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. and B2n−+1 In particular, when n = 2, B2n−1 j is a lens-shaped 3 ball and L is obtained from this ball by identifying its two curved disk faces via ρr, which may be described as the composition of the reﬂection across the plane con- taining the rim of the lens, taking one face of the lens to the other, followed by a rotation of this face through the angle 2π ℓ/m where ℓ = r ℓ1. The ﬁgure illustrates the case (m, ℓ) = (7, 2), with the two dots indicating a typical pair of identiﬁed points in the upper and lower faces of the lens. Since the lens space L is determined by the rotation angle 2π ℓ/m, it is conveniently written Lℓ/m. Clearly only the mod m value of ℓ matters. It is a classical theorem of Reidemeister from the 1930s that Lℓ/m is homeomorphic to Lℓ′/m′ iﬀ m′ = m and ℓ′ ≡ ±ℓ±1 mod m. For example, when m = 7 there are only two distinct lens spaces L1/7 and L2/7. The ‘if’ part of this theorem is easy: Reﬂecting the lens through a mirror shows that Lℓ/m ≈ L−ℓ/m, and by interchanging the roles of the two C factors of C2 one obtains Lℓ/m ≈ Lℓ−1/m. In the converse direction, Lℓ/m ≈ Lℓ′/m′ clearly implies m = m′ since π1(Lℓ/m) ≈ Zm. The rest of the theorem takes considerably more work, involving either special 3 dimensional tech- niques or more algebraic methods that generalize to classify the higher-dimensional lens spaces as well. The latter approach is explained in [Cohen 1973]. Returning to the construction of a CW structure on Lm(ℓ1, ···, ℓn), observe that the (2n − 3) dimensional lens space Lm(ℓ1, ···, ℓn−1) sits in
Lm(ℓ1, ···, ℓn) as the quotient of S 2n−3, and Lm(ℓ1, ···, ℓn) is obtained from this subspace by attaching two cells, of dimensions 2n − 2 and 2n − 1, coming from the interiors of B2n−1 and and B2n−2 its two identiﬁed faces B2n−2 j+1. Lm(ℓ1, ···, ℓn) with one cell ek in each dimension k ≤ 2n − 1. Inductively this gives a CW structure on j j The boundary maps in the associated cellular chain complex are computed as follows. The ﬁrst one, d2n−1, is zero since the identiﬁcation of the two faces of ﬁxing S 2n−3, followed by a rotaB2n−1 j is via a reﬂection (degree −1 ) across B2n−1 j 146 Chapter 2 Homology tion (degree +1 ), so d2n−1(e2n−1) = e2n−2 − e2n−2 = 0. The next boundary map d2n−2 takes e2n−2 to me2n−3 since the attaching map for e2n−2 is the quotient map S 2n−3→Lm(ℓ1, ···, ℓn−1) and the balls B2n−3 in S 2n−3 which project down onto e2n−3 are permuted cyclically by the rotation ρ of degree +1. Inductively, the subsequent boundary maps dk then alternate between 0 and multiplication by m. j Also of interest are the inﬁnite-dimensional lens spaces Lm(ℓ1, ℓ2, ···) = S ∞/Zm deﬁned in the same way as in the ﬁnite-dimensional case, starting from a sequence of integers ℓ1, ℓ2, ··· relatively prime to m. The space Lm(ℓ1, ℓ2, ···) is the union of the increasing sequence of ﬁnite-dimensional lens spaces Lm(ℓ1
, ···, ℓn) for n = 1, 2, ···, each of which is a subcomplex of the next in the cell structure we have just constructed, so Lm(ℓ1, ℓ2, ···) is also a CW complex. Its cellular chain complex consists of a Z in each dimension with boundary maps alternately 0 and m, so its reduced homology consists of a Zm in each odd dimension. In the terminology of §1.B, the inﬁnite-dimensional lens space Lm(ℓ1, ℓ2, ···) is an Eilenberg–MacLane space K(Zm, 1) since its universal cover S ∞ is contractible, as we showed there. By Theorem 1B.8 the homotopy type of Lm(ℓ1, ℓ2, ···) depends only on m, and not on the ℓi ’s. This is not true in the ﬁnite-dimensional case, when two lens spaces Lm(ℓ1, ···, ℓn) and Lm(ℓ′ n) have the same homotopy type iﬀ ℓ1 ··· ℓn ≡ ±knℓ′ 1 ··· ℓ′ n mod m for some integer k. A proof of this is outlined in Exercise 2 in §3.E and Exercise 29 in §4.2. For example, the 3 dimensional lens spaces L1/5 and L2/5 are not homotopy equivalent, though they have the same fundamental group and the same homology groups. On the other hand, L1/7 and L2/7 are homotopy equivalent but not homeomorphic. 1, ···, ℓ′ Euler Characteristic alternating sum For a ﬁnite CW complex X, the Euler characteristic χ (X) is deﬁned to be the n(−1)ncn where cn is the number of n cells of X, generalizing the familiar formula vertices − edges + faces for 2 dimensional complexes. The following result shows that χ (X) can be deﬁned purely in terms of homology, and hence depends only on the homotopy type of X. In particular, χ (X) is independent P
of the choice of CW structure on X. Theorem 2.44. χ (X) = n(−1)n rank Hn(X). Here the rank of a ﬁnitely generated abelian group is the number of Z summands P when the group is expressed as a direct sum of cyclic groups. We shall need the following fact, whose proof we leave as an exercise: If 0→A→B→C→0 is a short exact sequence of ﬁnitely generated abelian groups, then rank B = rank A + rank C. Proof of 2.44: This is purely algebraic. Let 0 -→ Ck dk------------→ Ck−1 -→ ··· -→ C1 d1------------→ C0 -→ 0 Computations and Applications Section 2.2 147 be a chain complex of ﬁnitely generated abelian groups, with cycles Zn = Ker dn, boundaries Bn = Im dn+1, and homology Hn = Zn/Bn. Thus we have short exact sequences 0→Zn→Cn→Bn−1→0 and 0→Bn→Zn→Hn→0, hence rank Cn = rank Zn + rank Bn−1 rank Zn = rank Bn + rank Hn Now substitute the second equation into the ﬁrst, multiply the resulting equation by n(−1)n rank Hn. Applying this n(−1)n rank Cn = (−1)n, and sum over n to get with Cn = Hn(X n, X n−1) then gives the theorem. ⊔⊓ P P For example, the surfaces Mg and Ng have Euler characteristics χ (Mg) = 2 − 2g and χ (Ng) = 2 − g. Thus all the orientable surfaces Mg are distinguished from each other by their Euler characteristics, as are the nonorientable surfaces Ng, and there are only the relations χ (Mg) = χ (N2g). Split Exact Sequences Suppose one has a retraction r : X→A, so r i = 11 where i : A→X is the inclusion. The induced map i∗ : Hn(A)→Hn(X) is then injective since r∗i∗ = 11. From
this it follows that the boundary maps in the long exact sequence for (X, A) are zero, so the long exact sequence breaks up into short exact sequences 0 -→ Hn(A) i∗-----→ Hn(X) j∗-----→ Hn(X, A) -→ 0 The relation r∗i∗ = 11 actually gives more information than this, by the following piece of elementary algebra: Splitting Lemma. For a short exact sequence 0 -→ A groups the following statements are equivalent : (a) There is a homomorphism p : B→A such that pi = 11 : A→A. (b) There is a homomorphism s : C→B such that js = 11 : C→C. (c) There is an isomorphism B ≈ A⊕ C making a commutative diagram as at the right, where i-----→ B j-----→ C -→ 0 of abelian the maps in the lower row are the obvious ones, a ֏ (a, 0) and (a, c) ֏ c. If these conditions are satisﬁed, the exact sequence is said to split. Note that (c) is symmetric: There is no essential diﬀerence between the roles of A and C. p(b), j(b) Sketch of Proof: For the implication (a) ⇒ (c) one checks that the map B→A⊕ C, b ֏, is an isomorphism with the desired properties. For (b) ⇒ (c) one uses instead the map A⊕ C→B, (a, c) ֏ i(a) + s(c). The opposite implications (c) ⇒ (a) and (c) ⇒ (b) are fairly obvious. If one wants to show (b) ⇒ (a) directly, one can deﬁne p(b) = i−1 ⊔⊓. Further details are left to the reader. b − sj(b) 148 Chapter 2 Homology Except for the implications (b) ⇒ (a) and (b) ⇒ (c), the proof works equally well for nonabelian groups. In the nonabelian case, (b) is deﬁnitely weaker than (a) and (c), and
short exact sequences satisfying (b) only determine B as a semidirect product of A and C. The diﬃculty is that s(C) might not be a normal subgroup of B. In the nonabelian case one deﬁnes ‘splitting’ to mean that (b) is satisﬁed. i-----→ B In both the abelian and nonabelian contexts, if C is free then every exact sequence j-----→ C→0 splits, since one can deﬁne s : C→B by choosing a basis {cα} 0→A for C and letting s(cα) be any element bα ∈ B such that j(bα) = cα. The converse is also true: If every short exact sequence ending in C splits, then C is free. This is because for every C there is a short exact sequence 0→A→B→C→0 with B free — choose generators for C and let B have a basis in one-to-one correspondence with these generators, then let B→C send each basis element to the corresponding generator — so if this sequence 0→A→B→C→0 splits, C is isomorphic to a subgroup of a free group, hence is free. From the Splitting Lemma and the remarks preceding it we deduce that a retraction r : X→A gives a splitting Hn(X) ≈ Hn(A)⊕ Hn(X, A). This can be used to show the nonexistence of such a retraction in some cases, for example in the situation of the Brouwer ﬁxed point theorem, where a retraction Dn→S n−1 would give an impossible splitting Hn−1(Dn) ≈ Hn−1(S n−1)⊕ Hn−1(Dn, S n−1). For a somewhat more subtle example, consider the mapping cylinder Mf of a degree m map f : S n→S n with m > 1. If Mf retracted onto the S n ⊂ Mf corresponding to the domain of f, we would have a split short exact sequence But this sequence does not split since Z is not isomorphic to Z⊕ Zm if m > 1, so the retraction cannot exist. In the simplest case of the degree 2 map S
1→S 1, z ֏ z2, this says that the M¨obius band does not retract onto its boundary circle. Homology of Groups In §1.B we constructed for each group G a CW complex K(G, 1) having a con- tractible universal cover, and we showed that the homotopy type of such a space K(G, 1) is uniquely determined by G. The homology groups Hn therefore depend only on G, and are usually denoted simply Hn(G). The calculations for lens spaces in Example 2.43 show that Hn(Zm) is Zm for odd n and 0 for even n > 0. Since S 1 is a K(Z, 1) and the torus is a K(Z× Z, 1), we also know the homology of these two groups. More generally, the homology of ﬁnitely generated abelian groups K(G, 1) can be computed from these examples using the K¨unneth formula in §3.B and the fact that a product K(G, 1)× K(H, 1) is a K(G× H, 1). Here is an application of the calculation of Hn(Zm) : Computations and Applications Section 2.2 149 Proposition 2.45. If a ﬁnite-dimensional CW complex X is a K(G, 1), then the group G = π1(X) must be torsionfree. This applies to quite a few manifolds, for example closed surfaces other than S 2 and RP2, and also many 3 dimensional manifolds such as complements of knots in S 3. Proof: If G had torsion, it would have a ﬁnite cyclic subgroup Zm for some m > 1, and the covering space of X corresponding to this subgroup of G = π1(X) would be a K(Zm, 1). Since X is a ﬁnite-dimensional CW complex, the same would be true of its covering space K(Zm, 1), and hence the homology of the K(Zm, 1) would be nonzero in only ﬁnitely many dimensions. But this contradicts the fact that Hn(Zm) ⊔⊓ is nonzero for inﬁnitely many values of
n. Reﬂecting the richness of group theory, the homology of groups has been studied quite extensively. A good starting place for those wishing to learn more is the text- book [Brown 1982]. At a more advanced level the books [Adem & Milgram 1994] and [Benson 1992] treat the subject from a mostly topological viewpoint. Mayer–Vietoris Sequences In addition to the long exact sequence of homology groups for a pair (X, A), there is another sort of long exact sequence, known as a Mayer–Vietoris sequence, which is equally powerful but is sometimes more convenient to use. For a pair of subspaces A, B ⊂ X such that X is the union of the interiors of A and B, this exact sequence has the form ··· -→ Hn(A ∩ B) ------------→ Hn(A) ⊕ Hn(B) ------------→ Hn(X) ∂------------→ Hn−1(A ∩ B) -→ ··· Φ Ψ ··· -→ H0(X) -→ 0 In addition to its usefulness for calculations, the Mayer–Vietoris sequence is also ap- plied frequently in induction arguments, where one might know that a certain state- ment is true for A, B, and A ∩ B by induction and then deduce that it is true for A ∪ B by the exact sequence. The Mayer–Vietoris sequence is easy to derive from the machinery of §2.1. Let Cn(A + B) be the subgroup of Cn(X) consisting of chains that are sums of chains in A and chains in B. The usual boundary map ∂ : Cn(X)→Cn−1(X) takes Cn(A + B) to Cn−1(A + B), so the Cn(A + B) ’s form a chain complex. According to Proposition 2.21, the inclusions Cn(A + B) ֓ Cn(X) induce isomorphisms on homology groups. The Mayer–Vietoris sequence is then the long exact sequence of homology groups asso- ciated to the short exact sequence of chain complexes formed by the short exact sequences 0 -→ Cn(A ∩ B) ϕ------------→ Cn(A) ⊕ Cn(
B) ψ------------→ Cn(A + B) -→ 0 150 Chapter 2 Homology where ϕ(x) = (x, −x) and ψ(x, y) = x + y. The exactness of this short exact sequence can be checked as follows. First, Ker ϕ = 0 since a chain in A ∩ B that is zero as a chain in A (or in B ) must be the zero chain. Next, Im ϕ ⊂ Ker ψ since ψϕ = 0. Also, Ker ψ ⊂ Im ϕ since for a pair (x, y) ∈ Cn(A)⊕ Cn(B) the condition x + y = 0 implies x = −y, so x is a chain in both A and B, that is, x ∈ Cn(A ∩ B), and (x, y) = (x, −x) ∈ Im ϕ. Finally, exactness at Cn(A + B) is immediate from the deﬁnition of Cn(A + B). The boundary map ∂ : Hn(X)→Hn−1(A ∩ B) can easily be made explicit. A class α ∈ Hn(X) is represented by a cycle z, and by barycentric subdivision or some other method we can choose z to be a sum x +y of chains in A and B, respectively. It need not be true that x and y are cycles individually, but ∂x = −∂y since ∂(x + y) = 0, and the element ∂α ∈ Hn−1(A ∩ B) is represented by the cycle ∂x = −∂y, as is clear from the deﬁnition of the boundary map in the long exact sequence of homology groups associated to a short exact sequence of chain complexes. There is also a formally identical Mayer–Vietoris sequence for reduced homology groups, obtained by augmenting the previous short exact sequence of chain complexes in the obvious way: Mayer–Vietoris sequences can be viewed as analogs of the van Kampen theorem since if A∩B is path-connected, the H1 terms of the reduced Mayer–Vietoris sequence yield an isomorphism H1(X) ≈. This is exactly the abelianized statement
of the van Kampen theorem, and H1 is the abelianization of π1 for pathconnected spaces, as we show in §2.A. H1(A)⊕ H1(B) / Im Φ There are also Mayer–Vietoris sequences for decompositions X = A ∪ B such that A and B are deformation retracts of neighborhoods U and V with U ∩V deformation retracting onto A ∩ B. Under these assumptions the ﬁve-lemma implies that the maps Cn(A + B)→Cn(U + V ) induce isomorphisms on homology, and hence so do the maps Cn(A + B)→Cn(X), which was all that we needed to obtain a Mayer–Vietoris sequence. For example, if X is a CW complex and A and B are subcomplexes, then we can choose for U and V neighborhoods of the form Nε(A) and Nε(B) constructed in the Appendix, which have the property that Nε(A) ∩ Nε(B) = Nε(A ∩ B). Hi(B) are zero, so we obtain isomorphisms Example 2.46. Take X = S n with A and B the northern and southern hemispheres, so that A ∩ B = S n−1. Then in the reduced Mayer–Vietoris sequence the terms Hi−1(S n−1). This gives Hi(A)⊕ another way of calculating the homology groups of S n by induction. e Example 2.47. We can decompose the Klein bottle K as the union of two M¨obius bands A and B glued together by a homeomorphism between their boundary circles. Hi(S n) ≈ e e e Computations and Applications Section 2.2 151 Then A, B, and A ∩ B are homotopy equivalent to circles, so the interesting part of the reduced Mayer–Vietoris sequence for the decomposition K = A ∪ B is the segment 0 -→ H2(K) -→ H1(A ∩ B) -----→ H1(A)⊕ H1(B) -→ H1(K) -→ 0 Φ The map is Z→Z⊕ Z, 1֏(2, −2), since the boundary circle
of a M¨obius band wraps is injective we obtain H2(K) = 0. Furthermore, twice around the core circle. Since we have H1(K) ≈ Z⊕ Z2 since we can choose (1, 0) and (1, −1) as a basis for Z⊕ Z. All the higher homology groups of K are zero from the earlier part of the Mayer–Vietoris Φ Φ sequence. Example 2.48. Let us describe an exact sequence which is somewhat similar to the Mayer–Vietoris sequence and which in some cases generalizes it. If we are given two maps f, g : X→Y then we can form a quotient space Z of the disjoint union of X × I and Y via the identiﬁcations (x, 0) ∼ f (x) and (x, 1) ∼ g(x), thus attaching one end of X × I to Y by f and the other end by g. For example, if f and g are each the identity map X→X then Z = X × S 1. If only one of f and g, say f, is the identity map, then Z is homeomorphic to what is called the mapping torus of g, the quotient space of X × I under the identiﬁcations (x, 0) ∼ (g(x), 1). The Klein bottle is an example, with g a reﬂection S 1→S 1. The exact sequence we want has the form (∗) ··· ----→ Hn(X) f∗−g∗ ----------------------------→ Hn(Y ) i∗---------→ Hn(Z) ----→ Hn−1(X) f∗−g∗ ----------------------------→ Hn−1(Y ) ----→ ··· where i is the evident inclusion Y ֓ Z. To derive this exact sequence, consider the map q : (X × I, X × ∂I)→(Z, Y ) that is the restriction to X × I of the quotient map X × I ∐ Y →Z. The map q induces a map of long exact sequences: In the upper row the middle term is the direct sum of two copies of Hn(X), and the map i∗ is surjective since X × I deformation retracts onto X
× {0} and X × {1}. Surjectivity of the maps i∗ in the upper row implies that the next maps are 0, which in turn implies that the maps ∂ are injective. Thus the map ∂ in the upper row gives an isomorphism of Hn+1(X × I, X × ∂I) onto the kernel of i∗, which consists of the pairs (α, −α) for α ∈ Hn(X). This kernel is a copy of Hn(X), and the middle vertical map q∗ takes (α, −α) to f∗(α) − g∗(α). The left-hand q∗ is an isomorphism since these are good pairs and q induces a homeomorphism of quotient spaces (X × I)/(X × ∂I)→Z/Y. Hence if we replace Hn+1(Z, Y ) in the lower exact sequence by the isomorphic group Hn(X) ≈ Ker i∗ we obtain the long exact sequence we want. In the case of the mapping torus of a reﬂection g : S 1→S 1, with Z a Klein bottle, the interesting portion of the exact sequence (∗) is 152 Chapter 2 Homology Thus H2(Z) = 0 and we have a short exact sequence 0→Z2→H1(Z)→Z→0. This splits since Z is free, so H1(Z) ≈ Z2 ⊕ Z. Other examples are given in the Exercises. If Y is the disjoint union of spaces Y1 and Y2, with f : X→Y1 and g : X→Y2, then Z consists of the mapping cylinders of these two maps with their domain ends identiﬁed. For example, suppose we have a CW complex decomposed as the union of two subcomplexes A and B and we take f and g to be the inclusions A ∩ B ֓ A and A ∩ B ֓ B. Then the double mapping cylinder Z is homotopy equivalent to A ∪ B since we can view Z as (A ∩ B)× I with A and B attached at the two ends, and then slide the attaching of A down to the B end to produce A ∪ B with (A ∩ B)
× I attached at one of its ends. By Proposition 0.18 the sliding operation preserves homotopy type, so we obtain a homotopy equivalence Z ≃ A ∪ B. The exact sequence (∗) in this case is the Mayer–Vietoris sequence. A relative form of the Mayer–Vietoris sequence is sometimes useful. If one has a pair of spaces (X, Y ) = (A ∪ B, C ∪ D) with C ⊂ A and D ⊂ B, such that X is the union of the interiors of A and B, and Y is the union of the interiors of C and D in Y, then there is a relative Mayer–Vietoris sequence ··· -→ Hn(A ∩ B, C ∩ D) ------------→ Hn(A, C) ⊕ Hn(B, D) ------------→ Hn(X, Y ) ∂------------→ ··· To derive this, consider the commutative diagram Φ Ψ where Cn(A + B, C + D) is the quotient of the subgroup Cn(A + B) ⊂ Cn(X) by its subgroup Cn(C + D) ⊂ Cn(Y ). Thus the columns of the diagram are exact. We have seen that the ﬁrst two rows are exact, and we claim that the third row is exact also, with the maps ϕ and ψ induced from the ϕ and ψ in the second row. Since ψϕ = 0 in the second row, this holds also in the third row, so the third row is at least a chain complex. Viewing the three rows as chain complexes, the diagram then represents a short exact sequence of chain complexes. The associated long exact sequence of homology groups has two out of every three terms zero since the ﬁrst two rows of the diagram are exact. Hence the remaining homology groups are zero and the third row is exact. Computations and Applications Section 2.2 153 The third column maps to 0→Cn(Y )→Cn(X)→Cn(X, Y )→0, inducing maps of homology groups that are isomorphisms for the X and Y terms as we have seen above. So by the ﬁve-lemma the maps Cn(A+B, C
+D)→Cn(X, Y ) also induce isomorphisms on homology. The relative Mayer–Vietoris sequence is then the long exact sequence of homology groups associated to the short exact sequence of chain complexes given by the third row of the diagram. Homology with Coeﬃcients There is an easy generalization of the homology theory we have considered so far that behaves in a very similar fashion and sometimes oﬀers technical advantages. i niσi where each σi is The generalization consists of using chains of the form a singular n simplex in X as before, but now the coeﬃcients ni are taken to lie in a ﬁxed abelian group G rather than Z. Such n chains form an abelian group Cn(X; G), and there is the expected relative version Cn(X, A; G) = Cn(X; G)/Cn(A; G). The old formula for the boundary maps ∂ can still be used for arbitrary G, namely vj, ···, vn]. Just as before, a calculation shows ∂ that ∂2 = 0, so the groups Cn(X; G) and Cn(X, A; G) form chain complexes. The P resulting homology groups Hn(X; G) and Hn(X, A; G) are called homology groups with coeﬃcients in G. Reduced groups Hn(X; G) are deﬁned via the augmented chain complex ··· -→ C0(X; G) ε-----→ G -→ 0 with ε again deﬁned by summing coeﬃcients. i,j (−1)jniσi \|\| [v0, ···, i niσi P P = b The case G = Z2 is particularly simple since one is just considering sums of singular simplices with coeﬃcients 0 or 1, so by discarding terms with coeﬃcient 0 e one can think of chains as just ﬁnite ‘unions’ of singular simplices. The boundary formulas also simplify since one no longer has to worry about signs. Since signs are an algebraic representation of orientation considerations, one can also ignore orientations. This means that homology with Z2 coeﬃ
cients is often the most natural tool in the absence of orientability. All the theory we developed in §2.1 for Z coeﬃcients carries over directly to general coeﬃcient groups G with no change in the proofs. The same is true for Mayer– Vietoris sequences. Diﬀerences between Hn(X; G) and Hn(X) begin to appear only when one starts making calculations. When X is a point, the method used to compute Hn(X) shows that Hn(X; G) is G for n = 0 and 0 for n > 0. From this it follows just as for G = Z that Hn(S k; G) is G for n = k and 0 otherwise. Cellular homology also generalizes to homology with coeﬃcients, with the cellular chain group Hn(X n, X n−1) replaced by Hn(X n, X n−1; G), which is a direct sum of G ’s, one for each n cell. The proof that the cellular homology groups H CW n (X) agree with singular homology Hn(X) extends immediately to give H CW n (X; G) ≈ Hn(X; G). The cellular boundary maps are given by the same formula as for Z coeﬃcients, dn needed to know that the coeﬃcients dαβ are the same as before:. The old proof applies, but the following result is α,β dαβnαen−1 α nαen P P = e α β 154 Chapter 2 Homology Lemma 2.49. If f : S k→S k has degree m, then f∗ : Hk(S k; G)→Hk(S k; G) is multiplication by m. Proof: As a preliminary observation, note that a homomorphism ϕ : G1→G2 induces maps ϕ♯ : Cn(X, A; G1)→Cn(X, A; G2) commuting with boundary maps, so there are induced homomorphisms ϕ∗ : Hn(X, A; G1)→Hn(X, A; G2). These have various naturality properties. For example, they give a commutative diagram mapping the long exact sequence of homology for the
pair (X, A) with G1 coeﬃcients to the corresponding sequence with G2 coeﬃcients. Also, the maps ϕ∗ commute with homomorphisms f∗ induced by maps f : (X, A)→(Y, B). Now let f : S k→S k have degree m and let ϕ : Z→G take 1 to a given element g ∈ G. Then we have a commutative diagram as at the right, where commu- tativity of the outer two squares comes from the inductive calculation of these homology groups, reducing to the case k = 0 when the commutativity is obvious. Since the diagram commutes, the assumption that the map across the top takes 1 to m implies that the map across the bottom takes g to mg. ⊔⊓ Example 2.50. It is instructive to see what happens to the homology of RPn when the coeﬃcient group G is chosen to be a ﬁeld F. The cellular chain complex is ··· 0-----→ F 2-----→ F 0-----→ F 2-----→ F 0-----→ F -→ 0 Hence if F has characteristic 2, for example if F = Z2, then Hk(RPn; F ) ≈ F for 0 ≤ k ≤ n, a more uniform answer than with Z coeﬃcients. On the other hand, if 2-----→ F are isomorF has characteristic diﬀerent from 2 then the boundary maps F phisms, hence Hk(RPn; F ) is F for k = 0 and for k = n odd, and is zero otherwise. In §3.A we will see that there is a general algebraic formula expressing homology with arbitrary coeﬃcients in terms of homology with Z coeﬃcients. Some easy special cases that give much of the ﬂavor of the general result are included in the Exercises. In spite of the fact that homology with Z coeﬃcients determines homology with other coeﬃcient groups, there are many situations where homology with a suitably chosen coeﬃcient group can provide more information than homology with Z coefﬁcients. A good example of this is the proof of the Borsuk–Ulam theorem using Z2 coe
ﬃcients in §2.B. As another illustration, we will now give an example of a map f : X→Y with the property that the induced maps f∗ are trivial for homology with Z coeﬃcients but not for homology with Zm coeﬃcients for suitably chosen m. Thus homology with Zm coeﬃcients tells us that f is not homotopic to a constant map, which we would not know using only Z coeﬃcients. Computations and Applications Section 2.2 155 Example 2.51. Let X be a Moore space M(Zm, n) obtained from S n by attaching a cell en+1 by a map of degree m. The quotient map f : X→X/S n = S n+1 induces trivial homomorphisms on reduced homology with Z coeﬃcients since the nonzero reduced homology groups of X and S n+1 occur in diﬀerent dimensions. But with Zm coeﬃcients the story is diﬀerent, as we can see by considering the long exact sequence of the pair (X, S n), which contains the segment 0 = Hn+1(S n; Zm) -→ Hn+1(X; Zm) f∗-----→ Hn+1(X/S n; Zm) e Exactness says that f∗ is injective, hence nonzero since lular boundary map Hn+1(X n+1, X n; Zm)→Hn(X n, X n−1; Zm) being Zm e e Hn+1(X; Zm) is Zm, the cel- m-----→ Zm. e Exercises 1. Prove the Brouwer ﬁxed point theorem for maps f : Dn→Dn by applying degree theory to the map S n→S n that sends both the northern and southern hemispheres of S n to the southern hemisphere via f. [This was Brouwer’s original proof.] 2. Given a map f : S 2n→S 2n, show that there is some point x ∈ S 2n with either f (x) = x or f (x) = −x. Deduce that every map RP2n→RP2n has a
ﬁxed point. Construct maps RP2n−1→RP2n−1 without ﬁxed points from linear transformations R2n→R2n without eigenvectors. 3. Let f : S n→S n be a map of degree zero. Show that there exist points x, y ∈ S n with f (x) = x and f (y) = −y. Use this to show that if F is a continuous vector ﬁeld deﬁned on the unit ball Dn in Rn such that F (x) ≠ 0 for all x, then there exists a point on ∂Dn where F points radially outward and another point on ∂Dn where F points radially inward. 4. Construct a surjective map S n→S n of degree zero, for each n ≥ 1. 5. Show that any two reﬂections of S n across diﬀerent n dimensional hyperplanes are homotopic, in fact homotopic through reﬂections. [The linear algebra formula for a reﬂection in terms of inner products may be helpful.] 6. Show that every map S n→S n can be homotoped to have a ﬁxed point if n > 0. 7. For an invertible linear transformation f : Rn→Rn show that the induced map on Hn(Rn, Rn − {0}) ≈ Hn−1(Rn − {0}) ≈ Z is 11 or −11 according to whether the determinant of f is positive or negative. [Use Gaussian elimination to show that the matrix of f can be joined by a path of invertible matrices to a diagonal matrix with ±1 ’s on the diagonal.] 8. A polynomial f (z) with complex coeﬃcients, viewed as a map C→C, can always f : S 2→S 2. Show be extended to a continuous map of one-point compactiﬁcations f equals the degree of f as a polynomial. Show also that the local that the degree of e b degree of f at a root of f is the multiplicity of the root. b b 156 Chapter 2 Homology 9. Compute the homology groups of the following 2 complexes: (a) The quotient of S 2 obtained by
identifying north and south poles to a point. (b) S 1 × (S 1 ∨ S 1). (c) The space obtained from D2 by ﬁrst deleting the interiors of two disjoint subdisks in the interior of D2 and then identifying all three resulting boundary circles together via homeomorphisms preserving clockwise orientations of these circles. (d) The quotient space of S 1 × S 1 obtained by identifying points in the circle S 1 × {x0} that diﬀer by 2π /m rotation and identifying points in the circle {x0}× S 1 that diﬀer by 2π /n rotation. 10. Let X be the quotient space of S 2 under the identiﬁcations x ∼ −x for x in the equator S 1. Compute the homology groups Hi(X). Do the same for S 3 with antipodal points of the equatorial S 2 ⊂ S 3 identiﬁed. 11. In an exercise for §1.2 we described a 3 dimensional CW complex obtained from the cube I3 by identifying opposite faces via a one-quarter twist. Compute the homology groups of this complex. 12. Show that the quotient map S 1 × S 1→S 2 collapsing the subspace S 1 ∨ S 1 to a point is not nullhomotopic by showing that it induces an isomorphism on H2. On the other hand, show via covering spaces that any map S 2→S 1 × S 1 is nullhomotopic. 13. Let X be the 2 complex obtained from S 1 with its usual cell structure by attaching two 2 cells by maps of degrees 2 and 3, respectively. (a) Compute the homology groups of all the subcomplexes A ⊂ X and the corre- sponding quotient complexes X/A. (b) Show that X ≃ S 2 and that the only subcomplex A ⊂ X for which the quotient map X→X/A is a homotopy equivalence is the trivial subcomplex, the 0 cell. 14. A map f : S n→S n satisfying f (x) = f (−x) for all x is called an even map. Show that an even map S n→S n must have even degree, and that the degree must in fact be zero when n is even. When
n is odd, show there exist even maps of any given even [Hints: If f is even, it factors as a composition S n→RPn→S n. Using the degree. calculation of Hn(RPn) in the text, show that the induced map Hn(S n)→Hn(RPn) sends a generator to twice a generator when n is odd. It may be helpful to show that the quotient map RPn→RPn/RPn−1 induces an isomorphism on Hn when n is odd.] 15. Show that if X is a CW complex then Hn(X n) is free by identifying it with the kernel of the cellular boundary map Hn(X n, X n−1)→Hn−1(X n−1, X n−2). 16. Let [vi0, ···, vik ∆ groups i( n = [v0, ···, vn] have its natural complex structure with k simplices ] for i0 < ··· < ik. Compute the ranks of the simplicial (or cellular) chain n) and the subgroups of cycles and boundaries. [Hint: Pascal’s triangle.] equal n has homology groups n)k ∆ ( Apply this to show that the k skeleton of Hi ∆ ∆ to 0 for i < k, and free of rank n k+1 for i = k. ∆ e ∆ Computations and Applications Section 2.2 157 n (X)→H CW 17. Show the isomorphism between cellular and singular homology is natural in the following sense: A map f : X→Y that is cellular — satisfying f (X n) ⊂ Y n for all n — induces a chain map f∗ between the cellular chain complexes of X and Y, and the map f∗ : H CW n (Y ) induced by this chain map corresponds to f∗ : Hn(X)→Hn(Y ) under the isomorphism H CW 18. For a CW pair (X, A) show there is a relative cellular chain complex formed by the groups Hi(X i, X i−1 ∪ Ai), having homology groups isomorphic to Hn(X, A). 19. Compute Hi(RPn/RPm) for m < n by cellular homology, using the standard CW structure on RPn with
RPm as its m skeleton. n ≈ Hn. 20. For ﬁnite CW complexes X and Y, show that χ (X × Y ) = χ (X) χ (Y ). If a ﬁnite CW complex X is the union of subcomplexes A and B, show that 21. χ (X) = χ (A) + χ (B) − χ (A ∩ B). 22. For X a ﬁnite CW complex and p : χ ( X) = n χ (X). X→X an n sheeted covering space, show that e e 23. Show that if the closed orientable surface Mg of genus g is a covering space of Mh, then g = n(h − 1) + 1 for some n, namely, n is the number of sheets in the covering. [Conversely, if g = n(h − 1) + 1 then there is an n sheeted covering Mg→Mh, as we saw in Example 1.41.] 24. Suppose we build S 2 from a ﬁnite collection of polygons by identifying edges in pairs. Show that in the resulting CW structure on S 2 the 1 skeleton cannot be either of the two graphs shown, with ﬁve and six vertices. [This is one step in a proof that neither of these graphs embeds in R2.] 25. Show that for each n ∈ Z there is a unique function ϕ assigning an integer to each ﬁnite CW complex, such that (a) ϕ(X) = ϕ(Y ) if X and Y are homeomorphic, (b) ϕ(X) = ϕ(A) + ϕ(X/A) if A is a subcomplex of X, and (c) ϕ(S 0) = n. For such a function ϕ, show that ϕ(X) = ϕ(Y ) if X ≃ Y. 26. For a pair (X, A), let X ∪ CA be X with a cone on A attached. (a) Show that X is a retract of X ∪ CA iﬀ A is contractible in X : There is a homotopy ft : A→X with f0 the inclusion A ֓ X and f1 a constant map. (b)
Show that if A is contractible in X then Hn(X, A) ≈ fact that (X ∪ CA)/X is the suspension SA of A. Hn(X)⊕ Hn−1(A), using the 27. The short exact sequences 0→Cn(A)→Cn(X)→Cn(X, A)→0 always split, but why does this not always yield splittings Hn(X) ≈ Hn(A)⊕ Hn(X, A)? 28. (a) Use the Mayer–Vietoris sequence to compute the homology groups of the space obtained from a torus S 1 × S 1 by attaching a M¨obius band via a homeomorphism from the boundary circle of the M¨obius band to the circle S 1 × {x0} in the torus. (b) Do the same for the space obtained by attaching a M¨obius band to RP2 via a homeomorphism of its boundary circle to the standard RP1 ⊂ RP2. e e 158 Chapter 2 Homology 29. The surface Mg of genus g, embedded in R3 in the standard way, bounds a compact region R. Two copies of R, glued together by the identity map between their boundary surfaces Mg, form a closed 3-manifold X. Compute the homology groups of X via the Mayer–Vietoris sequence for this decomposition of X into two copies of R. Also compute the relative groups Hi(R, Mg). 30. For the mapping torus Tf of a map f : X→X, we constructed in Example 2.48 a 11−f∗ ---------------------→ Hn(X) -→ Hn(Tf ) -→ Hn−1(X) -→ ···. Use long exact sequence ··· -→ Hn(X) this to compute the homology of the mapping tori of the following maps: (a) A reﬂection S 2→S 2. (b) A map S 2→S 2 of degree 2. (c) The map S 1 × S 1→S 1 × S 1 that is the identity on one factor and a reﬂection on the other. (d) The map S 1 × S 1→S 1 × S 1 that is a reﬂection on each factor
. (e) The map S 1 × S 1→S 1 × S 1 that interchanges the two factors and then reﬂects one of the factors. Hn(X ∨ Y ) ≈ Hn(Y ) if the basepoints of X and Y that are identiﬁed in X ∨ Y are defor- 31. Use the Mayer–Vietoris sequence to show there are isomorphisms Hn(X)⊕ mation retracts of neighborhoods U ⊂ X and V ⊂ Y. e 32. For SX the suspension of X, show by a Mayer–Vietoris sequence that there are e e isomorphisms Hn(SX) ≈ Hn−1(X) for all n. 33. Suppose the space X is the union of open sets A1, ···, An such that each interis either empty or has trivial reduced homology groups. Show section Ai1 Hi(X) = 0 for i ≥ n − 1, and give an example showing this inequality is best that e ∩ ··· ∩ Aik e possible, for each n. e 34. [Deleted — see the errata for comments.] 35. Use the Mayer–Vietoris sequence to show that a nonorientable closed surface, or more generally a ﬁnite simplicial complex X for which H1(X) contains torsion, cannot be embedded as a subspace of R3 in such a way as to have a neighborhood homeomorphic to the mapping cylinder of some map from a closed orientable surface to X. [This assumption on a neighborhood is in fact not needed if one deduces the result from Alexander duality in §3.3.] 36. Show that Hi(X × S n) ≈ Hi(X)⊕ Hi−n(X) for all i and n, where Hi = 0 for i < 0 by deﬁnition. Namely, show Hi(X × S n) ≈ Hi(X) ⊕ Hi(X × S n, X × {x0}) and Hi(X × S n, X × {x0}) ≈ Hi−1(X × S n−1, X × {x0}). [For the latter isomorphism the relative Mayer–Vietoris sequence yields an easy proof.] 37. Give an elementary derivation for the
Mayer–Vietoris sequence in simplicial ho- mology for a complex X decomposed as the union of subcomplexes A and B. ∆ Computations and Applications Section 2.2 159 38. Show that a commutative diagram with the two sequences across the top and bottom exact, gives rise to an exact sequence ··· -→ En+1 -→ Bn -→ Cn ⊕ Dn -→ En -→ Bn−1 -→ ··· where the maps are obtained from those in the previous diagram in the obvious way, except that Bn→Cn ⊕ Dn has a minus sign in one coordinate. 39. Use the preceding exercise to derive relative Mayer–Vietoris sequences for CW pairs (X, Y ) = (A ∪ B, C ∪ D) with A = B or C = D. 40. From the long exact sequence of homology groups associated to the short exn-----→ Ci(X) -→ Ci(X; Zn) -→ 0 deduce act sequence of chain complexes 0 -→ Ci(X) immediately that there are short exact sequences 0 -→ Hi(X)/nHi(X) -→ Hi(X; Zn) -→ n-Torsion(Hi−1(X)) -→ 0 n-----→ G, g ֏ ng. Use this to show that where n-Torsion(G) is the kernel of the map G Hi(X) is a vector space over Q for all i. Hi(X; Zp) = 0 for all i and all primes p iﬀ 41. For X a ﬁnite CW complex and F a ﬁeld, show that the Euler characteristic χ (X) e e n(−1)n dim Hn(X; F ), the alternating can also be computed by the formula χ (X) = sum of the dimensions of the vector spaces Hn(X; F ). P 42. Let X be a ﬁnite connected graph having no vertex that is the endpoint of just one edge, and suppose that H1(X; Z) is free abelian of rank n > 1, so the group of automorphisms of H1(X; Z) is GLn(Z), the group of invertible n× n matrices with integer entries whose inverse matrix also has integer entries
. Show that if G is a ﬁnite group of homeomorphisms of X, then the homomorphism G→GLn(Z) assigning to g : X→X the induced homomorphism g∗ : H1(X; Z)→H1(X; Z) is injective. Show the same result holds if the coeﬃcient group Z is replaced by Zm with m > 2. What goes wrong when m = 2? 43. (a) Show that a chain complex of free abelian groups Cn splits as a direct sum of subcomplexes 0→Ln+1→Kn→0 with at most two nonzero terms. [Show the short exact sequence 0→ Ker ∂→Cn→ Im ∂→0 splits and take Kn = Ker ∂.] (b) In case the groups Cn are ﬁnitely generated, show there is a further splitting into summands 0→Z→0 and 0 -→ Z m-----→ Z -→ 0. [Reduce the matrix of the boundary map Ln+1→Kn to echelon form by elementary row and column operations.] (c) Deduce that if X is a CW complex with ﬁnitely many cells in each dimension, then Hn(X; G) is the direct sum of the following groups: a copy of G for each Z summand of Hn(X) a copy of G/mG for each Zm summand of Hn(X) a copy of the kernel of G m-----→ G for each Zm summand of Hn−1(X) 160 Chapter 2 Homology Sometimes it is good to step back from the forest of details and look for gen- eral patterns. In this rather brief section we will ﬁrst describe the general pattern of homology by axioms, then we will look at some common formal features shared by many of the constructions we have made, using the language of categories and functors which has become common in much of modern mathematics. Axioms for Homology For simplicity let us restrict attention to CW complexes and focus on reduced ho- mology to avoid mentioning relative homology. A (reduced) homology theory assigns hn(X) and to each map to each nonempty CW complex X a sequence of abelian groups f : X→Y between CW complexes a sequence of homomorphisms f
∗ : hn(X)→ hn(Y ) such that (f g)∗ = f∗g∗ and 11∗ = 11, and so that the following three axioms are satisﬁed. e e e hn(X)→ (1) If f ≃ g : X→Y, then f∗ = g∗ : (2) There are boundary homomorphisms ∂ : hn(Y ). hn(X/A)→ e pair (X, A), ﬁtting into an exact sequence e q∗------------→ hn(X/A) hn(X) hn(A) i∗------------→ ∂------------→ ··· e e ∂------------→ hn−1(A) deﬁned for each CW hn−1(A) i∗------------→ ··· e e where i is the inclusion and q is the quotient map. Furthermore the boundary maps are natural: For f : (X, A)→(Y, B) inducing a quotient map f : X/A→Y /B, there are commutative diagrams e e (3) For a wedge sum X = hn(Xα)→ α iα∗ : α Xα with inclusions iα : Xα ֓ X, the direct sum map W hn(X) is an isomorphism for each n. L α L e e Negative values for the subscripts n are permitted. Ordinary singular homology is zero in negative dimensions by deﬁnition, but interesting homology theories with nontrivial groups in negative dimensions do exist. The third axiom may seem less substantial than the ﬁrst two, and indeed for ﬁnite wedge sums it can be deduced from the ﬁrst two axioms, though not in general for inﬁnite wedge sums, as an example in the Exercises shows. It is also possible, and not much more diﬃcult, to give axioms for unreduced homology theories. One supposes one has relative groups hn(X, A) deﬁned, specializing to absolute groups by setting hn(X) = hn(X, ∅). Axiom (1) is replaced by its The Formal View
point Section 2.3 161 obvious relative form, and axiom (2) is broken into two parts, the ﬁrst hypothesizing a long exact sequence involving these relative groups, with natural boundary maps, the second stating some version of excision, for example hn(X, A) ≈ hn(X/A, A/A) if one is dealing with CW pairs. In axiom (3) the wedge sum is replaced by disjoint union. These axioms for unreduced homology are essentially the same as those origi- nally laid out in the highly inﬂuential book [Eilenberg & Steenrod 1952], except that axiom (3) was omitted since the focus there was on ﬁnite complexes, and there was another axiom specifying that the groups hn(point) are zero for n ≠ 0, as is true for singular homology. This axiom was called the ‘dimension axiom’, presumably be- cause it speciﬁes that a point has nontrivial homology only in dimension zero. It can be regarded as a normalization axiom, since one can trivially deﬁne a homology theory where it fails by setting hn(X, A) = Hn+k(X, A) for a ﬁxed nonzero integer k. At the time there were no interesting homology theories known for which the dimension axiom did not hold, but soon thereafter topologists began studying a homology theory called ‘bordism’ having the property that the bordism groups of a point are nonzero in inﬁnitely many dimensions. Axiom (3) seems to have appeared ﬁrst in [Milnor 1962]. e Reduced and unreduced homology theories are essentially equivalent. From an unreduced theory h one gets a reduced theory hn(X) equal to the kernel of the canonical map hn(X)→hn(point). In the other direction, one sets hn(X) = hn(X+) where X+ is the disjoint union of X with a point. We leave it as an exercise to show that these two transformations between reduced and unre- h by setting e e duced homology are inverses of each other. Just as with ordinary homology, one has hn(X)�
� hn(x0) for any point x0 ∈ X, since the long exact sequence of the hn(X) ≈ hn(x0) = 0 for all n, pair (X, x0) splits via the retraction of X onto x0. Note that as can be seen by looking at the long exact sequence of reduced homology groups of the pair (x0, x0). e e The groups hn(x0) ≈ hn(S 0) are called the coeﬃcients of the homology theories h and h, by analogy with the case of singular homology with coeﬃcients. One can trivially realize any sequence of abelian groups Gi as the coeﬃcient groups of a homology theory by setting hn(X, A) = i Hn−i(X, A; Gi). e e In general, homology theories are not uniquely determined by their coeﬃcient L groups, but this is true for singular homology: If h is a homology theory deﬁned for CW pairs, whose coeﬃcient groups hn(x0) are zero for n ≠ 0, then there are natural isomorphisms hn(X, A) ≈ Hn(X, A; G) for all CW pairs (X, A) and all n, where G = h0(x0). This will be proved in Theorem 4.59. We have seen how Mayer–Vietoris sequences can be quite useful for singular ho- mology, and in fact every homology theory has Mayer–Vietoris sequences, at least for CW complexes. These can be obtained directly from the axioms in the follow- 162 Chapter 2 Homology ing way. For a CW complex X = A ∪ B with A and B subcomplexes, the inclusion (B, A ∩ B) ֓ (X, A) induces a commutative diagram of exact sequences The vertical maps between relative groups are isomorphisms since B/(A ∩ B) = X/A. Then it is a purely algebraic fact, whose proof is Exercise 38 at the end of the previous section, that a diagram such as this with every third vertical map an isomorphism gives rise to a long exact sequence involving the remaining nonisomorphic terms. In the present case this takes
the form of a Mayer-Vietoris sequence ··· -→ hn(A ∩ B) ϕ-----→ hn(A) ⊕ hn(B) ψ-----→ hn(X) ∂-----→ hn−1(A ∩ B) -→ ··· Categories and Functors Formally, singular homology can be regarded as a sequence of functions Hn that assign to each space X an abelian group Hn(X) and to each map f : X→Y a homomorphism Hn(f ) = f∗ : Hn(X)→Hn(Y ), and similarly for relative homology groups. This sort of situation arises quite often, and not just in algebraic topology, so it is useful to introduce some general terminology for it. Roughly speaking, ‘functions’ like Hn are called ‘functors’, and the domains and ranges of these functors are called ‘categories’. Thus for Hn the domain category consists of topological spaces and continuous maps, or in the relative case, pairs of spaces and continuous maps of pairs, and the range category consists of abelian groups and homomorphisms. A key point is that one is interested not only in the objects in the category, for example spaces or groups, but also in the maps, or ‘morphisms’, between these objects. Now for the precise deﬁnitions. A category C consists of three things: (1) A collection Ob(C) of objects. (2) Sets Mor(X, Y ) of morphisms for each pair X, Y ∈ Ob(C), including a distin- guished ‘identity’ morphism 11 = 11X ∈ Mor(X, X) for each X. (3) A ‘composition of morphisms’ function ◦ : Mor(X, Y )× Mor(Y, Z)→Mor(X, Z) for each triple X, Y, Z ∈ Ob(C), satisfying f ◦ 11 = f, 11 ◦ f = f, and (f ◦ g) ◦ h = f ◦ (g ◦ h). There are plenty of obvious examples, such as: The category of topological spaces, with continuous maps as the morphisms. Or we could restrict to special classes of spaces
such as CW complexes, keeping continuous maps as the morphisms. We could also restrict the morphisms, for example to homeomorphisms. The category of groups, with homomorphisms as morphisms. Or the subcategory of abelian groups, again with homomorphisms as the morphisms. Generalizing The Formal Viewpoint Section 2.3 163 this is the category of modules over a ﬁxed ring, with morphisms the module homomorphisms. The category of sets, with arbitrary functions as the morphisms. Or the mor- phisms could be restricted to injections, surjections, or bijections. There are also many categories where the morphisms are not simply functions, for example: Any group G can be viewed as a category with only one object and with G as the morphisms of this object, so that condition (3) reduces to two of the three axioms for a group. If we require only these two axioms, associativity and a left and right identity, we have a ‘group without inverses’, usually called a monoid since it is the same thing as a category with one object. A partially ordered set (X, ≤) can be considered a category where the objects are the elements of X and there is a unique morphism from x to y whenever x ≤ y. The relation x ≤ x gives the morphism 11 and transitivity gives the composition Mor(x, y)× Mor(y, z)→Mor(x, z). The condition that x ≤ y and y ≤ x implies x = y says that there is at most one morphism between any two objects. There is a ‘homotopy category’ whose objects are topological spaces and whose morphisms are homotopy classes of maps, rather than actual maps. This uses the fact that composition is well-deﬁned on homotopy classes: f0g0 ≃ f1g1 if f0 ≃ f1 and g0 ≃ g1. Chain complexes are the objects of a category, with chain maps as morphisms. This category has various interesting subcategories, obtained by restricting the objects. For example, we could take chain complexes whose groups are zero in negative dimensions, or zero outside a ﬁnite range. Or we could restrict to exact sequences, or short exact sequences. In each case we take morphisms to be chain maps, which are commutative
diagrams. Going a step further, there is a category whose objects are short exact sequences of chain complexes and whose morphisms are commutative diagrams of maps between such short exact sequences. F (X), F (Y ) A functor F from a category C to a category D assigns to each object X in C an object F (X) in D and to each morphism f ∈ Mor(X, Y ) in C a morphism F (f ) ∈ in D, such that F (11) = 11 and F (f ◦ g) = F (f ) ◦ F (g). In the case of Mor the singular homology functor Hn, the latter two conditions are the familiar properties 11∗ = 11 and (f g)∗ = f∗g∗ of induced maps. Strictly speaking, what we have just deﬁned is a covariant functor. A contravariant functor would diﬀer from this by assigning to f ∈ Mor(X, Y ) a ‘backwards’ morphism F (f ) ∈ Mor with F (11) = 11 and F (f ◦ g) = F (g) ◦ F (f ). A classical example of this is the dual vector F (Y ), F (X) space functor, which assigns to a vector space V over a ﬁxed scalar ﬁeld K the dual vector space F (V ) = V ∗ of linear maps V→K, and to each linear transformation 164 Chapter 2 Homology f : V→W the dual map F (f ) = f ∗ : W ∗→V ∗, going in the reverse direction. In the next chapter we will study the contravariant version of homology, called cohomology. A number of the constructions we have studied in this chapter are functors: The singular chain complex functor assigns to a space X the chain complex of singular chains in X and to a map f : X→Y the induced chain map. This is a functor from the category of spaces and continuous maps to the category of chain complexes and chain maps. The algebraic homology functor assigns to a chain complex its sequence of ho- mology groups and to a chain map the induced homomorphisms on homology. This is a functor from the category of chain complexes and chain maps to the
category whose objects are sequences of abelian groups and whose morphisms are sequences of homomorphisms. The composition of the two preceding functors is the functor assigning to a space its singular homology groups. The ﬁrst example above, the singular chain complex functor, can itself be re- garded as the composition of two functors. The ﬁrst functor assigns to a space X its singular complex S(X), a complex, and the second functor assigns to ∆ a complex its simplicial chain complex. This is what the two functors do on ∆ objects, and what they do on morphisms can be described in the following way. A map of spaces f : X→Y induces a map f∗ : S(X)→S(Y ) by composing singular complexes taking the simplices n→X with f. The map f∗ is a map between distinguished characteristic maps in the domain characteristic maps in the target ∆ complex to the distinguished ∆ complex. Call such maps D maps and let map incomplexes. Note that a ∆ them be the morphisms in the category of duces a chain map between simplicial chain complexes, taking basis elements to basis elements, so we have a simplicial chain complex functor taking the category ∆ ∆ ∆ of complexes and maps to the category of chain complexes and chain maps. There is a functor assigning to a pair of spaces (X, A) the associated long exact ∆ ∆ sequence of homology groups. Morphisms in the domain category are maps of pairs, and in the target category morphisms are maps between exact sequences forming commutative diagrams. This functor is the composition of two functors, the ﬁrst assigning to (X, A) a short exact sequence of chain complexes, the sec- ond assigning to such a short exact sequence the associated long exact sequence of homology groups. Morphisms in the intermediate category are the evident commutative diagrams. Another sort of process we have encountered is the transformation of one functor into another, for example: Boundary maps Hn(X, A)→Hn−1(A) in singular homology, or indeed in any homology theory. The Formal Viewpoint Section 2.3 165 Change-of-coeﬃcient homomorphisms Hn(X; G1)→Hn(X; G2) induced by a hom
omorphism G1→G2, as in the proof of Lemma 2.49. In general, if one has two functors F, G : C→D then a natural transformation T from F to G assigns a morphism TX : F (X)→G(X) to each object X ∈ C, in such a way that for each morphism f : X→Y in C the square at the right commutes. The case that F and G are contravariant rather than covariant is similar. We have been describing the passage from topology to the abstract world of cat- egories and functors, but there is also a nice path in the opposite direction: ∆ complex BC called the classifying To each category C there is associated a space of C, whose n simplices are the strings X0→X1→ ··· →Xn of morphisms in C. The faces of this simplex are obtained by deleting an Xi, and then composing the two adjacent morphisms if i ≠ 0, n. Thus when n = 2 the three faces of X0→X1→X2 are X0→X1, X1→X2, and the composed morphism X0→X2. In case C has a single object and the morphisms of C form a group G, then BC is complex BG constructed in Example 1B.7, a K(G, 1). In genthe same as the eral, the space BC need not be a K(G, 1), however. For example, if we start with a complex X and regard its set of simplices as a partially ordered set C(X) under the relation of inclusion of faces, then BC(X) is the barycentric subdivision of X. ∆ A functor F : C→D induces a map BC→BD. This is the map that sends an n simplex X0→X1→ ··· →Xn to the n simplex F (X0)→F (X1)→ ··· →F (Xn). A natural transformation from a functor F to a functor G induces a homotopy ∆ ∆ between the induced maps of classifying spaces. We leave this for the reader to n × I into (n + 1) simplices described make explicit, using the subdivision of earlier in the chapter. ∆ Exercises 1. If Tn(X, A) denotes
the torsion subgroup of Hn(X, A; Z), show that the functors (X, A) ֏ Tn(X, A), with the obvious induced homomorphisms Tn(X, A)→Tn(Y, B) and boundary maps Tn(X, A)→Tn−1(A), do not deﬁne a homology theory. Do the same for the ‘mod torsion’ functor MTn(X, A) = Hn(X, A; Z)/Tn(X, A). 2. Deﬁne a candidate for a reduced homology theory on CW complexes by i dimensional, but is not identically zero, for example for X = Q hn(X) = hn(X) is independent of n and is zero if X is ﬁnitei S i. Show that the Hi(X). Thus i L Hi(X) e e e e axioms for a homology theory are satisﬁed except that the wedge axiom fails. W 3. Show that if h is a reduced homology theory, then hn(point ) = 0 for all n. Deduce that there are suspension isomorphisms e hn(X) ≈ hn+1(SX) for all n. e 4. Show that the wedge axiom for homology theories follows from the other axioms e e in the case of ﬁnite wedge sums. 166 Chapter 2 Homology There is a close connection between H1(X) and π1(X), arising from the fact that a map f : I→X can be viewed as either a path or a singular 1 simplex. If f is a loop, with f (0) = f (1), this singular 1 simplex is a cycle since ∂f = f (1) − f (0). Theorem 2A.1. By regarding loops as singular 1 cycles, we obtain a homomorphism h : π1(X, x0)→H1(X). If X is path-connected, then h is surjective and has kernel the commutator subgroup of π1(X), so h induces an isomorphism from the abelianization of π1(X) onto H1(X
). Proof: Recall the notation f ≃ g for the relation of homotopy, ﬁxing endpoints, between paths f and g. Regarding f and g as chains, the notation f ∼ g will mean that f is homologous to g, that is, f − g is the boundary of some 2 chain. Here are some facts about this relation. (i) If f is a constant path, then f ∼ 0. Namely, f is a cycle since it is a loop, and since H1(point ) = 0, f must then be a boundary. Explicitly, f is the boundary of the constant singular 2 simplex σ having the same image as f since ∂σ = σ \|\| [v1, v2] − σ \|\| [v0, v2] + σ \|\| [v0, v1ii) If f ≃ g then f ∼ g. To see this, consider a homotopy F : I × I→X from f to g. This yields a pair of singular 2 simplices σ1 and σ2 in X by subdividing the square I × I into two triangles [v0, v1, v3] and [v0, v2, v3] as shown in the ﬁgure. When one computes ∂(σ1 − σ2), the two restrictions of F to the diagonal of the square cancel, leaving f − g together with two constant sin- gular 1 simplices from the left and right edges of the square. By (i) these are boundaries, so f − g is also a boundary. f and g. For if σ : (iii) f g ∼ f + g, where f g denotes the product of the paths 2→X is the composition of orthogonal 2 = [v0, v1, v2] onto the edge [v0, v2] followed projection of by f g : [v0, v2]→X, then ∂iv) f ∼ −f, where f is the inverse path of f. This follows from the preceding three observations, which give f + f ∼ f f ∼ 0. Applying (ii) and (iii) to loops, it follows that we have a well-deﬁned homomorphism h : π1(X, x0)→H1(X) sending the homotopy
class of a loop f to the homology class of the 1 cycle f. Homology and Fundamental Group Section 2.A 167 P To show h is surjective when X is path-connected, let i niσi be a 1 cycle representing a given element of H1(X). After relabeling the σi ’s we may assume each ni is ±1. By (iv) we may in fact take each ni to be +1, so our 1 cycle is i σi. If = 0 means there must be another some σi is not a loop, then the fact that ∂ σj such that the composed path σi σj is deﬁned. By (iii) we may then combine the terms σi and σj into a single term σi σj. Iterating this, we reduce to the case that each σi is a loop. Since X is path-connected, we may choose a path γi from x0 to the basepoint of σi. We have γi σi γi ∼ σi by (iii) and (iv), so we may assume all σi ’s are loops at x0. Then we can combine all the σi ’s into a single σ by (iii). This says the given element of H1(X) is in the image of h. i σi P P The commutator subgroup of π1(X) is contained in the kernel of h since H1(X) is abelian. To obtain the reverse inclusion we will show that every class [f ] in the kernel of h is trivial in the abelianization π1(X)ab of π1(X). ary of a 2 chain If an element [f ] ∈ π1(X) is in the kernel of h, then f, as a 1 cycle, is the boundi niσi. Again we may assume each ni is ±1. As in the discussion i niσi a 2 dimensional preceding Proposition 2.6, we can associate to the chain P complex K by taking a 2 simplex 2 i for each σi and identiP fying certain pairs of edges of these 2 simplices. Namely, if we ∆ apply the usual boundary formula to write ∂σi = τ
i0 − τi1 + τi2 for singular 1 simplices τij, then the formula ∆ f = ∂ i niσi = i ni∂σi = i,j(−1)j niτij P P P implies that we can group all but one of the τij ’s into pairs for which the two coeﬃcients (−1)jni in each pair are +1 and −1. The one remaining τij is equal to 2 j ’s corresponding to the paired τij ’s, preserving f. We then identify edges of the orientations of these edges so that we obtain a complex K. ∆ The maps σi ﬁt together to give a map σ : K→X. We can deform σ, staying ﬁxed on the edge corresponding to f, so that each vertex maps to the basepoint x0, in the following way. Paths from the images of these vertices to x0 deﬁne such a homotopy on the union of the 0 skeleton of K with the edge corresponding to f, ∆ and then we can appeal to the homotopy extension property in Proposition 0.16 to extend this homotopy to all of K. Alternatively, it is not hard to construct such an 2 i, we obtain a new chain extension by hand. Restricting the new σ to the simplices i niσi with boundary equal to f and with all τij ’s loops at x0. ∆ P Using additive notation in the abelian group π1(X)ab, we have the formula [f ] = i,j (−1)jni[τij ] because of the canceling pairs of τij ’s. We can rewrite the summai ni[∂σi] where [∂σi] = [τi0] − [τi1] + [τi2]. Since σi tion P gives a nullhomotopy of the composed loop τi0 − τi1 + τi2, we conclude that [f ] = 0 P ⊔⊓ in π1(X)ab. i,j(−1)j ni[τij ] as P 168 Chapter 2 Homology The end of this proof can be illuminated by looking more closely at the geometry. The complex K is in fact a compact surface
with boundary consisting of a single circle formed by the edge corresponding to f. This is because any pattern of identiﬁcations of pairs of edges of a ﬁnite collection of disjoint 2 simplices produces a compact sur- formula f = ∂ face with boundary. We leave it as an exercise for the reader to check that the algebraic with each ni = ±1 implies that K is an orientable surface. The component of K containing i niσi P the boundary circle is a standard closed orientable surface of some genus g with an open disk removed, by the basic structure theorem for compact orientable surfaces. Giving this surface the cell structure indicated in the ﬁgure, it then becomes obvious that f is homotopic to a product of g commutators in π1(X). The map h : π1(X, x0)→H1(X) can also be deﬁned by h([f ]) = f∗(α) where f : S 1→X represents a given element of π1(X, x0), f∗ is the induced map on H1, and α is the generator of H1(S 1) ≈ Z represented by the standard map σ : I→S 1, σ (s) = e2π is. This is because both [f ] ∈ π1(X, x0) and f∗(α) ∈ H1(X) are represented by the loop f σ : I→X. A consequence of this deﬁnition is that h([f ]) = h([g]) if f and g are homotopic maps S 1→X, since f∗ = g∗ by Theorem 2.10. Example 2A.2. For the closed orientable surface M of genus g, the abelianization of π1(M) is Z2g, the product of 2g copies of Z, and a basis for H1(M) consists of the 1 cycles represented by the 1 cells of M in its standard CW structure. We can also represent a basis by the loops αi and βi shown in the ﬁgure below since these loops are homotopic to the loops represented by the 1 cells, as one can see in the picture of the cell structure in Chapter 0. The loops γi, on
the other hand, are trivial in homology since the portion of M on one side of γi is a compact surface bounded by γi, so γi is homotopic to a loop that is a product of commutators, as we saw a couple paragraphs earlier. The loop α′ i represents the same homology class as αi since the region between γi and αi ∪ α′ i provides a homotopy between γi and a product of two loops homotopic to αi and the inverse of α′ i ∼ γi ∼ 0, hence αi ∼ α′ i. i, so αi − α′ Classical Applications Section 2.B 169 In this section we use homology theory to prove several interesting results in topology and algebra whose statements give no hint that algebraic topology might be involved. To begin, we calculate the homology of complements of embedded spheres and disks in a sphere. Recall that an embedding is a map that is a homeomorphism onto its image. Proposition 2B.1. (a) For an embedding h : Dk→S n, (b) For an embedding h : S k→S n with k < n, and 0 otherwise. Hi Hi S n − h(S k) e S n − h(Dk) = 0 for all i. is Z for i = n − k − 1 e As a special case of (b) we have the Jordan curve theorem: A subspace of S 2 homeomorphic to S 1 separates S 2 into two complementary components, or equivalently, path-components since open subsets of S n are locally path-connected. One could just as well use R2 in place of S 2 here since deleting a point from an open set in S 2 does not aﬀect its connectedness. More generally, (b) says that a subspace of S n homeomorphic to S n−1 separates it into two components, and these components have the same homology groups as a point. Somewhat surprisingly, there are embed- dings where these complementary components are not simply-connected as they are for the standard embedding. An example is the Alexander horned sphere in S 3 which we describe in detail following the proof of the proposition. These complications involving embedded S n−1 ’s in S n are all local in nature since it is known that any locally nicely embedded S n−1 in
S n is equivalent to the standard S n−1 ⊂ S n, equivalent in the sense that there is a homeomorphism of S n taking the given embedded S n−1 onto the standard S n−1. In particular, both complementary regions are homeomorphic to open balls. See [Brown 1960] for a precise statement and proof. When n = 2 it is a classical theorem of Schoenﬂies that all embeddings S 1 ֓ S 2 are equivalent. By contrast, when we come to embeddings of S n−2 in S n, even locally nice embeddings need not be equivalent to the standard one. This is the subject of knot theory, including the classical case of knotted embeddings of S 1 in S 3 or R3. For embeddings of S n−2 in S n the complement always has the same homology as S 1, according to the theorem, but the fundamental group can be quite diﬀerent. In spite of the fact that the homology of a knot complement does not detect knottedness, it is still possible to use homology to distinguish diﬀerent knots by looking at the homology of covering spaces of their complements. Proof: We prove (a) by induction on k. When k = 0, S n − h(D0) is homeomorphic to Rn, so this case is trivial. For the induction step it will be convenient to replace the domain disk Dk of h by the cube Ik. Let A = S n − h(Ik−1 × [0, 1/2]) and let 170 Chapter 2 Homology B = S n − h(Ik−1 × [1/2, 1]), so A ∩ B = S n − h(Ik) and A ∪ B = S n − h(Ik−1 × {1/2}). By Hi(A ∪ B) = 0 for all i, so the Mayer–Vietoris sequence gives isomorphisms induction Hi Hi(B) for all i. Modulo signs, the two components of Hi(A)⊕ → : e are induced by the inclusions S n − h(Ik) ֓ A and S n − h(Ik) ֓ B, so if there exists Φ Φ an i dimensional cycle α in S n − h(Ik) that is not a
boundary in S n − h(Ik), then α is also not a boundary in at least one of A and B. (When i = 0 the word ‘cycle’ e e S n − h(Ik) e here is to be interpreted in the sense of augmented chain complexes since we are dealing with reduced homology.) By iteration we can then produce a nested sequence of closed intervals I1 ⊃ I2 ⊃ ··· in the last coordinate of Ik shrinking down to a point p ∈ I, such that α is not a boundary in S n − h(Ik−1 × Im) for any m. On the other hand, by induction on k we know that α is the boundary of a chain β in S n − h(Ik−1 × {p}). This β is a ﬁnite linear combination of singular simplices with compact image in S n − h(Ik−1 × {p}). The union of these images is covered by the nested sequence of open sets S n − h(Ik−1 × Im), so by compactness β must actually be a chain in S n − h(Ik−1 × Im) for some m. This contradiction shows that α must be a boundary in S n − h(Ik), ﬁnishing the induction step. Part (b) is also proved by induction on k, starting with the trivial case k = 0 when S n − h(S 0) is homeomorphic to S n−1 × R. For the induction step, write S k as the − intersecting in S k−1. The Mayer–Vietoris sequence union of hemispheres Dk + and Dk for A = S n −h(Dk +) and B = S n −h(Dk −), both of which have trivial reduced homology ⊔⊓ by part (a), then gives isomorphisms Hi S n − h(S k−1) S n − h(S k) Hi+1 ≈. e e e e e H0 H0(B)→ S n − h(S n−1) H0(B) are zero, so exactness would imply that If we apply the last part of this proof to an embedding h : S n→S n, the Mayer→0. Both H0(A)⊕ Vietoris sequence
ends with the terms = 0 H0(A) and which appears to contradict the fact that S n − h(S n−1) has two path-components. e The only way out of this dilemma is for h to be surjective, so that A ∩ B is empty and H−1(∅) which is Z rather than 0. In particular, this shows that S n cannot be embedded in Rn since this would yield a nonsurjective embedding in S n. A consequence is that there is no embedding Rm ֓ Rn for m > n since this would restrict to an embedding of S n ⊂ Rm into Rn. More generally there is no continuous injection Rm→Rn for m > n since this too would give an embedding S n ֓ Rn. the 0 at the end of the Mayer-Vietoris sequence is S n − h(S n−1) H0 e e e Example 2B.2: The Alexander Horned Sphere. This is a subspace S ⊂ R3 homeomorphic to S 2 such that the unbounded component of R3 −S is not simply-connected as it is for the standard S 2 ⊂ R3. We will construct S by deﬁning a sequence of compact subspaces X0 ⊃ X1 ⊃ ··· of R3 whose intersection is homeomorphic to a ball, and then S will be the boundary sphere of this ball. We begin with X0 a solid torus S 1 × D2 obtained from a ball B0 by attaching In the ﬁgure this handle is shown as the union of a handle I × D2 along ∂I × D2. Classical Applications Section 2.B 171 two ‘horns’ attached to the ball, together with a shorter handle drawn as dashed lines. To form the space X1 ⊂ X0 we delete part of the short handle, so that what remains is a pair of linked handles attached to the ball B1 that is the union of B0 with the two horns. To form X2 the process is repeated: Decompose each of the second stage handles as a pair of horns and a short handle, then delete a part of the short handle. In the same way Xn is constructed inductively from Xn−1. Thus Xn is a ball Bn with 2n handles attached,
and Bn is obtained from Bn−1 by attaching 2n horns. There are homeomorphisms hn : Bn−1→Bn that are the identity outside a small neighborhood of Bn − Bn−1. As n goes to inﬁnity, the composition hn ··· h1 approaches a map f : B0→R3 which is continuous since the convergence is uniform. The set of points in B0 where f is not equal to hn ··· h1 for large n is a Cantor set, whose image under f is the intersection of all the handles. It is not hard to see that f is one-to-one. By compactness it follows that f is a homeomorphism onto its image, a ball B ⊂ R3 whose boundary sphere f (∂B0) is S, the Alexander horned sphere. Now we compute π1(R3 −B). Note that B is the intersection of the Xn ’s, so R3 −B is the union of the complements Yn of the Xn ’s, which form an increasing sequence Y0 ⊂ Y1 ⊂ ···. We will show that the groups π1(Yn) also form an increasing sequence of successively larger groups, whose union is π1(R3−B). To begin we have π1(Y0) ≈ Z since X0 is a solid torus embedded in R3 in a standard way. To compute π1(Y1), let Y 0 be the closure of Y0 in Y1, so Y 0 − Y0 is an open annulus A and π1(Y 0) is also Z. We obtain Y1 from Y 0 by attaching the space Z = Y1 − Y0 along A. The group π1(Z) is the free group F2 on two generators α1 and α2 represented by loops linking the two handles, since Z − A is homeomorphic to an open ball with two straight tubes deleted. A loop α generating π1(A) represents the commutator [α1, α2], as one can see by noting that the closure of Z is obtained from Z by adjoining two disjoint surfaces, each homeomorphic to a torus with an open disk removed; the boundary of this disk is homotopic to α and is also homotopic to the
commutator of meridian and longitude circles in the torus, which correspond to α1 and α2. Van Kampen’s theorem now implies that the inclusion Y0 ֓ Y1 induces an injection of π1(Y0) into π1(Y1) as the inﬁnite cyclic subgroup generated by [α1, α2]. In a similar way we can regard Yn+1 as being obtained from Yn by adjoining 2n copies of Z. Assuming inductively that π1(Yn) is the free group F2n with generators represented by loops linking the 2n smallest handles of Xn, then each copy of Z ad- 172 Chapter 2 Homology joined to Yn changes π1(Yn) by making one of the generators into the commutator of two new generators. Note that adjoining a copy of Z induces an injection on π1 since the induced homomorphism is the free product of the injection π1(A)→π1(Z) with the identity map on the complementary free factor. Thus the map π1(Yn)→π1(Yn+1) is an injection F2n→F2n+1. The group π1(R3 − B) is isomorphic to the union of this increasing sequence of groups by a compactness argument: Each loop in R3 − B has compact image and hence must lie in some Yn, and similarly for homotopies of loops. In particular we see explicitly why π1(R3 − B) has trivial abelianization, because each of its generators is exactly equal to the commutator of two other generators. This inductive construction in which each generator of a free group is decreed to be the commutator of two new generators is perhaps the simplest way of building a nontrivial group with trivial abelianization, and for the construction to have such a nice geometric interpretation is something to marvel at. From a naive viewpoint it may seem a little odd that a highly nonfree group can be built as a union of an increasing sequence of free groups, but this can also easily happen for abelian groups, as Q for example is the union of an increasing sequence of inﬁnite cyclic subgroups. The next theorem says that for subspaces of Rn, the property of being open is a topological invari
ant. This result is known classically as Invariance of Domain, the word ‘domain’ being an older designation for an open set in Rn. Theorem 2B.3. If U is an open set in Rn and h : U→Rn is an embedding, or more generally just a continuous injection, then the image h(U) is an open set in Rn. Proof: Viewing S n as the one-point compactiﬁcation of Rn, an equivalent statement is that h(U) is open in S n, and this is what we will prove. Each x ∈ U is the center point of a disk Dn ⊂ U. It will suﬃce to prove that h(Dn − ∂Dn) is open in S n. The hypothesis on h implies that its restrictions to Dn and ∂Dn are embeddings. By the previous proposition S n−h(∂Dn) has two path-components. These path-components are h(Dn − ∂Dn) and S n − h(Dn) since these two subspaces are disjoint and pathconnected, the ﬁrst since it is homeomorphic to Dn − ∂Dn and the second by the proposition. Since S n − h(∂Dn) is open in S n, its path-components are the same as its components. The components of a space with ﬁnitely many components are open, so h(Dn − ∂Dn) is open in S n − h(∂Dn) and hence also in S n. ⊔⊓ Here is an application involving the notion of an n manifold, which is a Hausdorﬀ space locally homeomorphic to Rn : Corollary 2B.4. If M is a compact n manifold and N is a connected n manifold, then an embedding h : M→N must be surjective, hence a homeomorphism. Proof: h(M) is closed in N since it is compact and N is Hausdorﬀ. Since N is connected it suﬃces to show h(M) is also open in N, and this is immediate from the theorem. ⊔⊓ Classical Applications Section 2.B 173 The Invariance of Domain and the
n dimensional generalization of the Jordan curve theorem were ﬁrst proved by Brouwer around 1910, at a very early stage in the development of algebraic topology. Division Algebras Here is an algebraic application of homology theory due to H. Hopf: Theorem 2B.5. R and C are the only ﬁnite-dimensional division algebras over R which are commutative and have an identity. By deﬁnition, an algebra structure on Rn is simply a bilinear multiplication map Rn × Rn→Rn, (a, b) ֏ ab. Thus the product satisﬁes left and right distributivity, a(b +c) = ab +ac and (a+b)c = ac +bc, and scalar associativity, α(ab) = (αa)b = a(αb) for α ∈ R. Commutativity, full associativity, and an identity element are not assumed. An algebra is a division algebra if the equations ax = b and xa = b are always solvable whenever a ≠ 0. In other words, the linear transformations x ֏ ax and x֏xa are surjective when a ≠ 0. These are linear maps Rn→Rn, so surjectivity is equivalent to having trivial kernel, which means there are no zero-divisors. The four classical examples are R, C, the quaternion algebra H, and the octonion algebra O. Frobenius proved in 1877 that R, C, and H are the only ﬁnite-dimensional associative division algebras over R with an identity element. If the product satisﬁes \|ab\| = \|a\|\|b\| as in the classical examples, then Hurwitz showed in 1898 that the dimension of the algebra must be 1, 2, 4, or 8, and others subsequently showed that the only examples with an identity element are the classical ones. A full discussion of all this, including some examples showing the necessity of the hypothesis of an identity element, can be found in [Ebbinghaus 1991]. As one would expect, the proofs of these results are algebraic, but if one drops the condition that \|ab\| = \|a\|\|b\| it seems that more topological proofs are required. We will show in Theorem 3.21
that a ﬁnite-dimensional division algebra over R must have dimension a power of 2. The fact that the dimension can be at most 8 is a famous theorem of [Bott & Milnor 1958] and [Kervaire 1958]. See §4.B for a few more comments on this. Proof: Suppose ﬁrst that Rn has a commutative division algebra structure. Deﬁne a map f : S n−1→S n−1 by f (x) = x2/\|x2\|. This is well-deﬁned since x ≠ 0 implies x2 ≠ 0 in a division algebra. The map f is continuous since the multiplication map Rn × Rn→Rn is bilinear, hence continuous. Since f (−x) = f (x) for all x, f induces a quotient map f : RPn−1→S n−1. The following argument shows that f is injective. An equality f (x) = f (y) implies x2 = α2y 2 for α = (\|x2\|/\|y 2\|)1/2 > 0. Thus we have x2 − α2y 2 = 0, which factors as (x + αy)(x − αy) = 0 using commutativity and the fact that α is a real scalar. Since there are no divisors of zero, we deduce that x = ±αy. Since x and y are unit vectors and α is real, this yields x = ±y, so x and y determine the same point of RPn−1, which means that f is injective. 174 Chapter 2 Homology Since f is an injective map of compact Hausdorﬀ spaces, it must be a homeo- morphism onto its image. By Corollary 2B.4, f must in fact be surjective if we are not in the trivial case n = 1. Thus we have a homeomorphism RPn−1 ≈ S n−1. This implies n = 2 since if n > 2 the spaces RPn−1 and S n−1 have diﬀerent homology groups (or diﬀerent fundamental groups). It remains to show that a 2 dimensional commutative division algebra A with identity is isomorphic to C. This is elementary algebra: If j ∈ A is not a real scalar
multiple of the identity element 1 ∈ A and we write j2 = a + bj for a, b ∈ R, then (j − b/2)2 = a + b2/4 so by rechoosing j we may assume that j2 = a ∈ R. If a ≥ 0, say a = c2, then j2 = c2 implies (j + c)(j − c) = 0, so j = ±c, but this contradicts the choice of j. So j2 = −c2 and by rescaling j we may assume j2 = −1, hence A is isomorphic to C. ⊔⊓ Leaving out the last paragraph, the proof shows that a ﬁnite-dimensional com- mutative division algebra, not necessarily with an identity, must have dimension at most 2. Oddly enough, there do exist 2 dimensional commutative division algebras without identity elements, for example C with the modiﬁed multiplication z·w = zw, the bar denoting complex conjugation. The Borsuk–Ulam Theorem In Theorem 1.10 we proved the 2 dimensional case of the Borsuk–Ulam theorem, and now we will give a proof for all dimensions, using the following theorem of Borsuk: Proposition 2B.6. An odd map f : S n→S n, satisfying f (−x) = −f (x) for all x, must have odd degree. The corresponding result that even maps have even degree is easier, and was an exercise for §2.2. The proof will show that using homology with a coeﬃcient group other than Z can sometimes be a distinct advantage. The main ingredient will be a certain exact sequence associated to a two-sheeted covering space p : X→X, ··· -→ Hn(X; Z2) τ∗-----→ Hn( X; Z2) p∗-----→ Hn(X; Z2) -→ Hn−1(X; Z2) -→ ··· e This is the long exact sequence of homology groups associated to a short exact se- e quence of chain complexes consisting of short exact sequences of chain groups 0 -→ Cn(X; Z2) τ-----→ Cn( X; Z2) p♯-----→ Cn(X; Z2) -→
0 n→X always lift to The map p♯ is surjective since singular simplices σ : is simply-connected. Each σ has in fact precisely two lifts σ1 and σ2. Because we e are using Z2 coeﬃcients, the kernel of p♯ is generated by the sums σ1 + σ2. So if we e n, then the image of deﬁne τ to send each σ : e τ is the kernel of p♯. Obviously τ is injective, so we have the short exact sequence e n→X to the sum of its two lifts to X, as ∆ ∆ e e n ∆ e∆ Classical Applications Section 2.B 175 indicated. Since τ and p♯ commute with boundary maps, we have a short exact sequence of chain complexes, yielding the long exact sequence of homology groups. The map τ∗ is a special case of more general transfer homomorphisms considered in §3.G, so we will refer to the long exact sequence involving the maps τ∗ as the transfer sequence. This sequence can also be viewed as a special case of the Gysin sequences discussed in §4.D. There is a generalization of the transfer sequence to homology with other coeﬃcients, but this uses a more elaborate form of homology called homology with local coeﬃcients, as we show in §3.H. Proof of 2B.6: The proof will involve the transfer sequence for the covering space p : S n→RPn. This has the following form, where to simplify notation we abbreviate RPn to P n and we let the coeﬃcient group Z2 be implicit: The initial 0 is Hn+1(P n; Z2), which vanishes since P n is an n dimensional CW complex. The other terms that are zero are Hi(S n) for 0 < i < n. We assume n > 1, leaving the minor modiﬁcations needed for the case n = 1 to the reader. All the terms that are not zero are Z2, by cellular homology. Alternatively, this exact sequence can be used to compute the homology groups Hi(RPn; Z2) if one does not already know them. Since all the nonzero groups in the sequence are Z2, exactness forces the maps to be isomorph
isms or zero as indicated. An odd map f : S n→S n induces a quotient map f : RPn→RPn. These two maps induce a map from the transfer sequence to itself, and we will need to know that the squares in the resulting diagram commute. This follows from the naturality of the long exact sequence of homology associated to a short exact sequence of chain complexes, once we verify commutativity of the diagram Here the right-hand square commutes since pf = f p. The left-hand square comσ2, the two lifts of mutes since for a singular i simplex σ : i→P n with lifts σ1 and f σ are f σ1 and f σ2 since f takes antipodal points to antipodal points. ∆ e e Now we can see that all the maps f∗ and f ∗ in the commutative diagram of transfer sequences are isomorphisms by induction on dimension, using the evident e e fact that if three maps in a commutative square are isomorphisms, so is the fourth. The induction starts with the trivial fact that f∗ and f ∗ are isomorphisms in dimension zero. 176 Chapter 2 Homology In particular we deduce that the map f∗ : Hn(S n; Z2)→Hn(S n; Z2) is an isomorphism. By Lemma 2.49 this map is multiplication by the degree of f mod 2, so the degree of f must be odd. ⊔⊓ The fact that odd maps have odd degree easily implies the Borsuk–Ulam theorem: Corollary 2B.7. For every map g : S n→Rn there exists a point x ∈ S n with g(x) = g(−x). Proof: Let f (x) = g(x) − g(−x), so f is odd. We need to show that f (x) = 0 for If this is not the case, we can replace f (x) by f (x)/\|f (x)\| to get a new some x. map f : S n→S n−1 which is still odd. The restriction of this f to the equator S n−1 then has odd degree by the proposition. But this restriction is nullhomotopic via the restriction of f to one of the hemispheres bounded by
S n−1. ⊔⊓ Exercises 1. Compute Hi(S n − X) when X is a subspace of S n homeomorphic to S k ∨ S ℓ or to S k ∐ S ℓ. 2. Show that Hi(S n − X) ≈ Hn−i−1(X) when X is homeomorphic to a ﬁnite connected graph. [First do the case that the graph is a tree.] e e 3. Let (D, S) ⊂ (Dn, S n−1) be a pair of subspaces homeomorphic to (Dk, S k−1), with D ∩ S n−1 = S. Show the inclusion S n−1 − S ֓ Dn − D induces an isomorphism on homology. [Glue two copies of (Dn, D) to the two ends of (S n−1 × I, S × I) to produce a k sphere in S n and look at a Mayer–Vietoris sequence for the complement of this k sphere.] 4. In the unit sphere S p+q−1 ⊂ Rp+q let S p−1 and S q−1 be the subspheres consisting of points whose last q and ﬁrst p coordinates are zero, respectively. (a) Show that S p+q−1 − S p−1 deformation retracts onto S q−1, and is in fact homeomorphic to S q−1× Rp. (b) Show that S p−1 and S q−1 are not the boundaries of any pair of disjointly embedded disks Dp and Dq in Dp+q. [The preceding exercise may be useful.] 5. Let S be an embedded k sphere in S n for which there exists a disk Dn ⊂ S n intersecting S in the disk Dk ⊂ Dn deﬁned by the ﬁrst k coordinates of Dn. Let Dn−k ⊂ Dn be the disk deﬁned by the last n − k coordinates, with boundary sphere S n−k−1. Show that the inclusion S n−k−1 ֓ S n − S induces an isomorphism on homology groups. 6. Modify the construction of the Alexander horned sphere to
produce an embedding S 2 ֓ R3 for which neither component of R3 − S 2 is simply-connected. Simplicial Approximation Section 2.C 177 7. Analyze what happens when the number of handles in the basic building block for the Alexander horned sphere is doubled, as in the ﬁgure at the right. 8. Show that R2n+1 is not a division algebra over R if n > 0 by considering how the determinant of the linear map x ֏ ax given by the multiplication in a division algebra structure would vary as a moves along a path in R2n+1 − {0} joining two antipodal points. 9. Make the transfer sequence explicit in the case of a trivial covering X = X × S 0. 10. Use the transfer sequence for the covering S ∞→RP∞ to compute Hn(RP∞; Z2). e 11. Use the transfer sequence for the covering X × S ∞→X × RP∞ to produce isomorphisms Hn(X × RP∞; Z2) ≈ i≤n Hi(X; Z2) for all n. e X→X, where L Many spaces of interest in algebraic topology can be given the structure of sim- plicial complexes, and early in the history of the subject this structure was exploited as one of the main technical tools. Later, CW complexes largely superseded simplicial complexes in this role, but there are still some occasions when the extra structure of simplicial complexes can be quite useful. This will be illustrated nicely by the proof of the classical Lefschetz ﬁxed point theorem in this section. One of the good features of simplicial complexes is that arbitrary continuous maps between them can always be deformed to maps that are linear on the simplices of some subdivision of the domain complex. This is the idea of ‘simplicial approxi- mation’, developed by Brouwer and Alexander before 1920. Here is the relevant definition: If K and L are simplicial complexes, then a map f : K→L is simplicial if it sends each simplex of K to a simplex of L by a linear map taking vertices to vertices. In barycentric coordinates, a linear map of a simplex [v0, ···, vn] has the form i tif (vi). Since a linear map from a
simplex to a simplex is uniquely determined by its values on vertices, this means that a simplicial map is uniquely P determined by its values on vertices. It is easy to see that a map from the vertices i tivi ֏ P of K to the vertices of L extends to a simplicial map iﬀ it sends the vertices of each simplex of K to the vertices of some simplex of L. Here is the most basic form of the Simplicial Approximation Theorem: Theorem 2C.1. If K is a ﬁnite simplicial complex and L is an arbitrary simplicial complex, then any map f : K→L is homotopic to a map that is simplicial with respect to some iterated barycentric subdivision of K. 178 Chapter 2 Homology To see that subdivision of K is essential, consider the case of maps S n→S n. With ﬁxed simplicial structures on the domain and range spheres there are only ﬁnitely many simplicial maps since there are only ﬁnitely many ways to map vertices to ver- tices. Hence only ﬁnitely many degrees are realized by maps that are simplicial with respect to ﬁxed simplicial structures in both the domain and range spheres. This re- mains true even if the simplicial structure on the range sphere is allowed to vary, since if the range sphere has more vertices than the domain sphere then the map cannot be surjective, hence must have degree zero. Before proving the simplicial approximation theorem we need some terminology and a lemma. The star St σ of a simplex σ in a simplicial complex X is deﬁned to be the subcomplex consisting of all the simplices of X that contain σ. Closely related to this is the open star st σ, which is the union of the interiors of all simplices containing σ, where the interior of a simplex τ is by deﬁnition τ − ∂τ. Thus st σ is an open set in X whose closure is St σ. Lemma 2C.2. For vertices v1, ···, vn of a simplicial complex X, the intersection st v1 ∩ ··· ∩ st vn is empty unless v1, ···, vn are the vertices of a simplex
σ of X, in which case st v1 ∩ ··· ∩ st vn = st σ. Proof: The intersection st v1 ∩ ··· ∩ st vn consists of the interiors of all simplices τ whose vertex set contains {v1, ···, vn}. If st v1 ∩ ··· ∩ st vn is nonempty, such a τ exists and contains the simplex σ = [v1, ···, vn] ⊂ X. The simplices τ containing ⊔⊓ {v1, ···, vn} are just the simplices containing σ, so st v1 ∩ ··· ∩ st vn = st σ. plex Proof of 2C.1: Choose a metric on K that restricts to the standard Euclidean metric on each simplex of K. For example, K can be viewed as a subcomplex of a simN whose vertices are all the vertices of K, and we can restrict a standard metN to give a metric on K. Let ε be a Lebesgue number for the open cover \| w is a vertex of L } of K. After iterated barycentric subdivision of K we ric on ∆ { f −1 may assume that each simplex has diameter less than ε/2. The closed star of each st w ∆ vertex v of K then has diameter less than ε, hence this closed star maps by f to the open star of some vertex g(v) of L. The resulting map g : K0→L0 thus satisﬁes f (St v) ⊂ st g(v) for all vertices v of K. To see that g extends to a simplicial map g : K→L, consider the problem of extending g over a simplex [v1, ···, vn] of K. An interior point x of this simplex lies in st vi for each i, so f (x) lies in st g(vi) for each i, since f (st vi) ⊂ st g(vi) by the deﬁnition of g(vi). Thus st g(v1) ∩ ··· ∩ st g(vn) ≠ ∅, so [g(v1), ···, g(vn)] is a simplex of L by
the lemma, and we can extend g linearly over [v1, ···, vn]. Both f (x) and g(x) lie in a single simplex of L since g(x) lies in [g(v1), ···, g(vn)] and f (x) lies in the star of this simplex. So taking the linear path (1−t)f (x)+tg(x), 0 ≤ t ≤ 1, in the simplex containing f (x) and g(x) deﬁnes a homotopy from f to g. To check continuity of this homotopy it suﬃces to restrict to the simplex [v1, ···, vn], where Simplicial Approximation Section 2.C 179 continuity is clear since f (x) varies continuously in the star of [g(v1), ···, g(vn)] ⊔⊓ and g(x) varies continuously in [g(v1), ···, g(vn)]. Notice that if f already sends some vertices of K to vertices of L then we may choose g to equal to f on these vertices, and hence the homotopy from f to g will be stationary on these vertices. This is convenient if one is in a situation where one wants maps and homotopies to preserve basepoints. The proof makes it clear that the simplicial approximation g can be chosen not just homotopic to f but also close to f if we allow subdivisions of L as well as K. The Lefschetz Fixed Point Theorem This very classical application of homology is a considerable generalization of the Brouwer ﬁxed point theorem. It is also related to the Euler characteristic formula. P For a homomorphism ϕ : Zn→Zn with matrix [aij ], the trace tr ϕ is deﬁned i aii, the sum of the diagonal elements of [aij ]. Since tr([aij ][bij ]) = to be tr([bij ][aij ]), conjugate matrices have the same trace, and it follows that tr ϕ is independent of the choice of basis for Zn. For a homomorphism ϕ : A→A of a ﬁnitely generated abelian group A we can then
deﬁne tr ϕ to be the trace of the induced homomorphism ϕ : A/Torsion→A/Torsion. For a map f : X→X of a ﬁnite CW complex X, or more generally any space whose homology groups are ﬁnitely generated and vanish in high dimensions, the Lefschetz number τ(f ) is deﬁned to be n(−1)n tr is the identity, or is homotopic to the identity, then τ(f ) is the Euler characteristic χ (X) since the trace of the n× n identity matrix is n. P f∗ : Hn(X)→Hn(X). In particular, if f Here is the Lefschetz ﬁxed point theorem: Theorem 2C.3. If X is a ﬁnite simplicial complex, or more generally a retract of a ﬁnite simplicial complex, and f : X→X is a map with τ(f ) ≠ 0, then f has a ﬁxed point. As we show in Theorem A.7 in the Appendix, every compact, locally contractible space that can be embedded in Rn for some n is a retract of a ﬁnite simplicial complex. This includes compact manifolds and ﬁnite CW complexes, for example. The compactness hypothesis is essential, since a translation of R has τ = 1 but no ﬁxed points. For an example showing that local properties are also signiﬁcant, let X be the compact subspace of R2 consisting of two concentric circles together with a copy of R between them whose two ends spiral in to the two circles, wrapping around them inﬁnitely often, and let f : X→X be a homeomorphism translating the copy of R along itself and rotating the circles, with no ﬁxed points. Since f is homotopic to the identity, we have τ(f ) = χ (X), which equals 1 since the three path components of X are two circles and a line. 180 Chapter 2 Homology If X has the same homology groups as a point, at least modulo torsion, then the theorem says that every map X→X has a ﬁxed point. This holds
for example for RPn if n is even. The case of projective spaces is interesting because of its connection with linear algebra. An invertible linear transformation f : Rn→Rn takes lines through 0 to lines through 0, hence induces a map f : RPn−1→RPn−1. Fixed points of f are equivalent to eigenvectors of f. The characteristic polyno- mial of f has odd degree if n is odd, hence has a real root, so an eigenvector ex- ists in this case. This is in agreement with the observation above that every map RP2k→RP2k has a ﬁxed point. On the other hand the rotation of R2k deﬁned by f (x1, ···, x2k) = (x2, −x1, x4, −x3, ···, x2k, −x2k−1) has no eigenvectors and its projectivization f : RP2k−1→RP2k−1 has no ﬁxed points. Similarly, in the complex case an invertible linear transformation f : Cn→Cn induces f : CPn−1→CPn−1, and this always has a ﬁxed point since the characteristic polynomial always has a complex root. Nevertheless, as in the real case there is a map CP2k−1→CP2k−1 without ﬁxed points. Namely, consider f : C2k→C2k deﬁned by f (z1, ···, z2k) = (z2, −z1, z4, −z3, ···, z2k, −z2k−1). This map is only ‘conjugatelinear’ over C, but this is still good enough to imply that f induces a well-deﬁned map f on CP2k−1, and it is easy to check that f has no ﬁxed points. The similarity between the real and complex cases persists in the fact that every map CP2k→CP2k has a ﬁxed point, though to deduce this from the Lefschetz ﬁxed point theorem re- quires more structure than homology has, so this will be left as an exercise for §3
.2, using cup products in cohomology. One could go further and consider the quaternionic case. The antipodal map of S 4 = HP1 has no ﬁxed points, but every map HPn→HPn with n > 1 does have a ﬁxed point. This is shown in Example 4L.4 using considerably heavier machinery. Proof of 2C.3: The general case easily reduces to the case of ﬁnite simplicial complexes, for suppose r : K→X is a retraction of a ﬁnite simplicial complex K onto X. For a map f : X→X, the composition f r : K→X ⊂ K then has exactly the same ﬁxed points as f. Since r∗ : Hn(K)→Hn(X) is projection onto a direct summand, we have tr(f∗r∗) = tr(f∗) and hence τ(f r ) = τ(f ). For X a ﬁnite simplicial complex, suppose that f : X→X has no ﬁxed points. We claim there is a subdivision L of X, a further subdivision K of L, and a simplicial map g : K→L homotopic to f such that g(σ )∩σ = ∅ for each simplex σ of K. To see this, ﬁrst choose a metric d on X as in the proof of the simplicial approximation theorem. Since f has no ﬁxed points, d x, f (x) > 0 for all x ∈ X, so by the compactness of X there is an ε > 0 such that d x, f (x) > ε for all x. Choose a subdivision L of X so that the stars of all simplices have diameter less than ε/2. Applying the simplicial approximation theorem, there is a subdivision K of L and a simplicial map g : K→L homotopic to f. By construction, g has the property that for each simplex σ of K, f (σ ) is contained in the star of the simplex g(σ ). Then g(σ ) ∩ σ = ∅ Simplicial Approximation Section 2.C 181 for each simplex σ of K since for any choice of x ∈ �
� we have d x, f (x) > ε, while g(σ ) lies within distance ε/2 of f (x) and σ lies within distance ε/2 of x, as a consequence of the fact that σ is contained in a simplex of L, K being a subdivision of L. The Lefschetz numbers τ(f ) and τ(g) are equal since f and g are homotopic. Since g is simplicial, it takes the n skeleton Kn of K to the n skeleton Ln of L, for each n. Since K is a subdivision of L, Ln is contained in Kn, and hence g(Kn) ⊂ Kn for all n. Thus g induces a chain map of the cellular chain complex {Hn(Kn, Kn−1)} to itself. This can be used to compute τ(g) according to the formula τ(g) = (−1)n tr Xn g∗ : Hn(Kn, Kn−1)→Hn(Kn, Kn−1) This is the analog of Theorem 2.44 for trace instead of rank, and is proved in precisely the same way, based on the elementary algebraic fact that trace is additive for endo- morphisms of short exact sequences: Given a com- mutative diagram as at the right with exact rows, then tr β = tr α + tr γ. This algebraic fact can be proved by reducing to the easy case that A, B, and C are free by ﬁrst factoring out the torsion in B, hence also the torsion in A, then eliminating any remaining torsion in C by replacing A by a larger subgroup A′ ⊂ B, with A having ﬁnite index in A′. The details of this argument are left to the reader. Finally, note that g∗ : Hn(Kn, Kn−1)→Hn(Kn, Kn−1) has trace 0 since the matrix for g∗ has zeros down the diagonal, in view of the fact that g(σ ) ∩ σ = ∅ for each ⊔⊓ n simplex σ. So τ(f ) = τ(g) = 0. Example 2C.4. Let us verify the theorem in an example. Let X be the closed orientable surface of genus 3 as shown in the
ﬁgure below, with f : X→X the 180 degree rotation about a vertical axis passing through the central hole of X. Since f has no ﬁxed points, we should have τ(f ) = 0. The induced map f∗ : H0(X)→H0(X) is the identity, as always for a path-connected space, so this contributes 1 to τ(f ). For H1(X) we saw in Example 2A.2 that the six loops αi and βi represent a basis. The map f∗ interchanges the homology classes of α1 and α3, and likewise for β1 and β3, while β2 is sent to itself and α2 is sent to α′ 2 which is homologous to α2 as we saw in Example 2A.2. So f∗ : H1(X)→H1(X) contributes −2 to τ(f ). It remains to check that f∗ : H2(X)→H2(X) is the identity, which we do by the commutative diagram at the right, where x is a point of X in the central torus and y = f (x). We can see that the 182 Chapter 2 Homology left-hand vertical map is an isomorphism by considering the long exact sequence of the triple (X, X − {x}, X 1) where X 1 is the 1 skeleton of X in its usual CW structure and x is chosen in X − X 1, so that X − {x} deformation retracts onto X 1 and Hn(X − {x}, X 1) = 0 for all n. The same reasoning shows the right-hand vertical map is an isomorphism. There is a similar commutative diagram with f replaced by a homeomorphism g that is homotopic to the identity and equals f in a neighbor- hood of x, with g the identity outside a disk in X containing x and y. Since g is homotopic to the identity, it induces the identity across the top row of the diagram, and since g equals f near x, it induces the same map as f in the bottom row of the diagram, by excision. It follows that the map f∗ in the upper row is the identity. This example generalizes to surfaces of any odd genus by adding symmetric pairs of tori at the left and right
. Examples for even genus are described in one of the exercises. Fixed point theory is a well-developed side branch of algebraic topology, but we touch upon it only occasionally in this book. For a nice introduction see [Brown 1971]. Simplicial Approximations to CW Complexes The simplicial approximation theorem allows arbitrary continuous maps to be replaced by homotopic simplicial maps in many situations, and one might wonder about the analogous question for spaces: Which spaces are homotopy equivalent to simplicial complexes? We will show this is true for the most common class of spaces in algebraic topology, CW complexes. In the Appendix the question is answered for a few other classes of spaces as well. Theorem 2C.5. Every CW complex X is homotopy equivalent to a simplicial complex, which can be chosen to be of the same dimension as X, ﬁnite if X is ﬁnite, and countable if X is countable. We will build a simplicial complex Y ≃ X inductively as an increasing union of subcomplexes Yn homotopy equivalent to the skeleta X n. For the inductive step, assuming we have already constructed Yn ≃ X n, let en+1 be an (n + 1) cell of X attached by a map ϕ : S n→X n. The map S n→Yn corresponding to ϕ under the homotopy equivalence Yn ≃ X n is homotopic to a simplicial map f : S n→Yn by the simplicial approximation theorem, and it is not hard to see that the spaces X n ∪ϕ en+1 and Yn ∪f en+1 are homotopy equivalent, where the subscripts denote attaching en+1 via ϕ and f, respectively; see Proposition 0.18 for a proof. We can view Yn ∪f en+1 as the mapping cone Cf, obtained from the mapping cylinder of f by collapsing the domain end to a point. If we knew that the mapping cone of a simplicial map was a simplicial complex, then by performing the same construction for all the (n + 1) cells of X we would have completed the induction step. Unfortunately, and somewhat surprisingly, mapping cones and mapping cylinders are rather awkward objects in the Simplicial Approximation Section 2.C 183 simplicial category. To avoid this awkwardness we will instead construct simplicial analogs of mapping cones
and cylinders that have all the essential features of actual mapping cones and cylinders. ∆ 2→ Let us ﬁrst construct the simplicial analog of a mapping cylinder. For a simplicial map f : K→L this will be a simplicial complex M(f ) containing both L and the barycentric subdivision K′ of K as subcomplexes, and such that there is a deformation retraction rt of M(f ) onto L with r1 \|\| K′ = f. The ﬁgure shows the case that f is a simplicial surjection 1. The construction proceeds one simplex of K at a time, by induction on dimension. To begin, the ∆ ordinary mapping cylinder of f : K0→L suﬃces for M(f \|\| K0). Assume inductively that we have already constructed M(f \|\| Kn−1). Let σ be an n simplex of K and let τ = f (σ ), a simplex of L of dimension n or less. By the inductive hypothesis we have already constructed M(f : ∂σ→τ) with the desired properties, and we let M(f : σ→τ) be the cone on M(f : ∂σ→τ), as shown in the ﬁgure. The space M(f : ∂σ→τ) is contractible since by induction it deformation retracts onto τ which is contractible. The cone M(f : σ→τ) is of course contractible, so the inclusion of M(f : ∂σ→τ) into M(f : σ→τ) is a homotopy equivalence. This implies that M(f : σ→τ) deformation retracts onto M(f : ∂σ→τ) by Corollary 0.20, or one can give a direct argument using the fact that M(f : ∂σ→τ) is contractible. By attaching M(f : σ→τ) to M(f \|\| Kn−1) along M(f : ∂σ→τ) ⊂ M(f \|\| Kn−1) for all n simplices σ of K we obtain M(f \|\| Kn) with a deformation retraction onto M(f \|\| Kn−1). Taking the union over all n yields M(f ) with a deformation retraction rt
onto L, the inﬁnite concatenation of the previous deformation retractions, with the deformation retraction of M(f \|\| Kn) onto M(f \|\| Kn−1) performed in the t interval [1/2n+1, 1/2n]. The map r1 \|\| K may not equal f, but it is homotopic to f via the linear homotopy tf +(1−t)r1, which is deﬁned since r1(σ ) ⊂ f (σ ) for all simplices σ of K. By applying the homotopy extension property to the homotopy of r1 that equals tf + (1 − t)r1 on K and the identity map on L, we can improve our deformation retraction of M(f ) onto L so that its restriction to K at time 1 is f. From the simplicial analog M(f ) of a mapping cylinder we construct the simplicial ‘mapping cone’ C(f ) by attaching the ordinary cone on K′ to the subcomplex K′ ⊂ M(f ). Proof of 2C.5: We will construct for each n a CW complex Zn containing X n as a deformation retract and also containing as a deformation retract a subcomplex Yn that is a simplicial complex. Beginning with Y0 = Z0 = X 0, suppose inductively that we have already constructed Yn and Zn. Let the cells en+1 of X be attached by maps ϕα : S n→X n. Using the simplicial approximation theorem, there is a homotopy from ϕα to a simplicial map fα : S n→Yn. The CW complex Wn = Zn α M(fα) contains a α S 184 Chapter 2 Homology α homeomorphic to S n at one end of M(fα), and the homeosimplicial subcomplex S n morphism S n ≈ S n α is homotopic in Wn to the map fα, hence also to ϕα. Let Zn+1 be obtained from Zn by attaching Dn+1 α × I ’s via these homotopies between the ϕα ’s and α ֓Wn. Thus Zn+1 contains X n+1 at one end, and at the other end we the inclusions S n
have a simplicial complex Yn+1 = Yn α C(fα), where C(fα) is obtained from M(fα) α. Since Dn+1 × I deformation retracts onto by attaching a cone on the subcomplex S n ∂Dn+1 × I ∪ Dn+1 × {1}, we see that Zn+1 deformation retracts onto Zn ∪ Yn+1, which in turn deformation retracts onto Yn ∪ Yn+1 = Yn+1 by induction. Likewise, Zn+1 deformation retracts onto X n+1 ∪ Wn which deformation retracts onto X n+1 ∪ Zn and hence onto X n+1 ∪ X n = X n+1 by induction. n Yn and Z = n Zn. The deformation retractions of Zn onto X n give deformation retractions of X ∪ Zn onto X, and the inﬁnite concatenation of the latter deformation retractions is a deformation retraction of Z onto X. Similarly, Z Let Y = S S S deformation retracts onto Y. ⊔⊓ Exercises 1. What is the minimum number of edges in simplicial complex structures K and L on S 1 such that there is a simplicial map K→L of degree n? 2. Use the Lefschetz ﬁxed point theorem to show that a map S n→S n has a ﬁxed point unless its degree is equal to the degree of the antipodal map x ֏ −x. 3. Verify that the formula f (z1, ···, z2k) = (z2, −z1, z4, −z3, ···, z2k, −z2k−1) deﬁnes a map f : C2k→C2k inducing a quotient map CP2k−1→CP2k−1 without ﬁxed points. 4. If X is a ﬁnite simplicial complex and f : X→X is a simplicial homeomorphism, show that the Lefschetz number τ(f ) equals the Euler characteristic of the set of ﬁxed points of f. In particular, τ(f ) is the number of �
��xed points if the ﬁxed points are isolated. [Hint: Barycentrically subdivide X to make the ﬁxed point set a subcomplex.] 5. Let M be a closed orientable surface embedded in R3 in such a way that reﬂection across a plane P deﬁnes a homeomorphism r : M→M ﬁxing M ∩ P, a collection of circles. Is it possible to homotope r to have no ﬁxed points? 6. Do an even-genus analog of Example 2C.4 by replacing the central torus by a sphere letting f be a homeomorphism that restricts to the antipodal map on this sphere. 7. Verify that the Lefschetz ﬁxed point theorem holds also when τ(f ) is deﬁned using homology with coeﬃcients in a ﬁeld F. 8. Let X be homotopy equivalent to a ﬁnite simplicial complex and let Y be homotopy equivalent to a ﬁnite or countably inﬁnite simplicial complex. Using the simplicial ap- proximation theorem, show that there are at most countably many homotopy classes of maps X→Y. 9. Show that there are only countably many homotopy types of ﬁnite CW complexes. Cohomology is an algebraic variant of homology, the result of a simple dualization in the deﬁnition. Not surprisingly, the cohomology groups H i(X) satisfy axioms much like the axioms for homology, except that induced homomorphisms go in the opposite direction as a result of the dualization. The basic distinction between homol- ogy and cohomology is thus that cohomology groups are contravariant functors while homology groups are covariant. In terms of intrinsic information, however, there is not a big diﬀerence between homology groups and cohomology groups. The homol- ogy groups of a space determine its cohomology groups, and the converse holds at least when the homology groups are ﬁnitely generated. What is a little surprising is that contravariance leads to extra structure in co- homology. This ﬁrst appears in
a natural product, called cup product, which makes the cohomology groups of a space into a ring. This is an extremely useful piece of additional structure, and much of this chapter is devoted to studying cup products, which are considerably more subtle than the additive structure of cohomology. How does contravariance lead to a product in cohomology that is not present in homology? Actually there is a natural product in homology, but it takes the somewhat diﬀerent form of a map Hi(X)× Hj(Y ) -→ Hi+j (X × Y ) called the cross product. If both X and Y are CW complexes, this cross product in homology is induced from a map of cellular chains sending a pair (ei, ej) consisting of a cell of X and a cell of Y to the product cell ei × ej in X × Y. The details of the construction are described in §3.B. Taking X = Y, we thus have the ﬁrst half of a hypothetical product Hi(X)× Hj(X) -→ Hi+j(X × X) -→ Hi+j(X) The diﬃculty is in deﬁning the second map. The natural thing would be for this to be induced by a map X × X→X. The multiplication map in a topological group, or more generally an H–space, is such a map, and the resulting Pontryagin product can be quite useful when studying these spaces, as we show in §3.C. But for general X, the only 186 Chapter 3 Cohomology natural maps X × X→X are the projections onto one of the factors, and since these projections collapse the other factor to a point, the resulting product in homology is rather trivial. With cohomology, however, the situation is better. One still has a cross product H i(X)× H j(Y ) -→ H i+j(X × Y ) constructed in much the same way as in homology, so one can again take X = Y and get the ﬁrst half of a product H i(X)× H j(X) -→ H i+j(X × X) -→ H i+j(X) But now by contravariance the second map would be induced by a map X→X × X, (x) = (x
, x). This and there is an obvious candidate for this map, the diagonal map turns out to work very nicely, giving a well-behaved product in cohomology, the cup ∆ product. Another sort of extra structure in cohomology whose existence is traceable to contravariance is provided by cohomology operations. These make the cohomology groups of a space into a module over a certain rather complicated ring. Cohomology operations lie at a depth somewhat greater than the cup product structure, so we defer their study to §4.L. The extra layer of algebra in cohomology arising from the dualization in its def- inition may seem at ﬁrst to be separating it further from topology, but there are many topological situations where cohomology arises quite naturally. One of these is Poincar´e duality, the topic of the third section of this chapter. Another is obstruction theory, covered in §4.3. Characteristic classes in vector bundle theory (see [Milnor & Stasheﬀ 1974] or [VBKT]) provide a further instance. From the viewpoint of homotopy theory, cohomology is in some ways more basic than homology. As we shall see in §4.3, cohomology has a description in terms of homotopy classes of maps that is very similar to, and in a certain sense dual to, the deﬁnition of homotopy groups. There is an analog of this for homology, described in §4.F, but the construction is more complicated. The Idea of Cohomology Let us look at a few low-dimensional examples to get an idea of how one might be led naturally to consider cohomology groups, and to see what properties of a space they might be measuring. For the sake of simplicity we consider simplicial cohomology of complexes, rather than singular cohomology of more general spaces. ∆ Taking the simplest case ﬁrst, let X be a 1 dimensional complex, or in other tices of X to G also forms an abelian group, which we denote by words an oriented graph. For a ﬁxed abelian group G, the set of all functions from ver0(X; G). Similarly the set of all functions assigning an element of G to each edge of X forms an abelian 1(X; G) 1(X; G)
whose value on an oriented 1(X; G). We will be interested in the homomorphism δ : 0(X; G) to the function δϕ ∈ ∆ 0(X; G)→ ∆ group sending ϕ ∈ ∆ ∆ ∆ ∆ ∆ The Idea of Cohomology 187 edge [v0, v1] is the diﬀerence ϕ(v1) − ϕ(v0). For example, X might be the graph formed by a system of trails on a mountain, with vertices at the junctions between trails. The function ϕ could then assign to each junction its elevation above sea level, in which case δϕ would measure the net change in elevation along the trail from one junction to the next. Or X might represent a simple electrical circuit with ϕ mea- suring voltages at the connection points, the vertices, and δϕ measuring changes in Regarding the map δ : voltage across the components of the circuit, represented by edges. 0(X; G)→ 1(X; G) as a chain complex with 0 ’s before and after these two terms, the homology groups of this chain complex are by deﬁnition 0(X; G) and the simplicial cohomology groups of X, namely H 0(X; G) = Ker δ ⊂ 1(X; G)/ Im δ. For simplicity we are using here the same notation as will H 1(X; G) = be used for singular cohomology later in the chapter, in anticipation of the theorem ∆ ∆ ∆ that the two theories coincide for complexes, as we show in §3.1. ∆ The group H 0(X; G) is easy to describe explicitly. A function ϕ ∈ 0(X; G) has δϕ = 0 iﬀ ϕ takes the same value at both ends of each edge of X. This is equivalent to saying that ϕ is constant on each component of X. So H 0(X; G) is the group of all functions from the set of components of X to G. This is a direct product of copies ∆ ∆ of G, one for each component of X. δϕ = ψ has a solution ϕ ∈ The cohomology group H 1(X; G)
= 1(X; G)/ Im δ will be trivial iﬀ the equation 1(X; G). Solving this equation 0(X; G) for each ψ ∈ ∆ means deciding whether specifying the change in ϕ across each edge of X determines ∆ 0(X; G). This is rather like the calculus problem of ﬁnding a an actual function ϕ ∈ function having a speciﬁed derivative, with the diﬀerence operator δ playing the role ∆ of diﬀerentiation. As in calculus, if a solution of δϕ = ψ exists, it will be unique up ∆ to adding an element of the kernel of δ, that is, a function that is constant on each component of X. The equation δϕ = ψ is always solvable if X is a tree since if we choose arbitrarily a value for ϕ at a basepoint vertex v0, then if the change in ϕ across each edge of X is speciﬁed, this uniquely determines the value of ϕ at every other vertex v by induction along the unique path from v0 to v in the tree. When X is not a tree, we ﬁrst choose a maximal tree in each component of X. Then, since every vertex lies in one of these maximal trees, the values of ψ on the edges of the maximal trees determine ϕ uniquely up to a constant on each component of X. But in order for the equation δϕ = ψ to hold, the value of ψ on each edge not in any of the maximal trees must equal the diﬀerence in the already-determined values of ϕ at the two ends of the edge. This condition need not be satisﬁed since ψ can have arbitrary values on these edges. Thus we see that the cohomology group H 1(X; G) is a direct product of copies of the group G, one copy for each edge of X not in one of the chosen maximal trees. This can be compared with the homology group H1(X; G) which consists of a direct sum of copies of G, one for each edge of X not in one of the maximal trees. 188 Chapter 3 Cohomology Note that the relation between H 1(X; G) and H1(X; G) is the same
as the relation between H 0(X; G) and H0(X; G), with H 0(X; G) being a direct product of copies of G and H0(X; G) a direct sum, with one copy for each component of X in either case. ∆ ∆ ∆ ∆ Deﬁne 0(X; G) and 1(X; G)→ ∆ to the abelian group G, and deﬁne Now let us move up a dimension, taking X to be a 2 dimensional complex. 1(X; G) as before, as functions from vertices and edges of X 2(X; G) to be the functions from 2 simplices of 2(X; G) is deﬁned by δψ([v0, v1, v2]) = X to G. A homomorphism δ : ψ([v0, v1]) + ψ([v1, v2]) − ψ([v0, v2]), a signed sum of the values of ψ on the three ∆ 0(X; G) was a edges in the boundary of [v0, v1, v2], just as δϕ([v0, v1]) for ϕ ∈ signed sum of the values of ϕ on the boundary of [v0, v1]. The two homomorphisms 0(X; G) we = 0. Extending this δ-----→ 0(X; G) have δδϕ = ∆ chain complex by 0 ’s on each end, the resulting homology groups are by deﬁnition the cohomology groups H i(X; G). The formula for the map δ : ∆ 2(X; G) form a chain complex since for ϕ ∈ + 2(X; G) can be looked at from several diﬀerent viewpoints. Perhaps the simplest is the observation that δψ = 0 iﬀ ψ satisﬁes the additivity property ψ([v0, v2]) = ψ([v0, v1]) + ψ([v1, v2]), where we think of the edge [v0, v2] as the sum of the edges [v0, v1] and [v1, v2
]. Thus δψ measures the deviation of ψ from being additive. 1(X; G) ϕ(v1)−ϕ(v0) ∆ ∆ ϕ(v2)−ϕ(v0) ϕ(v2)−ϕ(v1) 1(X; G)→ δ-----→ − ∆ ∆ ∆ ∆ From another point of view, δψ can be regarded as an obstruction to ﬁnding 0(X; G) with ψ = δϕ, for if ψ = δϕ then δψ = 0 since δδϕ = 0 as we ϕ ∈ saw above. We can think of δψ as a local obstruction to solving ψ = δϕ since it depends only on the values of ψ within individual 2 simplices of X. If this local obstruction vanishes, then ψ deﬁnes an element of H 1(X; G) which is zero iﬀ ψ = δϕ has an actual solution. This class in H 1(X; G) is thus the global obstruction to solving ψ = δϕ. This situation is similar to the calculus problem of determining whether a given vector ﬁeld is the gradient vector ﬁeld of some function. The local obstruction here is the vanishing of the curl of the vector ﬁeld, and the global obstruction is the vanishing of all line integrals around closed loops in the domain of the vector ﬁeld. The condition δψ = 0 has an interpretation of a more geometric nature when X is a surface and the group G is Z or Z2. Consider ﬁrst the simpler case G = Z2. The condition δψ = 0 means that the number of times that ψ takes the value 1 on the edges of each 2 simplex is even, either 0 or 2. This means we can associate to ψ a collection Cψ of disjoint curves in X crossing the 1 skeleton transversely, such that the number of intersections of Cψ with each edge is equal to the value of ψ on that edge. If ψ = δϕ for some ϕ, then the curves of Cψ divide X into two regions X0 and X1 where the subscript indicates
the value of ϕ on all vertices in the region. The Idea of Cohomology 189 When G = Z we can reﬁne this construction by building Cψ from a number of arcs in each 2 simplex, each arc having a transverse orientation, the orientation which agrees or disagrees with the orientation of each edge according to the sign of the value of ψ on the edge, as in the ﬁgure at the right. The resulting collection Cψ of disjoint curves in X can be thought of as something like level curves for a function ϕ with δϕ = ψ, if such a func- tion exists. The value of ϕ changes by 1 each time a curve of Cψ is crossed. For example, if X is a disk then we will show that H 1(X; Z) = 0, so δψ = 0 implies ψ = δϕ for some ϕ, hence every transverse curve system Cψ forms the level curves of a function ϕ. On the other hand, if X is an annulus then this need no longer be true, as illustrated in the example shown in the ﬁgure at the left, where the equation ψ = δϕ obviously has no solution even though δψ = 0. By identifying the inner and outer boundary circles of this annulus we obtain a similar example on the torus. Even with G = Z2 the equation ψ = δϕ has no solution since the curve Cψ does not separate X into two regions X0 and X1. The key to relating cohomology groups to homology groups is the observation that a function from i simplices of X to G is equivalent to a homomorphism from the i(X) is free abelian with basis the i simplices of X, and a homomorphism with domain a free abelian group is uniquely i(X) to G. This is because simplicial chain group ∆ ∆ determined by its values on basis elements, which can be assigned arbitrarily. Thus we i(X), G) of homomorphisms i(X). There is also a simple relationship i(X; G)→ i+1(X; G) and the boundary ∆ i(X). The general formula for δ is have an identiﬁcation of i(X)→G, which is
called the dual group of of duality between the homomorphism δ : ∆ homomorphism ∂ : i(X; G) with the group Hom( i+1(X)→ ∆ ∆ δϕ([v0, ···, vi+1]) = ∆ ∆ Xj ∆ ∆ (−1)jϕ([v0, ···, vj, ···, vi+1]) b and the latter sum is just ϕ(∂[v0, ···, vi+1]). Thus we have δϕ = ϕ∂. In other words, ϕ-----→ G, which δ sends each ϕ ∈ Hom( i(X), G) to the composition i+1(X) i(X) ∂-----→ in the language of linear algebra means that δ is the dual map of ∂. ∆ ∆ ∆ Thus we have the algebraic problem of understanding the relationship between the homology groups of a chain complex and the homology groups of the dual complex obtained by applying the functor C֏Hom(C, G). This is the ﬁrst topic of the chapter. 190 Chapter 3 Cohomology Homology groups Hn(X) are the result of a two-stage process: First one forms a ∂-----→ Cn−1 -----→ ··· of singular, simplicial, or cellular chains, chain complex ··· -----→ Cn then one takes the homology groups of this chain complex, Ker ∂/ Im ∂. To obtain the cohomology groups H n(X; G) we interpolate an intermediate step, replacing the chain groups Cn by the dual groups Hom(Cn, G) and the boundary maps ∂ by their dual maps δ, before forming the cohomology groups Ker δ/ Im δ. The plan for this section is ﬁrst to sort out the algebra of this dualization process and show that the cohomology groups are determined algebraically by the homology groups, though in a somewhat subtle way. Then after this algebraic excursion we will deﬁne the cohomology groups of spaces and show that these satisfy basic properties very much like those for homology. The payoﬀ for all this formal work will begin to be apparent in subsequent sections. The Universal Coe�
�cient Theorem Let us begin with a simple example. Consider the chain complex where Z 2-----→ Z is the map x ֏ 2x. If we dualize by taking Hom(−, G) with G = Z, we obtain the cochain complex In the original chain complex the homology groups are Z ’s in dimensions 0 and 3, together with a Z2 in dimension 1. The homology groups of the dual cochain complex, which are called cohomology groups to emphasize the dualization, are again Z ’s in dimensions 0 and 3, but the Z2 in the 1 dimensional homology of the original complex has shifted up a dimension to become a Z2 in 2 dimensional cohomology. More generally, consider any chain complex of ﬁnitely generated free abelian groups. Such a chain complex always splits as the direct sum of elementary complexes of the forms 0→Z→0 and 0→Z m-----→ Z→0, according to Exercise 43 in §2.2. Applying Hom(−, Z) to this direct sum of elementary complexes, we obtain the direct sum of the corresponding dual complexes 0← Z← 0 and 0← Z m←------ Z← 0. Thus the cohomology groups are the same as the homology groups except that torsion is shifted up one dimension. We will see later in this section that the same relation between ho- mology and cohomology holds whenever the homology groups are ﬁnitely generated, even when the chain groups are not ﬁnitely generated. It would also be quite easy to Cohomology Groups Section 3.1 191 see in this example what happens if Hom(−, Z) is replaced by Hom(−, G), since the m←------ G←0. dual elementary cochain complexes would then be 0←G← 0 and 0← G Consider now a completely general chain complex C of free abelian groups ··· -----→ Cn+1 ∂------------→ Cn ∂------------→ Cn−1 -----→ ··· To dualize this complex we replace each chain group Cn by its dual cochain group n = Hom(Cn, G), the group of homomorphisms Cn→G, and we replace each boundC ∗ ary map ∂ : Cn→Cn−1 by its dual coboundary map �
� = ∂∗ : C ∗ n. The reason why δ goes in the opposite direction from ∂, increasing rather than decreasing dimension, is purely formal: For a homomorphism α : A→B, the dual homomorphism ϕ-----→ G to the α∗ : Hom(B, G)→Hom(A, G) is deﬁned by α∗(ϕ) = ϕα, so α∗ sends B ϕ-----→ G. Dual homomorphisms obviously satisfy (αβ)∗ = β∗α∗, composition A 11∗ = 11, and 0∗ = 0. In particular, since ∂∂ = 0 it follows that δδ = 0, and the cohomology group H n(C; G) can be deﬁned as the ‘homology group’ Ker δ/ Im δ at C ∗ n−1→C ∗ n in the cochain complex α-----→ B ··· ←--------- C ∗ n+1 δ←---------------- C ∗ n δ←---------------- C ∗ n−1←--------- ··· Our goal is to show that the cohomology groups H n(C; G) are determined solely by G and the homology groups Hn(C) = Ker ∂/ Im ∂. A ﬁrst guess might be that H n(C; G) is isomorphic to Hom(Hn(C), G), but this is overly optimistic, as shown by the example above where H2 was zero while H 2 was nonzero. Nevertheless, there is a natural map h : H n(C; G)→Hom(Hn(C), G), deﬁned as follows. Denote the cycles and boundaries by Zn = Ker ∂ ⊂ Cn and Bn = Im ∂ ⊂ Cn. A class in H n(C; G) is represented by a homomorphism ϕ : Cn→G such that δϕ = 0, that is, ϕ∂ = 0, or in other words, ϕ vanishes on Bn. The restriction ϕ0 = ϕ \|\| Zn then induces a quotient homomorphism ϕ0 : Zn/Bn→G, an element of Hom(H
n(C), G). If ϕ is in Im δ, say ϕ = δψ = ψ∂, then ϕ is zero on Zn, so ϕ0 = 0 and hence also ϕ0 = 0. Thus there is a well-deﬁned quotient map h : H n(C; G)→Hom(Hn(C), G) sending the cohomology class of ϕ to ϕ0. Obviously h is a homomorphism. It is not hard to see that h is surjective. The short exact sequence 0 -→ Zn -→ Cn ∂-----→ Bn−1 -→ 0 splits since Bn−1 is free, being a subgroup of the free abelian group Cn−1. Thus there is a projection homomorphism p : Cn→Zn that restricts to the identity on Zn. Composing with p gives a way of extending homomorphisms ϕ0 : Zn→G to homomorphisms ϕ = ϕ0p : Cn→G. In particular, this extends homomorphisms Zn→G that vanish on Bn to homomorphisms Cn→G that still vanish on Bn, or in other words, it extends homomorphisms Hn(C)→G to elements of Ker δ. Thus we have a homomorphism Hom(Hn(C), G)→ Ker δ. Composing this with the quotient map Ker δ→H n(C; G) gives a homomorphism from Hom(Hn(C), G) to H n(C; G). If we 192 Chapter 3 Cohomology follow this map by h we get the identity map on Hom(Hn(C), G) since the eﬀect of composing with h is simply to undo the eﬀect of extending homomorphisms via p. This shows that h is surjective. In fact it shows that we have a split short exact sequence 0 -→ Ker h -→ H n(C; G) h-----→ Hom(Hn(C), G) -→ 0 The remaining task is to analyze Ker h. A convenient way to start the process is to consider not just the chain complex C, but also its subcomplexes consisting of the cycles and the boundaries. Thus we consider the commutative diagram of short exact sequences
(i) where the vertical boundary maps on Zn+1 and Bn are the restrictions of the boundary map in the complex C, hence are zero. Dualizing (i) gives a commutative diagram (ii) The rows here are exact since, as we have already remarked, the rows of (i) split, and the dual of a split short exact sequence is a split short exact sequence because of the natural isomorphism Hom(A ⊕ B, G) ≈ Hom(A, G) ⊕ Hom(B, G). We may view (ii), like (i), as part of a short exact sequence of chain complexes. n complexes are zero, the associated long n and B∗ Since the coboundary maps in the Z ∗ exact sequence of homology groups has the form (iii) ···←------ B∗ n ←------ Z ∗ n ←------ H n(C; G)←------ B∗ n−1←------ Z ∗ n−1←------ ··· n→B∗ The ‘boundary maps’ Z ∗ n in this long exact sequence are in fact the dual maps n of the inclusions in : Bn→Zn, as one sees by recalling how these boundary maps i∗ are deﬁned: In (ii) one takes an element of Z ∗ n, applies δ to get an element of C ∗ n. The ﬁrst of these steps extends a homomorphism ϕ0 : Zn→G to ϕ : Cn→G, the second step composes this ϕ with ∂, and the third step undoes this composition and restricts ϕ to Bn. The net eﬀect is just to restrict ϕ0 from Zn to Bn. n+1, then pulls this back to B∗ n, pulls this back to C ∗ A long exact sequence can always be broken up into short exact sequences, and doing this for the sequence (iii) yields short exact sequences (iv) 0←------ Ker i∗ n←------ H n(C; G)←------ Coker i∗ n−1←------ 0 The group Ker i∗ n can be identiﬁed naturally with Hom(Hn(C), G) since elements of n are homomorphisms Zn
→G that vanish on the subgroup Bn, and such homoKer i∗ morphisms are the same as homomorphisms Zn/Bn→G. Under this identiﬁcation of Cohomology Groups Section 3.1 193 n with Hom(Hn(C), G), the map H n(C; G)→ Ker i∗ Ker i∗ considered earlier. Thus we can rewrite (iv) as a split short exact sequence n in (iv) becomes the map h (v) 0 -→ Coker i∗ n−1 -→ H n(C; G) h-----→ Hom(Hn(C), G) -→ 0 Our objective now is to show that the more mysterious term Coker i∗ n−1 depends only on Hn−1(C) and G, in a natural, functorial way. First let us observe that Coker i∗ n−1 would be zero if it were always true that the dual of a short exact sequence was exact, since the dual of the short exact sequence (vi) is the sequence (vii) 0 -----→ Bn−1 in−1 -----------------→ Zn−1 -----→ Hn−1(C) -----→ 0 0←------ B∗ n−1 i∗ n−1←------------------- Z ∗ n−1←------ Hn−1(C)∗←------ 0 and if this were exact at B∗ n−1 would be zero. This argument does apply if Hn−1(C) happens to be free, since (vi) splits in this case, which implies that (vii) is also split exact. So in this case the map h n−1 would be surjective, hence Coker i∗ n−1, then i∗ in (v) is an isomorphism. However, in the general case it is easy to ﬁnd short exact sequences whose duals are not exact. For example, if we dualize 0→Z n-----→ Z→Zn→0 by applying Hom(−, Z) we get 0← Z n←------ Z← 0← 0 which fails to be exact at the left-hand Z, precisely the place we are interested in for Coker i∗ n−1. We might mention in passing that the loss of exact
ness at the left end of a short exact sequence after dualization is in fact all that goes wrong, in view of the following: Exercise. If A→B→C→0 is exact, then dualizing by applying Hom(−, G) yields an exact sequence A∗← B∗← C ∗← 0. However, we will not need this fact in what follows. The exact sequence (vi) has the special feature that both Bn−1 and Zn−1 are free, so (vi) can be regarded as a free resolution of Hn−1(C), where a free resolution of an abelian group H is an exact sequence ··· -----→ F2 f2------------→ F1 f1------------→ F0 f0------------→ H -----→ 0 with each Fn free. may lose exactness, but at least we get a chain complex — or perhaps we should If we dualize this free resolution by applying Hom(−, G), we say ‘cochain complex’, but algebraically there is no diﬀerence. This dual complex has the form ···←------ F ∗ 2 f ∗ 2←------------- F ∗ 1 f ∗ 1←------------- F ∗ 0 f ∗ 0←------------- H ∗←------ 0 Let us use the temporary notation H n(F ; G) for the homology group Ker f ∗ n+1/ Im f ∗ n of this dual complex. Note that the group Coker i∗ n−1 that we are interested in is H 1(F ; G) where F is the free resolution in (vi). Part (b) of the following lemma therefore shows that Coker i∗ n−1 depends only on Hn−1(C) and G. 194 Chapter 3 Cohomology Lemma 3.1. (a) Given free resolutions F and F ′ of abelian groups H and H ′, then every homomorphism α : H→H ′ can be extended to a chain map from F to F ′ : Furthermore, any two such chain maps extending α are chain homotopic. (b) For any two free resolutions F and F ′ of H, there are canonical isomorphisms H n(F ; G) ≈ H n(F ′; G) for all n. Proof: The αi �
�s will be constructed inductively. Since the Fi ’s are free, it suﬃces to deﬁne each αi on a basis for Fi. To deﬁne α0, observe that surjectivity of f ′ 0 implies that for each basis element x of F0 there exists x′ ∈ F ′ 0(x′) = αf0(x), so we deﬁne α0(x) = x′. We would like to deﬁne α1 in the same way, sending a basis 1(x′) = α0f1(x). Such an x′ will element x ∈ F1 to an element x′ ∈ F ′ exist if α0f1(x) lies in Im f ′ 1 = Ker f ′ 0α0f1 = αf0f1 = 0. The same procedure deﬁnes all the subsequent αi ’s. 1 such that f ′ 0, which it does since f ′ 0 such that f ′ i : Fi→F ′ If we have another chain map extending α given by maps α′ i, then the i deﬁne a chain map extending the zero map β : H→H ′. It i+1 deﬁning a chain homotopy from βi to 0, i+1λi + λi−1fi. The λi ’s are constructed inductively by a procedure 0 be zero, 1λ0. We can achieve this by letting 1(x′) = β0(x). Such 0β0(x) = βf0(x) = 0. For the inductive diﬀerences βi = αi − α′ will suﬃce to construct maps λi : Fi→F ′ that is, with βi = f ′ much like the construction of the αi ’s. When i = 0 we let λ−1 : H→F ′ and then the desired relation becomes β0 = f ′ λ0 send a basis element x to an element x′ ∈ F ′ an x′ exists since Im f ′ step we wish to deﬁne λi to take a basis element x ∈ Fi to an element x′ ∈ F ′ such
that f ′ lies in Im f ′ i βi = βi−1fi and the relation βi−1 = f ′ f ′ have i+1 i+1(x′) = βi(x) − λi−1fi(x). This will be possible if βi(x) − λi−1fi(x) i+1 = Ker f ′ i (βi − λi−1fi) = 0. Using the relation i λi−1 + λi−2fi−1 which holds by induction, we i, which will hold if f ′ 1 such that f ′ 1 = Ker f ′ 0 and f ′ i (βi − λi−1fi) = f ′ f ′ i βi − f ′ = βi−1fi − f ′ i λi−1fi i λi−1fi = (βi−1 − f ′ i λi−1)fi = λi−2fi−1fi = 0 as desired. This ﬁnishes the proof of (a). The maps αn constructed in (a) dualize to maps α∗ n forming a chain map between the dual complexes F ′∗ and F ∗. Therefore we have induced homomorphisms on cohomology α∗ : H n(F ′; G)→H n(F ; G). These do not depend on the choice of αn ’s since any other choices α′ n are chain homotopic, say via chain homotopies λn, and then α∗ n since the dual of the relation αi − α′ i−1. n are chain homotopic via the dual maps λ∗ i λ∗ n and α′∗ i = f ′ i+1λi + λi−1fi is α∗ i+1 + f ∗ i = λ∗ i − α′∗ n : F ′∗ n →F ∗ i f ′∗ The induced homomorphisms α∗ : H n(F ′; G)→H n(F ; G) satisfy (βα)∗ = α∗β∗ α-----→ H ′ β-----→ H ′′
with a free resolution F ′′ of H ′′ also given, since for a composition H Cohomology Groups Section 3.1 195 one can choose the compositions βnαn of extensions αn of α and βn of β as an extension of βα. In particular, if we take α to be an isomorphism and β to be its inverse, with F ′′ = F, then α∗β∗ = (βα)∗ = 11, the latter equality coming from the obvious extension of 11 : H→H by the identity map of F. The same reasoning shows β∗α∗ = 11, so α∗ is an isomorphism. Finally, if we specialize further, taking α to be the identity but with two diﬀerent free resolutions F and F ′, we get a canonical isomorphism 11∗ : H n(F ′; G)→H n(F ; G). ⊔⊓ Every abelian group H has a free resolution of the form 0→F1→F0→H→0, with Fi = 0 for i > 1, obtainable in the following way. Choose a set of generators for H and let F0 be a free abelian group with basis in one-to-one correspondence with these generators. Then we have a surjective homomorphism f0 : F0→H sending the basis elements to the chosen generators. The kernel of f0 is free, being a subgroup of a free abelian group, so we can let F1 be this kernel with f1 : F1→F0 the inclusion, and we can then take Fi = 0 for i > 1. For this free resolution we obviously have H n(F ; G) = 0 for n > 1, so this must also be true for all free resolutions. Thus the only interesting group H n(F ; G) is H 1(F ; G). As we have seen, this group depends only on H and G, and the standard notation for it is Ext(H, G). This notation arises from the fact that Ext(H, G) has an interpretation as the set of isomorphism classes of extensions of G by H, that is, short exact sequences 0→G→J→H→0, with a natural deﬁnition of isomorphism between such exact sequences. This is explained
in books on homological algebra, for example [Brown 1982], [Hilton & Stammbach 1970], or [MacLane 1963]. However, this interpretation of Ext(H, G) is rarely needed in algebraic topology. Summarizing, we have established the following algebraic result: Theorem 3.2. If a chain complex C of free abelian groups has homology groups Hn(C), then the cohomology groups H n(C; G) of the cochain complex Hom(Cn, G) are determined by split exact sequences 0 -→ Ext(Hn−1(C), G) -→ H n(C; G) h-----→ Hom(Hn(C), G) -→ 0 ⊔⊓ This is known as the universal coeﬃcient theorem for cohomology because it is formally analogous to the universal coeﬃcient theorem for homology in §3.A which expresses homology with arbitrary coeﬃcients in terms of homology with Z coeﬃcients. Computing Ext(H, G) for ﬁnitely generated H is not diﬃcult using the following three properties: Ext(H ⊕ H ′, G) ≈ Ext(H, G)⊕ Ext(H ′, G). Ext(H, G) = 0 if H is free. Ext(Zn, G) ≈ G/nG. The ﬁrst of these can be obtained by using the direct sum of free resolutions of H and If H is free, the free resolution 0→H→H→0 H ′ as a free resolution for H ⊕ H ′. 196 Chapter 3 Cohomology yields the second property, while the third comes from dualizing the free resolution 0 -→ Z n-----→ Z -→ Zn -→ 0 to produce an exact sequence In particular, these three properties imply that Ext(H, Z) is isomorphic to the torsion subgroup of H if H is ﬁnitely generated. Since Hom(H, Z) is isomorphic to the free part of H if H is ﬁnitely generated, we have: Corollary 3.3. If the homology groups Hn and Hn−1 of a chain complex C of free abelian groups are ﬁn
itely generated, with torsion subgroups Tn ⊂ Hn and Tn−1 ⊂ Hn−1, then H n(C; Z) ≈ (Hn/Tn)⊕ Tn−1. ⊔⊓ It is useful in many situations to know that the short exact sequences in the universal coeﬃcient theorem are natural, meaning that a chain map α between chain complexes C and C ′ of free abelian groups induces a commutative diagram This is apparent if one just thinks about the construction; one obviously obtains a map between the short exact sequences (iv) containing Ker i∗ n−1, the identiﬁcation Ker i∗ n = Hom(Hn(C), G) is certainly natural, and the proof of Lemma 3.1 shows that Ext(H, G) depends naturally on H. n and Coker i∗ However, the splitting in the universal coeﬃcient theorem is not natural since it depends on the choice of the projections p : Cn→Zn. An exercise at the end of the section gives a topological example showing that the splitting in fact cannot be natural. The naturality property together with the ﬁve-lemma proves: Corollary 3.4. If a chain map between chain complexes of free abelian groups induces an isomorphism on homology groups, then it induces an isomorphism on co- homology groups with any coeﬃcient group G. ⊔⊓ One could attempt to generalize the algebraic machinery of the universal coeﬃ- cient theorem by replacing abelian groups by modules over a chosen ring R and Hom by HomR, the R module homomorphisms. The key fact about abelian groups that was needed was that subgroups of free abelian groups are free. Submodules of free R modules are free if R is a principal ideal domain, so in this case the generalization is automatic. One obtains natural split short exact sequences 0 -→ ExtR(Hn−1(C), G) -→ H n(C; G) h-----→ HomR(Hn(C), G) -→ 0 Cohomology Groups Section 3.1 197 where C is a chain complex of free R modules with boundary maps R module ho- momorphisms,
and the coeﬃcient group G is also an R module. If R is a ﬁeld, for example, then R modules are always free and so the ExtR term is always zero since we may choose free resolutions of the form 0→F0→H→0. It is interesting to note that the proof of Lemma 3.1 on the uniqueness of free res- olutions is valid for modules over an arbitrary ring R. Moreover, every R module H has a free resolution, which can be constructed in the following way. Choose a set of generators for H as an R module, and let F0 be a free R module with basis in one-toone correspondence with these generators. Thus we have a surjective homomorphism f0 : F0→H sending the basis elements to the chosen generators. Now repeat the process with Ker f0 in place of H, constructing a homomorphism f1 : F1→F0 sending a basis for a free R module F1 onto generators for Ker f0. And inductively, construct fn : Fn→Fn−1 with image equal to Ker fn−1 by the same procedure. R(H, G) can be nonzero for n > 1. resolution F. The standard notation for H n(F ; G) is Extn complicated rings R the groups Extn more advanced topics in algebraic topology these Extn By Lemma 3.1 the groups H n(F ; G) depend only on H and G, not on the free R(H, G). For suﬃciently In certain R groups play an essential role. A ﬁnal remark about the deﬁnition of Extn R (H, G) : By the Exercise stated earlier, 0 ← H ∗← 0. This means exactness of F1→F0→H→0 implies exactness of F ∗ that H 0(F ; G) as deﬁned above is zero. Rather than having Ext0 R(H, G) be automatically zero, it is better to deﬁne H n(F ; G) as the nth homology group of the complex ··· ← F ∗ 0 ← 0 with the term H ∗ omitted. This can be viewed as deﬁning the groups H n(F ; G) to be unreduced cohomology groups.
With this slightly modiﬁed R(H, G) = H 0(F ; G) = H ∗ = HomR(H, G) by the exactness of deﬁnition we have Ext0 0 ← H ∗← 0. The real reason why unreduced Ext groups are better than re1 ← F ∗ F ∗ duced groups is perhaps to be found in certain exact sequences involving Ext and 1 ← F ∗ 1 ← F ∗ Hom derived in §3.F, which would not work with the Hom terms replaced by zeros. Cohomology of Spaces Now we return to topology. Given a space X and an abelian group G, we deﬁne the group C n(X; G) of singular n cochains with coeﬃcients in G to be the dual group Hom(Cn(X), G) of the singular chain group Cn(X). Thus an n cochain ϕ ∈ C n(X; G) n→X a value ϕ(σ ) ∈ G. Since the singular assigns to each singular n simplex σ : n simplices form a basis for Cn(X), these values can be chosen arbitrarily, hence ∆ n cochains are exactly equivalent to functions from singular n simplices to G. The coboundary map δ : C n(X; G)→C n+1(X; G) is the dual ∂∗, so for a cochain ϕ-----→ G. This ϕ ∈ C n(X; G), its coboundary δϕ is the composition Cn+1(X) means that for a singular (n + 1) simplex σ : ∂-----→ Cn(X) n+1→X we have δϕ(σ ) = (−1)iϕ(σ \|\| [v0, ···, ∆ vi, ···, vn+1]) Xi b 198 Chapter 3 Cohomology It is automatic that δ2 = 0 since δ2 is the dual of ∂2 = 0. Therefore we can deﬁne the cohomology group H n(X; G) with coeﬃcients in G to be the quotient Ker δ/ Im δ at C n(X;
G) in the cochain complex ···←------ C n+1(X; G) δ←------------- C n(X; G) δ←------------- C n−1(X; G)←------ ···←------ C 0(X; G)←------ 0 Elements of Ker δ are cocycles, and elements of Im δ are coboundaries. For a cochain ϕ to be a cocycle means that δϕ = ϕ∂ = 0, or in other words, ϕ vanishes on boundaries. Since the chain groups Cn(X) are free, the algebraic universal coeﬃcient theorem takes on the topological guise of split short exact sequences 0 -→ Ext(Hn−1(X), G) -→ H n(X; G) -→ Hom(Hn(X), G) -→ 0 which describe how cohomology groups with arbitrary coeﬃcients are determined purely algebraically by homology groups with Z coeﬃcients. For example, if the ho- mology groups of X are ﬁnitely generated then Corollary 3.3 tells how to compute the cohomology groups H n(X; Z) from the homology groups. When n = 0 there is no Ext term, and the universal coeﬃcient theorem reduces to an isomorphism H 0(X; G) ≈ Hom(H0(X), G). This can also be seen directly from the deﬁnitions. Since singular 0 simplices are just points of X, a cochain in C 0(X; G) is an arbitrary function ϕ : X→G, not necessarily continuous. For this to be a cocycle means that for each singular 1 simplex σ : [v0, v1]→X we have δϕ(σ ) = ϕ(∂σ ) = = 0. This is equivalent to saying that ϕ is constant on pathϕ components of X. Thus H 0(X; G) is all the functions from path-components of X to G. This is the same as Hom(H0(X), G). σ (v0) σ (v1) − ϕ Likewise in the case of H 1(X; G) the universal coeﬃcient
theorem gives an isomorphism H 1(X; G) ≈ Hom(H1(X), G) since Ext(H0(X), G) = 0, the group H0(X) being free. If X is path-connected, H1(X) is the abelianization of π1(X) and we can identify Hom(H1(X), G) with Hom(π1(X), G) since G is abelian. The universal coeﬃcient theorem has a simpler form if we take coeﬃcients in In §2.2 we deﬁned the homology a ﬁeld F for both homology and cohomology. groups Hn(X; F ) as the homology groups of the chain complex of free F modules Cn(X; F ), where Cn(X; F ) has basis the singular n simplices in X. The dual complex HomF (Cn(X; F ), F ) of F module homomorphisms is the same as Hom(Cn(X), F ) since both can be identiﬁed with the functions from singular n simplices to F. Hence the homology groups of the dual complex HomF (Cn(X; F ), F ) are the cohomology groups H n(X; F ). In the generalization of the universal coeﬃcient theorem to the case of modules over a principal ideal domain, the ExtF terms vanish since F is a ﬁeld, so we obtain isomorphisms H n(X; F ) ≈ HomF (Hn(X; F ), F ) Cohomology Groups Section 3.1 199 Thus, with ﬁeld coeﬃcients, cohomology is the exact dual of homology. Note that when F = Zp or Q we have HomF (H, G) = Hom(H, G), the group homomorphisms, for arbitrary F modules G and H. For the remainder of this section we will go through the main features of singular homology and check that they extend without much diﬃculty to cohomology. e e e e theorem identiﬁes H 0(X; G) with Hom( Reduced Groups. Reduced cohomology groups the augmented chain complex ··· →C0(X) hom
ology, this gives H n(X; G) can be deﬁned by dualizing ε-----→ Z→0, then taking Ker / Im. As with H n(X; G) = H n(X; G) for n > 0, and the universal coeﬃcient H0(X), G). We can describe the diﬀerence beH 0(X; G) and H 0(X; G) more explicitly by using the interpretation of H 0(X; G) tween as functions X→G that are constant on path-components. Recall that the augmentation map ε : C0(X)→Z sends each singular 0 simplex σ to 1, so the dual map ε∗ ϕ-----→ G, which is sends a homomorphism ϕ : Z→G to the composition C0(X) the function σ ֏ ϕ(1). This is a constant function X→G, and since ϕ(1) can be any element of G, the image of ε∗ consists of precisely the constant functions. Thus H 0(X; G) is all functions X→G that are constant on path-components modulo the functions that are constant on all of X. e Relative Groups and the Long Exact Sequence of a Pair. To deﬁne relative groups H n(X, A; G) for a pair (X, A) we ﬁrst dualize the short exact sequence ε-----→ Z e 0 -→ Cn(A) i-----→ Cn(X) j-----→ Cn(X, A) -→ 0 by applying Hom(−, G) to get 0←------ C n(A; G) i∗←------ C n(X; G) j∗←------ C n(X, A; G)←------ 0 where by deﬁnition C n(X, A; G) = Hom(Cn(X, A), G). This sequence is exact by the following direct argument. The map i∗ restricts a cochain on X to a cochain on A. Thus for a function from singular n simplices in X to G, the image of this function under i∗ is obtained by restricting the domain of the function to singular n simplices in A.
Every function from singular n simplices in A to G can be extended to be deﬁned on all singular n simplices in X, for example by assigning the value 0 to all singular n simplices not in A, so i∗ is surjective. The kernel of i∗ consists of cochains taking the value 0 on singular n simplices in A. Such cochains are the same as homomorphisms Cn(X, A) = Cn(X)/Cn(A)→G, so the kernel of i∗ is exactly C n(X, A; G) = Hom(Cn(X, A), G), giving the desired exactness. Notice that we can view C n(X, A; G) as the functions from singular n simplices in X to G that vanish on simplices in A, since the basis for Cn(X) consisting of singular n simplices in X is the disjoint union of the simplices with image contained in A and the simplices with image not contained in A. Relative coboundary maps δ : C n(X, A; G)→C n+1(X, A; G) are obtained as restrictions of the absolute δ ’s, so relative cohomology groups H n(X, A; G) are deﬁned. The 200 Chapter 3 Cohomology fact that the relative cochain group is a subgroup of the absolute cochains, namely the cochains vanishing on chains in A, means that relative cohomology is conceptually a little simpler than relative homology. The maps i∗ and j∗ commute with δ since i and j commute with ∂, so the preceding displayed short exact sequence of cochain groups is part of a short exact sequence of cochain complexes, giving rise to an associated long exact sequence of cohomology groups ··· -→ H n(X, A; G) j∗-----→ H n(X; G) i∗-----→ H n(A; G) δ-----→ H n+1(X, A; G) -→ ··· By similar reasoning one obtains a long exact sequence of reduced cohomology groups H n(X, A; G) = H n(X, A; G) for all n, as in for a pair (X, A) with A nonempty, where homology. Taking
A to be a point x0, this exact sequence gives an identiﬁcation of H n(X; G) with H n(X, x0; G). e More generally there is a long exact sequence for a triple (X, A, B) coming from e the short exact sequences 0←------ C n(A, B; G) i∗←------ C n(X, B; G) j∗←------ C n(X, A; G)←------ 0 The long exact sequence of reduced cohomology can be regarded as the special case that B is a point. As one would expect, there is a duality relationship between the connecting homomorphisms δ : H n(A; G)→H n+1(X, A; G) and ∂ : Hn+1(X, A)→Hn(A). This takes the form of the commutative diagram shown at the right. To verify commu- tativity, recall how the two connecting homomorphisms are deﬁned, via the diagrams The connecting homomorphisms are represented by the dashed arrows, which are well-deﬁned only when the chain and cochain groups are replaced by homology and cohomology groups. To show that hδ = ∂∗h, start with an element α ∈ H n(A; G) represented by a cocycle ϕ ∈ C n(A; G). To compute δ(α) we ﬁrst extend ϕ to a cochain ϕ ∈ C n(X; G), say by letting it take the value 0 on singular simplices not in A. Then we compose ϕ with ∂ : Cn+1(X)→Cn(X) to get a cochain ϕ∂ ∈ C n+1(X; G), which actually lies in C n+1(X, A; G) since the original ϕ was a cocycle in A. This cochain ϕ∂ ∈ C n+1(X, A; G) represents δ(α) in H n+1(X, A; G). Now we apply the map h, which simply restricts the domain of ϕ∂ to relative cycles in Cn+1(X, A), that is, (n
+ 1) chains in X whose boundary lies in A. On such chains we have ϕ∂ = ϕ∂ since the extension of ϕ to ϕ is irrelevant. The net result of all this is that hδ(α) Cohomology Groups Section 3.1 201 is represented by ϕ∂. Let us compare this with ∂∗h(α). Applying h to ϕ restricts its domain to cycles in A. Then applying ∂∗ composes with the map which sends a relative (n + 1) cycle in X to its boundary in A. Thus ∂∗h(α) is represented by ϕ∂ just as hδ(α) was, and so the square commutes. Induced Homomorphisms. Dual to the chain maps f♯ : Cn(X)→Cn(Y ) induced by f : X→Y are the cochain maps f ♯ : C n(Y ; G)→C n(X; G). The relation f♯∂ = ∂f♯ dualizes to δf ♯ = f ♯δ, so f ♯ induces homomorphisms f ∗ : H n(Y ; G)→H n(X; G). In the relative case a map f : (X, A)→(Y, B) induces f ∗ : H n(Y, B; G)→H n(X, A; G) by the same reasoning, and in fact f induces a map between short exact sequences of cochain complexes, hence a map between long exact sequences of cohomology groups, with commuting squares. The properties (f g)♯ = g♯f ♯ and 11♯ = 11 imply (f g)∗ = g∗f ∗ and 11∗ = 11, so X ֏ H n(X; G) and (X, A) ֏ H n(X, A; G) are contravariant functors, the ‘contra’ indicating that induced maps go in the reverse direction. The algebraic universal coeﬃcient theorem applies also to relative cohomology since the relative chain groups Cn(X, A) are free, and there is a naturality statement: A map f : (X, A)→(Y, B) induces a
commutative diagram This follows from the naturality of the algebraic universal coeﬃcient sequences since the vertical maps are induced by the chain maps f♯ : Cn(X, A)→Cn(Y, B). When the subspaces A and B are empty we obtain the absolute forms of these results. Homotopy Invariance. The statement is that if f ≃ g : (X, A)→(Y, B), then f ∗ = g∗ : H n(Y, B)→H n(X, A). This is proved by direct dualization of the proof for homology. From the proof of Theorem 2.10 we have a chain homotopy P satisfying g♯ − f♯ = ∂P + P ∂. This relation dualizes to g♯ − f ♯ = P ∗δ + δP ∗, so P ∗ is a chain homotopy between the maps f ♯, g♯ : C n(Y ; G)→C n(X; G). This restricts also to a chain homotopy between f ♯ and g♯ on relative cochains, the cochains vanishing on singular simplices in the subspaces B and A. Since f ♯ and g♯ are chain homotopic, they induce the same homomorphism f ∗ = g∗ on cohomology. Excision. For cohomology this says that for subspaces Z ⊂ A ⊂ X with the closure of Z contained in the interior of A, the inclusion i : (X − Z, A − Z) ֓ (X, A) induces isomorphisms i∗ : H n(X, A; G)→H n(X − Z, A − Z; G) for all n. This follows from the corresponding result for homology by the naturality of the universal coeﬃcient theorem and the ﬁve-lemma. Alternatively, if one wishes to avoid appealing to the universal coeﬃcient theorem, the proof of excision for homology dualizes easily to cohomology by the following argument. In the proof for homology there were chain maps ι : Cn(A + B)→Cn(X) and ρ : Cn(X)→Cn(
A + B) such that ρι = 11 and 11 − ιρ = ∂D + D∂ for a chain homotopy D. Dualizing by taking Hom(−, G), we have maps 202 Chapter 3 Cohomology ρ∗ and ι∗ between C n(A + B; G) and C n(X; G), and these induce isomorphisms on cohomology since ι∗ρ∗ = 11 and 11 − ρ∗ι∗ = D∗δ + δD∗. By the ﬁve-lemma, the maps C n(X, A; G)→C n(A + B, A; G) also induce isomorphisms on cohomology. There is an obvious identiﬁcation of C n(A+B, A; G) with C n(B, A∩B; G), so we get isomorphisms H n(X, A; G)) ≈ H n(B, A ∩ B; G) induced by the inclusion (B, A ∩ B) ֓ (X, A). Axioms for Cohomology. These are exactly dual to the axioms for homology. Restrict- contravariant functors ing attention to CW complexes again, a (reduced) cohomology theory is a sequence of hn from CW complexes to abelian groups, together with nathn+1(X/A) for CW pairs (X, A), satis- ural coboundary homomorphisms δ : hn(A)→ fying the following axioms: e (1) If f ≃ g : X→Y, then f ∗ = g∗ : (2) For each CW pair (X, A) there is a long exact sequence hn(Y )→ hn(X). e e e e i∗------------→ ··· δ------------→ hn(X/A) q∗------------→ hn(X) hn(A) δ------------→ hn+1(X/A) q∗------------→ ··· Q e α Q e where i is the inclusion and q is the quotient map. e e e e (3) For a wedge sum X = hn(X)→ α X
α with inclusions iα : Xα ֓ X, the product map W hn(Xα) is an isomorphism for each n. α i∗ α : We have already seen that the ﬁrst axiom holds for singular cohomology. The sec- ond axiom follows from excision in the same way as for homology, via isomorphisms H n(X/A; G) ≈ H n(X, A; G). Note that the third axiom involves direct product, rather than the direct sum appearing in the homology version. This is because of the nate α Hom(Aα, G), which implies that the cochain ural isomorphism Hom( α Xα is the direct product of the cochain complexes complex of a disjoint union of the individual Xα ’s, and this direct product splitting passes through to cohomology groups. The same argument applies in the relative case, so we get isomorphisms α H n(Xα, Aα; G). The third axiom is obtained by taking the H n( Aα ’s to be basepoints xα and passing to the quotient α Aα; G) ≈ αAα, G) ≈ α xα = α Xα/ α Xα. α Xα, L ` Q ` Q ` The relation between reduced and unreduced cohomology theories is the same as ` ` W for homology, as described in §2.3. simplicial chain groups Simplicial Cohomology. If X is a n(X, A) dualize to simplicial cochain groups complex and A ⊂ X is a subcomplex, then the n(X, A; G) = n(X, A), G), and the resulting cohomology groups are by deﬁnition the simn(X, A) ⊂ Cn(X, A) (X, A; G). Since the inclusions n(X, A) ≈ Hn(X, A), Corollary 3.4 implies that the dual maps ∆ Hom( plicial cohomology groups H n induce isomorphisms H C n(X, A; G)→ n(X, A; G) also induce isomorphisms H n(X, A; G) ≈ H n (X
, A; G). ∆ ∆ ∆ ∆ ∆ ∆ Cellular Cohomology. For a CW complex X this is deﬁned via the cellular cochain ∆ ∆ complex formed by the horizontal sequence in the following diagram, where coeﬃcients in a given group G are understood, and the cellular coboundary maps dn are Cohomology Groups Section 3.1 203 the compositions δnjn, making the triangles commute. Note that dndn−1 = 0 since jnδn−1 = 0. Theorem 3.5. H n(X; G) ≈ Ker dn/ Im dn−1. Furthermore, the cellular cochain complex {H n(X n, X n−1; G), dn} is isomorphic to the dual of the cellular chain complex, obtained by applying Hom(−, G). Proof: The universal coeﬃcient theorem implies that H k(X n, X n−1; G) = 0 for k ≠ n. The long exact sequence of the pair (X n, X n−1) then gives isomorphisms H k(X n; G) ≈ H k(X n−1; G) for k ≠ n, n − 1. Hence by induction on n we obtain H k(X n; G) = 0 if k > n. Thus the diagonal sequences in the preceding diagram are exact. The universal coeﬃcient theorem also gives H k(X, X n+1; G) = 0 for k ≤ n + 1, so H n(X; G) ≈ H n(X n+1; G). The diagram then yields isomorphisms H n(X; G) ≈ H n(X n+1; G) ≈ Ker δn ≈ Ker dn/ Im δn−1 ≈ Ker dn/ Im dn−1 For the second statement in the theorem we have the diagram The cellular coboundary map is the composition across the top, and we want to see that this is the same as the composition across the bottom. The ﬁrst and third vertical maps are isomorphisms by the universal coeﬃcient theorem, so it suﬃces to show the diagram commutes. The ﬁrst square commutes by natur
ality of h, and commu- tativity of the second square was shown in the discussion of the long exact sequence of cohomology groups of a pair (X, A). ⊔⊓ Mayer–Vietoris Sequences. In the absolute case these take the form ··· -→ H n(X; G) -----→ H n(A; G) ⊕ H n(B; G) -----→ H n(A ∩ B; G) -→ H n+1(X; G) -→ ··· where X is the union of the interiors of A and B. This is the long exact sequence Ψ Φ associated to the short exact sequence of cochain complexes 0 -→ C n(A + B; G) ψ-----→ C n(A; G) ⊕ C n(B; G) ϕ-----→ C n(A ∩ B; G) -→ 0 204 Chapter 3 Cohomology Here C n(A + B; G) is the dual of the subgroup Cn(A + B) ⊂ Cn(X) consisting of sums of singular n simplices lying in A or in B. The inclusion Cn(A + B) ⊂ Cn(X) is a chain homotopy equivalence by Proposition 2.21, so the dual restriction map C n(X; G)→C n(A + B; G) is also a chain homotopy equivalence, hence induces an isomorphism on cohomology as shown in the discussion of excision a couple pages back. The map ψ has coordinates the two restrictions to A and B, and ϕ takes the diﬀerence of the restrictions to A ∩ B, so it is obvious that ϕ is onto with kernel the image of ψ. There is a relative Mayer–Vietoris sequence ··· -→ H n(X, Y ; G) -→ H n(A, C; G) ⊕ H n(B, D; G) -→ H n(A ∩ B, C ∩ D; G) -→ ··· for a pair (X, Y ) = (A ∪ B, C ∪ D) with C ⊂ A and D ⊂ B such that X is the union of the interiors of A and B while Y is the union of the inter
iors of C and D. To derive this, consider ﬁrst the map of short exact sequences of cochain complexes Here C n(A + B, C + D; G) is deﬁned as the kernel of C n(A + B; G) -→ C n(C + D; G), the restriction map, so the second sequence is exact. The vertical maps are restrictions. The second and third of these induce isomorphisms on cohomology, as we have seen, so by the ﬁve-lemma the ﬁrst vertical map also induces isomorphisms on cohomology. The relative Mayer–Vietoris sequence is then the long exact sequence associated to the short exact sequence of cochain complexes 0 -→ C n(A + B, C + D; G) ψ-----→ C n(A, C; G) ⊕ C n(B, D; G) ϕ-----→ C n(A ∩ B, C ∩ D; G) -→ 0 This is exact since it is the dual of the short exact sequence 0 -→ Cn(A ∩ B, C ∩ D) -→ Cn(A, C) ⊕ Cn(B, D) -→ Cn(A + B, C + D) -→ 0 constructed in §2.2, which splits since Cn(A + B, C + D) is free with basis the singular n simplices in A or in B that do not lie in C or in D. Exercises 1. Show that Ext(H, G) is a contravariant functor of H for ﬁxed G, and a covariant functor of G for ﬁxed H. 2. Show that the maps G n-----→ G and H n-----→ H multiplying each element by the integer n induce multiplication by n in Ext(H, G). 3. Regarding Z2 as a module over the ring Z4, construct a resolution of Z2 by free modules over Z4 and use this to show that Extn Z4 (Z2, Z2) is nonzero for all n. Cohomology Groups Section 3.1 205 4. What happens if one deﬁnes homology groups hn(X; G) as the homology groups of the chain complex ··· →Hom → ···
? More specifG, Cn(X) G, Cn−1(X) ically, what are the groups hn(X; G) when G = Z, Zm, and Q? 5. Regarding a cochain ϕ ∈ C 1(X; G) as a function from paths in X to G, show that if ϕ is a cocycle, then →Hom (a) ϕ(f g) = ϕ(f ) + ϕ(g), (b) ϕ takes the value 0 on constant paths, (c) ϕ(f ) = ϕ(g) if f ≃ g, (d) ϕ is a coboundary iﬀ ϕ(f ) depends only on the endpoints of f, for all f. [In particular, (a) and (c) give a map H 1(X; G)→Hom(π1(X), G), which the universal coeﬃcient theorem says is an isomorphism if X is path-connected.] (a) Directly from the deﬁnitions, compute the simplicial cohomology groups of 6. S 1 × S 1 with Z and Z2 coeﬃcients, using the (b) Do the same for RP2 and the Klein bottle. 7. Show that the functors hn(X) = Hom(Hn(X), Z) do not deﬁne a cohomology theory on the category of CW complexes. complex structure given in §2.1. ∆ 8. Many basic homology arguments work just as well for cohomology even though maps go in the opposite direction. Verify this in the following cases: (a) Compute H i(S n; G) by induction on n in two ways: using the long exact sequence of a pair, and using the Mayer–Vietoris sequence. (b) Show that if A is a closed subspace of X that is a deformation retract of some neighborhood, then the quotient map X→X/A induces isomorphisms H n(X, A; G) ≈ H n(X/A; G) for all n. (c) Show that if A is a retract of X then H n(X; G) ≈ H n(A; G)⊕ H n(X, A; G
). e 9. Show that if f : S n→S n has degree d then f ∗ : H n(S n; G)→H n(S n; G) is multiplication by d. 10. For the lens space Lm(ℓ1, ···, ℓn) deﬁned in Example 2.43, compute the cohomology groups using the cellular cochain complex and taking coeﬃcients in Z, Q, Zm, and Zp for p prime. Verify that the answers agree with those given by the universal coeﬃcient theorem. 11. Let X be a Moore space M(Zm, n) obtained from S n by attaching a cell en+1 by a map of degree m. (a) Show that the quotient map X→X/S n = S n+1 induces the trivial map on Hi(−; Z) for all i, but not on H n+1(−; Z). Deduce that the splitting in the universal coeﬃcient theorem for cohomology cannot be natural. (b) Show that the inclusion S n ֓ X induces the trivial map on not on Hn(−; Z). 12. Show H k(X, X n; G) = 0 if X is a CW complex and k ≤ n, by using the cohomology version of the second proof of the corresponding result for homology in Lemma 2.34. H i(−; Z) for all i, but e e 206 Chapter 3 Cohomology 13. Let hX, Y i denote the set of basepoint-preserving homotopy classes of basepointpreserving maps X→Y. Using Proposition 1B.9, show that if X is a connected CW complex and G is an abelian group, then the map hX, K(G, 1)i→H 1(X; G) sending a map f : X→K(G, 1) to the induced homomorphism f∗ : H1(X)→H1 ≈ G is a bijection, where we identify H 1(X; G) with Hom(H1(X), G) via the universal coeﬃcient theorem. K(G, 1) In the introduction to this chapter we sketched a deﬁnition of cup product in terms

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