deleted.) The complex EG is contractible by the homotopy ht that slides each point x ∈ [g0, ···, gn] along the line segment in [e, g0, ···, gn] from x to the vertex [e], where e is the identity element of G. This is well-deﬁned in EG since when we restrict to a face [g0, ···, gi, ···, gn] we have the linear deformation to [e] gi, ···, gn]. Note that ht carries [e] around the loop [e, e], so ht is not in [e, g0, ···, actually a deformation retraction of EG onto
[e]. (The notation ∆ b b b The group G acts on EG by left multiplication, an element g ∈ G taking the simplex [g0, ···, gn] linearly onto the simplex [gg0, ···, ggn]. Only the identity e takes any simplex to itself, so by an exercise at the end of this section, the action of G on EG is a covering space action. Hence the quotient map EG→EG/G is the universal cover of the orbit space BG = EG/G, and BG is a K(G, 1). b Since G acts on EG by freely permuting simplices, BG inherits a complex ∆ structure from EG. The action of G on EG identiﬁes all the vertices of EG, so BG has just one vertex. To describe the complex structure on BG explicitly, note ﬁrst that every n simplex of EG can be written uniquely in the form ∆ [g0, g0g1, g0g1g2, ···, g0g1 ··· gn] = g0[e, g1, g1g2, ···, g1 ··· gn] The image of this simplex in BG may be denoted unambiguously by the symbol [g1\|g2\| ··· \|gn]. In this ‘bar’ notation the gi ’s and their ordered products can be used to label edges, viewing an edge label as the ratio between the two labels on the vertices at the endpoints of the edge, as indicated in the ﬁgure. With this notation, the boundary of a simplex [g1\| ··· \|gn] of BG consists of the simplices [g2\| ··· \|gn], [g1\| ··· \|gn−1], and [g1\| ··· \|gigi+1\| ··· \|gn] for i = 1, ···, n − 1. This construction of a K(G, 1) produces a rather large space, since BG is al- ways inﬁnite-dimensional, and if G is inﬁnite, BG has an inﬁnite number of cells in each positive dimension. For example, BZ is much bigger than S 1, the most eﬃcient K(Z, 1). On
the other hand, BG has the virtue of being functorial: A homomorphism f : G→H induces a map Bf : BG→BH sending a simplex [g1\| ··· \|gn] to the simplex [f (g1)\| ··· \|f (gn)]. A diﬀerent construction of a K(G, 1) is given in §4.2. Here one starts with any 2 dimensional complex having fundamental group G, for example 90 Chapter 1 The Fundamental Group the complex XG associated to a presentation of G, and then one attaches cells of dimension 3 and higher to make the universal cover contractible without aﬀecting π1. In general, it is hard to get any control on the number of higher-dimensional cells needed in this construction, so it too can be rather ineﬃcient. Indeed, ﬁnding an eﬃcient K(G, 1) for a given group G is often a diﬃcult problem. It is a curious and almost paradoxical fact that if G contains any elements of ﬁnite order, then every K(G, 1) CW complex must be inﬁnite-dimensional. This is shown In particular the inﬁnite-dimensional lens space K(Zm, 1) ’s in in Proposition 2.45. Example 1B.4 cannot be replaced by any ﬁnite-dimensional complex. In spite of the great latitude possible in the construction of K(G, 1) ’s, there is a very nice homotopical uniqueness property that accounts for much of the interest in K(G, 1) ’s: Theorem 1B.8. The homotopy type of a CW complex K(G, 1) is uniquely determined by G. Having a unique homotopy type of K(G, 1) ’s associated to each group G means that algebraic invariants of spaces that depend only on homotopy type, such as ho- mology and cohomology groups, become invariants of groups. This has proved to be a quite fruitful idea, and has been much studied both from the algebraic and topological viewpoints. The discussion following Proposition 2.45 gives a few references. The preceding theorem will follow easily from: Proposition 1B.9. Let X be a connected CW complex and let Y be a
K(G, 1). Then every homomorphism π1(X, x0)→π1(Y, y0) is induced by a map (X, x0)→(Y, y0) that is unique up to homotopy ﬁxing x0. To deduce the theorem from this, let X and Y be CW complex K(G, 1) ’s with isomorphic fundamental groups. The proposition gives maps f : (X, x0)→(Y, y0) and g : (Y, y0)→(X, x0) inducing inverse isomorphisms π1(X, x0) ≈ π1(Y, y0). Then f g and gf induce the identity on π1 and hence are homotopic to the identity maps. Proof of 1B.9: Let us ﬁrst consider the case that X has a single 0 cell, the basepoint x0. Given a homomorphism ϕ : π1(X, x0)→π1(Y, y0), we begin the construction of a map f : (X, x0)→(Y, y0) with f∗ = ϕ by setting f (x0) = y0. Each 1 cell e1 α of X has closure a circle determining an element [e1 α] ∈ π1(X, x0), and we let f on the closure of e1 α]). If i : X 1 ֓ X denotes be a map representing ϕ([e1 the inclusion, then ϕi∗ = f∗ since π1(X 1, x0) is generated by the elements [e1 α]. α To extend f over a cell e2 β with attaching map ψβ : S 1→X 1, all we need is for the composition f ψβ to be nullhomotopic. Choosing a basepoint s0 ∈ S 1 and a path in X 1 from ψβ(s0) to x0, ψβ determines an element [ψβ] ∈ π1(X 1, x0), and the existence K(G,1) Spaces and Graphs of Groups Section 1.B 91 of a nullhomotopy of f ψβ is equivalent to f∗([�
�β]) being zero in π1(Y, y0). We have i∗([ψβ]) = 0 since the cell e2 β provides a nullhomotopy of ψβ in X. Hence f∗([ψβ]) = ϕi∗([ψβ]) = 0, and so f can be extended over e2 β. γ with n > 2 is possible since the attaching maps ψγ : S n−1→X n−1 have nullhomotopic compositions f ψγ : S n−1→Y. This is because f ψγ lifts to the universal cover of Y if n > 2, and this cover is contractible by hypothesis, so the lift of f ψγ is nullhomotopic, hence also f ψγ itself. Extending f inductively over cells en Turning to the uniqueness statement, if two maps f0, f1 : (X, x0)→(Y, y0) induce the same homomorphism on π1, then we see immediately that their restrictions to X 1 are homotopic, ﬁxing x0. To extend the resulting map X 1× I ∪ X × ∂I→Y over the remaining cells en × (0, 1) of X × I we can proceed just as in the preceding paragraph since these cells have dimension n + 1 > 2. Thus we obtain a homotopy ft : (X, x0)→(Y, y0), ﬁnishing the proof in the case that X has a single 0 cell. α] ∈ π1(X, x0), and we let f on the closure of e1 The case that X has more than one 0 cell can be treated by a small elaboration on this argument. Choose a maximal tree T ⊂ X. To construct a map f realizing a given ϕ, begin by setting f (T ) = y0. Then each edge e1 α in X − T determines an element [e1 α be a map representing ϕ([e1 α]). Extending f over higher-dimensional cells then proceeds just as before. Constructing a homotopy ft joining two given maps f0 and f1 with f0∗ = f1∗ also has an extra step. Let ht : X 1→X 1 be a homotopy starting with h0 = 11
and restricting to a deformation retraction of T onto x0. (It is easy to extend such a deformation retraction to a homotopy deﬁned on all of X 1.) We can construct a homotopy from f0\|X 1 to f1\|X 1 by ﬁrst deforming f0\|X 1 and f1\|X 1 to take T to y0 by composing with ht, then applying the earlier argument to obtain a homotopy between the modiﬁed f0\|X 1 and f1\|X 1. Having a homotopy f0\|X 1 ≃ f1\|X 1 we extend this over all of X in ⊔⊓ the same way as before. The ﬁrst part of the preceding proof also works for the 2 dimensional complexes XG associated to presentations of groups. Thus every homomorphism G→H is realized as the induced homomorphism of some map XG→XH. However, there is no uniqueness statement for this map, and it can easily happen that diﬀerent presentations of a group G give XG ’s that are not homotopy equivalent. Graphs of Groups As an illustration of how K(G, 1) spaces can be useful in group theory, we shall describe a procedure for assembling a collection of K(G, 1) ’s together into a K(G, 1) for a larger group G. Group-theoretically, this gives a method for assembling smaller groups together to form a larger group, generalizing the notion of free products. Let be a graph that is connected and oriented, that is, its edges are viewed as arrows, each edge having a speciﬁed direction. Suppose that at each vertex v of Γ we Γ 92 Chapter 1 The Fundamental Group graph of groups. Now build a space B place a group Gv and along each edge e of we put a homomorphism ϕe from the group at the tail of the edge to the group at the head of the edge. We call this data a by putting the space BGv from Example 1B.7 and then ﬁlling in a mapping cylinder of the map Bϕe along, identifying the two ends of the mapping cylinder with the two BGv ’s is then a CW complex since the maps Bϕe can be take n cells homeomorphically onto n cells
. In fact, the cell structure on B at the ends of e. The resulting space B at each vertex v of each edge e of Γ Γ Γ Γ canonically subdivided into a complex structure using the prism construction from the proof of Theorem 2.10, but we will not need to do this here. Γ ∆ More generally, instead of BGv one could take any CW complex K(Gv, 1) at the vertex v, and then along edges put mapping cylinders of maps realizing the homomorphisms ϕe. We leave it for the reader to check that the resulting space K constructed above. homotopy equivalent to the B is Γ Γ Example 1B.10. Suppose consists of one central vertex with a number of edges Γ radiating out from it, and the group Gv at this central vertex is trivial, hence also all the edge homomorphisms. Then van Kampen’s theorem implies that π1(K ) is the free product of the groups at all the outer vertices. Γ Γ In view of this example, we shall call π1(K the graph product of the vertex groups Gv with respect to the edge homomorphisms ϕe. The name for π1(K ) that is generally used in the literature is the rather awkward phrase, ‘the fundamental group of the graph of groups’. ) for a general graph of groups Γ Γ Here is the main result we shall prove about graphs of groups: Γ Theorem 1B.11. If all the edge homomorphisms ϕe are injective, then K K(G, 1) and the inclusions K(Gv, 1) ֓ K induce injective maps on π1. is a Γ Before giving the proof, let us look at some interesting special cases: Γ Example 1B.12: Free Products with Amalgamation. Suppose the graph of groups is A← C→B, with the two maps monomorphisms. One can regard this data as specifying embeddings of C as subgroups of A and B. Applying van Kampen’s theorem to the decomposition of K into its two mapping cylinders, we see that π1(K ) is the quotient of A ∗ B obtained by identifying the subgroup C ⊂ A with the subgroup C ⊂ B. The standard notation
for this group is A ∗C B, the free product of A and B amalgamated along the subgroup C. According to the theorem, A ∗C B contains both A and B as subgroups. Γ Γ For example, a free product with amalgamation Z ∗Z Z can be realized by mapping cylinders of the maps S 1← S 1→S 1 that are m sheeted and n sheeted covering spaces, respectively. We studied this case in Examples 1.24 and 1.35 where we showed that the complex K is a deformation retract of the complement of a torus knot in S 3 if m and n are relatively prime. It is a basic result in 3 manifold theory that the Γ K(G,1) Spaces and Graphs of Groups Section 1.B 93 complement of every smooth knot in S 3 can be built up by iterated graph of groups constructions with injective edge homomorphisms, starting with free groups, so the theorem implies that these knot complements are K(G, 1) ’s. Their universal covers are all R3, in fact. Example 1B.13: HNN Extensions. Consider a graph of groups with ϕ and ψ both monomorphisms. This is analogous to the previous case A← C→B, ), but with the two groups A and B coalesced to a single group. The group π1(K which was denoted A ∗C B in the previous case, is now denoted A∗C. To see what as being obtained from K(A, 1) by attaching this group looks like, let us regard K Γ K(C, 1)× I along the two ends K(C, 1)× ∂I via maps realizing the monomorphisms ϕ and ψ. Using a K(C, 1) with a single 0 cell, we see that K K(A, 1) ∨ S 1 by attaching cells of dimension two and greater, so π1(K ) is a quotient of A ∗ Z, and it is not hard to ﬁgure out that the relations deﬁning this quotient are of the form tϕ(c)t−1 = ψ(c) where t is a generator of the Z factor and c ranges over C, or a set of generators for C. We leave the veriﬁcation of this
for the Exercises. can be obtained from Γ Γ Γ Γ As a very special case, taking ϕ = ψ = 11 gives A∗A = A× Z since we can take = K(A, 1)× S 1 in this case. More generally, taking ϕ = 11 with ψ an arbitrary K automorphism of A, we realize any semidirect product of A and Z as A∗A. For example, the Klein bottle occurs this way, with ϕ realized by the identity map of S 1 and ψ by a reﬂection. In these cases when ϕ = 11 we could realize the same group π1(K a single edge, labeled ψ. ) using a slightly simpler graph of groups, with a single vertex, labeled A, and Here is another special case. Suppose we take a torus, delete a small open disk, then identify the resulting boundary circle with a longitudinal circle of the torus. This produces a space X that happens to be homeomorphic to a subspace of the standard picture of a Klein bottle in R3 ; see Exercise 12 of §1.2. The fundamental group π1(X) has the form (Z ∗ Z) ∗Z Z with the deﬁning relation tb±1t−1 = aba−1b−1 where a is a meridional loop and b is a longitudinal loop on the torus. The sign of the exponent in the term b±1 is immaterial since the two ways of glueing the boundary circle to the longitude produce homeomorphic spaces. The group π1(X) = a, b, t \|\|\|\| tbt−1aba−1b−1 abelianizes to Z× Z, but to show that π1(X) is not isomorphic to Z ∗ Z takes some work. There is a surjection π1(X)→Z ∗ Z obtained by setting b = 1. This has nontrivial kernel since b is nontrivial in π1(X) by the preceding theorem. If π1(X) were isomorphic to Z ∗ Z we would then have a surjective homomorphism Z ∗ Z→Z ∗ Z that was not an isomorphism. However, it is a theorem in group theory that a free group F is hopﬁan — every sur
so we have injective of A. To see this, note that p : X) = 0, hence maps π1( A) = 0. For example, if X is the torus S 1 × S 1 and A is the circle S 1 × {x0}, then π1( p−1(A) consists of inﬁnitely many parallel lines in R2, each a universal cover of A. Mf →Mf be the Mf is itself the mapping cylinder universal cover of the mapping cylinder Mf. Then f : p−1(A)→p−1(B) since the line segments in the mapping cylinder strucof a map ture on Mf lift to line segments in Mf deﬁning a mapping cylinder structure. Since Mf is a mapping cylinder, it deformation retracts onto p−1(B), so p−1(B) is also simply-connected, hence is the universal cover of B. If f induces an injection on π1, f then the remarks in the preceding paragraph apply, and the components of p−1(A) are universal covers of A. If we assume further that A and B are K(G, 1) ’s, then Mf and the components of p−1(A) are contractible, and we claim that Mf deformation f Mf is a homoretracts onto each component topy equivalence since both spaces are contractible, and then Corollary 0.20 implies A of p−1(A). Namely, the inclusion A֓ f f f f e e that Mf deformation retracts onto A since the pair ( Mf, extension property, as shown in Example 0.15. f Now we can describe the construction of the covering space f e e the union of an increasing sequence of spaces let Γ K1 be the universal cover of one of the mapping cylinders Mf of K K1 ⊂ e e K of K. It will be K2 ⊂ ···. For the ﬁrst stage,. By the e f A) satisﬁes the homotopy e e e Γ K(G,1) Spaces and Graphs of Groups Section 1.B 95 preceding remarks, this contains various disjoint copies of universal covers of the K1 by attaching to each of these two K(Gv, 1
) ’s at the ends of Mf. We build universal covers of K(Gv, 1) ’s a copy of the universal cover of each mapping cylinder meeting Mf at the end of Mf in question. Now repeat the process to Mg of K construct K3 by attaching universal covers of mapping cylinders at all the universal Γ covers of K(Gv, 1) ’s created in the previous step. In the same way, we construct Kn+1 from Kn for all n, and then we set K2 from K = e e e n e pieces to Note that Kn since it is formed by attaching Kn that deformation retract onto the subspaces along which they attach, K is contractible since we can deformation Kn during the time interval [1/2n+1, 1/2n], and then ﬁnish with a It follows that by our earlier remarks. e Kn+1 onto retract e e e e Kn. Kn+1 deformation retracts onto S e e contraction of K1 to a point during the time interval [1/2, 1]. K→K is clearly a covering space, so this ﬁnishes the e e The natural projection e is a K(G, 1). proof that K e Γ Γ The remaining statement that each inclusion K(Gv, 1)֓ K induces an injection on π1 can easily be deduced from the preceding constructions. For suppose a loop Γ γ : S 1→K(Gv, 1) is nullhomotopic in K. By the lifting criterion for covering spaces, γ : S 1→ there is a lift cover of K(Gv, 1), so e e nullhomotopic in K(Gv, 1). K. This has image contained in one of the copies of the universal γ is nullhomotopic in this universal cover, and hence γ is ⊔⊓ Γ e The various mapping cylinders that make up the universal cover of K are ar- ranged in a treelike pattern. The tree in question, call it T copy of a universal cover of a K(Gv, 1) in K, and two vertices are joined by an edge whenever the two universal covers of K(Gv, 1) ’s corresponding to these vertices are connected by a line segment lifting a line segment in the mapping cylinder
structure of, has one vertex for each e Γ Γ a mapping cylinder of K construction of T Corresponding to. The inductive construction of K is reﬂected in an inductive as a union of an increasing sequence of subtrees T1 ⊂ T2 ⊂ ···. consisting of a central vertex with a number e Γ K1 is a subtree T1 ⊂ T Γ e K1 to of edges radiating out from it, an ‘asterisk’ with possibly an inﬁnite number of edges. When we enlarge K2, T1 is correspondingly enlarged to a tree T2 by attaching a similar asterisk at the end of each outer vertex of T1, and each subsequent enlargee ment is handled in the same way. The action of π1(K K as deck transformations induces an action on T, permuting its vertices and edges, and the orbit space of T ) on e Γ under this action is just the original graph be a free action since the elements of a subgroup Gv ⊂ π1(K corresponding to one of the universal covers of K(Gv, 1).. The action on T Γ Γ Γ will not generally Γ ) ﬁx the vertex of T Γ There is in fact an exact correspondence between graphs of groups and groups Γ Γ e acting on trees. See [Scott & Wall 1979] for an exposition of this rather nice theory. From the viewpoint of groups acting on trees, the deﬁnition of a graph of groups is 96 Chapter 1 The Fundamental Group usually taken to be slightly more restrictive than the one we have given here, namely, one considers only oriented graphs obtained from an unoriented graph by subdividing each edge by adding a vertex at its midpoint, then orienting the two resulting edges outward, away from the new vertex. Exercises 1. Suppose a group G acts simplicially on a complex X, where ‘simplicially’ means that each element of G takes each simplex of X onto another simplex by a linear homeomorphism. If the action is free, show it is a covering space action. ∆ 2. Let X be a connected CW complex and G a group such that every homomorphism π1(X)→G is trivial. Show that every map X→K(G, 1) is
nullhomotopic. 3. Show that every graph product of trivial groups is free. 4. Use van Kampen’s theorem to compute A∗C as a quotient of A ∗ Z, as stated in the text. having one vertex, Z, and one edge, the map Z→Z 5. Consider the graph of groups that is multiplication by 2, realized by the 2 sheeted covering space S 1→S 1. Show Γ a, b \|\|\|\| bab−1a−2 that π1(K and describe the universal cover of K explicitly as a product T × R with T a tree. [The group π1(K ) has presentation ) is the ﬁrst in a family of groups called Baumslag-Solitar groups, having presentations of the form a, b \|\|\|\| bamb−1a−n. These are HNN extensions Z∗Z.] Γ Γ Γ 6. Show that for a graph of groups all of whose edge homomorphisms are injective maps Z→Z, we can choose K to have universal cover a product T × R with T a tree. Work out in detail the case that the graph of groups is the inﬁnite sequence Z 2-----→ Z 3-----→ Z 4-----→ Z -→ ··· where the map Z n-----→ Z is multiplication by n. Show ) is isomorphic to Q in this case. How would one modify this example that π1(K ) isomorphic to the subgroup of Q consisting of rational numbers with to get π1(K Γ denominator a power of 2? Γ 7. Show that every graph product of groups can be realized by a graph whose vertices are partitioned into two subsets, with every oriented edge going from a vertex in the ﬁrst subset to a vertex in the second subset. 8. Show that a ﬁnite graph product of ﬁnitely generated groups is ﬁnitely generated, and similarly for ﬁnitely presented groups. 9. If is a ﬁnite graph of ﬁnite groups with injective edge homomorphisms, show that the graph product of the groups has a free subgroup of ﬁnite index by constructing a suitable ﬁnite-sheeted covering space of K from universal covers of the mapping cylinders in K. [The con
verse is also true: A ﬁnitely generated group having a free subgroup of ﬁnite index is isomorphic to such a graph product. For a proof of this Γ see [Scott & Wall 1979], Theorem 7.3.] Γ Γ Γ The fundamental group π1(X) is especially useful when studying spaces of low dimension, as one would expect from its deﬁnition which involves only maps from low-dimensional spaces into X, namely loops I→X and homotopies of loops, maps I × I→X. The deﬁnition in terms of objects that are at most 2 dimensional manifests itself for example in the fact that when X is a CW complex, π1(X) depends only on the 2 skeleton of X. In view of the low-dimensional nature of the fundamental group, we should not expect it to be a very reﬁned tool for dealing with high-dimensional spaces. Thus it cannot distinguish between spheres S n with n ≥ 2. This limitation to low dimensions can be removed by considering the natural higher-dimensional analogs of π1(X), the homotopy groups πn(X), which are deﬁned in terms of maps of the n dimensional cube In into X and homotopies In × I→X of such maps. Not surprisingly, when X is a CW complex, πn(X) depends only on the (n + 1) skeleton of X. And as one might hope, homotopy groups do indeed distinguish spheres of all dimensions since πi(S n) is 0 for i < n and Z for i = n. However, the higher-dimensional homotopy groups have the serious drawback that they are extremely diﬃcult to compute in general. Even for simple spaces like spheres, the calculation of πi(S n) for i > n turns out to be a huge problem. Fortunately there is a more computable alternative to homotopy groups: the homology groups Hn(X). Like πn(X), the homology group Hn(X) for a CW complex X depends only on the (n + 1) skeleton. For spheres, the homology groups Hi(S n) are isomorphic to the homotopy groups πi(S n) in the range 1 ≤ i ≤ n, but homology groups
have the advantage that Hi(S n) = 0 for i > n. The computability of homology groups does not come for free, unfortunately. The deﬁnition of homology groups is decidedly less transparent than the deﬁnition of homotopy groups, and once one gets beyond the deﬁnition there is a certain amount of technical machinery to be set up before any real calculations and applications can be given. In the exposition below we approach the deﬁnition of Hn(X) by two preliminary stages, ﬁrst giving a few motivating examples nonrigorously, then constructing 98 Chapter 2 Homology a restricted model of homology theory called simplicial homology, before plunging into the general theory, known as singular homology. After the deﬁnition of singular homology has been assimilated, the real work of establishing its basic properties be- gins. This takes close to 20 pages, and there is no getting around the fact that it is a substantial eﬀort. This takes up most of the ﬁrst section of the chapter, with small digressions only for two applications to classical theorems of Brouwer: the ﬁxed point theorem and ‘invariance of dimension’. The second section of the chapter gives more applications, including the ho- mology deﬁnition of Euler characteristic and Brouwer’s notion of degree for maps S n→S n. However, the main thrust of this section is toward developing techniques for calculating homology groups eﬃciently. The maximally eﬃcient method is known as cellular homology, whose power comes perhaps from the fact that it is ‘homology squared’ — homology deﬁned in terms of homology. Another quite useful tool is Mayer–Vietoris sequences, the analog for homology of van Kampen’s theorem for the fundamental group. An interesting feature of homology that begins to emerge after one has worked with it for a while is that it is the basic properties of homology that are used most often, and not the actual deﬁnition itself. This suggests that an axiomatic approach to homology might be possible. This is indeed the case, and in the third section of the chapter we list axioms which completely characterize homology groups for CW complexes
. One could take the viewpoint that these rather algebraic axioms are all that really matters about homology groups, that the geometry involved in the deﬁnition of homology is secondary, needed only to show that the axiomatic theory is not vacuous. The extent to which one adopts this viewpoint is a matter of taste, and the route taken here of postponing the axioms until the theory is well-established is just one of several possible approaches. The chapter then concludes with three optional sections of Additional Topics. The ﬁrst is rather brief, relating H1(X) to π1(X), while the other two contain a selection of classical applications of homology. These include the n dimensional version of the Jordan curve theorem and the ‘invariance of domain’ theorem, both due to Brouwer, along with the Lefschetz ﬁxed point theorem. The Idea of Homology The diﬃculty with the higher homotopy groups πn is that they are not directly computable from a cell structure as π1 is. For example, the 2-sphere has no cells in dimensions greater than 2, yet its n dimensional homotopy group πn(S 2) is nonzero for inﬁnitely many values of n. Homology groups, by contrast, are quite directly related to cell structures, and may indeed be regarded as simply an algebraization of the ﬁrst layer of geometry in cell structures: how cells of dimension n attach to cells of dimension n − 1. The Idea of Homology 99 Let us look at some examples to see what the idea is. Consider the graph X1 shown in the ﬁgure, consisting of two vertices joined by four edges. When studying the fundamental group of X1 we consider loops formed by sequences of edges, starting and ending at a ﬁxed basepoint. For example, at the basepoint x, the loop ab−1 travels forward along the edge a, then backward along b, as indicated by the exponent −1. A more complicated loop would be ac−1bd−1ca−1. A salient feature of the fundamental group is that it is generally nonabelian, which both enriches and compli- cates the theory. Suppose we simplify matters by abelianizing. Thus for example the two loops ab−1 and
b−1a are to be regarded as equal if we make a commute with b−1. These two loops ab−1 and b−1a are really the same circle, just with a diﬀerent choice of starting and ending point: x for ab−1 and y for b−1a. The same thing happens for all loops: Rechoosing the basepoint in a loop just permutes its letters cyclically, so a byproduct of abelianizing is that we no longer have to pin all our loops down to a ﬁxed basepoint. Thus loops become cycles, without a chosen basepoint. Having abelianized, let us switch to additive notation, so cycles become linear combinations of edges with integer coeﬃcients, such as a − b + c − d. Let us call these linear combinations chains of edges. Some chains can be decomposed into cycles in several diﬀerent ways, for example (a − c) + (b − d) = (a − d) + (b − c), and if we adopt an algebraic viewpoint then we do not want to distinguish between these diﬀerent decompositions. Thus we broaden the meaning of the term ‘cycle’ to be simply any linear combination of edges for which at least one decomposition into cycles in the previous more geometric sense exists. What is the condition for a chain to be a cycle in this more algebraic sense? A geometric cycle, thought of as a path traversed in time, is distinguished by the prop- erty that it enters each vertex the same number of times that it leaves the vertex. For an arbitrary chain ka + ℓb + mc + nd, the net number of times this chain enters y is k + ℓ + m + n since each of a, b, c, and d enters y once. Similarly, each of the four edges leaves x once, so the net number of times the chain ka + ℓb + mc + nd enters x is −k − ℓ − m − n. Thus the condition for ka + ℓb + mc + nd to be a cycle is simply k + ℓ + m + n = 0. To describe this result in a way that would generalize to all graphs, let C1 be the free abelian group with basis the edges a, b, c, d
and let C0 be the free abelian group with basis the vertices x, y. Elements of C1 are chains of edges, or 1 dimensional chains, and elements of C0 are linear combinations of vertices, or 0 dimensional chains. Deﬁne a homomorphism ∂ : C1→C0 by sending each basis element a, b, c, d to y − x, the vertex at the head of the edge minus the vertex at the tail. Thus we have ∂(ka + ℓb + mc + nd) = (k + ℓ + m + n)y − (k + ℓ + m + n)x, and the cycles are precisely the kernel of ∂. It is a simple calculation to verify that a−b, b −c, and c −d 100 Chapter 2 Homology form a basis for this kernel. Thus every cycle in X1 is a unique linear combination of these three most obvious cycles. By means of these three basic cycles we convey the geometric information that the graph X1 has three visible ‘holes’, the empty spaces between the four edges. Let us now enlarge the preceding graph X1 by attaching a 2 cell A along the cycle a − b, producing a 2 dimensional cell complex X2. If we think of the 2 cell A as being oriented clockwise, then we can regard its boundary as the cycle a − b. This cycle is now homotopically trivial since we can contract it to a point Algebraically, we can deﬁne now a pair of homomorphisms C2 by sliding over A. In other words, it no longer encloses a hole in X2. This suggests that we form a quotient of the group of cycles in the preceding example by factoring out the subgroup generated by a − b. In this quotient the cycles a − c and b − c, for example, become equivalent, consistent with the fact that they are homotopic in X2. ∂1------------→ C0 where C2 is the inﬁnite cyclic group generated by A and ∂2(A) = a − b. The map ∂1 is the boundary homomorphism in the previous example. The quotient group we are interested in is Ker ∂1/ Im ∂2, the kernel of ∂1 modulo the image of ∂2, or in other words, the
1 dimensional cycles modulo those that are boundaries, the multiples of a − b. This quotient group is the homology group H1(X2). The previous example can be ﬁt into this scheme too by taking C2 to be zero since there are no 2 cells in X1, so in this case H1(X1) = Ker ∂1/ Im ∂2 = Ker ∂1, which as we saw was free abelian on three generators. In the present example, H1(X2) is free abelian on two generators, b − c and c − d, expressing the geometric fact that by ﬁlling in the 2 cell A we have ∂2------------→ C1 reduced the number of ‘holes’ in our space from three to two. Suppose we enlarge X2 to a space X3 by attaching a second 2 cell B along the same cycle a − b. This gives a 2 dimensional chain group C2 consisting of linear combinations of A and B, and the boundary homomorphism ∂2 : C2→C1 sends both A and B to a−b. The homology group H1(X3) = Ker ∂1/ Im ∂2 is the same as for X2, but now ∂2 has a nontrivial kernel, the inﬁnite cyclic group generated by A − B. We view A − B as a 2 dimensional cycle, generating the homology group H2(X3) = Ker ∂2 ≈ Z. Topologically, the cycle A − B is the sphere formed by the cells A and B together with their common boundary circle. This spherical cycle detects the presence of a ‘hole’ in X3, the missing interior of the sphere. However, since this hole is enclosed by a sphere rather than a circle, it is of a diﬀerent sort from the holes detected by H1(X3) ≈ Z× Z, which are detected by the cycles b − c and c − d. Let us continue one more step and construct a complex X4 from X3 by attaching a 3 cell C along the 2 sphere formed by A and B. This creates a chain group C3 The Idea of Homology 101 generated by this 3 cell C, and we deﬁne a boundary homomorphism ∂3 : C3→C2 sending C to A − B
since the cycle A − B should be viewed as the boundary of C in the same way that the 1 dimensional cycle a − b is the boundary of A. Now we ∂1------------→ C0 and have a sequence of three boundary homomorphisms C3 the quotient H2(X4) = Ker ∂2/ Im ∂3 has become trivial. Also H3(X4) = Ker ∂3 = 0. The group H1(X4) is the same as H1(X3), namely Z× Z, so this is the only nontrivial homology group of X4. ∂3------------→ C2 ∂2------------→ C1 It is clear what the general pattern of the examples is. For a cell complex X one has chain groups Cn(X) which are free abelian groups with basis the n cells of X, and there are boundary homomorphisms ∂n : Cn(X)→Cn−1(X), in terms of which one deﬁnes the homology group Hn(X) = Ker ∂n/ Im ∂n+1. The major diﬃculty is how to deﬁne ∂n in general. For n = 1 this is easy: The boundary of an oriented edge is the vertex at its head minus the vertex at its tail. The next case n = 2 is also not hard, at least for cells attached along cycles that are simply loops of edges, for then the boundary of the cell is this cycle of edges, with the appropriate signs taking orientations into account. But for larger n, matters become more complicated. Even if one restricts attention to cell complexes formed from polyhedral cells with nice attaching maps, there is still the matter of orientations to sort out. The best solution to this problem seems to be to adopt an indirect approach. Arbitrary polyhedra can always be subdivided into special polyhedra called simplices (the triangle and the tetrahedron are the 2 dimensional and 3 dimensional instances) so there is no loss of generality, though initially there is some loss of eﬃciency, in re- stricting attention entirely to simplices. For simplices there is no diﬃculty in deﬁning boundary maps or in handling orientations. So one obtains a homology theory, called simplicial homology, for cell complexes built from simpl
ices. Still, this is a rather restricted class of spaces, and the theory itself has a certain rigidity that makes it awkward to work with. The way around these obstacles is to step back from the geometry of spaces decomposed into simplices and to consider instead something which at ﬁrst glance seems wildly more complicated, the collection of all possible continuous maps of simplices into a given space X. These maps generate tremendously large chain groups Cn(X), but the quotients Hn(X) = Ker ∂n/ Im ∂n+1, called singular homology groups, turn out to be much smaller, at least for reasonably nice spaces X. In particular, for spaces like those in the four examples above, the singular homology groups coincide with the homology groups we computed from the cellular chains. And as we shall see later in this chapter, singular homology allows one to deﬁne these nice cellular homology groups for all cell complexes, and in particular to solve the problem of deﬁning the boundary maps for cellular chains. ∆ 102 Chapter 2 Homology The most important homology theory in algebraic topology, and the one we shall be studying almost exclusively, is called singular homology. Since the technical appa- ratus of singular homology is somewhat complicated, we will ﬁrst introduce a more primitive version called simplicial homology in order to see how some of the apparatus works in a simpler setting before beginning the general theory. The natural domain of deﬁnition for simplicial homology is a class of spaces we call complexes, which are a mild generalization of the more classical notion of a simplicial complex. Historically, the modern deﬁnition of singular homology was ﬁrst given in [Eilenberg 1944], and complexes were introduced soon thereafter in ∆ [Eilenberg-Zilber 1950] where they were called semisimplicial complexes. Within a few years this term came to be applied to what Eilenberg and Zilber called complete semisimplicial complexes, and later there was yet another shift in terminology as the latter objects came to be called simplicial sets. In theory this frees up the term semisimplicial complex to have its original meaning, but to avoid potential confusion it seems best to introduce a new name, and the term complex has at least the virtue of brevity. D–Complexes ∆ The tor
us, the projective plane, and the Klein bottle can each be obtained from a square by identifying opposite edges in the way indicated by the arrows in the follow- ing ﬁgures: Cutting a square along a diagonal produces two triangles, so each of these surfaces can also be built from two triangles by identifying their edges in pairs. In similar fashion a polygon with any number of sides can be cut along diagonals into triangles, so in fact all closed surfaces can be constructed from triangles by identifying edges. Thus we have a single building block, the triangle, from which all surfaces can be constructed. Using only triangles we could also construct a large class of 2 dimensional spaces that are not surfaces in the strict sense, by allowing more than two edges to be identiﬁed together at a time. Simplicial and Singular Homology Section 2.1 103 The idea of a complex is to generalize constructions like these to any number of dimensions. The n dimensional analog of the triangle is the n simplex. This is the ∆ smallest convex set in a Euclidean space Rm containing n + 1 points v0, ···, vn that do not lie in a hyperplane of dimen- sion less than n, where by a hyperplane we mean the set of solutions of a system of linear equations. An equivalent condi- tion would be that the diﬀerence vectors v1 − v0, ···, vn − v0 are linearly independent. The points vi are the vertices of the simplex, and the simplex itself is denoted [v0, ···, vn]. For example, there is the standard n simplex n = (t0, ···, tn) ∈ Rn+1 \|\|\|\| iti = 1 and ti ≥ 0 for all i whose vertices are the unit vectors along the coordinate axes. P ∆ For purposes of homology it will be important to keep track of the order of the vertices of a simplex, so ‘ n simplex’ will really mean ‘ n simplex with an ordering of its vertices’. A by-product of ordering the vertices of a simplex [v0, ···, vn] is that this determines orientations of the edges [vi, vj] according to increasing subscripts, as shown in the two preceding ﬁgures. Specifying the ordering of the
vertices also determines a canonical linear homeomorphism from the standard n simplex n onto any other n simplex [v0, ···, vn], preserving the order of vertices, namely, i tivi. The coeﬃcients ti are the barycentric coordinates of the point (t0, ···, tn)֏ ∆ i tivi in [v0, ···, vn]. P If we delete one of the n + 1 vertices of an n simplex [v0, ···, vn], then the P remaining n vertices span an (n − 1) simplex, called a face of [v0, ···, vn]. We adopt the following convention: The vertices of a face, or of any subsimplex spanned by a subset of the vertices, will always be ordered according to their order in the larger simplex. The union of all the faces of simplex ◦ n is n − ∂ n is the boundary of n. n, the interior of ∆ ∆ A D complex structure on a space X is a collection of maps σα : ∆ ∆ ∆ depending on the index α, such that: ∆ n, written ∂ n. The open ∆ n→X, with n ∆ n is injective, and each point of X is in the image of exactly (i) The restriction σα ◦ \|\| one such restriction σα ◦ n. \|\| ∆ (ii) Each restriction of σα to a face of n with ∆ are identifying the face of n−1→X. Here we n is one of the maps σβ : n−1 by the canonical linear homeomorphism ∆ ∆ between them that preserves the ordering of the vertices. ∆ (iii) A set A ⊂ X is open iﬀ σ −1 α (A) is open in ∆ n for each σα. ∆ 104 Chapter 2 Homology Among other things, this last condition rules out trivialities like regarding all the points of X as individual vertices. The earlier decompositions of the torus, projective plane, and Klein bottle into two triangles, three edges, and one or two vertices deﬁne complex structures with a total of six σα ’s for the torus and Klein
bottle and seven for the projective plane. The orientations on the edges in the pictures are compatible ∆ with a unique ordering of the vertices of each simplex, and these orderings determine the maps σα. of disjoint simplices A consequence of (iii) is that X can be built as a quotient space of a collection n→X, the quotient space obtained by n α, one for each σα : n−1 n corresponding to the restriction σβ of identifying each face of a α with the β σα to the face in question, as in condition (ii). One can think of building the quotient space inductively, starting with a discrete set of vertices, then attaching edges to ∆ ∆ ∆ ∆ these to produce a graph, then attaching 2 simplices to the graph, and so on. From complex can be described purely this viewpoint we see that the data specifying a n α for each n together with functions combinatorially as collections of n simplices associating to each face of each n simplex ∆ n α an (n − 1) simplex ∆ n−1 β. ∆ ∆ More generally, complexes can be built from collections of disjoint simplices by identifying various subsimplices spanned by subsets of the vertices, where the iden- tiﬁcations are performed using the canonical linear homeomorphisms that preserve ∆ the orderings of the vertices. The earlier complex structures on a torus, projective plane, or Klein bottle can be obtained in this way, by identifying pairs of edges of ∆ two 2 simplices. If one starts with a single 2 simplex and identiﬁes all three edges to a single edge, preserving the orientations given by the ordering of the vertices, this produces a complex known as the ‘dunce hat’. By contrast, if the three edges of a 2 simplex are identiﬁed preserving a cyclic orientation of the three edges, as in ∆ the ﬁrst ﬁgure at the right, this does not produce a complex structure, although if the 2 simplex is subdivided into three smaller 2 simplices about a ∆ central vertex, then one does obtain a complex structure on the quotient space. ∆ Thinking of a complex X as a quotient space of a collection of disjoint sim- ∆
plices, it is not hard to see that X must be a Hausdorﬀ space. Condition (iii) then n is a homeomorphism onto its image, which is implies that each restriction σα It follows from Proposition A.2 in the Appendix that thus an open simplex in X. α of a CW complex structure on X with these open simplices σα( the σα ’s as characteristic maps. We will not need this fact at present, however. n) are the cells en ∆ \|\| ◦ ◦ ∆ Simplicial Homology Our goal now is to deﬁne the simplicial homology groups of a complex X. Let n(X) be the free abelian group with basis the open n simplices en α of X. Elements ∆ ∆ Simplicial and Singular Homology Section 2.1 105 α nαen ∆ ∆ P P α nασα where σα : α, with image the closure of en n(X), called n chains, can be written as ﬁnite formal sums α with con→X is the α as described above. Such a α nασα can be thought of as a ﬁnite collection, or ‘chain’, of n simplices in X of eﬃcients nα ∈ Z. Equivalently, we could write characteristic map of en sum with integer multiplicities, the coeﬃcients nα. P As one can see in the next ﬁgure, the boundary of the n simplex [v0, ···, vn] convi, ···, vn], where the ‘hat’ sists of the various (n−1) dimensional simplices [v0, ···, over vi indicates that this vertex is deleted from the sequence v0, ···, vn. symbol In terms of chains, we might then wish to say that the boundary of [v0, ···, vn] is the vi, ···, vn]. However, it turns (n − 1) chain formed by the sum of the faces [v0, ···, out to be better to insert certain signs and instead let the boundary of [v0, ···, vn] be vi, ···, vn]. Heuristically, the signs are inserted to take
orientations into account, so that all the faces of a simplex are coherently oriented, as indicated in P the following ﬁgure: i(−1)i[v0, ···, b b b b ∂[v0, v1] = [v1] − [v0] ∂[v0, v1, v2] = [v1, v2] − [v0, v2] + [v0, v1] ∂[v0, v1, v2, v3] = [v1, v2, v3] − [v0, v2, v3] + [v0, v1, v3] − [v0, v1, v2] In the last case, the orientations of the two hidden faces are also counterclockwise when viewed from outside the 3 simplex. With this geometry in mind we deﬁne for a general complex X a boundary homomorphism ∂n : n(X)→ n−1(X) by specifying its values on basis elements: ∆ ∂n(σα) = ∆ ∆ (−1)iσα \|\| [v0, ···, vi, ···, vn] Xi Note that the right side of this equation does indeed lie in tion σα \|\| [v0, ···, vi, ···, vn] is the characteristic map of an (n − 1) simplex of X. n−1(X) since each restric- b ∆ ∂n−1 ------------------→ Lemma 2.1. The composition b Proof: We have ∂n(σ ) = ∂n−1∂n(σ ) = ∂n------------→ n(X) i(−1)iσ \|\| [v0, ···, vi, ···, vn], and hence ∆ n−1(X) ∆ ∆ n−2(X) is zero. P (−1)i(−1)j σ \|\|[v0, ···, b vj, ···, vi, ···, vn] Xj<i + (−1)i(−1)j−1σ \|\|[v0, ···, b b vi, ···, vj, ···, vn] Xj>i b b 106
Chapter 2 Homology The latter two summations cancel since after switching i and j in the second sum, it becomes the negative of the ﬁrst. ⊔⊓ The algebraic situation we have now is a sequence of homomorphisms of abelian groups ··· -→ Cn+1 ∂n+1 -----------------→ Cn ∂n------------→ Cn−1 -→ ··· -→ C1 ∂1------------→ C0 ∂0------------→ 0 with ∂n∂n+1 = 0 for each n. Such a sequence is called a chain complex. Note that we have extended the sequence by a 0 at the right end, with ∂0 = 0. The equation ∂n∂n+1 = 0 is equivalent to the inclusion Im ∂n+1 ⊂ Ker ∂n, where Im and Ker denote image and kernel. So we can deﬁne the nth homology group of the chain complex to be the quotient group Hn = Ker ∂n/ Im ∂n+1. Elements of Ker ∂n are called cycles and elements of Im ∂n+1 are called boundaries. Elements of Hn are cosets of Im ∂n+1, called homology classes. Two cycles representing the same homology class are said to be homologous. This means their diﬀerence is a boundary. Returning to the case that Cn = n(X), the homology group Ker ∂n/ Im ∂n+1 will be denoted H n(X) and called the nth simplicial homology group of X. ∆ ∆ Example 2.2. X = S 1, with one vertex v and one edge e. Then and 0(S 1) 1(S 1) are both Z and the boundary map ∂1 is zero since ∂e = v −v. n(S 1) are 0 for n ≥ 2 since there are no simplices in these ∆ The groups ∆ dimensions. Hence ∆ H n(S 1) ≈ ∆ Z 0 for n = 0, 1 for n ≥ 2 This is an illustration of the general fact that if the boundary maps in a chain complex are all zero, then the homology groups of the complex are isomorphic to the chain groups themselves. complex structure pictured earlier, having Example 2.3. X = T
, the torus with the one vertex, three edges a, b, and c, and two 2 simplices U and L. As in the previous 0 (T ) ≈ Z. Since ∂2U = a + b − c = ∂2L and {a, b, a + b − c} is example, ∂1 = 0 so H 1 (T ) ≈ Z⊕ Z with basis the homology classes [a] ∆ a basis for ∆ and [b]. Since there are no 3 simplices, H 2 (T ) is equal to Ker ∂2, which is inﬁnite cyclic generated by U − L since ∂(pU + qL) = (p + q)(a + b − c) = 0 only if p = −q. ∆ 1(T ), it follows that H ∆ ∆ Thus H n( for n = 1 for n = 0, 2 for n ≥ 3  Example 2.4. X = RP2, as pictured earlier, with two vertices v and w, three edges a, b, and c, and two 2 simplices U and L. Then Im ∂1 is generated by w − v, so 0 (X) ≈ Z with either vertex as a generator. Since ∂2U = −a+b+c and ∂2L = a−b+c, H 2 (X) = 0. Further, Ker ∂1 ≈ Z⊕ Z with basis a − b and ∆ we see that ∂2 is injective, so H c, and Im ∂2 is an index-two subgroup of Ker ∂1 since we can choose c and a − b + c ∆ Simplicial and Singular Homology Section 2.1 107 as a basis for Ker ∂1 and a − b + c and 2c = (a − b + c) + (−a + b + c) as a basis for Im ∂2. Thus H 1 (X) ≈ Z2. ∆ Example 2.5. We can obtain a and identifying their boundaries via the identity map. Labeling these two n simplices ∆ U and L, then it is obvious that Ker ∂n is inﬁnite cyclic generated by U − L. Thus complex structure on S n. Computing the other homology H
complex structure on S n by taking two copies of n(S n) ≈ Z for this ∆ n groups would be more diﬃcult. ∆ ∆ Many similar examples could be worked out without much trouble, such as the other closed orientable and nonorientable surfaces. However, the calculations do tend to increase in complexity before long, particularly for higher-dimensional complexes. Some obvious general questions arise: Are the groups H the choice of complex structure on X? In other words, if two n(X) independent of ∆ complexes are homeomorphic, do they have isomorphic homology groups? More generally, do they have isomorphic homology groups if they are merely homotopy equivalent? To answer such questions and to develop a general theory it is best to leave the rather rigid simplicial realm and introduce the singular homology groups. These have the added advantage that they are deﬁned for all spaces, not just complexes. At the end of this section, after some theory has been developed, we will show that simplicial and singular homology groups coincide for complexes. ∆ ∆ ∆ ∆ Traditionally, simplicial homology is deﬁned for simplicial complexes, which are ∆ the complexes whose simplices are uniquely determined by their vertices. This amounts to saying that each n simplex has n + 1 distinct vertices, and that no other n simplex has this same set of vertices. Thus a simplicial complex can be described combinatorially as a set X0 of vertices together with sets Xn of n simplices, which are (n + 1) element subsets of X0. The only requirement is that each (k + 1) element subset of the vertices of an n simplex in Xn is a k simplex, in Xk. From this combicomplex X can be constructed, once we choose a partial ordering natorial data a of the vertices X0 that restricts to a linear ordering on the vertices of each simplex in Xn. For example, we could just choose a linear ordering of all the vertices. This might perhaps involve invoking the Axiom of Choice for large vertex sets. ∆ An exercise at the end of this section is to show that every complex can be subdivided to be a simplicial complex. In particular, every complex is then homeo- ∆ morphic to a simplicial complex. Compared with simplicial complexes, complexes have the
advantage of simpler ∆ computations since fewer simplices are required. For example, to put a simplicial ∆ complex structure on the torus one needs at least 14 triangles, 21 edges, and 7 vertices, and for RP2 one needs at least 10 triangles, 15 edges, and 6 vertices. This would slow down calculations considerably! 108 Chapter 2 Homology Singular Homology A singular n simplex in a space X is by deﬁnition just a map σ : n→X. The word ‘singular’ is used here to express the idea that σ need not be a nice embedding but can have ‘singularities’ where its image does not look at all like a simplex. All that is required is that σ be continuous. Let Cn(X) be the free abelian group with basis the set of singular n simplices in X. Elements of Cn(X), called n chains, or more n→X. precisely singular n chains, are ﬁnite formal sums A boundary map ∂n : Cn(X)→Cn−1(X) is deﬁned by the same formula as before: i niσi for ni ∈ Z and σi : P ∆ ∆ ∂n(σ ) = (−1)iσ \|\| [v0, ···, vi, ···, vn] Xi Implicit in this formula is the canonical identiﬁcation of [v0, ···, b n−1, preserving the ordering of vertices, so that σ \|\| [v0, ···, n−1→X, that is, a singular (n − 1) simplex. as a map ∆ vi, ···, vn] with vi, ···, vn] is regarded b b ∆ Often we write the boundary map ∂n from Cn(X) to Cn−1(X) simply as ∂ when this does not lead to serious ambiguities. The proof of Lemma 2.1 applies equally well to singular simplices, showing that ∂n∂n+1 = 0 or more concisely ∂2 = 0, so we can deﬁne the singular homology group Hn(X) = Ker ∂n/ Im ∂n+1. It is evident from the de�
�nition that homeomorphic spaces have isomorphic singular homology groups Hn, in contrast with the situation for H n. On the other hand, ∆ since the groups Cn(X) are so large, the number of singular n simplices in X usually being uncountable, it is not at all clear that for a complex X with ﬁnitely many simplices, Hn(X) should be ﬁnitely generated for all n, or that Hn(X) should be zero for n larger than the dimension of X — two properties that are trivial for H ∆ Though singular homology looks so much more general than simplicial homology, n(X). ∆ it can actually be regarded as a special case of simplicial homology by means of the complex with one n simplex with restrictions of σ to the various (n − 1) simplices in ∂ following construction. For an arbitrary space X, deﬁne the singular complex S(X) n→X, to be the n σ attached in the obvious way to the (n − 1) simplices of S(X) that are the n. It is clear from the deﬁni∆ is identical with Hn(X) for all n, and in this sense the singular tions that H homology group Hn(X) is a special case of a simplicial homology group. One can complex model for X, although it is usually an extremely large regard S(X) as a n σ for each singular n simplex σ : S(X) n ∆ ∆ ∆ ∆ ∆ object compared to X. ∆ Cycles in singular homology are deﬁned algebraically, but they can be given a somewhat more geometric interpretation in terms of maps from ﬁnite complexes. To see this, note ﬁrst that a singular n chain ξ can always be written in the form i εiσi with εi = ±1, allowing repetitions of the singular n simplices σi. Given such i εiσi, when we compute ∂ξ as a sum of singular (n − 1) simplices an n chain ξ = P with signs ±1, there may be some canceling pairs consisting of two identical singular (n − 1) simplices with opposite signs. Choosing a maximal collection of
such P ∆ Simplicial and Singular Homology Section 2.1 109 ∆ ∆ canceling pairs, construct an n dimensional complex Kξ from a disjoint union of n i, one for each σi, by identifying the pairs of (n−1) dimensional faces n simplices corresponding to the chosen canceling pairs. The σi ’s then induce a map Kξ→X. If n ξ is a cycle, all the (n − 1) dimensional faces of the i ’s are identiﬁed in pairs. Thus Kξ is a manifold, locally homeomorphic to Rn, near all points in the complement of the (n − 2) skeleton Kn−2 of Kξ. All the n simplices of Kξ can be coherently oriented by taking the signs of the σi ’s into account, so Kξ − Kn−2 is actually an oriented manifold. A closer inspection shows that Kξ is also a manifold near points in the interiors of (n − 2) simplices, so the nonmanifold points of Kξ in fact lie in the (n − 3) skeleton. However, near points in the interiors of (n − 3) simplices it can very well happen that Kξ is not a manifold. ∆ ξ ξ In particular, elements of H1(X) are represented by collections of oriented loops in X, and elements of H2(X) are represented by maps of closed oriented surfaces α→X is into X. With a bit more work it can be shown that an oriented 1 cycle zero in H1(X) iﬀ it extends to a map of a compact oriented surface with boundary α into X. The analogous statement for 2 cycles is also true. In the early days of homology theory it may have been believed, or at least hoped, that this close connec` αS 1 αS 1 ` tion with manifolds continued in all higher dimensions, but this has turned out not to be the case. There is a sort of homology theory built from manifolds, called bordism, but it is quite a bit more complicated than the homology theory we are studying here. After these preliminary remarks let us begin to see what can be proved about singular homology. Proposition 2.6. Corresponding to the decomposition of a space X into its pathα Hn(
Xα). components Xα there is an isomorphism of Hn(X) with the direct sum Proof: Since a singular simplex always has path-connected image, Cn(X) splits as the direct sum of its subgroups Cn(Xα). The boundary maps ∂n preserve this direct sum decomposition, taking Cn(Xα) to Cn−1(Xα), so Ker ∂n and Im ∂n+1 split similarly as ⊔⊓ direct sums, hence the homology groups also split, Hn(X) ≈ α Hn(Xα). L Proposition 2.7. If X is nonempty and path-connected, then H0(X) ≈ Z. Hence for any space X, H0(X) is a direct sum of Z ’s, one for each path-component of X. L Proof: By deﬁnition, H0(X) = C0(X)/ Im ∂1 since ∂0 = 0. Deﬁne a homomorphism ε : C0(X)→Z by ε i ni. This is obviously surjective if X is nonempty. The claim is that Ker ε = Im ∂1 if X is path-connected, and hence ε induces an isomorphism H0(X) ≈ Z. i niσi P P = To verify the claim, observe ﬁrst that Im ∂1 ⊂ Ker ε since for a singular 1 simplex 1→X we have ε∂1(. For the reverse σ : i ni = 0. The σi ’s are singular inclusion Ker ε ⊂ Im ∂1, suppose ε 0 simplices, which are simply points of X. Choose a path τi : I→X from a basepoint σ \|\| [v1] − σ \|\| [v0] = 0, so i niσi P P ∆ 110 Chapter 2 Homology x0 to σi(v0) and let σ0 be the singular 0 simplex with image x0. We can view τi as a singular 1 simplex, a map τi : [v0, v1]→X, and then we have ∂τi = σi − σ0. i niσ
i is a Hence ∂ ⊔⊓ boundary, which shows that Ker ε ⊂ Im ∂1. P P i ni = 0. Thus i niσi since i niσ0 = i niσi − i niτi P P P P = Proposition 2.8. If X is a point, then Hn(X) = 0 for n > 0 and H0(X) ≈ Z. Proof: In this case there is a unique singular n simplex σn for each n, and ∂(σn) = i(−1)iσn−1, a sum of n + 1 terms, which is therefore 0 for n odd and σn−1 for n even, n ≠ 0. Thus we have the chain complex P ··· -→ Z ≈------------→ Z 0------------→ Z ≈------------→ Z 0------------→ Z -→ 0 with boundary maps alternately isomorphisms and trivial maps, except at the last Z. The homology groups of this complex are trivial except for H0 ≈ Z. ⊔⊓ It is often very convenient to have a slightly modiﬁed version of homology for which a point has trivial homology groups in all dimensions, including zero. This is Hn(X) to be the homology groups done by deﬁning the reduced homology groups of the augmented chain complex e ··· -→ C2(X) ∂2------------→ C1(X) ∂1------------→ C0(X) ε------------→ Z -→ 0 = P i niσi where ε i ni as in the proof of Proposition 2.7. Here we had better require X to be nonempty, to avoid having a nontrivial homology group in dimension −1. Since ε∂1 = 0, ε vanishes on Im ∂1 and hence induces a map H0(X)→Z with H0(X)⊕ Z. Obviously Hn(X) ≈ kernel H0(X), so H0(X) ≈ Hn(X) for n > 0. P e Formally, one can think of the extra Z in the augmented chain complex as generated by the unique map [∅]→X where [∅] is the empty simplex, with no vertices. v0] = [�
�]. The augmentation map ε is then the usual boundary map since ∂[v0] = [ e e Readers who know about the fundamental group π1(X) may wish to make a detour here to look at §2.A where it is shown that H1(X) is the abelianization of π1(X) whenever X is path-connected. This result will not be needed elsewhere in the chapter, however. b Homotopy Invariance The ﬁrst substantial result we will prove about singular homology is that ho- motopy equivalent spaces have isomorphic homology groups. This will be done by showing that a map f : X→Y induces a homomorphism f∗ : Hn(X)→Hn(Y ) for each n, and that f∗ is an isomorphism if f is a homotopy equivalence. For a map f : X→Y, an induced homomorphism f♯ : Cn(X)→Cn(Y ) is deﬁned n→X with f to get a singular n simplex by composing each singular n simplex σ : ∆ Simplicial and Singular Homology Section 2.1 111 f♯(σ ) = f σ : i niσi i nif σi. The maps f♯ : Cn(X)→Cn(Y ) satisfy f♯∂ = ∂f♯ since n→Y, then extending f♯ linearly via f♯ P = i nif♯(σi) = P P ∆ f♯∂(σ ) = f♯ i(−1)iσ \|\|[v0, ···, = i(−1)if σ \|\|[v0, ···, P vi, ···, vn] = ∂f♯(σ ) vi, ···, vn] b Thus we have a diagram P b such that in each square the composition f♯∂ equals the composition ∂f♯. A diagram of maps with the property that any two compositions of maps starting at one point in the diagram and ending at another are equal is called a commutative diagram. In the present case commutativity of the diagram is equivalent to the commutativity relation f♯∂ = ∂f♯, but comm
utative diagrams can contain commutative triangles, pentagons, etc., as well as commutative squares. The fact that the maps f♯ : Cn(X)→Cn(Y ) satisfy f♯∂ = ∂f♯ is also expressed by saying that the f♯ ’s deﬁne a chain map from the singular chain complex of X to that of Y. The relation f♯∂ = ∂f♯ implies that f♯ takes cycles to cycles since ∂α = 0 implies ∂(f♯α) = f♯(∂α) = 0. Also, f♯ takes boundaries to boundaries since f♯(∂β) = ∂(f♯β). Hence f♯ induces a homomorphism f∗ : Hn(X)→Hn(Y ). An algebraic statement of what we have just proved is: Proposition 2.9. A chain map between chain complexes induces homomorphisms ⊔⊓ between the homology groups of the two complexes. Two basic properties of induced homomorphisms which are important in spite of being rather trivial are: (i) (f g)∗ = f∗g∗ for a composed mapping X g-----→ Y associativity of compositions n σ-----→ X f-----→ Z. This follows from g-----→ Y f-----→ Z. (ii) 11∗ = 11 where 11 denotes the identity map of a space or a group. ∆ Less trivially, we have: Theorem 2.10. If two maps f, g : X→Y are homotopic, then they induce the same homomorphism f∗ = g∗ : Hn(X)→Hn(Y ). In view of the formal properties (f g)∗ = f∗g∗ and 11∗ = 11, this immediately implies: Corollary 2.11. The maps f∗ : Hn(X)→Hn(Y ) induced by a homotopy equivalence f : X→Y are isomorphisms for all n. ⊔⊓ For example, if X is contractible then Hn(X) = 0 for all n. e 112 Chapter 2 Homology ∆ ∆ ∆ ∆ ∆ In and n × I, let ∆ n ×
I→ cases n = 1, 2. Proof of 2.10: The essential ingredient is a procedure for n × I into simplices. The ﬁgure shows the subdividing n × {0} = [v0, ···, vn] n × {1} = [w0, ···, wn], where vi and wi have the n. We can pass same image under the projection from [v0, ···, vn] to [w0, ···, wn] by interpolating a sequence of n simplices, each obtained from the preceding one by moving one vertex vi up to wi, starting with vn and working backwards to v0. Thus the ﬁrst step is to move [v0, ···, vn] up to [v0, ···, vn−1, wn], then the second step is to move this up to [v0, ···, vn−2, wn−1, wn], In the typical step [v0, ···, vi, wi+1, ···, wn] and so on. moves up to [v0, ···, vi−1, wi, ···, wn]. The region between these two n simplices is exactly the (n+1) simplex [v0, ···, vi, wi, ···, wn] which has [v0, ···, vi, wi+1, ···, wn] as its lower face and n × I is the union of the [v0, ···, vi−1, wi, ···, wn] as its upper face. Altogether, (n + 1) simplices [v0, ···, vi, wi, ···, wn], each intersecting the next in an n simplex face. ∆ Given a homotopy F : X × I→Y from f to g and a singular simplex σ : n→X, n × I→X × I→Y. Using this, we can deﬁne we can form the composition F ◦ (σ × 11) : prism operators P : Cn(X)→Cn+1(Y ) by the following formula: ∆ ∆ P (σ ) = (−1)iF ◦ (σ × 11) \|\| [v
0, ···, vi, wi, ···, wn] Xi We will show that these prism operators satisfy the basic relation ∂P = g♯ − f♯ − P ∂ Geometrically, the left side of this equation represents the boundary of the prism, and n × {0}, and the three terms on the right side represent the top n × {1}, the bottom the sides ∂ n × I of the prism. To prove the relation we calculate ∂P (σ ) = ∆ (−1)i(−1)jF ◦(σ × 11)\|\|[v0, ···, vj, ···, vi, wi, ···, wn] Xj≤i + (−1)i(−1)j+1F ◦(σ × 11)\|\|[v0, ···, vi, wi, ···, b wj, ···, wn] ∆ ∆ Xj≥i c The terms with i = j in the two sums cancel except for F ◦ (σ × 11) \|\| [ which is g ◦ σ = g♯(σ ), and −F ◦ (σ × 11) \|\| [v0, ···, vn, The terms with i ≠ j are exactly −P ∂(σ ) since v0, w0, ···, wn], wn], which is −f ◦ σ = −f♯(σ ). b P ∂(σ ) = (−1)i(−1)jF ◦(σ × 11)\|\|[v0, ···, vi, wi, ···, wj, ···, wn] c Xi<j + Xi>j (−1)i−1(−1)j F ◦(σ × 11)\|\|[v0, ···, c vj, ···, vi, wi, ···, wn] b Simplicial and Singular Homology Section 2.1 113 Now we can ﬁnish the proof of the theorem. If α ∈ Cn(X) is a cycle, then we have g♯(α) − f♯(α) = ∂P (α) + P ∂(α) = ∂P (α) since ∂α = 0. Thus g�
�(α) − f♯(α) is a boundary, so g♯(α) and f♯(α) determine the same homology class, which means ⊔⊓ that g∗ equals f∗ on the homology class of α. The relationship ∂P + P ∂ = g♯ − f♯ is expressed by saying P is a chain homotopy between the chain maps f♯ and g♯. We have just shown: Proposition 2.12. Chain-homotopic chain maps induce the same homomorphism on ⊔⊓ homology. There are also induced homomorphisms f∗ : Hn(Y ) for reduced homology groups since f♯ε = εf♯ where f♯ is the identity map on the added groups Z in the augmented chain complexes. The properties of induced homomorphisms we proved e e Hn(X)→ above hold equally well in the setting of reduced homology, with the same proofs. Exact Sequences and Excision If there was always a simple relationship between the homology groups of a space X, a subspace A, and the quotient space X/A, then this could be a very useful tool in understanding the homology groups of spaces such as CW complexes that can be built inductively from successively more complicated subspaces. Perhaps the simplest possible relationship would be if Hn(X) contained Hn(A) as a subgroup and the quotient group Hn(X)/Hn(A) was isomorphic to Hn(X/A). While this does hold in some cases, if it held in general then homology theory would collapse totally since every space X can be embedded as a subspace of a space with trivial homology groups, namely the cone CX = (X × I)/(X × {0}), which is contractible. It turns out that this overly simple model does not have to be modiﬁed too much to get a relationship that is valid in fair generality. The novel feature of the actual relationship is that it involves the groups Hn(X), Hn(A), and Hn(X/A) for all values of n simultaneously. In practice this is not as bad as it might sound, and in addition it has the pleasant side eﬀect of sometimes allowing higher-dimensional homology groups
to be computed in terms of lower-dimensional groups which may already be known, for example by induction. In order to formulate the relationship we are looking for, we need an algebraic deﬁnition which is central to algebraic topology. A sequence of homomorphisms ··· ---------→ An+1 αn+1 -------------------------→ An αn------------------→ An−1 ---------→ ··· is said to be exact if Ker αn = Im αn+1 for each n. The inclusions Im αn+1 ⊂ Ker αn are equivalent to αnαn+1 = 0, so the sequence is a chain complex, and the opposite inclusions Ker αn ⊂ Im αn+1 say that the homology groups of this chain complex are trivial. 114 Chapter 2 Homology A number of basic algebraic concepts can be expressed in terms of exact se- quences, for example: (i) 0 -→ A (ii) A (iii) 0 -→ A (iv) 0 -→ A α-----→ B is exact iﬀ Ker α = 0, i.e., α is injective. α-----→ B -→ 0 is exact iﬀ Im α = B, i.e., α is surjective. α-----→ B -→ 0 is exact iﬀ α is an isomorphism, by (i) and (ii). α-----→ B β-----→ C -→ 0 is exact iﬀ α is injective, β is surjective, and Ker β = Im α, so β induces an isomorphism C ≈ B/ Im α. This can be written C ≈ B/A if we think of α as an inclusion of A as a subgroup of B. An exact sequence 0→A→B→C→0 as in (iv) is called a short exact sequence. Exact sequences provide the right tool to relate the homology groups of a space, a subspace, and the associated quotient space: Theorem 2.13. If X is a space and A is a nonempty closed subspace that is a deformation retract of some neighborhood in X, then there is an exact sequence ··· -----→ Hn(A) i∗------------→ Hn(X) j∗------------→ Hn(X/A) ∂-----→ Hn−1(A) e
e e e i∗------------→ ··· -----→ e Hn−1(X) -----→ ··· H0(X/A) -----→ 0 where i is the inclusion A ֓ X and j is the quotient map X→X/A. e The map ∂ will be constructed in the course of the proof. The idea is that an Hn(X/A) can be represented by a chain α in X with ∂α a cycle in A element x ∈ whose homology class is ∂x ∈ e Hn−1(A). e Pairs of spaces (X, A) satisfying the hypothesis of the theorem will be called good pairs. For example, if X is a CW complex and A is a nonempty subcomplex, then (X, A) is a good pair by Proposition A.5 in the Appendix. Corollary 2.14. Hn(S n) ≈ Z and Hi(S n) = 0 for i ≠ n. e e Proof: For n > 0 take (X, A) = (Dn, S n−1) so X/A = S n. The terms Hi(Dn) in the long exact sequence for this pair are zero since Dn is contractible. Exactness of the Hi−1(S n−1) are isomorphisms for Hi(S n) sequence then implies that the maps H0(S n) = 0. The result now follows by induction on n, starting with ⊔⊓ i > 0 and that the case of S 0 where the result holds by Propositions 2.6 and 2.8. ∂-----→ e e e e As an application of this calculation we have the following classical theorem of Brouwer, the 2 dimensional case of which was proved in §1.1. Corollary 2.15. ∂Dn is not a retract of Dn. Hence every map f : Dn→Dn has a ﬁxed point. Proof: If r : Dn→∂Dn is a retraction, then r i = 11 for i : ∂Dn→Dn the inclusion map. Hn−1(∂Dn) is then the identity map The composition Hn−1(∂Dn) Hn−1(Dn) r∗------------→ i∗------------→ e e
e Simplicial and Singular Homology Section 2.1 115 on Hn−1(∂Dn) ≈ Z. But i∗ and r∗ are both 0 since contradiction. The statement about ﬁxed points follows as in Theorem 1.9. e Hn−1(Dn) = 0, and we have a ⊔⊓ e The derivation of the exact sequence of homology groups for a good pair (X, A) will be rather a long story. We will in fact derive a more general exact sequence which holds for arbitrary pairs (X, A), but with the homology groups of the quotient space X/A replaced by relative homology groups, denoted Hn(X, A). These turn out to be quite useful for many other purposes as well. Relative Homology Groups It sometimes happens that by ignoring a certain amount of data or structure one obtains a simpler, more ﬂexible theory which, almost paradoxically, can give results not readily obtainable in the original setting. A familiar instance of this is arithmetic mod n, where one ignores multiples of n. Relative homology is another example. In this case what one ignores is all singular chains in a subspace of the given space. Relative homology groups are deﬁned in the following way. Given a space X and a subspace A ⊂ X, let Cn(X, A) be the quotient group Cn(X)/Cn(A). Thus chains in A are trivial in Cn(X, A). Since the boundary map ∂ : Cn(X)→Cn−1(X) takes Cn(A) to Cn−1(A), it induces a quotient boundary map ∂ : Cn(X, A)→Cn−1(X, A). Letting n vary, we have a sequence of boundary maps ··· -→ Cn(X, A) ∂------------→ Cn−1(X, A) -→ ··· The relation ∂2 = 0 holds for these boundary maps since it holds before passing to quotient groups. So we have a chain complex, and the homology groups Ker ∂/ Im ∂ of this chain complex are by deﬁnition the relative homology groups Hn(X, A). By considering the de�
�nition of the relative boundary map we see: Elements of Hn(X, A) are represented by relative cycles: n chains α ∈ Cn(X) such that ∂α ∈ Cn−1(A). A relative cycle α is trivial in Hn(X, A) iﬀ it is a relative boundary: α = ∂β + γ for some β ∈ Cn+1(X) and γ ∈ Cn(A). These properties make precise the intuitive idea that Hn(X, A) is ‘homology of X modulo A ’. The quotient Cn(X)/Cn(A) could also be viewed as a subgroup of Cn(X), the n→X whose image is not consubgroup with basis the singular n simplices σ : tained in A. However, the boundary map does not take this subgroup of Cn(X) to the corresponding subgroup of Cn−1(X), so it is usually better to regard Cn(X, A) as a quotient rather than a subgroup of Cn(X). ∆ Our goal now is to show that the relative homology groups Hn(X, A) for any pair (X, A) ﬁt into a long exact sequence ··· -→ Hn(A) -→ Hn(X) -→ Hn(X, A) -→ Hn−1(A) -→ Hn−1(X) -→ ··· ··· -→ H0(X, A) -→ 0 116 Chapter 2 Homology This will be entirely a matter of algebra. To start the process, consider the diagram where i is inclusion and j is the quotient map. The diagram is commutative by the def- inition of the boundary maps. Letting n vary, and drawing these short exact sequences vertically rather than horizontally, we have a large commutative diagram of the form shown at the right, where the columns are exact and the rows are chain complexes which we denote A, B, and C. Such a diagram is called a short exact sequence of chain com- plexes. We will show that when we pass to homology groups, this short exact sequence of chain complexes stretches out into a long exact sequence of homol- ogy groups ··· -→ Hn(A)
i∗------------→ Hn(B) j∗------------→ Hn(C) ∂-----→ Hn−1(A) i∗------------→ Hn−1(B) -→ ··· where Hn(A) denotes the homology group Ker ∂/ Im ∂ at An in the chain complex A, and Hn(B) and Hn(C) are deﬁned similarly. The commutativity of the squares in the short exact sequence of chain complexes means that i and j are chain maps. These therefore induce maps i∗ and j∗ on homology. To deﬁne the boundary map ∂ : Hn(C)→Hn−1(A), let c ∈ Cn be a cycle. Since j is onto, c = j(b) for some b ∈ Bn. The element ∂b ∈ Bn−1 is in Ker j since j(∂b) = ∂j(b) = ∂c = 0. So ∂b = i(a) for some a ∈ An−1 since Ker j = Im i. Note that ∂a = 0 since i(∂a) = ∂i(a) = ∂∂b = 0 and i is injective. We deﬁne ∂ : Hn(C)→Hn−1(A) by sending the homology class of c to the homology class of a, ∂[c] = [a]. This is well-deﬁned since: The element a is uniquely determined by ∂b since i is injective. A diﬀerent choice b′ for b would have j(b′) = j(b), so b′ − b is in Ker j = Im i. Thus b′ − b = i(a′) for some a′, hence b′ = b + i(a′). The eﬀect of replacing b by b + i(a′) is to change a to the homologous element a + ∂a′ since i(a + ∂a′) = i(a) + i(∂a′) = ∂b + ∂i(a′) = ∂(b + i(a′)). A diﬀerent choice of c
within its homology class would have the form c + ∂c′. Since c′ = j(b′) for some b′, we then have c + ∂c′ = c + ∂j(b′) = c + j(∂b′) = j(b + ∂b′), so b is replaced by b + ∂b′, which leaves ∂b and therefore also a unchanged. Simplicial and Singular Homology Section 2.1 117 The map ∂ : Hn(C)→Hn−1(A) is a homomorphism since if ∂[c1] = [a1] and ∂[c2] = [a2] via elements b1 and b2 as above, then j(b1 + b2) = j(b1) + j(b2) = c1 + c2 and i(a1 + a2) = i(a1) + i(a2) = ∂b1 + ∂b2 = ∂(b1 + b2), so ∂([c1] + [c2]) = [a1] + [a2]. Theorem 2.16. The sequence of homology groups i∗------------→ Hn(B) ··· -→ Hn(A) j∗------------→ Hn(C) ∂-----→ Hn−1(A) i∗------------→ Hn−1(B) -→ ··· is exact. Proof: There are six things to verify: Im i∗ ⊂ Ker j∗. This is immediate since ji = 0 implies j∗i∗ = 0. Im j∗ ⊂ Ker ∂. We have ∂j∗ = 0 since in this case ∂b = 0 in the deﬁnition of ∂. Im ∂ ⊂ Ker i∗. Here i∗∂ = 0 since i∗∂ takes [c] to [∂b] = 0. Ker j∗ ⊂ Im i∗. A homology class in Ker j∗ is represented by a cycle b ∈ Bn with j(b) a boundary, so j(b) = ∂c′ for some c′ ∈ Cn+1. Since j is surjective,
c′ = j(b′) for some b′ ∈ Bn+1. We have j(b − ∂b′) = j(b) − j(∂b′) = j(b) − ∂j(b′) = 0 since ∂j(b′) = ∂c′ = j(b). So b − ∂b′ = i(a) for some a ∈ An. This a is a cycle since i(∂a) = ∂i(a) = ∂(b − ∂b′) = ∂b = 0 and i is injective. Thus i∗[a] = [b − ∂b′] = [b], showing that i∗ maps onto Ker j∗. Ker ∂ ⊂ Im j∗. In the notation used in the deﬁnition of ∂, if c represents a homology class in Ker ∂, then a = ∂a′ for some a′ ∈ An. The element b − i(a′) is a cycle since ∂(b − i(a′)) = ∂b − ∂i(a′) = ∂b − i(∂a′) = ∂b − i(a) = 0. And j(b − i(a′)) = j(b) − ji(a′) = j(b) = c, so j∗ maps [b − i(a′)] to [c]. Ker i∗ ⊂ Im ∂. Given a cycle a ∈ An−1 such that i(a) = ∂b for some b ∈ Bn, then ⊔⊓ j(b) is a cycle since ∂j(b) = j(∂b) = ji(a) = 0, and ∂ takes [j(b)] to [a]. This theorem represents the beginnings of the subject of homological algebra. The method of proof is sometimes called diagram chasing. Returning to topology, the preceding algebraic theorem yields a long exact se- quence of homology groups: ··· -→ Hn(A) i∗------------→ Hn(X) j∗------------→ Hn(X, A) ∂-----→ Hn−1(A) i�
�------------→ Hn−1(X) -→ ··· ··· -→ H0(X, A) -→ 0 The boundary map ∂ : Hn(X, A)→Hn−1(A) has a very simple description: If a class [α] ∈ Hn(X, A) is represented by a relative cycle α, then ∂[α] is the class of the cycle ∂α in Hn−1(A). This is immediate from the algebraic deﬁnition of the boundary homomorphism in the long exact sequence of homology groups associated to a short exact sequence of chain complexes. This long exact sequence makes precise the idea that the groups Hn(X, A) measure the diﬀerence between the groups Hn(X) and Hn(A). In particular, exactness 118 Chapter 2 Homology implies that if Hn(X, A) = 0 for all n, then the inclusion A֓X induces isomorphisms Hn(A) ≈ Hn(X) for all n, by the remark (iii) following the deﬁnition of exactness. The converse is also true according to an exercise at the end of this section. There is a completely analogous long exact sequence of reduced homology groups for a pair (X, A) with A ≠ ∅. This comes from applying the preceding algebraic ma- chinery to the short exact sequence of chain complexes formed by the short exact sequences 0→Cn(A)→Cn(X)→Cn(X, A)→0 in nonnegative dimensions, augmented by the short exact sequence 0 -→ Z 11-----→ Z -→ 0 -→ 0 in dimension −1. In particular this means that Hn(X, A) is the same as Hn(X, A) for all n, when A ≠ ∅. Example 2.17. In the long exact sequence of reduced homology groups for the pair e Hi−1(S n−1) are isomorphisms for all i > 0 (Dn, ∂Dn), the maps Hi(Dn, ∂Dn) Hi(Dn) are zero for all i. Thus we obtain the calculation since the remaining terms ∂-----→ e Hi(Dn, ∂Dn) ≈ e Z for i
= n 0 otherwise Example 2.18. Applying the long exact sequence of reduced homology groups to a pair (X, x0) with x0 ∈ X yields isomorphisms Hn(X, x0) ≈ Hn(X) for all n since Hn(x0) = 0 for all n. e There are induced homomorphisms for relative homology just as there are in the e nonrelative, or ‘absolute’, case. A map f : X→Y with f (A) ⊂ B, or more concisely f : (X, A)→(Y, B), induces homomorphisms f♯ : Cn(X, A)→Cn(Y, B) since the chain map f♯ : Cn(X)→Cn(Y ) takes Cn(A) to Cn(B), so we get a well-deﬁned map on quotients, f♯ : Cn(X, A)→Cn(Y, B). The relation f♯∂ = ∂f♯ holds for relative chains since it holds for absolute chains. By Proposition 2.9 we then have induced homomorphisms f∗ : Hn(X, A)→Hn(Y, B). Proposition 2.19. If two maps f, g : (X, A)→(Y, B) are homotopic through maps of pairs (X, A)→(Y, B), then f∗ = g∗ : Hn(X, A)→Hn(Y, B). Proof: The prism operator P from the proof of Theorem 2.10 takes Cn(A) to Cn+1(B), hence induces a relative prism operator P : Cn(X, A)→Cn+1(Y, B). Since we are just passing to quotient groups, the formula ∂P + P ∂ = g♯ − f♯ remains valid. Thus the maps f♯ and g♯ on relative chain groups are chain homotopic, and hence they induce ⊔⊓ the same homomorphism on relative homology groups. An easy generalization of the long exact sequence of a pair (X, A) is the long exact sequence of a triple (X, A, B), where
B ⊂ A ⊂ X : ··· -→ Hn(A, B) -→ Hn(X, B) -→ Hn(X, A) -→ Hn−1(A, B) -→ ··· This is the long exact sequence of homology groups associated to the short exact sequence of chain complexes formed by the short exact sequences 0 -→ Cn(A, B) -→ Cn(X, B) -→ Cn(X, A) -→ 0 Simplicial and Singular Homology Section 2.1 119 For example, taking B to be a point, the long exact sequence of the triple (X, A, B) becomes the long exact sequence of reduced homology for the pair (X, A). Excision A fundamental property of relative homology groups is given by the following Excision Theorem, describing when the relative groups Hn(X, A) are unaﬀected by deleting, or excising, a subset Z ⊂ A. Theorem 2.20. Given subspaces Z ⊂ A ⊂ X such that the closure of Z is contained in the interior of A, then the inclusion (X − Z, A − Z) ֓ (X, A) induces isomorphisms Hn(X − Z, A − Z)→Hn(X, A) for all n. Equivalently, for subspaces A, B ⊂ X whose interiors cover X, the inclusion (B, A ∩ B) ֓ (X, A) induces isomorphisms Hn(B, A ∩ B)→Hn(X, A) for all n. The translation between the two versions is obtained by setting B = X − Z and Z = X − B. Then A ∩ B = A − Z and the condition cl Z ⊂ int A is equivalent to X = int A ∪ int B since X − int B = cl Z. The proof of the excision theorem will involve a rather lengthy technical detour involving a construction known as barycentric subdivision, which allows homology groups to be computed using small singular simplices. In a metric space ‘smallness’ can be deﬁned in terms of diameters, but for general spaces it will be deﬁned in terms of covers. form an open cover of X, and
let C For a space X, let U = {Uj} be a collection of subspaces of X whose interiors U n (X) be the subgroup of Cn(X) consisting of i niσi such that each σi has image contained in some set in the cover U. The chains U boundary map ∂ : Cn(X)→Cn−1(X) takes C n (X) form a chain complex. We denote the homology groups of this chain complex by U n−1(X), so the groups C U n (X) to C P H U n (X). Proposition 2.21. The inclusion ι : C lence, that is, there is a chain map ρ : Cn(X)→C homotopic to the identity. Hence ι induces isomorphisms H U n (X) ֓ Cn(X) is a chain homotopy equivaU n (X) such that ιρ and ρι are chain U n (X) ≈ Hn(X) for all n. Proof: The barycentric subdivision process will be performed at four levels, beginning with the most geometric and becoming increasingly algebraic. i tivi with (1) Barycentric Subdivision of Simplices. The points of a simplex [v0, ···, vn] are the i ti = 1 and ti ≥ 0 for each i. The barycenter or linear combinations ‘center of gravity’ of the simplex [v0, ···, vn] is the point b = i tivi whose barycentric coordinates ti are all equal, namely ti = 1/(n + 1) for each i. The barycentric subdivision of [v0, ···, vn] is the decomposition of [v0, ···, vn] into the n simplices [b, w0, ···, wn−1] where, inductively, [w0, ···, wn−1] is an (n − 1) simplex in the P P P 120 Chapter 2 Homology vi, ···, vn]. The induction starts with the barycentric subdivision of a face [v0, ···, case n = 0 when the barycentric subdivision of [v0] is deﬁned to be just [v0] itself.
b The next two cases n = 1, 2 and part of the case n = 3 are shown in the ﬁgure. It follows from the inductive deﬁnition that the ver- tices of simplices in the barycentric subdivision of [v0, ···, vn] are exactly the barycenters of all ] of [v0, ···, vn] for 0 ≤ k ≤ n. When k = 0 this the k dimensional faces [vi0, ···, vik gives the original vertices vi since the barycenter of a 0 simplex is itself. The barycenter of [vi0, ···, vik ] has barycentric coordinates ti = 1/(k + 1) for i = i0, ···, ik and ti = 0 otherwise. The n simplices of the barycentric subdivision of do in fact form a complex structure on ∆ though we shall not need to know this in what follows. ∆ ∆ n, together with all their faces, n, indeed a simplicial complex structure, A fact we will need is that the diameter of each simplex of the barycentric subdivision of [v0, ···, vn] is at most n/(n+1) times the diameter of [v0, ···, vn]. Here the diameter of a simplex is by deﬁnition the maximum distance between any two of its points, and we are using the metric from the ambient Euclidean space Rm containing [v0, ···, vn]. The diameter of a simplex equals the maximum distance between any i tivi of [v0, ···, vn] of its vertices because the distance between two points v and satisﬁes the inequality P v − itivi = P P iti(v − vi) P P ≤ iti\|v − vi\| ≤ iti maxj\|v − vj\| = maxj\|v − vj\| To obtain the bound n/(n + 1) on the ratio of diameters, we therefore need to verify that the distance between any two vertices wj and wk of a simplex [w0, ···, wn] of the barycentric subdivision of [v0, ···, vn] is at most n/(n+1) times the diameter of [v0
, ···, vn]. If neither wj nor wk is the barycenter b of [v0, ···, vn], then these two points lie in a proper face of [v0, ···, vn] and we are done by induction on n. So we may suppose wj, say, is the barycenter b, and then by the previous displayed inequalvi, ···, vn], ity we may take wk to be a vertex vi. Let bi be the barycenter of [v0, ···, with all barycentric coordinates equal to 1/n except for ti = 0. Then we have b = 1 n+1 bi. The sum of the two coeﬃcients is 1, so b lies on the line segment [vi, bi] from vi to bi, and the distance from b to vi is n/(n + 1) times the length of [vi, bi]. Hence the distance from b to vi is bounded by n/(n + 1) times the diameter of [v0, ···, vn]. n+1 vi + n b The signiﬁcance of the factor n/(n+1) is that by repeated barycentric subdivision r approaches we can produce simplices of arbitrarily small diameter since n/(n+1) Simplicial and Singular Homology Section 2.1 121 0 as r goes to inﬁnity. It is important that the bound n/(n + 1) does not depend on the shape of the simplex since repeated barycentric subdivision produces simplices of many diﬀerent shapes. (2) Barycentric Subdivision of Linear Chains. The main part of the proof will be to construct a subdivision operator S : Cn(X)→Cn(X) and show this is chain homotopic to the identity map. First we will construct S and the chain homotopy in a more restricted linear setting. For a convex set Y in some Euclidean space, the linear maps n→Y generate a subgroup of Cn(Y ) that we denote LCn(Y ), the linear chains. The boundary map ∂ : Cn(Y )→Cn−1(Y ) takes LCn(Y ) to LCn−1(Y ), so the linear chains form a subcomplex of the singular chain complex of Y
. We can uniquely designate a linear map n→Y by [w0, ···, wn] where wi is the image under λ of the ith vertex of n. To avoid having to make exceptions for 0 simplices it will be convenient to augment the complex LC(Y ) by setting LC−1(Y ) = Z generated by the empty simplex [∅], with ∂[w0] = [∅] for all 0 simplices [w0]. λ : ∆ ∆ ∆ Each point b ∈ Y determines a homomorphism b : LCn(Y )→LCn+1(Y ) deﬁned on basis elements by b([w0, ···, wn]) = [b, w0, ···, wn]. Geometrically, the homomorphism b can be regarded as a cone operator, sending a linear chain to the cone having the linear chain as the base of the cone and the point b as the tip of the cone. Applying the usual formula for ∂, we obtain the relation ∂b([w0, ···, wn]) = [w0, ···, wn] − b(∂[w0, ···, wn]). By linearity it follows that ∂b(α) = α − b(∂α) for all α ∈ LCn(Y ). This expresses algebraically the geometric fact that the boundary of a cone consists of its base together with the cone on the boundary of its base. The relation ∂b(α) = α−b(∂α) can be rewritten as ∂b +b∂ = 11, so b is a chain homotopy on n. Let λ : between the identity map and the zero map on the augmented chain complex LC(Y ). Now we deﬁne a subdivision homomorphism S : LCn(Y )→LCn(Y ) by induction n→Y be a generator of LCn(Y ) and let bλ be the image of the n under λ. Then the inductive formula for S is S(λ) = bλ(S∂λ) barycenter of ∆ where bλ : LCn−1(Y )→LCn(Y ) is the cone operator deﬁned in the preceding paragraph. The induction starts with S
([∅]) = [∅], so S is the identity on LC−1(Y ). It is also the identity on LC0(Y ), since when n = 0 the formula for S becomes S([w0]) = w0(S∂[w0]) = w0(S([∅])) = w0([∅]) = [w0]. When λ is an embedding, with image a genuine n simplex [w0, ···, wn], then S(λ) is the sum of the n simplices in the barycentric subdivision of [w0, ···, wn], with certain signs that could be computed explicitly. This is apparent by comparing the inductive deﬁnition ∆ of S with the inductive deﬁnition of the barycentric subdivision of a simplex. Let us check that the maps S satisfy ∂S = S∂, and hence give a chain map from the chain complex LC(Y ) to itself. Since S = 11 on LC0(Y ) and LC−1(Y ), we certainly have ∂S = S∂ on LC0(Y ). The result for larger n is given by the following calculation, in which we omit some parentheses to unclutter the formulas: 122 Chapter 2 Homology ∂Sλ = ∂bλ(S∂λ) = S∂λ − bλ∂(S∂λ) since ∂bλ = 11 − bλ∂ = S∂λ − bλS(∂∂λ) since ∂S(∂λ) = S∂(∂λ) by induction on n = S∂λ since ∂∂ = 0 We next build a chain homotopy T : LCn(Y )→LCn+1(Y ) between S and the iden- tity, ﬁtting into a diagram We deﬁne T on LCn(Y ) inductively by setting T = 0 for n = −1 and letting T λ = bλ(λ − T ∂λ) for n ≥ 0. The geometric motivation for this formula is an inductively deﬁned subdivision of to the barycenter of joining all simplices in n × I obtained by n × I n × {0} ∪ �
� ∆ n × {1}, as indicated in the ﬁgure in the case n = 2. What T ∆ actually does is take the image of this subn. division under the projection n × I→ ∆ ∆ The chain homotopy formula ∂T + T ∂ = 11 − S is trivial on LC−1(Y ) where T = 0 and S = 11. Verifying the formula on LCn(Y ) with n ≥ 0 is done by the calculation ∆ ∆ ∂T λ = ∂bλ(λ − T ∂λ) = λ − T ∂λ − bλ∂(λ − T ∂λ) since ∂bλ = 11 − bλ∂ = λ − T ∂λ − bλ ∂λ − ∂T (∂λ) = λ − T ∂λ − bλ S(∂λ) + T ∂(∂λ) by induction on n = λ − T ∂λ − Sλ since ∂∂ = 0 and Sλ = bλ(S∂λ) Now we can discard the group LC−1(Y ) and the relation ∂T + T ∂ = 11 − S still holds since T was zero on LC−1(Y ). (3) Barycentric Subdivision of General Chains. Deﬁne S : Cn(X)→Cn(X) by setting n→X. Since S n for a singular n simplex σ : n is the sum of the Sσ = σ♯S n, with certain signs, Sσ is the corren simplices in the barycentric subdivision of sponding signed sum of the restrictions of σ to the n simplices of the barycentric ∆ ∆ ∆ subdivision of n. The operator S is a chain map since ∆ ∂Sσ = ∂σ♯S ∆ = σ♯S n i n = σ♯∂S i(−1)i ∆ i(−1)iσ♯S P i(−1)iS(σ \|\| ∆ i(−1)iσ \|\| ♯S∂ where ∆ n i is the ith face of n ∆ ∆ = S(∂σ ) Simplicial and Singular Homology
Section 2.1 123 In similar fashion we deﬁne T : Cn(X)→Cn+1(X) by T σ = σ♯T n, and this gives a chain homotopy between S and the identity, since the formula ∂T + T ∂ = 11 − S holds by the calculation ∂T σ = ∂σ♯T n = σ♯∂T n = σ♯( ) = σ − Sσ − σ♯T ∂ = σ − Sσ − T (∂σ ) ∆ n ∆ ∆ where the last equality follows just as in the previous displayed calculation, with S replaced by T. (4) Iterated Barycentric Subdivision. A chain homotopy between 11 and the iterate Sm is given by the operator Dm = 0≤i<m T S i since ∂Dm + Dm∂ = P ∂T S i + T S i∂ = ∂T S i + T ∂S i = X0≤i<m S i = ∂T + T ∂ 11 − S X0≤i<m S i = S i − S i+1 X0≤i<m X0≤i<m X0≤i<m = 11 − Sm For each singular n simplex σ : U n (X) since the diameter of the simplices of Sm( C number of the cover of n→X there exists an m such that Sm(σ ) lies in n) will be less than a Lebesgue n by the open sets σ −1(int Uj) if m is large enough. (Recall that a Lebesgue number for an open cover of a compact metric space is a number ∆ ∆ ε > 0 such that every set of diameter less than ε lies in some set of the cover; such a ∆ number exists by an elementary compactness argument.) We cannot expect the same number m to work for all σ ’s, so let us deﬁne m(σ ) to be the smallest m such that Smσ is in C U n (X). n simplex σ : We now deﬁne D : Cn(X)→Cn+1(X) by setting Dσ = Dm(
σ )σ for each singular n→X. For this D we would like to ﬁnd a chain map ρ : Cn(X)→Cn(X) U n (X) satisfying the chain homotopy equation with image in C ∆ (∗) ∂D + D∂ = 11 − ρ A quick way to do this is simply to regard this equation as deﬁning ρ, so we let ρ = 11 − ∂D − D∂. It follows easily that ρ is a chain map since ∂ρ(σ ) = ∂σ − ∂2Dσ − ∂D∂σ = ∂σ − ∂D∂σ and ρ(∂σ ) = ∂σ − ∂D∂σ − D∂2σ = ∂σ − ∂D∂σ To check that ρ takes Cn(X) to C U n (X) we compute ρ(σ ) more explicitly: ρ(σ ) = σ − ∂Dσ − D(∂σ ) = σ − ∂Dm(σ )σ − D(∂σ ) = Sm(σ )σ + Dm(σ )(∂σ ) − D(∂σ ) U The term Sm(σ )σ lies in C n (X) by the deﬁnition of m(σ ). The remaining terms Dm(σ )(∂σ ) − D(∂σ ) are linear combinations of terms Dm(σ )(σj ) − Dm(σj )(σj) for σj n, so m(σj ) ≤ m(σ ) and hence the diﬀerence the restriction of σ to a face of since ∂Dm + Dm∂ = 11 − Sm ∆ 124 Chapter 2 Homology U n (X) since T takes C U n−1(X) to C Viewing ρ as a chain map Cn(X)→C Dm(σ )(σj ) − Dm(σj )(σj) consists of terms T S i(σj) with i ≥ m(σj), and these terms U n (X). lie in C U n (X), the equation (∗) says that ∂D
+ D∂ = U n (X)֓Cn(X) the inclusion. Furthermore, ρι = 11 since D is identically U n (X), hence the summation deﬁning Dσ is ⊔⊓ empty. Thus we have shown that ρ is a chain homotopy inverse for ι. U n (X), as m(σ ) = 0 if σ is in C 11−ιρ for ι : C zero on C Proof of the Excision Theorem: We prove the second version, involving a decomposition X = A ∪ B. For the cover U = {A, B} we introduce the suggestive notation U Cn(A + B) for C n (X), the sums of chains in A and chains in B. At the end of the preceding proof we had formulas ∂D + D∂ = 11 − ιρ and ρι = 11. All the maps ap- pearing in these formulas take chains in A to chains in A, so they induce quotient maps when we factor out chains in A. These quotient maps automatically satisfy the same two formulas, so the inclusion Cn(A + B)/Cn(A) ֓ Cn(X)/Cn(A) induces an isomorphism on homology. The map Cn(B)/Cn(A ∩ B)→Cn(A + B)/Cn(A) induced by inclusion is obviously an isomorphism since both quotient groups are free with basis the singular n simplices in B that do not lie in A. Hence we obtain the desired isomorphism Hn(B, A ∩ B) ≈ Hn(X, A) induced by inclusion. ⊔⊓ All that remains in the proof of Theorem 2.13 is to replace relative homology groups with absolute homology groups. This is achieved by the following result. Proposition 2.22. For good pairs (X, A), the quotient map q : (X, A)→(X/A, A/A) induces isomorphisms q∗ : Hn(X, A)→Hn(X/A, A/A) ≈ Proof: Let V be a neighborhood of A in X that deformation retracts onto A. We have a commutative diagram Hn(X/A)
for all n. e The upper left horizontal map is an isomorphism since in the long exact sequence of the triple (X, V, A) the groups Hn(V, A) are zero for all n, because a deformation retraction of V onto A gives a homotopy equivalence of pairs (V, A) ≃ (A, A), and Hn(A, A) = 0. The deformation retraction of V onto A induces a deformation retraction of V /A onto A/A, so the same argument shows that the lower left horizontal map is an isomorphism as well. The other two horizontal maps are isomorphisms directly from excision. The right-hand vertical map q∗ is an isomorphism since q restricts to a homeomorphism on the complement of A. From the commutativity of the diagram it follows that the left-hand q∗ is an isomorphism. ⊔⊓ This proposition shows that relative homology can be expressed as reduced abso- lute homology in the case of good pairs (X, A), but in fact there is a way of doing this Simplicial and Singular Homology Section 2.1 125 for arbitrary pairs. Consider the space X ∪ CA where CA is the cone (A× I)/(A× {0}) whose base A× {1} we identify with A ⊂ X. Using terminology introduced in Chapter 0, X ∪CA can also be described as the mapping cone of the inclusion A ֓ X. The assertion is that Hn(X, A) Hn(X ∪ CA) for all n via the sequence of isois isomorphic to morphisms e Hn(X ∪ CA) ≈ Hn(X ∪ CA, CA) ≈ Hn(X ∪ CA − {p}, CA − {p}) ≈ Hn(X, A) where p ∈ CA is the tip of the cone. The ﬁrst isomorphism comes from the exact e sequence of the pair, using the fact that CA is contractible. The second isomorphism is excision, and the third comes from a deformation retraction of CA − {p} onto A. Here is an application of the preceding proposition: n, ∂ n), we will show by induction on n that the identity map in : Example 2.23.
Let us ﬁnd explicit cycles representing generators of the inﬁnite Hn(S n). Replacing (Dn, ∂Dn) by the equivalent pair cyclic groups Hn(Dn, ∂Dn) and n→ n, viewed ( n). That it is a cycle is clear as a singular n simplex, is a cycle generating Hn( ∆ since we are considering relative homology. When n = 0 it certainly represents a n be the union of all but one of the generator. For the induction step, let nn − 1) dimensional faces of n. Then we claim there are isomorphisms Hn( n, ∂ n) ≈------------→ Hn−1(∂ ∆ Λ n, ∆ ≈←------------- Hn−1( ) n−1, ∂ n−1, ∆ n, n, ∂ onto ∆ i : ∆ ), whose third terms Hi( ) ≃ ( The ﬁrst isomorphism is a boundary map in the long exact sequence of the triple n deformation retracts ). The second isomorphism is induced by the inclusion. When n = 1, i induces an isomorphism n−1, Λ n as the face not contained in Λ n,, hence ( Λ ∆ n−1→∂ Λ Λ ∆ ) are zero since on relative homology since this is true already at the chain level. When n > 1, ∂ n−1/∂ n−1 ≈ ∂ quotients sent under the ﬁrst isomorphism to the cycle ∂in which equals ±in−1 in Cn−1(∂ Λ Hn(S n) let us regard S n as two n simplices Λ is nonempty so we are dealing with good pairs and i induces a homeomorphism of. The induction step then follows since the cycle in is n, ). n 1 and ∆ Λ n 2 with their boundaries identiﬁed in the obvious way, preserving the ordering of e n 2, viewed as a singular n chain, is then a cycle, and we Hn(S n). To see this, consider the isomorphisms vertices. The diﬀerence ∆ claim it represents a generator of ∆ To ﬁnd a cycle
generating n 1 − n/ ∆ ∆ ∆ ∆ ∆ ∆ Hn(S n) ∆ ≈------------→ Hn(S n, e n 2 ) ≈←------------- Hn( n 1, ∂ n 1 ) e where the ﬁrst isomorphism comes from the long exact sequence of the pair (S n, n 2 ) ∆ and the second isomorphism is justiﬁed in the nontrivial cases n > 0 by passing to n 2 in the ﬁrst group n 1 in the third group, which represents a generator of this n 1 − quotients as before. Under these isomorphisms the cycle ∆ n 2 represents a generator of group as we have seen, so corresponds to the cycle ∆ Hn(S n 126 Chapter 2 Homology The preceding proposition implies that the excision property holds also for sub- complexes of CW complexes: Corollary 2.24. If the CW complex X is the union of subcomplexes A and B, then the inclusion (B, A ∩ B)֓ (X, A) induces isomorphisms Hn(B, A ∩ B)→Hn(X, A) for all n. Proof: Since CW pairs are good, Proposition 2.22 allows us to pass to the quotient spaces B/(A ∩ B) and X/A which are homeomorphic, assuming we are not in the ⊔⊓ trivial case A ∩ B = ∅. Here is another application of the preceding proposition: Corollary 2.25. For a wedge sum Hn(Xα)→ α iα∗ : morphism at basepoints xα ∈ Xα such that the pairs (Xα, xα) are good. α Xα, the inclusions iα : Xα֓ Hn( W α Xα induce an isoα Xα), provided that the wedge sum is formed α L L W W e e Proof: Since reduced homology is the same as homology relative to a basepoint, this ⊔⊓ follows from the proposition by taking (X, A) = ( αXα, α{xα}). ` ` Here is an application of the machinery we have developed, a classical result of Brouwer from around 1910 known as ‘invariance of dimension
’, which says in particular that Rm is not homeomorphic to Rn if m ≠ n. Theorem 2.26. If nonempty open sets U ⊂ Rm and V ⊂ Rn are homeomorphic, then m = n. Proof: For x ∈ U we have Hk(U, U − {x}) ≈ Hk(Rm, Rm − {x}) by excision. From the long exact sequence for the pair (Rm, Rm − {x}) we get Hk(Rm, Rm − {x}) ≈ Hk−1(Rm − {x}). Since Rm − {x} deformation retracts onto a sphere Sm−1, we conclude that Hk(U, U − {x}) is Z for k = m and 0 otherwise. By the same reasoning, e Hk(V, V − {y}) is Z for k = n and 0 otherwise. Since a homeomorphism h : U→V induces isomorphisms Hk(U, U − {x})→Hk(V, V − {h(x)}) for all k, we must have ⊔⊓ m = n. Generalizing the idea of this proof, the local homology groups of a space X at a point x ∈ X are deﬁned to be the groups Hn(X, X −{x}). For any open neighborhood U of x, excision gives isomorphisms Hn(X, X − {x}) ≈ Hn(U, U − {x}) assuming points are closed in X, and thus the groups Hn(X, X − {x}) depend only on the local topology of X near x. A homeomorphism f : X→Y must induce isomorphisms Hn(X, X − {x}) ≈ Hn(Y, Y − {f (x)}) for all x and n, so the local homology groups can be used to tell when spaces are not locally homeomorphic at certain points, as in the preceding proof. The exercises give some further examples of this. Simplicial and Singular Homology Section 2.1 127 Naturality The exact sequences we have been constructing have an extra property that will become important later at key points in many arguments, though at ﬁrst glance this property may seem just an idle technical
ity, not very interesting. We shall discuss the property now rather than interrupting later arguments to check it when it is needed, but the reader may prefer to postpone a careful reading of this discussion. The property is called naturality. For example, to say that the long exact sequence of a pair is natural means that for a map f : (X, A)→(Y, B), the diagram is commutative. Commutativity of the squares involving i∗ and j∗ follows from the obvious commutativity of the corresponding squares of chain groups, with Cn in place of Hn. For the other square, when we deﬁned induced homomorphisms we saw that f♯∂ = ∂f♯ at the chain level. Then for a class [α] ∈ Hn(X, A) represented by a relative cycle α, we have f∗∂[α] = f∗[∂α] = [f♯∂α] = [∂f♯α] = ∂[f♯α] = ∂f∗[α]. Alternatively, we could appeal to the general algebraic fact that the long exact sequence of homology groups associated to a short exact sequence of chain complexes is natural: For a commutative diagram of short exact sequences of chain complexes the induced diagram ∗α∗ and γj = j′β implies γ∗j∗ = j′ is commutative. Commutativity of the ﬁrst two squares is obvious since βi = i′α implies β∗i∗ = i′ ∗β∗. For the third square, recall that the map ∂ : Hn(C)→Hn−1(A) was deﬁned by ∂[c] = [a] where c = j(b) and i(a) = ∂b. Then ∂[γ(c)] = [α(a)] since γ(c) = γj(b) = j′(β(b)) and i′(α(a)) = βi(a) = β∂(b) = ∂β(b). Hence ∂γ∗[c] = α∗[a] = α∗∂[c]. 128 Chapter 2 Homology This algebraic
fact also implies naturality of the long exact sequence of a triple and the long exact sequence of reduced homology of a pair. Finally, there is the naturality of the long exact sequence in Theorem 2.13, that is, commutativity of the diagram where i and q denote inclusions and quotient maps, and f : X/A→Y /B is induced by f. The ﬁrst two squares commute since f i = if and f q = qf. The third square expands into We have already shown commutativity of the ﬁrst and third squares, and the second square commutes since f q = qf. The Equivalence of Simplicial and Singular Homology We can use the preceding results to show that the simplicial and singular homol- ogy groups of complexes are always isomorphic. For the proof it will be convenient to consider the relative case as well, so let X be a complex with A ⊂ X a sub- chains groups H ∆ n(X)/ complex. Thus A is the complex formed by any union of simplices of X. Relative n(X, A) can be deﬁned in the same way as for singular homology, via relative n(X, A) = ∆ n(A), and this yields a long exact sequence of simplicial homology groups for the pair (X, A) by the same algebraic argument as for singular n(X, A)→Hn(X, A) induced by the n(X, A)→Cn(X, A) sending each n simplex of X to its characteristic n→X. The possibility A = ∅ is not excluded, in which case the relative homology. There is a canonical homomorphism H chain map map σ : ∆ ∆ ∆ ∆ ∆ ∆ groups reduce to absolute groups. ∆ ∆ Theorem 2.27. The homomorphisms H all n and all complex pairs (X, A). n(X, A)→Hn(X, A) are isomorphisms for ∆ Proof: First we do the case that X is ﬁnite-dimensional and A is empty. For X k the k skeleton of X, consisting of all simplices of dimension k or less, we have a ∆ commutative diagram of exact sequences: Simplicial and Singular Homology Section
2.1 129 ∆ istic maps The simplicial chain group Let us ﬁrst show that the ﬁrst and fourth vertical maps are isomorphisms for all n. n(X k, X k−1) is zero for n ≠ k, and is free abelian with n(X k, X k−1) has exactly the same basis the k simplices of X when n = k. Hence H description. The corresponding singular homology groups Hn(X k, X k−1) can be comα)→(X k, X k−1) formed by the character: puted by considering the map k→X for all the k simplices of X. Since induces a homeomorphism ` α ≈ X k/X k−1, it induces isomorphisms on all singuk of quotient spaces lar homology groups. Thus Hn(X k, X k−1) is zero for n ≠ k, while for n = k this group is free abelian with basis represented by the relative cycles given by the chark) is acteristic maps of all the k simplices of X, in view of the fact that Hk( k, as we showed in Example 2.23. Therefore the generated by the identity map Φ α∂ k α, ∂ k, ∂ k α/ α→ map H k (X k, X k−1)→Hk(X k, X k−1) is an isomorphism. ∆ ∆ By induction on k we may assume the second and ﬁfth vertical maps in the pre- ∆ ∆ ∆ ceding diagram are isomorphisms as well. The following frequently quoted basic alge- braic lemma will then imply that the middle vertical map is an isomorphism, ﬁnishing the proof when X is ﬁnite-dimensional and A = ∅. The Five-Lemma. In a commutative diagram of abelian groups as at the right, if the two rows are exact and α, β, δ, and ε are isomorphisms, then γ is an isomorphism also. Proof: It suﬃces to show: (a) γ is surjective if β and δ are surjective and ε is injective. (b) γ is injective
if β and δ are injective and α is surjective. The proofs of these two statements are straightforward diagram chasing. There is really no choice about how the argument can proceed, and it would be a good exercise for the reader to close the book now and reconstruct the proofs without looking. To prove (a), start with an element c′ ∈ C ′. Then k′(c′) = δ(d) for some d ∈ D since δ is surjective. Since ε is injective and εℓ(d) = ℓ′δ(d) = ℓ′k′(c′) = 0, we deduce that ℓ(d) = 0, hence d = k(c) for some c ∈ C by exactness of the upper row. The diﬀerence c′ − γ(c) maps to 0 under k′ since k′(c′) − k′γ(c) = k′(c′) − δk(c) = k′(c′) − δ(d) = 0. Therefore c′ − γ(c) = j′(b′) for some b′ ∈ B′ by exactness. Since β is surjective, b′ = β(b) for some b ∈ B, and then γ(c + j(b)) = γ(c) + γj(b) = γ(c) + j′β(b) = γ(c) + j′(b′) = c′, showing that γ is surjective. To prove (b), suppose that γ(c) = 0. Since δ is injective, δk(c) = k′γ(c) = 0 implies k(c) = 0, so c = j(b) for some b ∈ B. The element β(b) satisﬁes j′β(b) = γj(b) = γ(c) = 0, so β(b) = i′(a′) for some a′ ∈ A′. Since α is surjective, a′ = α(a) for some a ∈ A. Since β is injective, β(i(a) − b) = βi(a) − β
(b) = i′α(a) − β(b) = i′(a′)−β(b) = 0 implies i(a)−b = 0. Thus b = i(a), and hence c = j(b) = ji(a) = 0 ⊔⊓ since ji = 0. This shows γ has trivial kernel. 130 Chapter 2 Homology Returning to the proof of the theorem, we next consider the case that X is inﬁnite- dimensional, where we will use the following fact: A compact set in X can meet only ﬁnitely many open simplices of X, that is, simplices with their proper faces deleted. This is a general fact about CW complexes proved in the Appendix, but here is a direct proof for complexes. If a compact set C intersected inﬁnitely many open simplices, it would contain an inﬁnite sequence of points xi each lying in a diﬀerent ∆ j≠i{xj}, which are open since their preimages open simplex. Then the sets Ui = X − under the characteristic maps of all the simplices are clearly open, form an open cover S of C with no ﬁnite subcover. This can be applied to show the map H n(X)→Hn(X) is surjective. Represent a given element of Hn(X) by a singular n cycle z. This is a linear combination of ﬁnitely many singular simplices with compact images, meeting only ﬁnitely many open simn(X k)→Hn(X k) plices of X, hence contained in X k for some k. We have shown that H is an isomorphism, in particular surjective, so z is homologous in X k (hence in X ) to a simplicial cycle. This gives surjectivity. Injectivity is similar: If a simplicial n cycle ∆ ∆ z is the boundary of a singular chain in X, this chain has compact image and hence n(X k)→Hn(X k). must lie in some X k, so z represents an element of the kernel of H But we know this map is injective, so z is a simplicial boundary in X k, and therefore in X. ∆ It remains to
do the case of arbitrary X with A ≠ ∅, but this follows from the absolute case by applying the ﬁve-lemma to the canonical map from the long exact sequence of simplicial homology groups for the pair (X, A) to the corresponding long exact sequence of singular homology groups. ⊔⊓ is a We can deduce from this theorem that Hn(X) is ﬁnitely generated whenever X complex with ﬁnitely many n simplices, since in this case the simplicial chain n(X) is ﬁnitely generated, hence also its subgroup of cycles and therefore also the latter group’s quotient H n(X). If we write Hn(X) as the direct sum of cyclic groups, then the number of Z summands is known traditionally as the nth Betti ∆ group ∆ ∆ number of X, and integers specifying the orders of the ﬁnite cyclic summands are called torsion coeﬃcients. It is a curious historical fact that homology was not thought of originally as a sequence of groups, but rather as Betti numbers and torsion coeﬃcients. One can after all compute Betti numbers and torsion coeﬃcients from the simplicial boundary maps without actually mentioning homology groups. This computational viewpoint, with homology being numbers rather than groups, prevailed from when Poincar´e ﬁrst started serious work on homology around 1900, up until the 1920s when the more abstract viewpoint of groups entered the picture. During this period ‘homology’ meant primarily ‘simplicial homology’, and it was another 20 years before the shift to singular homology was complete, with the ﬁnal deﬁnition of singular homology emerging only Simplicial and Singular Homology Section 2.1 131 in a 1944 paper of Eilenberg, after contributions from quite a few others, particularly Alexander and Lefschetz. Within the next few years the rest of the basic structure of homology theory as we have presented it fell into place, and the ﬁrst deﬁnitive treatment appeared in the classic book [Eilenberg & Steenrod 1952]. Exercises 1. What familiar space is the quotient by identifying the edges [v0, v1
] and [v1, v2], preserving the ordering of vertices? complex of a 2 simplex [v0, v1, v2] obtained 3 by performing the order-preserving 2. Show that the edge identiﬁcations [v0, v1] ∼ [v1, v3] and [v0, v2] ∼ [v2, v3] deformation retracts onto a Klein bottle. Also, ﬁnd other pairs of identiﬁcations of edges that produce complex obtained from ∆ ∆ ∆ complexes deformation retracting onto a torus, a 2 sphere, and RP2. complex structure 3. Construct a ∆ on S n having vertices the two vectors of length 1 along each coordinate axis in Rn+1. complex structure on RPn as a quotient of a ∆ ∆ 4. Compute the simplicial homology groups of the triangular parachute obtained from 2 by identifying its three vertices to a single point. 5. Compute the simplicial homology groups of the Klein bottle using the ∆ structure described at the beginning of this section. complex ∆ 6. Compute the simplicial homology groups of the 2 0, ···, 2 simplices i > 0 identifying the edges [v0, v1] and [v1, v2] of [v0, v2] to the edge [v0, v1] of 2 n by identifying all three edges of 2 i−1. ∆ ∆ ∆ complex obtained from n + 1 2 0 to a single edge, and for ∆ 2 i to a single edge and the edge complex structure 7. Find a way of identifying pairs of faces of on S 3 having a single 3 simplex, and compute the simplicial homology groups of this ∆ 3 to produce a ∆ ∆ ∆ complex. 8. Construct a 3 dimensional complex X from n tetrahe∆ dra T1, ···, Tn by the following two steps. First arrange the ∆ tetrahedra in a cyclic pattern as in the ﬁgure, so that each Ti shares a common vertical face with its two neighbors Ti−1 and Ti+1, subscripts being taken mod n. Then identify the bottom face of Ti with the top face of Ti+1 for each i. Show the simplicial homology groups of X in dimensions 0,
1, 2, 3 are Z, Zn, 0, Z, respectively. [The space X is an example of a lens space; see Example 2.43 for the general case.] n by identi9. Compute the homology groups of the fying all faces of the same dimension. Thus X has a single k simplex for each k ≤ n. ∆ 10. (a) Show the quotient space of a ﬁnite collection of disjoint 2 simplices obtained by identifying pairs of edges is always a surface, locally homeomorphic to R2. (b) Show the edges can always be oriented so as to deﬁne a complex X obtained from complex structure on ∆ the quotient surface. [This is more diﬃcult.] ∆ 132 Chapter 2 Homology 11. Show that if A is a retract of X then the map Hn(A)→Hn(X) induced by the inclusion A ⊂ X is injective. 12. Show that chain homotopy of chain maps is an equivalence relation. 13. Verify that f ≃ g implies f∗ = g∗ for induced homomorphisms of reduced homology groups. 14. Determine whether there exists a short exact sequence 0→Z4→Z8 ⊕ Z2→Z4→0. More generally, determine which abelian groups A ﬁt into a short exact sequence 0→Zpm→A→Zpn→0 with p prime. What about the case of short exact sequences 0→Z→A→Zn→0? 15. For an exact sequence A→B→C→D→E show that C = 0 iﬀ the map A→B is surjective and D→E is injective. Hence for a pair of spaces (X, A), the inclusion A ֓ X induces isomorphisms on all homology groups iﬀ Hn(X, A) = 0 for all n. 16. (a) Show that H0(X, A) = 0 iﬀ A meets each path-component of X. (b) Show that H1(X, A) = 0 iﬀ H1(A)→H1(X) is surjective and each path-component of X contains at most one path-component of A. 17. (a) Compute the homology groups
Hn(X, A) when X is S 2 or S 1 × S 1 and A is a ﬁnite set of points in X. (b) Compute the groups Hn(X, A) and Hn(X, B) for X a closed orientable surface of genus two with A and B the circles shown. [What are X/A and X/B?] 18. Show that for the subspace Q ⊂ R, the relative homology group H1(R, Q) is free abelian and ﬁnd a basis. 19. Compute the homology groups of the subspace of I × I consisting of the four boundary edges plus all points in the interior whose ﬁrst coordinate is rational. 20. Show that Hn+1(SX) for all n, where SX is the suspension of X. More generally, thinking of SX as the union of two cones CX with their bases identiﬁed, Hn(X) ≈ compute the reduced homology groups of the union of any ﬁnite number of cones e e CX with their bases identiﬁed. 21. Making the preceding problem more concrete, construct explicit chain maps s : Cn(X)→Cn+1(SX) inducing isomorphisms Hn(X)→ Hn+1(SX). 22. Prove by induction on dimension the following facts about the homology of a ﬁnite-dimensional CW complex X, using the observation that X n/X n−1 is a wedge sum of n spheres: (a) If X has dimension n then Hi(X) = 0 for i > n and Hn(X) is free. (b) Hn(X) is free with basis in bijective correspondence with the n cells if there are e e no cells of dimension n − 1 or n + 1. (c) If X has k n cells, then Hn(X) is generated by at most k elements. Simplicial and Singular Homology Section 2.1 133 23. Show that the second barycentric subdivision of a complex is a simplicial complex. Namely, show that the ﬁrst barycentric subdivision produces a complex with the property that each simplex has all its vertices distinct, then show that for a ∆ ∆ complex with this property
, barycentric subdivision produces a simplicial complex. 24. Show that each n simplex in the barycentric subdivision of ∆ inequalities ti0 permutation of (0, ···, n). n is deﬁned by n in its barycentric coordinates, where (i0, ···, in) is a ≤ ··· ≤ tin ≤ ti1 ∆ H1(X/A) if X = [0, 1] and A is the 25. Find an explicit, noninductive formula for the barycentric subdivision operator S : Cn(X)→Cn(X). 26. Show that H1(X, A) is not isomorphic to sequence 1, 1/2, 1/3, ··· together with its limit 0. [See Example 1.25.] 27. Let f : (X, A)→(Y, B) be a map such that both f : X→Y and the restriction f : A→B are homotopy equivalences. (a) Show that f∗ : Hn(X, A)→Hn(Y, B) is an isomorphism for all n. (b) For the case of the inclusion f : (Dn, S n−1) ֓ (Dn, Dn − {0}), show that f is not a homotopy equivalence of pairs — there is no g : (Dn, Dn − {0})→(Dn, S n−1) such that f g and gf are homotopic to the identity through maps of pairs. [Observe that a homotopy equivalence of pairs (X, A)→(Y, B) is also a homotopy equivalence for the pairs obtained by replacing A and B by their closures.] e 3 to the barycenter of 28. Let X be the cone on the 1 skeleton of 3, the union of all line segments joining 3. Compute the local homology points in the six edges of groups Hn(X, X − {x}) for all x ∈ X. Deﬁne ∂X to be the subspace of points x such that Hn(X, X − {x}) = 0 for all n, and compute the local homology groups Hn(∂X, ∂X − {x}). Use these calculations to determine which subsets A �
�� X have the property that f (A) ⊂ A for all homeomorphisms f : X→X. 29. Show that S 1 × S 1 and S 1 ∨ S 1 ∨ S 2 have isomorphic homology groups in all dimensions, but their universal covering spaces do not. ∆ ∆ ∆ 30. In each of the following commutative diagrams assume that all maps but one are isomorphisms. Show that the remaining map must be an isomorphism as well. 31. Using the notation of the ﬁve-lemma, give an example where the maps α, β, δ, and ε are zero but γ is nonzero. This can be done with short exact sequences in which all the groups are either Z or 0. 134 Chapter 2 Homology Now that the basic properties of homology have been established, we can begin to move a little more freely. Our ﬁrst topic, exploiting the calculation of Hn(S n), is Brouwer’s notion of degree for maps S n→S n. Historically, Brouwer’s introduction of this concept in the years 1910–12 preceded the rigorous development of homology, so his deﬁnition was rather diﬀerent, using the technique of simplicial approximation which we explain in §2.C. The later deﬁnition in terms of homology is certainly more elegant, though perhaps with some loss of geometric intuition. More in the spirit of Brouwer’s deﬁnition is a third approach using diﬀerential topology, presented very lucidly in [Milnor 1965]. Degree For a map f : S n→S n with n > 0, the induced map f∗ : Hn(S n)→Hn(S n) is a homomorphism from an inﬁnite cyclic group to itself and so must be of the form f∗(α) = dα for some integer d depending only on f. This integer is called the degree of f, with the notation deg f. Here are some basic properties of degree: (a) deg 11 = 1, since 11∗ = 11. (b) deg f = 0 if f is not surjective. For if we choose a point x0 ∈ S n − f (S n) then f can be factored
as a composition S n→S n − {x0} ֓ S n and Hn(S n − {x0}) = 0 since S n − {x0} is contractible. Hence f∗ = 0. (c) If f ≃ g then deg f = deg g since f∗ = g∗. The converse statement, that f ≃ g if deg f = deg g, is a fundamental theorem of Hopf from around 1925 which we prove in Corollary 4.25. (d) deg f g = deg f deg g, since (f g)∗ = f∗g∗. As a consequence, deg f = ±1 if f is a homotopy equivalence since f g ≃ 11 implies deg f deg g = deg 11 = 1. (e) deg f = −1 if f is a reﬂection of S n, ﬁxing the points in a subsphere S n−1 and interchanging the two complementary hemispheres. For we can give S n a n 1 and 2 represents a generator of Hn(S n) as we saw in n 2 sends this generator to complex structure with these two hemispheres as its two n simplices n 2, and the n chain ∆ Example 2.23, so the reﬂection interchanging ∆ ∆ its negative. n 1 and n 1 − ∆ ∆ n (f) The antipodal map −11 : S n→S n, x ֏ −x, has degree (−1)n+1 since it is the composition of n + 1 reﬂections, each changing the sign of one coordinate in Rn+1. ∆ ∆ (g) If f : S n→S n has no ﬁxed points then deg f = (−1)n+1. For if f (x) ≠ x then the line segment from f (x) to −x, deﬁned by t ֏ (1 − t)f (x) − tx for 0 ≤ t ≤ 1, does not pass through the origin. Hence if f has no ﬁxed points, the formula ft(x) = [(1 − t)f (x) − tx]/\|(1 − t)f (x) − tx\| deﬁnes a homot
opy from f to Computations and Applications Section 2.2 135 the antipodal map. Note that the antipodal map has no ﬁxed points, so the fact that maps without ﬁxed points are homotopic to the antipodal map is a sort of converse statement. Here is an interesting application of degree: Theorem 2.28. S n has a continuous ﬁeld of nonzero tangent vectors iﬀ n is odd. Proof: Suppose x ֏ v(x) is a tangent vector ﬁeld on S n, assigning to a vector x ∈ S n the vector v(x) tangent to S n at x. Regarding v(x) as a vector at the origin instead of at x, tangency just means that x and v(x) are orthogonal in Rn+1. If v(x) ≠ 0 for all x, we may normalize so that \|v(x)\| = 1 for all x by replacing v(x) by v(x)/\|v(x)\|. Assuming this has been done, the vectors (cos t)x + (sin t)v(x) lie in the unit circle in the plane spanned by x and v(x). Letting t go from 0 to π, we obtain a homotopy ft(x) = (cos t)x + (sin t)v(x) from the identity map of S n to the antipodal map −11. This implies that deg(−11) = deg 11, hence (−1)n+1 = 1 and n must be odd. Conversely, if n is odd, say n = 2k − 1, we can deﬁne v(x1, x2, ···, x2k−1, x2k) = (−x2, x1, ···, −x2k, x2k−1). Then v(x) is orthogonal to x, so v is a tangent vector ﬁeld on S n, and \|v(x)\| = 1 for all x ∈ S n. ⊔⊓ For the much more diﬃcult problem of ﬁnding the maximum number of tangent vector ﬁelds on S n that are linearly independent at each point, see [VB
KT] or [Husemoller 1966]. Another nice application of degree, giving a partial answer to a question raised in Example 1.43, is the following result: Proposition 2.29. Z2 is the only nontrivial group that can act freely on S n if n is even. Recall that an action of a group G on a space X is a homomorphism from G to the group Homeo(X) of homeomorphisms X→X, and the action is free if the homeomorphism corresponding to each nontrivial element of G has no ﬁxed points. In the case of S n, the antipodal map x ֏ −x generates a free action of Z2. Proof: Since homeomorphisms have degree ±1, an action of a group G on S n determines a degree function d : G→{±1}. This is a homomorphism since deg f g = deg f deg g. If the action is free, d sends each nontrivial element of G to (−1)n+1 by property (g) above. Thus when n is even, d has trivial kernel, so G ⊂ Z2. ⊔⊓ Next we describe a technique for computing degrees which can be applied to most maps that arise in practice. Suppose f : S n→S n, n > 0, has the property that for some point y ∈ S n, the preimage f −1(y) consists of only ﬁnitely many points, say 136 Chapter 2 Homology x1, ···, xm. Let U1, ···, Um be disjoint neighborhoods of these points, mapped by f into a neighborhood V of y. Then f (Ui − xi) ⊂ V − y for each i, and we have a diagram where all the maps are the obvious ones, and in particular ki and pi are induced by inclusions, so the triangles and squares commute. The two isomorphisms in the upper half of the diagram come from excision, while the lower two isomorphisms come from exact sequences of pairs. Via these four isomorphisms, the top two groups in the diagram can be identiﬁed with Hn(S n) ≈ Z, and the top homomorphism f∗ becomes multiplication by an integer called the local degree of f at xi, written deg f \|\| xi. For example,
if f is a homeomorphism, then y can be any point and there is only one corresponding xi, so all the maps in the diagram are isomorphisms and deg f \|\| xi = deg f = ±1. More generally, if f maps each Ui homeomorphically onto V, then deg f \|\| xi = ±1 for each i. This situation occurs quite often in applications, and it is usually not hard to determine the correct signs. Here is the formula that reduces degree calculations to computing local degrees: Proposition 2.30. deg f = i deg f \|\| xi. P S n, S n − f −1(y) Proof: By excision, the central term Hn in the preceding diagram is the direct sum of the groups Hn(Ui, Ui − xi) ≈ Z, with ki the inclusion of the ith summand. The map pi is projection onto the ith summand since the upper triangle commutes and pikj = 0 for j ≠ i, as pikj factors through Hn(Uj, Uj) = 0. Identifying the outer groups in the diagram with Z as before, commutativity of the lower triangle says that pij(1) = 1, hence j(1) = (1, ···, 1) = i ki(1). Commutativity of the upper square says that the middle f∗ takes ki(1) to deg f \|\| xi, hence the sum P i deg f \|\| xi. Commutativity of the lower square then ⊔⊓ i ki(1) = j(1) is taken to gives the formula deg f = P i deg f \|\| xi. P P Example 2.31. We can use this result to construct a map S n→S n of any given degree, for each n ≥ 1. Let q : S n→ k S n be the quotient map obtained by collapsing the k S n→S n identify complement of k disjoint open balls Bi in S n to a point, and let p : all the summands to a single sphere. Consider the composition f = pq. For almost all W y ∈ S n we have f −1(y) consisting of one point xi in each Bi. The local degree of f at xi is ±1 since f is a homeomorphism near xi. By precomposing p with re
ﬂections k S n if necessary, we can make each local degree either +1 or of the summands of −1, whichever we wish. Thus we can produce a map S n→S n of degree ±k. W W Computations and Applications Section 2.2 137 Example 2.32. In the case of S 1, the map f (z) = zk, where we view S 1 as the unit circle in C, has degree k. This is evident in the case k = 0 since f is then constant. The case k < 0 reduces to the case k > 0 by composing with z ֏ z−1, which is a reﬂection, of degree −1. To compute the degree when k > 0, observe ﬁrst that for any y ∈ S 1, f −1(y) consists of k points x1, ···, xk near each of which f is a local homeomorphism, stretching a circular arc by a factor of k. This local stretching can be eliminated by a deformation of f near xi that does not change local degree, so the local degree at xi is the same as for a rotation of S 1. A rotation is a homeomorphism so its local degree at any point equals its global degree, which is +1 since a rotation is homotopic to the identity. Hence deg f \|\| xi = 1 and deg f = k. Another way of obtaining a map S n→S n of degree k is to take a repeated sus- pension of the map z ֏ zk in Example 2.32, since suspension preserves degree: Proposition 2.33. deg Sf = deg f, where Sf : S n+1→S n+1 is the suspension of the map f : S n→S n. Proof: Let CS n denote the cone (S n × I)/(S n × 1) with base S n = S n × 0 ⊂ CS n, so CS n/S n is the suspension of S n. The map f induces Cf : (CS n, S n)→(CS n, S n) with quotient Sf. The naturality of the boundary maps in the long exact sequence of the pair (CS n, S n) then gives commutativity of the diagram at the right. Hence if f∗ is multiplication by d, so is Sf∗. �
��⊓ Note that for f : S n→S n, the suspension Sf maps only one point to each of the two ‘poles’ of S n+1. This implies that the local degree of Sf at each pole must equal the global degree of Sf. Thus the local degree of a map S n→S n can be any integer if n ≥ 2, just as the degree itself can be any integer when n ≥ 1. Cellular Homology Cellular homology is a very eﬃcient tool for computing the homology groups of CW complexes, based on degree calculations. Before giving the deﬁnition of cellular homology, we ﬁrst establish a few preliminary facts: Lemma 2.34. If X is a CW complex, then : (a) Hk(X n, X n−1) is zero for k ≠ n and is free abelian for k = n, with a basis in one-to-one correspondence with the n cells of X. (b) Hk(X n) = 0 for k > n. In particular, if X is ﬁnite-dimensional then Hk(X) = 0 for k > dim X. (c) The map Hk(X n)→Hk(X) induced by the inclusion X n ֓ X is an isomorphism for k < n and surjective for k = n. Proof: Statement (a) follows immediately from the observation that (X n, X n−1) is a good pair and X n/X n−1 is a wedge sum of n spheres, one for each n cell of X. Here 138 Chapter 2 Homology we are using Proposition 2.22 and Corollary 2.25. Next consider the following part of the long exact sequence of the pair (X n, X n−1) : Hk+1(X n, X n−1) -→ Hk(X n−1) -→ Hk(X n) -→ Hk(X n, X n−1) If k ≠ n the last term is zero by part (a) so the middle map is surjective, while if k ≠ n − 1 then the ﬁrst term is zero so the middle map is injective. Now look at the inclusion-induced homomorphisms Hk(X 0) -
→ Hk(X 1) -→ ··· -→ Hk(X k−1) -→ Hk(X k) -→ Hk(X k+1) -→ ··· By what we have just shown these are all isomorphisms except that the map to Hk(X k) may not be surjective and the map from Hk(X k) may not be injective. The ﬁrst part of the sequence then gives statement (b) since Hk(X 0) = 0 when k > 0. Also, the last part of the sequence gives (c) when X is ﬁnite-dimensional. The proof of (c) when X is inﬁnite-dimensional requires more work, and this can be done in two diﬀerent ways. The more direct approach is to descend to the chain level and use the fact that a singular chain in X has compact image, hence meets only ﬁnitely many cells of X by Proposition A.1 in the Appendix. Thus each chain lies in a ﬁnite skeleton Xm. So a k cycle in X is a cycle in some Xm, and then by the ﬁnite-dimensional case of (c), the cycle is homologous to a cycle in X n if n ≥ k, so Hk(X n)→Hk(X) is surjective. Similarly for injectivity, if a k cycle in X n bounds a chain in X, this chain lies in some Xm with m ≥ n, so by the ﬁnite-dimensional case the cycle bounds a chain in X n if n > k. The other approach is more general. From the long exact sequence of the pair Hk(X/X n), (X, X n) it suﬃces to show Hk(X, X n) = 0 for k ≤ n. Since Hk(X, X n) ≈ this reduces the problem to showing: e (∗) Hk(X) = 0 for k ≤ n if the n skeleton of X is a point. When X is ﬁnite-dimensional, (∗) is immediate from the ﬁnite-dimensional case e of (c) which we have already shown. It will suﬃce therefore to reduce the inﬁnite- dimensional case
to the ﬁnite-dimensional case. This reduction will be achieved by stretching X out to a complex that is at least locally ﬁnite-dimensional, using a special case of the ‘mapping telescope’ construction described in greater generality in §3.F. Consider X × [0, ∞) with its product cell structure, where we give [0, ∞) the cell structure with the integer i X i × [i, ∞), a subcomplex points as 0 cells. Let T = of X × [0, ∞). The ﬁgure shows a schematic picture of T with [0, ∞) in the horizontal direction and the subcomplexes X i × [i, i + 1] as rectangles whose size increases with i since X i ⊂ X i+1. The line labeled R can be ignored for now. We claim that T ≃ X, hence Hk(X) ≈ Hk(T ) for all k. Since X is a deformation retract of X × [0, ∞), it suﬃces to show that X × [0, ∞) also deformation retracts onto T. Let Yi = T ∪. Then Yi deformation retracts onto Yi+1 since X × [i, i+1] deformation retracts onto X i × [i, i + 1] ∪ X × {i + 1} by Proposition 0.16. If we perform the X × [i, ∞) S Computations and Applications Section 2.2 139 deformation retraction of Yi onto Yi+1 during the t interval [1 − 1/2i, 1 − 1/2i+1], then this gives a deformation retraction ft of X × [0, ∞) onto T, with points in X i × [0, ∞) stationary under ft for t ≥ 1 − 1/2i+1. Continuity follows from the fact that CW complexes have the weak topology with respect to their skeleta, so a map is continuous if its restriction to each skeleton is continuous. Recalling that X 0 is a point, let R ⊂ T be the ray X 0× [0, ∞) and let Z ⊂ T be the union of this ray with all the subcomplexes X i × {i}. Then Z/R is homeomorphic
to i X i, a wedge sum of ﬁnite-dimensional complexes with n skeleton a point, so the ﬁnite-dimensional case of (∗) together with Corollary 2.25 describing the homology W Hk(Z/R) = 0 for k ≤ n. The same is therefore true for Z, of wedge sums implies that from the long exact sequence of the pair (Z, R), since R is contractible. Similarly, T /Z is a wedge sum of ﬁnite-dimensional complexes with (n + 1) skeleton a point, since if we ﬁrst collapse each subcomplex X i × {i} of T to a point, we obtain the inﬁnite sequence of suspensions SX i ‘skewered’ along the ray R, and then if we collapse R to X i is the reduced suspension of X i, obtained from a point we obtain SX i by collapsing the line segment X 0 × [i, i+1] to a point, so X i has (n+1) skeleton Hk(T /Z) = 0 for k ≤ n + 1. The long exact sequence of the pair (T, Z) a point. Thus ⊔⊓ Hk(T ) = 0 for k ≤ n, and we have proved (∗). then implies that X i where i W Σ Σ Σ e e Let X be a CW complex. Using Lemma 2.34, portions of the long exact sequences e for the pairs (X n+1, X n), (X n, X n−1), and (X n−1, X n−2) ﬁt into a diagram where dn+1 and dn are deﬁned as the compositions jn∂n+1 and jn−1∂n, which are just ‘relativizations’ of the boundary maps ∂n+1 and ∂n. The composition dndn+1 includes two successive maps in one of the exact sequences, hence is zero. Thus the horizontal row in the diagram is a chain complex, called the cellular chain complex of X since Hn(X n, X n−1) is free with basis in one-to-one correspondence with the n cells of X, so one can think of elements of Hn(X n, X
n−1) as linear combinations of n cells of X. The homology groups of this cellular chain complex are called the cellular homology groups of X. Temporarily we denote them H CW n (X). Theorem 2.35. H CW n (X) ≈ Hn(X). 140 Chapter 2 Homology Proof: From the diagram above, Hn(X) can be identiﬁed with Hn(X n)/ Im ∂n+1. Since jn is injective, it maps Im ∂n+1 isomorphically onto Im(jn∂n+1) = Im dn+1 and Hn(X n) isomorphically onto Im jn = Ker ∂n. Since jn−1 is injective, Ker ∂n = Ker dn. Thus jn induces an isomorphism of the quotient Hn(X n)/ Im ∂n+1 onto ⊔⊓ Ker dn/ Im dn+1. Here are a few immediate applications: (i) Hn(X) = 0 if X is a CW complex with no n cells. (ii) More generally, if X is a CW complex with k n cells, then Hn(X) is generated by at most k elements. For since Hn(X n, X n−1) is free abelian on k generators, the subgroup Ker dn must be generated by at most k elements, hence also the quotient Ker dn/ Im dn+1. (iii) If X is a CW complex having no two of its cells in adjacent dimensions, then Hn(X) is free abelian with basis in one-to-one correspondence with the n cells of X. This is because the cellular boundary maps dn are automatically zero in this case. This last observation applies for example to CPn, which has a CW structure with one cell of each even dimension 2k ≤ 2n as we saw in Example 0.6. Thus Z for i = 0, 2, 4, ···, 2n 0 otherwise Another simple example is S n × S n with n > 1, using the product CW structure consisting of a 0 cell, two n cells, and a 2n cell. Hi(CPn) ≈ It is possible to prove the statements (i)–(iii) for ﬁnite-dimensional CW complexes
by induction on the dimension, without using cellular homology but only the basic results from the previous section. However, the viewpoint of cellular homology makes (i)–(iii) quite transparent. Next we describe how the cellular boundary maps dn can be computed. When n = 1 this is easy since the boundary map d1 : H1(X 1, X 0)→H0(X 0) is the same as 0(X). In case X is connected and has only the simplicial boundary map one 0 cell, then d1 must be 0, otherwise H0(X) would not be Z. When n > 1 we will show that dn can be computed in terms of degrees: 1(X)→ ∆ ∆ Cellular Boundary Formula. dn(en map S n−1 P the quotient map collapsing X n−1 − en−1 α →X n−1→S n−1 α) = β that is the composition of the attaching map of en where dαβ is the degree of the α with β dαβen−1 β to a point. β Here we are identifying the cells en β summands of the cellular chain groups. The summation in the formula contains only ﬁnitely many terms since the attaching map of en α has compact image, so this image meets only ﬁnitely many cells en−1 with generators of the corresponding α and en−1. β To derive the cellular boundary formula, consider the commutative diagram Computations and Applications Section 2.2 141 where: α and ϕα is its attaching map. α is the characteristic map of the cell en q : X n−1→X n−1/X n−2 is the quotient map. Φ qβ : X n−1/X n−2→S n−1 resulting quotient sphere being identiﬁed with S n−1 acteristic map αβ : ∂Dn β β. α→S n−1 Φ β α followed by the quotient map X n−1→S n−1 in X n−1 to a point. β of en ∆ en−1 β collapses the complement of the cell en−1 to a point, the β = Dn−1 β β /∂Dn−1 β via the char- is the composition qβqϕα, in other words, the attaching map collapsing the
complement of Φ α] ∈ Hn(Dn α∗ takes a chosen generator [Dn α, ∂Dn The map summand of Hn(X n, X n−1) corresponding to en α. Letting en commutativity of the left half of the diagram then gives dn(en terms of the basis for Hn−1(X n−1, X n−2) corresponding to the cells en−1 is the projection of Commutativity of the diagram then yields the formula for dn given above. α) to a generator of the Z α denote this generator, α) = jn−1ϕα∗∂[Dn α]. In, the map qβ∗ Hn−1(X n−1/X n−2) onto its Z summand corresponding to en−1. β β e Example 2.36. Let Mg be the closed orientable surface of genus g with its usual CW structure consisting of one 0 cell, 2g 1 cells, and one 2 cell attached by the product of commutators [a1, b1] ··· [ag, bg]. The associated cellular chain complex is 0 -----→ Z d2------------→ Z2g d1------------→ Z -----→ 0 As observed above, d1 must be 0 since there is only one 0 cell. Also, d2 is 0 because each ai or bi appears with its inverse in [a1, b1] ··· [ag, bg], so the maps αβ are homotopic to constant maps. Since d1 and d2 are both zero, the homology groups of Mg are the same as the cellular chain groups, namely, Z in dimensions 0 and 2, and Z2g in dimension 1. ∆ Example 2.37. The closed nonorientable surface Ng of genus g has a cell structure with one 0 cell, g 1 cells, and one 2 cell attached by the word a2 g. Again d1 = 0, and d2 : Z→Zg is speciﬁed by the equation d2(1) = (2, ···, 2) since each ai appears in the attaching word of the 2 cell with total exponent 2, which means that αβ is homotopic to the map z ֏ z2, of degree 2. Since d2(1) = (2, ···
, 2), we each have d2 injective and hence H2(Ng) = 0. If we change the basis for Zg by replacing the last standard basis element (0, ···, 0, 1) by (1, ···, 1), we see that H1(Ng) ≈ Zg−1 ⊕ Z2. 2 ··· a2 1a2 ∆ 142 Chapter 2 Homology These two examples illustrate the general fact that the orientability of a closed connected manifold M of dimension n is detected by Hn(M), which is Z if M is orientable and 0 otherwise. This is shown in Theorem 3.26. Example 2.38: An Acyclic Space. Let X be obtained from S 1 ∨ S 1 by attaching two 2 cells by the words a5b−3 and b3(ab)−2. Then d2 : Z2→Z2 has matrix, with the two columns coming from abelianizing a5b−3 and b3(ab)−2 to 5a − 3b and −2a + b, in additive notation. The matrix has determinant −1, so d2 is an isomorphism and Hi(X) = 0 for all i. Such a space X is called acyclic. We can see that this acyclic space is not contractible by considering π1(X), which. There is a nontrivial homomorphism e has the presentation a, b \|\|\|\| a5b−3, b3(ab)−2 5 −2 −3 1 from this group to the group G of rotational symmetries of a regular dodecahedron, sending a to the rotation ρa through angle 2π /5 about the axis through the center of a pentagonal face, and b to the rotation ρb through angle 2π /3 about the axis through a vertex of this face. The composition ρaρb is a rotation through angle π about the axis through the midpoint of an edge abutting this vertex. Thus the relations a5 = b3 = (ab)2 deﬁning π1(X) become ρ5 b = (ρaρb)2 = 1 in G, which means there is a well-deﬁned homomorphism ρ : π1(X)→G
sending a to ρa and b to ρb. It is not hard to see that G is generated by ρa and ρb, so ρ is surjective. With more work one can compute that the kernel of ρ is Z2, generated by the element a5 = b3 = (ab)2, and this Z2 is in fact the center of π1(X). In particular, π1(X) has order 120 since G has order 60. a = ρ3 After these 2 dimensional examples, let us now move up to three dimensions, where we have the additional task of computing the cellular boundary map d3. Example 2.39. A 3 dimensional torus can be constructed from a cube by identifying each pair of opposite square faces as in the ﬁrst of the two ﬁgures. The second ﬁgure shows a slightly diﬀerent pattern of identiﬁcations of opposite faces, with the front and back faces now identiﬁed via a rotation of the cube around a horizontal left-right axis. The space produced by these identiﬁcations is the product K × S 1 of a Klein bottle and a circle. For both T 3 and K × S 1 we have a CW structure with one 3 cell, three 2 cells, three 1 cells, and one 0 cell. The cellular chain complexes thus have the form 0 -→ Z d3------------→ Z3 d2------------→ Z3 0-----→ Z -→ 0 In the case of the 3 torus T 3 the cellular boundary map d2 is zero by the same calculation as for the 2 dimensional torus. We claim that d3 is zero as well. This αβ : S 2→S 2 corresponding to the three 2 cells amounts to saying that the three maps ∆ ∆ Computations and Applications Section 2.2 143 have degree zero. Each αβ maps the interiors of two opposite faces of the cube homeomorphically onto the complement of a point in the target S 2 and sends the remaining four faces to this point. Computing local degrees at the center points of ∆ −1 at the other, since the restrictions of the two opposite faces, we see that the local degree is +1 at one of these points and αβ to these two faces diﬀer by a reﬂection of the boundary of the cube across the plane
midway between them, and a reﬂection ∆ has degree −1. Since the cellular boundary maps are all zero, we deduce that Hi(T 3) is Z for i = 0, 3, Z3 for i = 1, 2, and 0 for i > 3. For K × S 1, when we compute local degrees for the front and back faces we ﬁnd αβ on these two faces diﬀers not by a reﬂection but by a rotation of the boundary of the cube. that the degrees now have the same rather than opposite signs since the map The local degrees for the other faces are the same as before. Using the letters A, B, C to denote the 2 cells given by the faces orthogonal to the edges a, b, c, respectively, we have the boundary formulas d3e3 = 2C, d2A = 2b, d2B = 0, and d2C = 0. It follows that H3(K × S 1) = 0, H2(K × S 1) = Z⊕ Z2, and H1(K × S 1) = Z⊕ Z⊕ Z2. Many more examples of a similar nature, quotients of a cube or other polyhedron with faces identiﬁed in some pattern, could be worked out in similar fashion. But let us instead turn to some higher-dimensional examples. Example 2.40: Moore Spaces. Given an abelian group G and an integer n ≥ 1, we Hi(X) = 0 for i ≠ n. Such a will construct a CW complex X such that Hn(X) ≈ G and space is called a Moore space, commonly written M(G, n) to indicate the dependence on G and n. It is probably best for the deﬁnition of a Moore space to include the e condition that M(G, n) be simply-connected if n > 1. The spaces we construct will have this property. As an easy special case, when G = Zm we can take X to be S n with a cell en+1 attached by a map S n→S n of degree m. More generally, any ﬁnitely generated G can be realized by taking wedge sums of examples of this type for ﬁnite cyclic summands of G, together with copies of S n for in
ﬁnite cyclic summands of G. In the general nonﬁnitely generated case let F→G be a homomorphism of a free abelian group F onto G, sending a basis for F onto some set of generators of G. The kernel K of this homomorphism is a subgroup of a free abelian group, hence is itself free abelian. Choose bases {xα} for F and {yβ} for K, and write yβ = α dβαxα. Let X n = α, so Hn(X n) ≈ F via Corollary 2.25. We will construct X from X n by α S n via maps fβ : S n→X n such that the composition of fβ with the attaching cells en+1 W β projection onto the summand S n α has degree dβα. Then the cellular boundary map dn+1 will be the inclusion K ֓ F, hence X will have the desired homology groups. The construction of fβ generalizes the construction in Example 2.31 of a map α \|dβα\| S n→S n of given degree. Namely, we can let fβ map the complement of P P 144 Chapter 2 Homology disjoint balls in S n to the 0 cell of X n while sending \|dβα\| of the balls onto the summand S n α by maps of degree +1 if dβα > 0, or degree −1 if dβα < 0. Example 2.41. By taking a wedge sum of the Moore spaces constructed in the preceding example for varying n we obtain a connected CW complex with any prescribed sequence of homology groups in dimensions 1, 2, 3, ···. Example 2.42: Real Projective Space RPn. As we saw in Example 0.4, RPn has a CW structure with one cell ek in each dimension k ≤ n, and the attaching map for ek is the 2 sheeted covering projection ϕ : S k−1→RPk−1. To compute the boundary map dk we compute the degree of the composition S k−1 ϕ-----→ RPk−1 q-----→ RPk−1/RPk−2 = S k−1, with q the quotient map. The map qϕ restricts to a homeomorphism from each component of S k−1 − S k−2 onto
RPk−1 − RPk−2, and these two homeomorphisms are obtained from each other by precomposing with the antipodal map of S k−1, which has degree (−1)k. Hence deg qϕ = deg 11 + deg(−11) = 1 + (−1)k, and so dk is either 0 or multiplication by 2 according to whether k is odd or even. Thus the cellular chain complex for RPn is 0 -→ Z 2-----→ Z 0-----→ ··· 2-----→ Z 0-----→ Z 2-----→ Z 0-----→ Z -→ 0 0 -→ Z 0-----→ Z 2-----→ ··· 2-----→ Z 0-----→ Z 2-----→ Z 0-----→ Z -→ 0 if n is even if n is odd From this it follows that Hk(RPn) = Z Z2 0   for k = 0 and for k = n odd for k odd, 0 < k < n otherwise  Example 2.43: Lens Spaces. This example is somewhat more complicated. Given an integer m > 1 and integers ℓ1, ···, ℓn relatively prime to m, deﬁne the lens space L = Lm(ℓ1, ···, ℓn) to be the orbit space S 2n−1/Zm of the unit sphere S 2n−1 ⊂ Cn with the action of Zm generated by the rotation ρ(z1, ···, zn) = (e2π iℓ1/mz1, ···, e2π iℓn/mzn), rotating the j th C factor of Cn by the angle 2π ℓj/m. In particular, when m = 2, ρ is the antipodal map, so L = RP2n−1 in this case. In the general case, the projection S 2n−1→L is a covering space since the action of Zm on S 2n−1 is free: Only the identity element ﬁxes any point of S 2n−1 since each point of S 2n−1 has some coordinate zj nonzero and then e2π ikℓj /mzj ≠ zj for 0 < k < m, as a result of the assumption that ℓj
is relatively prime to m. We shall construct a CW structure on L with one cell ek for each k ≤ 2n − 1 and show that the resulting cellular chain complex is 0 -→ Z 0-----→ Z m------------→ Z 0-----→ ··· 0-----→ Z m------------→ Z 0-----→ Z -→ 0 with boundary maps alternately 0 and multiplication by m. Hence Hk Lm(ℓ1, ···, ℓn) =    for k = 0, 2n − 1 Z Zm for k odd, 0 < k < 2n − 1 0 otherwise Computations and Applications Section 2.2 145 j bounded by S 2n−3. Speciﬁcally, B2n−2 To obtain the CW structure, ﬁrst subdivide the unit circle C in the nth C factor of Cn by taking the points e2π ij/m ∈ C as vertices, j = 1, ···, m. Joining the j th vertex of C to the unit sphere S 2n−3 ⊂ Cn−1 by arcs of great circles in S 2n−1 yields a (2n − 2) dimensional ball B2n−2 consists of the points cos θ (0, ···, 0, e2π ij/m)+sin θ (z1, ···, zn−1, 0) for 0 ≤ θ ≤ π /2. Similarly, joining the j th edge of C to S 2n−3 gives a ball B2n−1 and B2n−2 j+1, subscripts being taken mod m. The rotation ρ carries S 2n−3 to itself and rotates C by the angle 2π ℓn/m, hence ρ permutes the B2n−2 ’s. A suitable power of ρ, namely ρr where r ℓn ≡ 1 mod m, takes each B2n−2 to the next one. Since ρr has order m, it is also a generator of the rotation group Zm, and hence we may obtain L as the quotient of one B2n−1 by identifying its two faces B2n−2 and B2n−2 bounded by B2n−2 ’s and the B2n−1 together via ρr

Subsets and Splits

No community queries yet

The top public SQL queries from the community will appear here once available.