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Y / be excisive in X Y. Then we have two boundary operators @0 W Hm..X; A/.Y; B//! Hm1.X B [ A Y; X B/ Hm1.A.Y; B//; 242 Chapter 9. Singular Homology @00 W Hm..X; A/.Y; B//! Hm1.X B [AY; AY / Hm1..X; A/B/: t is the interchange map t.u; v/ D.v; u/. Let C D fcg be a point and 1 2 H0.C I R/ be represented by c ˝ 1. 9.8.1 Properties of the homology product..f g/.x y/ D fx g.y/; @x y D @0.x y/; x @y D.1/jxj@00.x y/;.x y/ z D x.y z/; x y D.1/jxjjyjt.y x/; 1 y D y: The algebraic Künneth formula (11.10.1) yields (9.8.2) Theorem (Künneth Formula). Let R be an integral domain. Further, let.A Y; X B/ be excisive in X Y for singular homology. Then we have a natural exact sequence L iCj Dn L 0!! Hi.X; AI R/ ˝ Hj.Y; BI R/! Hn..X; A/.Y; B/I R/ Hi.X; AI R/ Hj.Y; BI R/! 0: iCj Dn1 The sequence splits, but the splitting is not natural in the variable.X; A/. For homology with coefﬁcients in a ﬁeld k we obtain an isomorphism H.X; AI k/ ˝k H.Y; BI k/ Š H..X; A/.Y; B/I k/ as a special case. This isomorphism holds for an arbitrary commutative ring R if the homology groups H.X; AI R/ are free R-modules. (9.8.3) Example. The homology class e 2 H1.R; R X 0I R/ Š R
represented by ˝ 1 with the singular simplex W 1! R,.t0; t1/ 7! 1 2t0 is a generator. The product y 7! e y with e is an isomorphism Hn.Y; BI N / Š HnC1..R; R X 0/.Y; B/I N / for each R-module N. This isomorphism can also be deduced from the axiomatic properties 9.8.1 (see a similar deduction (17.3.1) in the case of cohomology theories). The n-fold product en D e e 2 Hn.Rn; Rn X 0I R/ serves as a canonical generator (a homological R-orientation of Rn). Þ 9.8. The Homology Product 243 Problems 1. Let v0; : : : vn be afﬁnely independent points in Rn and suppose that the origin 0 2 Rn is contained in the interior of the afﬁne simplex v D Œv0; : : : ; vn. We then have the singular simplex W n! Rn determined by.ei / D vi. Show that represents a generator xv of Hn.Rn; Rn X0I Z/. We therefore have a relation xv D ˙en. Determine the sign, depending on v. You might ﬁrst consider the case n D 2 and make an intuitive guess. (This problem indicates that keeping track of signs can be a nuisance.) 2. Verify the properties 9.8.1 of the homology product. 3. Study the axioms for a multiplicative cohomology theory and use 9.8.1 to deﬁne multiplicative homology theories axiomatically. Chapter 10 Homology In this chapter we deﬁne homology theories via the axioms of Eilenberg and Steenrod. From these axioms we derive some classical results: the Jordan separation theorem; invariance of domain and dimension; degree and its determination by local data; the theorem of Borsuk–Ulam. The theorem of Borsuk–Ulam is used for a problem in combinatorics: the determination of the chromatic number of Kneser graphs. A second topic of the chapter is the derivation of some results of a general nature: reduced homology; add
itivity; suspension isomorphisms; Mayer–Vietoris sequences; compatibility of homology with colimits. 10.1 The Axioms of Eilenberg and Steenrod Recall the category TOP.2/ of pairs of topological spaces. We use the functor W TOP.2/! TOP.2/ which sends.X; A/ to.A; ;/ and f W.X; A/!.Y; B/ to the restriction f W.A; ;/!.B; ;/. A homology theory for pairs of spaces consists of a family.hn j n 2 Z/ of covariant functors hn W TOP.2/! R- MOD and a family.@n j n 2 Z/ of natural transformations @n W hn! hn1 ı. These data are required to satisfy the following axioms of Eilenberg and Steenrod [58], [59]: (1) Homotopy invariance. For each homotopy ft in TOP.2/ the equality hn.f0/ D hn.f1/ holds. (2) Exact sequence. For each pair.X; A/ the sequence! hnC1.X; A/ @! hn.A; ;/! hn.X; ;/! hn.X; A/ @! is exact. The undecorated homomorphisms are induced by the inclusions. (3) Excision. Let.X; A/ be a pair and U A such that xU Aı. Then the inclusion.X X U; A X U /!.X; A/ induces an excision isomorphism hn.X X U; A X U / Š hn.X; A/. The excision axiom can be expressed in a different form. Suppose Y1; Y2 are subspaces of Y such that Y D Y ı 2. Then the inclusion induces an isomorphism hn.Y1; Y1 \ Y2/ Š hn.Y; Y2/. 1 [Y ı The module hn.X; A/ is the n-th homology group (module) of.X; A/ in the homology theory (we also say in degree or in dimension n). We set hn.X; ;/ D hn.X/. The groups hn.X/
are the absolute groups and the groups hn.X; A/ are 10.1. The Axioms of Eilenberg and Steenrod 245 the relative groups. The homomorphisms @ D @n are the boundary operators. We often write hn.f / D f and call (as already above) f the induced morphism. The homology groups hn.P / for a point P are the coefﬁcient groups of the theory. (More precisely, a group hn together with a compatible family of isomorphisms "P W hn! hn.P / for each point P is given.) In the case that hn D 0 for n 6D 0, we say that the homology theory satisﬁes the dimension axiom and call the homology theory an ordinary or classical one. The exact sequence of a pair of spaces can be extended slightly to an exact sequence for triples B A X (see [59, p. 24], [189]). The boundary operator for a triple is deﬁned by @ W hn.X; A/! hn1.A/! hn1.A; B/I the ﬁrst map is the previous boundary operator and the second map is induced by the inclusion. (10.1.1) Proposition. For each triple.X; A; B/ the sequence! hnC1.X; A/ @! hn.A; B/! hn.X; B/! hn.X; A/ @! is exact. The undecorated homomorphisms are induced by the inclusions. We do not derive this proposition from the axioms right now (see 10.4.2 for a proof which uses the homotopy invariance). In most constructions of homology theories one veriﬁes this more general exact sequence directly from the deﬁnitions; so we can treat it as an extended axiom. A homology theory is called additive, if the homology groups are compatible with topological sums. We formulate this as another axiom. (4) Additivity. Let.Xj ; Aj /; j 2 J be a family of pairs of spaces. Denote by ij W.Xj ; Aj /!.qj Xj ; qj Aj / the canonical inclusions into the topological sum. Then the additivity ax
iom says that L j 2J hn.Xj ; Aj /! hn.qj Xj ; qj Aj /;.xj / 7! P j 2J hn.ij /.xj / is always an isomorphism. For ﬁnite families the additivity follows from the axioms (see (10.2.1)). Singular homology theory has further properties which may also be required in an axiomatic treatment. (5) Weak equivalence. A weak equivalence f W X! Y induces isomorphisms f W h.X/ Š h.Y / of the homology groups. (6) Compact support. For each x 2 hn.X; A/ there exists a map f W.K; L/!.X; A/ from a pair.K; L/ of compact Hausdorff spaces and z 2 hn.K; L/ with fz D x. 246 Chapter 10. Homology If Axiom (5) holds, then the theory is determined by its restriction to CW-complexes (actually to simplicial complexes). Sometimes we have to compare different homology theories. Let h D.hn; @n/ and k D.kn; @0 n/ be homology theories. A natural transformation'W h! k of homology theories consists of a family 'n W hn! kn of natural transformations which are compatible with the boundary operators @0 n ı'D 'n1 ı @n. Problems 1. Let.hn; @n/ be a homology theory with values in R- MOD. Let."n 2 R/ be a family of units of the ring R. Then.hn; "n@n/ is again a homology theory. It is naturally isomorphic to the original theory. 2. Given a homology theory.hn; @n/ we can deﬁne a new theory by shifting the indices kn D hnCt. 3. Let h be a homology theory. For a ﬁxed space Y we deﬁne a new homology theory whose ingredients are the groups kn.X; A/ D hn.X Y; A Y /. The boundary operators for the new theory are the boundary operators of the pair.X Y; A Y /. The projections pr W X Y! X induce a natural transformation k! h of homology theories. If
h is additive then k is additive. 4. Let h be a homology theory with values in R- MOD. Let M be a ﬂat R-module, i.e., the tensor product ˝RM preserves exact sequences. Then the hn./ ˝R M and @ ˝R M are the data of a new homology theory. Since the tensor product preserves direct sums, the new theory is additive if h was additive. In the case that R D Z one can take for M a subring T of the rational numbers Q, in particular Q itself. It turns out that the rationalized theories h./ ˝ Q are in many respects simpler than the original ones but still contain interesting information. 5. If j h is a family of homology theories.j 2 J /, then their direct sum a homology theory. One can combine this device with the shift of indices. j j h is again L 10.2 Elementary Consequences of the Axioms We assume given a homology theory h and derive some consequences of the axioms of Eilenberg and Steenrod. Suppose the inclusion A X induces for j D n; n C 1 an isomorphism hj.A/ Š hj.X/. Then hn.X; A/ D 0. In particular hn.X; X/ D 0 always, and hn.;/ D hn.;; ;/ D 0. This is an immediate consequence of the exact sequence. Let f W X! Y be an h-equivalence. Then f W hn.X/! hn.Y / is an isomorphism, by functoriality and homotopy invariance. If f is, in addition, an inclusion, then hn.X; Y / D 0. Let f W.X; A/!.Y; B/ be a map such that the components f W X! Y and f W A! B induce isomorphisms of homology groups, e.g., the components are 10.2. Elementary Consequences of the Axioms 247 h-equivalences. Then the induced maps f W hn.X; A/! hn.Y; B/ are isomorphisms. The map f induces a morphism from the homology sequence of.X; A/ into the homology sequence of.Y; B/, by functoriality and
naturality of @. The claim is then a consequence of the Five Lemma (11.1.4). If X and A are contractible, then hn.X; A/ D 0. If.X; A; B/ is a triple and X; B are contractible, then the exact sequence of a triple shows that @ W hn.X; A/! hn1.A; B/ is an isomorphism. Let the homology theory satisfy the dimension axiom. If X is contractible, then hk.X/ D 0 for k 6D 0. A null homotopic map Y! Y therefore induces the zero morphism on hk.Y / for k 6D 0. Let i W A X and suppose there exists r W X! A such that ri'id. From id D id D.ri/ D ri we see that r is a retraction of i, hence i is always injective. Therefore the exact homology sequence decomposes into split short exact sequences 0! hn.A/! hn.X/! hn.X; A/! 0. (10.2.1) Proposition (Finite Additivity). Let.Xj ; Aj /; j 2 J be a ﬁnite family of pairs of spaces. Denote by ij W.Xj ; Aj /!.qj Xj ; qj Aj / the canonical inclusions into the topological sum. Then L j 2J hn.Xj ; Aj /! hn.qj Xj ; qj Aj /;.xj / 7! P j 2J hn.ij /.xj / is an isomorphism. Proof. As a consequence of the excision axiom we see that the inclusion always induces an isomorphism hn.A; B/ Š hn.A C C; B C C /. It sufﬁces to consider the case J D f1; 2g and, by the Five Lemma, to deal with the absolute groups. One applies the Sum Lemma (11.1.2) with Ak D hn.Xk/; Bk D hn.X; Xk/; C D hn.X1 C X2/. (10.2.2) Proposition. The identity n of n is a cycle modulo @n in singular homology theory
. The group Hn.n; @n/ is isomorphic to Z and Œn is a generator. Proof. The proof is by induction on n. Denote by s.i/ D d n n. Consider i n1 the i-th face of hk1.n1; @n1/.d n i / Š hk1.@n; @n X s.i/ı/ @ Š hk.n; @n/: The space @n X s.i/ı is contractible, a linear homotopy contracts it to the point ei. Since n is also contractible, @ is an isomorphism. The inclusion d n i maps.n1; @n1/ into the complement of ei, and as such it is a deformation retraction of pairs. The excision of ei induces an isomorphism. Therefore.d n i / If we always work with the ﬁrst face is the composition of two isomorphisms. map d n 0, we obtain by iteration an isomorphism hkn.0/ Š hk.n; @n/. (So far we can work with any homology theory.) 248 Chapter 10. Homology Now consider the special case hn D Hn of singular homology with coefﬁcients i, since the d n j for 0, we see that the in Z. By deﬁnition of the boundary operator, @Œn D.1/i Œd n j 6D i are zero in the relative group. If we again work with d n isomorphism above sends the generator Œn1 to Œn. If we work with Hk for k 6D n, then the isomorphism above and the dimension axiom tell us that Hk.n; @n/ D 0. The pair.n; @n/ is homeomorphic to.Dn; S n1/. So we also see that Hk.Dn; S n1/ is zero for k 6D n and isomorphic to Z for k D n. In an additive homology theory we also have an additivity isomorphism for pointed spaces if the base points are well-behaved. For a ﬁnite number of summands we do not need the additivity axiom in (10.2.3). (10.2.3)
Proposition. Let.Xj ; Pj / be a family of pointed spaces and j Xj D.X; P / the pointed sum with embedding i j W Xj! X of a summand. Assume that the closure of P in X has a neighbourhood U such that h.U; P /! h.X; P / is the zero map. Then hi j j h.Xj ; Pj /! h.X; P / is an isomorphism. i W L W Proof. Set Uj D U \ Xj. Consider the diagram 0 h.X; P /.1/ h.X; U /.2/.3/ h.qXj ; qPj / h.qXj ; qUj / @ @ h.U; P /.4/ h.qUj ; qPj /: The vertical morphisms are induced by the quotient maps. The horizontal morphisms are part of the exact sequence of triples. The hypothesis implies that (1) is injective. We use the additivity in order to conclude that (2) and (4) are injective. This is due to the fact that we have the projections pj W X! Xj, and pk D ıkj. We show that (3) is an isomorphism. Consider the diagram i j h.X X P; U X P / D h.q.Xj X Pj /; q.Uj X Pj // Š h.X; U / Š.3/ h.qXj ; qUj /: The isomorphisms hold by excision; here we use the assumption xP U ı. Diagram chasing (Five Lemma) now shows that (2) is also surjective. (10.2.4) Remark. The hypothesis of (10.2.3) is satisﬁed if Pj is closed in Xj and has a neighbourhood Uj such that Uj Xj is pointed homotopic to the constant map. These conditions hold if the spaces.Xj ; Pj / are well-pointed; see (5.4.4). Þ 10.3. Jordan Curves. Invariance of Domain 249 10.2.5 Suspension. We deﬁne the homological suspension isomorphism D.X;A/ by the commut
ative diagram hn.X; A/ hnC1..I; @I /.X; A// Š @ Š hn.1 X; 1 B/ Š.1/ hn.@I X [ I A; 0 X [ I A/: (1) is an isomorphism: excise 0 X and then use an h-equivalence. The boundary operator @ for the triple sequence of.I X; @I X [ I A; 0 X [ I A/ is an isomorphism, since 0X [I A I X is an h-equivalence. We can interchange the roles of 0 and 1; let denote this suspension isomorphism. By applying the Hexagon Lemma (11.1.3) with center group hn.I X; I A [ @I X/ we obtain D (draw the appropriate diagram). For some purposes of homotopy theory one has to use a similar suspension isomorphism deﬁned with X I. The n-fold iteration of provides us in particular with an isomorphism Þ n W hk Š hk.I 0/ Š hkC1.I; @I / Š Š hkCn.I n; @I n/. 10.3 Jordan Curves. Invariance of Domain As a ﬁrst application of homology theory we prove a general duality theorem. We then apply this general result to prove classical results: A generalized Jordan separation theorem and the invariance of domain and dimension. For this section see [53]. The propositions (10.3.1) and (10.3.2) can be proved in a similar manner for an arbitrary homology (or, mutatis mutandis, cohomology) theory. For the applications one needs homology groups which determine the cardinality of 0.X/ for open subsets X of Euclidean spaces, and this holds, e.g., for singular homology H./. (If one knows a little analysis, one can use de Rham cohomology for open subsets of Euclidean spaces.) (10.3.1) Proposition. Let A Rn be a closed subset. Then Hk.Rn; Rn X A/ and HkC1.Rn R; Rn R X A 0/ are isomorphic. Proof. We use the open subsets of
RnC1 HC D.Rn X A/ 1; 1Œ [ A 0; 1Œ H D.Rn X A/ 1; 1Œ [ A 1; 0Œ HC [ H D RnC1 X A 0 HC \ H D.Rn X A/ 1; 1Œ: 250 Chapter 10. Homology These data occur in the diagram Hk.Rn; Rn X A/ HkC1.RnC1; RnC1 X A 0/.3/ @ Hk.RnC1; HC \ H/.2/ Hk.HC; HC \ H/.1/ Hk.HC [ H; H/: The map @ is the boundary operator of the triple.RnC1; HC [ H; H/. The maps (1), (2), and (3) are induced by the inclusions. We show that the morphisms in the diagram are isomorphisms. The proof uses the fact that HC and H are contractible; the homotopy ht W HC! HC,.x; s/ 7!.x; s C t/ starts at the identity and has an image in the contractible space Rn 0; 1Œ. The map @ is an isomorphism, because Hk.RnC1; H/ D 0, by contractibility of H. The map (1) is an excision. The maps (2) and (3) are isomorphisms by homotopy invariance (Rn D Rn 0). (10.3.2) Theorem (Duality Theorem). Let A and B be closed homeomorphic subsets of Rn. Then the groups Hk.Rn; Rn X A/ and Hk.Rn; Rn X B/ are isomorphic (k 2 Z). Proof. A homeomorphism f W A! B yields a homeomorphism ˛ W Rn Rn! Rn Rn, which sends A 0 via f 0 to B 0 (see (7.3.1)). We obtain an induced isomorphism HkCn.Rn Rn; Rn Rn X A 0/ Š HkCn.Rn Rn; Rn Rn X B 0/: Now apply (10.3.1) n times. (10.3.3) The
orem (Component Theorem). Let A and B be closed homeomorphic subsets of Rn. Then 0.Rn X A/ and 0.Rn X B/ have the same cardinality. Proof. We use the fact that H0.Rn X A/ is the free abelian group on 0.Rn X A/ and the algebraic fact a free abelian group determines the cardinality of a basis (D the rank). Suppose that A 6D Rn. Then we have an exact sequence 0! H1.Rn; Rn X A/! H0.Rn X A/! H0.Rn/! 0: This shows the rank of H1.Rn; Rn X A/ is one less than the rank of H0.Rn X A/. Hence if A and B are different from Rn, then the result follows from (10.3.2). We see in the next section that A D Rn implies that B is open in Rn and therefore, since Rn is connected, also equal to Rn. An injective continuous map f W S 1! R2 is an embedding, and its image is called a Jordan curve. 10.3. Jordan Curves. Invariance of Domain 251 (10.3.4) Theorem (Jordan Separation Theorem). Let S Rn be homeomorphic to S n1 (n 2). Then Rn X S has two path components, the bounded interior J and the unbounded exterior A. Moreover, S is the set of boundary points of J and of A. Proof. The assertion holds in the elementary case S D S n1. Hence, by (10.3.3), the complement of S has two components. It remains to study the boundary points. Let x 2 S and let V be an open neighbourhood of x in Rn. Then C D S X.S \V / is closed in S and homeomorphic to a proper closed subset D of S n1. Therefore Rn X D is path connected and hence, by (10.3.3), also Rn X C. Let p 2 J and q 2 A and w W Œ0; 1! Rn X C a path from p to q. Then w1.S/ 6D ;. Let t1 be the minimum and t2 the maximum of w1.S/. Then w.t1/ and w.
t2/ are contained in S \ V. Therefore w.t1/ is limit point of w.Œ0; t1Œ / J and w.t2/ limit point of w. t2; 1/ A. Hence there exist t3 2 Œ0; t1Œ with w.t3/ 2 J \ V and t4 2 t2; 1 with w.t4/ 2 A \ V. This shows that x is contained in the boundary of J and of A. (10.3.5) Remarks. For n D 2 one can improve the separation theorem. There holds the theorem of Schoenﬂies (for a topological proof see [141]): Let J R2 be a Jordan curve. Then there exists a homeomorphism f W R2! R2 which maps J onto the standard circle. There exists a stronger result. By the Riemann mapping theorem there exists a holomorphic isomorphism from the interior of J onto the interior of S 1; and one can show that this isomorphism can be extended to a homeomorphism of the closures. See e.g., [149]. For n 3 it is in general not true that the interior of an embedding S n1! Rn is homeomorphic to an n-cell. One can construct an embedding D3! R3 such that the complement is not simply connected. Under some regularity conditions on the embedding the situation still resembles the standard embedding. There holds the theorem of M. Brown [32] (see also [25, p. 236]): Let f W S n1 Œ1; 1! S n be an embedding (n 1). Then the closure of each component of S n X f.S n1 0/ is homeomorphic to Dn. From the duality theorem (18.3.3) one can conclude that both components of S n X S have for an arbitrary embedding the integral singular homology groups of Þ a point. (10.3.6) Theorem. Let A Rn be homeomorphic to Dk, k n. Then Rn X A is path connected.n > 1/. Proof. Dk is compact, hence A is compact too. Therefore A is closed in Rn and the assertion follows from (10.3.3), since Dk obviously has a path connected complement. (10.3.7) Theorem (
Invariance of Domain). Let U Rn be open and f W U! Rn an injective continuous map. Then f.U / is open in Rn, and f maps U homeomorphically onto f.U /. 252 Chapter 10. Homology Let V Rn be homeomorphic to an open subset of U Rn. Then V is open in Rn. Proof. It sufﬁces to show that f.U / is open. It then follows that f is open. Let D D fx 2 Rn j kx ak ı g U, and let S be the boundary of D. It sufﬁces to show that f.Dı/ is open. We consider the case n 2 and leave the case n D 1 as exercise. The set S as well as T D f.S/ are homeomorphic to S n1. Suppose U1, U2 are the (open) components of Rn X T. Let U1 be unbounded. By (10.3.6), Rn X f.D/ is path connected, and therefore contained in U1 or U2. Since f.D/ is compact, the complement is unbounded. This implies T [ U1 D Rn X U2 f.D/ and then U1 f.Dı/. Since Dı is connected, so is f.Dı/. The inclusion f.Dı/ U1 [ U2 shows that f.Dı/ U1. Therefore f.Dı/ D U1, and this set is open. The second statement follows from the ﬁrst, since by hypothesis there exists an injective continuous map f W U! Rn with image V. (10.3.8) Theorem (Invariance of Dimension). Let U Rm and V Rn be nonempty homeomorphic open subsets. Then m D n. Proof. Let m < n. Then, by (10.3.7), U Rm Rn is open in Rn, which is impossible. 10.4 Reduced Homology Groups The coefﬁcient groups of a homology theory are important data of the theory but they contain no information about spaces. We therefore split off these groups from the homology groups hn.X/. Let X be a non-empty space and r W X! P the unique map to a point. We set Qhn.X/ D
kernel.r W hn.X/! hn.P // and call this group a reduced homology group. The homomorphism hn.f / for a continuous map f W X! Y restricts to Qhn.f / W Qhn.X/! Qhn.Y /. In this way Qhn becomes a homotopy invariant functor TOP! R- MOD. Let.X; / be a pointed space and i W P D fg X be the inclusion of the base point. Then we have a short exact sequence 0! hn.P / i! hn.X/ j! hn.X; P /! 0 with a splitting r of i. Thus j induces isomorphisms Qhn.X/ Š hn.X; P / Š Coker.i/. By(11.1.1) we also have an isomorphism hn.X/ D Qhn.X/ ˚ i.hn.P // Š Qhn.X/ ˚ hn 10.4. Reduced Homology Groups 253 which is natural for pointed maps (but not, in general, for arbitrary maps), and the canonical diagram Qhn.X/ hn.X/ Coker i Š $ Š j is commutative. hn.X; P / (10.4.1) Proposition. Let A be non-empty. The image of the boundary operator @ W hn.X; A/! hn1.A/ is contained in Qhn1.A/. The images of hn.X/ and Qhn.X/ in hn.X; A/ coincide. The homology sequence for the reduced groups! Qhn.A/! Qhn.X/! hn.X; A/! Qhn1.A/! is exact. Proof. Map the exact sequence of.X; A/ into the exact sequence of.P; P / and perform diagram chasing. The exactness is also a special case of (11.3.2). From (10.4.1) we see: If A is contractible, then Qhn.A/ D 0 and Qhn.X/! hn.X; A/ is an isomorphism. If X is contractible, then @ W hn.X; A/! Qhn1.A/ is an isomorphism. 10.
4.2 Triple sequence. Let.X; A; / be a pointed pair. The reduced homology sequence of.X; A/ is canonically isomorphic to the homology sequence of the triple.X; A; /. Hence the latter is exact. Let CB denote the cone over B. The homology sequence of a triple.X; A; B/ is, via excision, isomorphic to the sequence of.X [ CB; A [ CB; CB/, and the latter, via h-equivalence isomorphic to the sequence of.X [ CB; A [ CB; /. This Þ proves the exactness of the triple sequence. L theory an isomorphism j mands we do not need the additivity axiom. We call Qhn.Xj /! Qhn respect to Qhn if the canonical map Under the hypothesis of (10.2.3) or (10.2.4) we have for an additive homology. For a ﬁnite number of sumj Xj decomposable with j Xj is an isomorphism. Qh.Xj / Š Qh L j W W j Xj W (10.4.3) Proposition. Suppose X _ Y is decomposable with respect to hn; hnC1. Then the homology sequence of.X Y; X _ Y / yields a split short exact sequence 0! Qhn.X _ Y /! Qhn.X Y /! hn.X Y; X _ Y /! 0: Proof. The projections onto the factors induce Qhn.X Y /! Qhn.X/ ˚ Qhn.Y /, and together with the assumed decomposition isomorphism we obtain a left inverse to the morphism Qhn.X _ Y /! Qhn.X Y /. $ 254 Chapter 10. Homology Let.C; / be a monoid in h-TOP, i.e., W C! C _ C is a pointed map such that the composition with the inclusions of the summands is pointed homotopic to the identity. Then we have the -sum in each homotopy set ŒC; Y 0, deﬁned by Œf C Œg D Œı.f _ g/ with the folding map ı D hid; id i W Y _ Y! Y. Let us write h D Q
hn. (10.4.4) Proposition. Assume that C _C is decomposable with respect to hn. Then the morphism! W ŒC; Y 0! Hom.h.C /; h.Y //; Œf 7! f is a homomorphism. Proof. Consider the commutative diagram h.C / h.C _ C / h.Y _ Y /.f _g/ # d.1/ Š.2/ h.C / ˚ h.C / f˚g h.Y / ˚ h.Y / h.Y / ı a with the diagonal d and the addition a.y; z/ D y C z. By our assumption about C, the isomorphism (1) is induced by the projections onto the summands, and by our assumption about, the left triangle commutes. The morphism (2) and the inverse of.1/ are induced by the injection of the summands; this shows that the rectangle and the right triangle commute. Now observe that a.f ˚ g/d D f C g. The hypothesis of (10.4.4) holds for the suspension C D †X of a well-pointed space X. We thus obtain in particular a homomorphism! W n.X/! Hom. Qhn.S n/; Qhn.X// for each pointed space X. (10.4.5) Proposition. Let i W A X be a coﬁbration and let p W.X; A/!.X=A; / be the map which identiﬁes A to a base point. Then p W hn.X; A/! hn.X=A; / is an isomorphism. We can write this isomorphism also in the form q W hn.X; A/ Š Qhn.X=A/. Proof. Let X [ CA D.CA C X/=.a; 1/ i.a/ be the mapping cone of i. The inclusion j W.X; A/!.X [ CA; CA/ induces an isomorphism in homology: Excise the cone point A 0 and apply a homotopy equivalence. For a coﬁbration we have an h-equivalence p W X [CA! X [CA=CA �
� X=A. Hence q D pj is the composition of two isomorphisms. # 10.4. Reduced Homology Groups 255 The isomorphism q also holds for A D ;. In this case X=A D X C fg and p is the inclusion.X; ;/!.X; fg/. We use q to modify the exact sequence (10.4.1) in the case of a coﬁbration A X! Qhn.A/! Qhn.X/! Qhn.X=A/! Qhn1.A/! : (10.4.6) Proposition. Let j W A X be a coﬁbration and attach X to B via f W A! B. Then the map hn.X; A/! hn.X [A B; B/ induced by the inclusion is an isomorphism. Proof. We apply (10.4.5) to the homeomorphism X=A! X [A B=B. (10.4.7) Proposition. Let f W X! Y be a pointed map between well-pointed spaces and X sequence Qhn.X/ f! Y f1! C.f / the beginning of the coﬁbre sequence. Then the Qhn.Y / Qhn.C.f // is exact. f1 f Proof. Let Z.f / D.X I C Y /=.x; 1/ f.x/ be the (unpointed) mapping cylinder of f and X Z.f /, x 7!.x; 0/ the canonical inclusion, a coﬁbration. Consider the commuting diagram Qhn.X/ Qhn.Z.f // hn.Z.f /; X/ D Š Š Qhn.X/ f Qhn.Y / c.f / Qhn.Z.f /=X/ with the canonical inclusion c.f / W Y Z.f /=X. The top row is the exact sequence of the pair. The isomorphisms hold by homotopy invariance and (10.4.5). Now we use that for a well-pointed pair the quotient map Z.f /=X! C.f / is a homotopy equivalence, since a unit interval which is embedded as a coﬁbration is identi�
�ed to a point. For a well-pointed space X the inclusion X @I [ fg I X I is a coﬁbration. The quotient is the suspension †X. From (10.4.5) we obtain an isomorphism q W hn..X; /.I; @I //! hn.†X; / Š Qhn.†X/ and a suspension isomorphism Q W Qhn.X/ Š QhnC1.†X/ which makes the diagram hn.X; / hnC1..X; /.I; @I // q q Qhn.X/ Q QhnC1.†X/ commutative. In particular, we obtain hm Š Qhm.I 0=@I 0/ Š hmCn.I n=@I n/. 256 Chapter 10. Homology Problems 1. Let fq W S n! S n be a map of degree q, and denote by M.q; n/ its mapping cone. Determine the groups and homomorphisms in the sequence (9.6.1) for the space M.q; n/. 2. Let M.q; 1/ D M.q/. Use the coﬁbre sequence of idX ^fq in order to derive an exact sequence 0! Qhn.X/ ˝ Z=q! QhnC1.X ^ M.q//! Tor. Qhn1.X/; Z=q/! 0: This suggests deﬁning for any homology theory Qh a theory with coefﬁcients Z=q by Qh. ^M.q//. Unfortunately the homotopy situation is more complicated than one would expect (or wish), see [8]. Spaces of the type M.q; n/ are sometimes called Moore spaces. 10.5 The Degree In 10.2.5 we determined the homology groups of spheres from the axioms of a homology theory h. We describe yet another variant. We use the standard subspaces Dn; S n1; En D Dn X S n1 of Rn and Dn ˙ D f.x1; : : : ; xnC1/ 2 S n j ˙xnC1 0g. The space D0 D fDg is a single
ton and S 1 D ;. We deﬁne a suspension isomorphism C as the composition C W Qhk1.S n1/ @ Š hk.Dn ; S n1/ sC Š hk.S n; Dn C/ j Š Qhk.S n/: The maps j and @ are isomorphisms, since Dn morphism (compare (6.4.4)). Iteration of C yields (suspension) isomorphisms ˙ is contractible, and sC is an iso-.n/ W hkn Š Qhkn.S 0/ Š Qhk.S n/; hk.S n/ Š hkn ˚ hn: The ﬁrst isomorphism is determined by hm! Qhm.S 0/ hm.S 0/ Š hm.C1/ ˚ hm.1/ Š hm ˚ hm; x 7!.x; x/: We also have an isomorphism @ W hk.Dn; S n1/ Š hk1.S n1; e/ Š Qhk1.S n1/. In the case of ordinary homology H.I G/ with coefﬁcients in G we obtain: Hk.S nI G/ Š k D n > 0; n > k D 0; G; G ˚ G; k D n D 0; 0; otherwise: Moreover Hn.Dn; S n1I G/ Š Hn.Rn; Rn X 0I G/ Š G. A generator of Hn.Dn; S n1I Z/ Š Z is called a homological orientation of Dn; and a generator of Hn.S n/, n 1 a homological orientation of S n. Given an orientation z 2 Hn.S n/ and a continuous map f W S n! S n, we deﬁne the 10.5. The Degree 257 (homological ) degree d.f / 2 Z of f by f.z/ D d.f /z. A different choice of a generator does not effect the degree. We also deﬁne the degree for n D 0: The identity has degree C1, the antipodal map
the degree 1 and a constant map the degree 0. From the deﬁnition we see directly some properties of the degree: d.f ı g/ D d.f /d.g/, d.id/ D 1; a homotopy equivalence has degree ˙1; a null homotopic map has degree zero. (10.5.1) Proposition. Let h be a homology theory such that h0.Point/ D Z. Then! W ŒS n; S n! Hom. Qhn.S n/; Qhn.S n//, f 7! f is an isomorphism (n 1). Proof. The suspension isomorphism and the hypothesis yield Qhn.S n/ Š Z. Thus the Hom-group is canonically isomorphic to Z. We now use that n.S n/ Š ŒS n; S n Š Z. The identity of S n is mapped to 1. By (10.4.4),! is a homo- morphism, hence necessarily an isomorphism. We have already deﬁned a (homotopical) degree. From (10.5.1) we see that the homotopical and the homological degree coincide. If one starts algebraic topology with (singular) homology, then one has in any case the important homotopy invariant “degree”. Proposition (10.5.4) is not immediate from the homological deﬁnition. (10.5.2) Proposition. Deﬁne an isomorphism W Qhk1.S n1/! Qhk.S n/ as in the case of C, but with the roles of Dn ˙ interchanged. Then C D. Proof. Consider the commutative diagram lC " hk.S n; Dn C/ & ’ %"""""""" ; S n1/ ŠsC jC i hk.Dn @C Qhk.S n/ l % ’!!!!!!!!! & j iC " @ hk.S n; S n1/ Š s @ hk.Dn C; S n1/ hk.S n; Dn / Qhk1.S n1/ and apply the Hexagon Lemma (11.1.3). (
10.5.3) Proposition. Let A 2 O.n C 1/ and l A W S n! S n, x 7! Ax. Then l A is on Qhk.S n/ the multiplication by det.A/. The antipodal map x 7! x on S n has the degree.1/nC1. " % & ’ ’ % & " 258 Chapter 10. Homology Proof. Let t W S n! S n change the sign of the ﬁrst coordinate. Then tC D. Hence t D id, by (10.5.2). The group O.n C 1/ has two path components. If A and B are contained in the same component, then l A and l B are homotopic. Representatives of the components are the unit matrix E and the diagonal matrix T D Diag.1; 1; : : : ; 1/. The relations l T D t and l E D id now ﬁnish the proof. (10.5.4) Proposition. Let f W S n! S n be a map of degree d. Then f induces on Qhk.S n/ the multiplication by d. Proof. The cases d D 1; 0 are clear and d D 1 follows from (10.5.3). The general case is then a consequence of (10.4.4) and our knowledge of n.S n/. (10.5.5) Proposition. Let A 2 GLn.R/ and l A W Rn! Rn, x 7! Ax. Then l A is on hk.Rn; Rn X 0/ the multiplication by the sign ".A/ D det.A/=j det.A/j of the determinant. @! Qhk1.Rn X 0/ Š Qhk1.S n1/ Proof. We have isomorphisms hk.Rn; Rn X 0/ which are compatible with the action of l A if A 2 O.n/. In this case the claim follows from (10.5.3). In the general case we use that GLn.R/ has two path com- ponents which are characterized by the sign of the determinant. Let S n hr;a W S n! S n winding number of f W S n rb ı f ı hr;a. r.a/ D fx
2 RnC1 j kx ak Dr g. We have a canonical homeomorphism r.a/, x 7! rx C a and rb W RnC1 X b! S n, x 7! N.x b/. The r.a/! RnC1 X b about b is deﬁned as the degree of 10.5.6 Local degree. Let K S n be compact and different from S n and let U be an open neighbourhood of K. Then we have an excision isomorphism Hn.U; U X K/ Š Hn.S n; S n X K/. For a continuous map f W S n! S n we let K D f 1.p/. Consider the diagram Hn.S n/ f Hn.S n/.1/ Hn.S n; S n X K/ f Hn.S n; S n X p/ Š Hn.U; U X K/ f U D Hn.S n; S n X p/ with the restriction f U of f. The exact sequence of the pair.S n; S n X p/ shows that (1) is an isomorphism (n 1). Let z 2 Hn.S n/ be a generator and z.U; K/ its image in Hn.U; U X K/. Commutativity of the diagram shows f U zU;K D d.f /zS n;p. Hence the degree only depends on the restriction f U. For any compact set L with f.L/ D fpg and open neighbourhood W of L we deﬁne the (partial) 10.5. The Degree 259 degree d.f; L/ by f W of W. z.W; L/ D d.f; L/z.S n; p/; it is independent of the choice (10.5.7) Lemma. Suppose U D U1 [ U2 is the disjoint union of open sets Uj. Set Kj D Uj \ K. The inclusions induce the additivity isomorphism hi1; i2 i W Hn.U1; U1 X K1/ ˚ Hn.U2; U2 X K2/! Hn.U; U X K/: Then the relation z.U; K/ D i1z.U
1; K1/ C i2z.U2; K2/ holds. Proof. Consider the diagram with M D S n: Hn.M / (############ Hn.M; M X K/ Š Hn.M; M X K2/ Š Hn.U; U X K/ j1 i2 Hn.U2; U2 X K2/ ( a1 Hn.U; U X K2/. There exist x1; x2 such that z.U; K/ D i1x1 C i2x2. We compute j1z.U; K/ D z.U; K2/ D a1z.U2; K2/ D j1i2x2 D a1x2: This proves x2 D z.U2; K2/. As a consequence of this lemma we obtain the additivity of the degree d.f / D d.f; K1/ C d.f; K2/. Suppose K is a ﬁnite set, then we can choose U as a disjoint union of open sets Ux; x 2 K such that Ux \ K D fxg. In that case L x2K Hn.Ux; Ux X x/ Š Hn.U; U X K/; and Hn.Ux; Ux X x/ Š Z, by excision and Hn.Dn; S n1/ Š Z. We call the integer d.f; x/ deﬁned by fzUx ;x D d.f; x/zS n;p the local degree of f at p. With this notation we therefore have Þ d.f / D P x2K d.f; x/. Let U Rn be an open neighbourhood of the origin. Then we have an excision isomorphism hk.U; U X 0/ Š hk.Rn; Rn X 0/. Suppose g W U! Rn is a map with the properties: (1) Continuously differentiable (a C 1-map); (2) g1.0/ D f0g; (3) the differential Dg.0/ is invertible. Under these conditions we show: (10.5.8) Proposition. The induced map hk.Rn; Rn
X 0/ Š hk.U; U X 0/ g! hk.Rn; Rn X 0/ is multiplication by the sign of the determinant of the differential Dg.0/. ( ( 260 Chapter 10. Homology Proof. There exist continuous maps hi W U! Rn with g.x/ D Dg.0/.x1; : : : ; xn/ D.Rn; Rn X 0/ from g to Dg.0/ P n iD1 xi hi.x/ and P n iD1 xi hi.0/. We deﬁne a homotopy ht W.U; U X 0/! ht.x/ D P n iD1 xi hi.tx/ D ( t 1g.tx/; Dg.0/; t > 0; t D 0: Now we use (10.5.5). (10.5.9) Example. Let f W S n! S n be continuously differentiable and p a regular value of f, i.e., the differential of f in each point x 2 f 1.p/ is bijective. Then d.f; x/ D ˙1, and the plus-sign holds, if the differential respects the orientation at x. For the proof express f in terms of orientation preserving local charts and Þ apply (10.5.8). If S is homeomorphic to S n and f a self map of S we choose a homeomorphism h W S n! S and deﬁne the degree of f as the degree of h1f h. Let g W Rn! Rn be a proper map. We deﬁne its degree as the degree of the extension of g to the one-point compactiﬁcation. A map f W S n S n! S n, n 1, has a bi-degree.a; b/ 2 Z Z where a is the degree of x 7! f.x; y/ for a ﬁxedy and b the degree of y 7! f.x; y/ for a ﬁxed x. Problems 1. If f W S n! S n is not surjective, then f is null homotopic and hence d.f / D 0. 2. Suppose f.x/ 6D x for each x 2 S n,
then h.x; t/ D tf.x/C.1t/x 6D 0 for t 2 Œ0; 1. We compose with N W x 7! x=kxk and obtain in F.x; t/ D N h.x; t/ a homotopy from the identity to f. If always f.x/ 6D x, then G.x; t/ D N.tx C.1 t/f.x// is a homotopy from f to the antipodal map. Thus if d.f / 6D ˙1, there exists x such that f.x/ D x or f.x/ D x. 3. A permutation of.t0; : : : ; tn/ induces an afﬁne homeomorphism of.n; @n/. The induced homomorphism in hk.n; @n/ is the multiplication with the sign of the permutation. The same holds for the linear permutation map l induced by on the vector space N D f.t0; : : : ; tn/ j i ti D 0g and hk.N; N X 0/. One can compute the determinant of l by using the decomposition RnC1 D N ˚ D with the diagonal D D f.t; : : : ; t/ j t 2 Rg. 4. The map S i! S i,.y; t/ 7!.2ty; 2t 2 1/, y 2 Ri, t 2 R has degree 1 C.1/iC1. The point.0; : : : ; 0; 1/ is a regular value. 5. Consider a complex polynomial as self-map of the Riemann sphere CP 1 Š S 2. Then the homological degree is the algebraic degree of the polynomial. A quotient f D p=q of two complex polynomials (without common divisor) p of degree m and q of degree n can be considered as a self-map of CP 1. Show that the homological degree is a D max.m; n/. In homogeneous coordinates f can be written as Œz; w 7! Œwap.z=w/; waq.z=w/. Suppose P 10.6. The Theorem of Borsuk and Ulam 261 c 2 C is such
that z 7! p.z/ cq.z/ has a pairwise different zeros. Then Œc; 1 is a regular value of f. 6. Consider S 3 as the topological group of quaternions of norm 1. Determine the degree of z 7! zk, k 2 Z. 7. The map S n S n! S n,.x; / 7! 2h x; ix has bi-degree.1 C.1/n1; 1/. If D.1; 0; : : : ; 0/ D e0, then e0 is a regular value of x 7! e0 2h e0; x ix. (Here h ; i is the standard inner product.) 8. Let a; b; p; q 2 N with ap bq D 1. Then f W C2! C2,.x; y/ 7!.xp xyq; xb C ya/ is proper and has degree 1. The point.1; 0/ is a regular value. 10.6 The Theorem of Borsuk and Ulam We describe another classical result which uses the homotopy notion in the presence of a symmetry. As a rather striking application to a problem in combinatorics we present the proof of Greene [74] for the determination of the chromatic number of the Kneser graphs. We have the antipodal symmetry x 7! x on the Euclidean spaces. A map f W A! B which is equivariant with respect to this symmetry, i.e., which satisﬁes f.x/ D f.x/, is called antipodal or odd; here A and B are subsets of Euclidean spaces that are invariant with respect to the antipodal symmetry. The additional presence of the symmetry has remarkable consequences: Classical theorems known under the name of Borsuk–Ulam theorems and Lusternik–Schnirelmann theorems. The basic result has a number of equivalent formulations. (10.6.1) Theorem. The following assertions are equivalent: (1) Let f W S n! Rn be continuous. Then there exists x 2 S n such that f.x/ D f.x/. (2) Let f W S n! Rn be antipodal. Then there exists x 2 S n such that f.x/ D 0. (
3) There does not exist an antipodal map f W S n! S n1. (4) There does not exist a continuous map f W Dn! S n1 which restricts to an antipodal map on the boundary. (5) An antipodal map S n1! S n1 is not null homotopic. (6) Suppose S n D F1 [ F2 [ [ FnC1 with non-empty closed sets Fj. Then at least one of the sets Fj contains an antipodal pair of points. (7) Let S n D A1 [ A2 [ [ AnC1 and assume that each Aj is either open or closed. Then at least one of the Aj contains an antipodal pair. Proof..1/ ).2/. By (1) there exists x with f.x/ D f.x/. Since f is antipodal, f.x/ D f.x/ and hence f.x/ D 0..2/ ).3/. The existence of an antipodal map contradicts (2)..3/ ).4/. Let Dn ˙ D f.x0; : : : ; xn/ 2 S n j ˙xn 0g. The projection C! Dn1 onto the ﬁrst n 1 coordinates is a homeomorphism which is the h W Dn 262 Chapter 10. Homology identity on the boundary. Suppose f W Dn! S n1 is antipodal on the boundary. Deﬁne g W S n! S n1 by ( g.x/ D x 2 Dn C; f h.x/; f h.x/; x 2 Dn : If x 2 S n1 D Dn, then h.x/ D x, h.x/ D x and f h.x/ D f.x/ D f.x/ D f h.x/. Hence g is well-deﬁned and continuous. One veriﬁes that g is antipodal. C \ Dn.4/,.5/. If an antipodal map S n1! S n1 were null homotopic, then we could extend this map to Dn, contradicting (4). Conversely, if a map of type (4) would exist, then the restriction to S n1 would contradict (5)..1
/ ).6/. We consider the function f W S n! Rn; x 7!.d.x; F1/; : : : ; d.x; Fn// deﬁned with the Euclidean distance d. By (1), there exists x such that f.x/ D f.x/ D y. If the i-th component of y is zero, then d.x; Fi / D d.x; Fi / D 0 and therefore x; x 2 Fi since Fi is closed. If all coordinates of y are non-zero, then x and x are not contained in n iD1 Fi, so they are contained in FnC1..6/ ).3/. There exists a closed covering F1; : : : ; FnC1 of S n1 such that no Fi contains an antipodal pair, e.g., project the standard simplex onto the sphere and take the images of the faces. Suppose f W S n! S n1 is antipodal. Then the covering by the f 1.Fi / contradicts (6). S.3/ ).2/. If f.x/ 6D 0 for all x, then x 7! kf.x/k1f.x/ is an antipodal map S n! S n1..2/ ).1/. Given f W S n! Rn. Then g.x/ D f.x/ f.x/ is antipodal; g.x/ D 0 implies f.x/ D f.x/..6/ ).7/. Suppose for the moment that the Aj are open. Then we can ﬁnd a closed shrinking and apply (6). In the general case let A1; : : : ; Aj be closed and Uj C1; : : : ; UnC1 open. Suppose there are no antipodal pairs in the Uj. Thicken the Ai to open "-neighbourhoods U".Ai /. Let " D n1. By the case of an open covering we ﬁnd an antipodal pair.xn; xn/ in some U".Ai /. By passing to a subsequence we ﬁnd an i j and lim n!1 d.xn; Ai / D lim n!1 d.xn; Ai / D 0: A convergent subsequence yields an antipodal
pair in Ai..7/ ).3/. As in the case.6/ ).3/. Since the identity is antipodal, we see that (10.6.1) implies the retraction theorem, see (6.6.1). Part (1) shows that Rn does not contain a subset homeomorphic to S n. 10.6. The Theorem of Borsuk and Ulam 263 (10.6.2) Lemma. Let F W S n! S n be an odd map. Then F is homotopic as odd map to a map g such that g.S i / S i for 0 i n. Proof. Let p W S n! RP n be the quotient map and f W RP n! RP n be induced by F on the orbit space. Choose a homotopy ft from f D f0 to a cellular map, i.e., f1.RP i / RP i for 0 i n. Lift the homotopy ft p D ht to a homotopy Ht W S n! S n with initial condition f. Then Ht is an odd map and H1 D g has the desired property. We obtain a proof of (10.6.1) from (10.6.3) Theorem. An odd map has odd degree. Proof. The proof is by induction on the dimension of the sphere. Let f W S n! S n be an odd map. By (10.6.2) we can deform f as odd map into a map g such that g.S n1/ S n1. The induction is now a consequence of (10.6.4). (10.6.4) Proposition. Let f W S n! S n be an odd map such that f.S n1/ S n1. Then we have the degree D.f / of f and the degree d.f / of its restriction to S n1. These degrees have the same parity. Proof. We study the diagram in the proof of (10.5.2) more closely for singular homology with coefﬁcients in Z. We ﬁx a generator z 2 QHn1.S n1/ and deﬁne other generators by @˙z˙ D z; s˙z˙ D w˙; l˙v˙ D w˙: We set iz
˙ D u˙. The Sum Lemma tells us that u˙ is a Z-basis of the group Hn.S n; S n1/. Let T W S n! S n be the antipodal map. Suppose given f W S n! S n such that f.S n1/ S n1 and Tf D f T. Then we have two degrees D.f / and d.f / deﬁned by f.vC/ D D.f /vC; f.z/ D d.f /z: Since u˙ is a Z-basis we can write f.uC/ D auC C bu. Using this notation, we show d.f / D a C b; D.f / D a b: Hence d.f / and D.f / are congruent modulo 2. From fz D f@uC D @fuC D @.auC Cbu/ D.aCb/z we obtain the ﬁrst assertion d.f / D aCb. Naturality of the boundary operator @T D T@C and Tz D.1/nz imply TzC D.1/nz and TuC D.1/nu. We conclude fu D.1/nfTuC D.1/nTfuC D.1/nT.auC C bu/ D au C buC: 264 Chapter 10. Homology The exactness of.; @/ shows that the image of is generated by uC u, hence.vC/ D ".uC u/ with " D ˙1. The computation jCuC D sCizC D lC D jCvC D "jC.uC u/ D "jCuC shows " D 1. The computation D.f /.uC uC/ D D.f /vC D fvC D fvC D f.uC u/ D.a b/.uC u/ ﬁnally yields the second assertion D.f / D a b. (10.6.5) Example. The map f W S 1! S 1, z 7! z2kC1 satisﬁes the hypothesis of (10.6.4). We know already that D.f / D 2k C 1 and d.f /
D 1, hence a D k C 1 and b D k. Let dC denote the singular 1-simplex represented by the path from 1 to 1 and d the 1-simplex running from 1 to 1 (both counter-clockwise). Then dC C d is a cycle and vC D ŒdC C d is a natural choice of a generator. Then d represents uC; wC, and zC; and z D Œ1 Œ1. By considering the simplex f d, the relation fuC D Œf d D.k C 1/Œd C kŒdC becomes apparent. For k D 1 say, f d runs counter-clockwise from 1 to 1 to 1 to 1. Let f W S 1! S 1 be any self-map which commutes with the antipodal map. We can multiply f by a constant such that the new map g satisﬁes g.1/ D 1. From (10.6.4) we see that g and hence f has odd degree, since g has degree 1 on S 0. Þ We now apply the Borsuk–Ulam theorem to a problem in combinatorics: The determination of the chromatic number of the so-called Kneser graphs. We begin by explaining the problem. Let Œn D f1; : : : ; ng and denote by Nk the set of subsets of Œn with k elements. A graph consists of a set E of vertices and a set K of edges. Each edge has two boundary points, and they are identiﬁed with one or two points in E. (In other terms: A graph is a 1-dimensional CW-complex.) The Kneser graph KGn;k has E D Nk. Vertices F1; F2 are connected by an edge if they represent disjoint subsets of Œn. Let C D Œk, and call the elements of C colours. A k-colouring of a graph.E; K/ is a map f W E! C such that f.e1/ 6D f.e2/ whenever e1 and e2 are connected by an edge. The chromatic number of a graph.E; K/ is the smallest k such that there exists a k-colouring. The following result was conjectured by Martin Kneser (1955)
[105]. This conjecture was proved by Lovász [114] with topological methods. The following ingenious proof was given by Greene [74]. (10.6.6) Theorem. Let k > 0 and n 2k 1. Then KGn;k has the chromatic number n 2k C 2. Proof. An explicit construction shows that the chromatic number is at most n2kC2. We associate to a set F with k elements the colour '.F / D min.min.F /; 10.7. Mayer–Vietoris Sequences 265 n 2k C 2/. Suppose '.F / D '.F 0/ D i < n 2k C 2. Then these sets are not disjoint, since they contain the element i. If their colour is n 2k C 2, then they are contained in fn 2k C 2; : : : ; ng, and then they cannot be disjoint. Now we come to the topological part. Let d D 2n 2k C 1. Choose a set X S d with n elements and such that no hyperplane (through the origin) contains more than d points of X. We consider the subsets of X with cardinality k as the vertices of the Kneser graph KGn;k. Suppose there exists a colouring with d D n 2k C 1 elements and we choose one. Let Ai D fx 2 S d j H.x/ D fyjhx; y i > 0g contains a k-tuple in X with colour ig and Ad C1 D S d X.A1 [ [ Ad /. The sets A1; : : : ; Ad are open and Ad C1 is closed. By (10.6.1), one of the sets Ai contains an antipodal pair x; x. If i d, then we have two disjoint k-tuples with colour i, one in H.x/ the other one in H.x/. This contradicts the deﬁnition of a colouring. Let i D d C1. Then, by deﬁnition of A1; : : : ; Ad, the half-space H.x/ contains at most k 1 points of X, and similarly for H.x/. The set S d X.H.x/ [ H.x// is contained in a hyperplane and contains at least n 2.k
1/ D d C 1 points, and this contradicts the choice of X. Problems 1. Let n be odd. Then ˙ id W S n! S n are homotopic as odd maps. 2. Let f; g W S n! S n be odd maps with the same degree. Then they are homotopic as odd maps. 3. Let d0; d1; : : : ; dn be a family of odd integers with d0 D ˙1. There exists an odd map f W S n! S n such that f.S i / S i and the map S i! S i induced by f has degree di. 10.7 Mayer–Vietoris Sequences We derive further exact sequences for homology from the axioms, the so-called Mayer–Vietoris sequences. Let h be a homology theory. Let.XI A; B/ be a triad, i.e., A; B X D A[B. The triad.XI A; B/ is said to be excisive for the homology theory if the inclusion induces an (excision) isomorphism h.A; A \ B/ Š h.X; B/. The condition is actually symmetric in A and B. We write AB D A \ B. (10.7.1) Proposition. The following are equivalent: (1).A [ BI A; B/ is excisive. (2).A [ BI B; A/ is excisive. (3) W h.A; AB/ ˚ h.B; AB/! h.A [ B; AB/ is an isomorphism. (4) W h.A [ B; AB/! h.A [ B; A/ ˚ h.A [ B; B/ is an isomorphism. 266 Chapter 10. Homology Proof. Apply the Sum Lemma (11.1.2) to the diagram h.A; A \ B/ h.A [ B; B/ % ’!!!!!!!! h.A [ B; A \ B/ ’!!!!!!!! % h.B; A \ B/ h.A [ B; A/. The morphisms are induced by the inclusions. The boundary operator of an excisive triad is deﬁned by W hn.X/! hn.X; B/ Š hn.A
; AB/ @! hn1.AB/: This operator is part of the Mayer–Vietoris sequence (D MVS) of the triad. (10.7.2) Theorem. Let.A [ BI A; B/ be an excisive triad and C AB. Then the sequence : : :! hn.AB; C /.1/! hn.A; C / ˚ hn.B; C /.2/! hn.A [ B; C /! hn1.AB/.1/! is exact. The inclusions i A W AB A and i B W AB B yield the ﬁrst map x/; and the inclusions j A W A A [ B and j B W B A [ B yield x 7!.i A the second map.a; b/ 7! j A b. If C D fg is a point, we obtain the MVS for reduced homology groups. a C j B x; i B There exists another relative MVS. Let.A[BI A; B/ be excisive and A[B X. We deﬁne a boundary operator by W hn.X; A [ B/ @! hn1.A [ B; A/ Š hn1.B; AB/! hn1.X; AB/: (10.7.3) Theorem. The sequence : : : hn.X; AB/.1/ hn.X; A/ ˚ hn.X; B/.2/ hn.X; A [ B/ hn1.X; AB/.1/ is exact. The maps.1/ and.2/ are deﬁned as in (10.7.2). Proof. The homology sequences of the triples of.B; A\B; C / and.X; A; C / yield a commutative diagram (a “ladder”) hn.AB; C / hn1.AB; C / hn.A; AB/ hn.A; C / hn.B; C / hn.X; C / hn.X; B/ hn1.B; C /. % ’ % ’ 10.7. Mayer–Vietoris Sequences 267 We apply (10
.7.4) to this diagram and obtain (10.7.2). There exists a similar diagram which compares the sequences of the triples.X; A; AB/ and.X; A [ B; B/, and (10.7.4) yields now (10.7.3). (10.7.4) Lemma. Suppose the following diagram of abelian groups and homomorphisms is commutative and has exact rows.! Ai ai! A0 i fi f 0 i gi g 0 i Bi bi B 0 i Ci ci C 0 i hi h0 i Ai1! ai1 A0 i1!. Assume moreover that the ci are isomorphisms. Then the sequence! Ai.fi ;ai / Bi ˚ A0 i h bi ;f 0 i i B 0 i hi c1 i g 0 i Ai1!. is exact ([17, p. 433]). We use abbreviations of the type I X D IX, @I X D @IX, 0 X D 0X. We associate to a triad.XI A; B/ the subspace N D N.A; B/ D 0A [ IAB [ 1B of I X. Let p W N.A; B/! X be the projection onto the second factor. (10.7.5) Proposition. The following are equivalent: (1) The triad.XI A; B/ is excisive. (2) p W h.N.A; B//! h.X/ is an isomorphism. Proof. We have isomorphisms h.A; AB/ ˚ h.B; AB/ Š h.0A C 1B; 0AB C 1AB/ Š h.N; IAB/; by additivity, excision and h-equivalence. It transforms p into the map of item (3) in (10.7.1). Hence (1) and (2) are equivalent. The excision axiom says that the triad is excisive if X D Aı [B ı. The auxiliary space N.A; B/ allows us to transfer the problem into homotopy theory: A triad is excisive for each homology theory, if p W N! X is an h-equivalence. Recall from Section 3.3: (10.7.6) Proposition
. Suppose the covering A, B of X is numerable. Then p is an Þ h-equivalence. We now give a second proof of the MVS; it uses the homotopy axiom but not lemma (10.7.4). Let.XI A; B/ be a triad. Via excision and h-invariance we see that the inclusion induces an isomorphism hn..I; @I / AB/ Š hn.N; 0 A [ 1 B/: 268 Chapter 10. Homology We rewrite the exact sequence of the pair.N; 0 A [ 1 B/. Using the suspension isomorphism and the additivity, we obtain an exact sequence! hn.A/ ˚ hn.B/! hn.N /! hn1.AB/! : If the triad is excisive, we can use (10.7.5) and replace h.N / by h.X/. It is an exercise to compare the boundary operators of the two constructions of the MVS. There exists a more general MVS for pairs of spaces. It comprises the previously discussed cases. (10.7.7) Theorem. Let.AI A0; A1/.XI X0; X1/ be two excisive triads. Set X01 D X0 \ X1, A01 D A0 \ A1. Then there exists an exact Mayer–Vietoris sequence of the form! hn.X01; A01/! hn.X0; A0/ ˚ hn.X1; A1/! hn.X; A/! : Proof. Let N.X; A/ D 0X0 [ IA01 [ 1X1. The sequence in question arises from a rewriting of the exact sequence of the triple.N.X/; N.X; A/; N.A//. We consider three typical terms. (1) p W h.N.X/; N.A// Š h.X; A/, by (10.7.5) and the hypotheses. (2) The inclusions.Xj ; Aj / Š fj g.X j ; Aj /!.N.X; A/; N.A// induce an isomorphism h.X0; A0/ ˚ h.X1; A1
/! h.N.X; A/; N.A//: For the proof one excises Œ1=3; 2=3 A01 and then uses an h-equivalence and additivity. (3) The group h.N.X/; N.X; A// is isomorphic to h.IX01; @IX01 [ A01/ via inclusion, and the latter via suspension isomorphic to h1.X01; A01/. For the proof one replaces N.X; A/ by the thickened space 0X0 [ Œ0; 1=4X01 [ IA01 [ Œ3=4X01 [ 1X1: Then one can excise 0X0 and 1X1 and use suitable h-equivalences. It remains to identify the morphisms in the resulting sequence. The map hn.N.X; A/; N.A//! hn.N.X/; N.A// becomes hj 0 ; j 1 i W hn.X0; A0/ ˚ hn.X1; A1/! hn.X; A/: The map @ W hnC1.N.X/; N.X; A//! hn.N.X; A/; N.A/ becomes, with our definition of the suspension isomorphism,.i 0 ; i 1 / W hn.X01; A01/! hn.X0; A0/ ˚ hn.X1; A1/: The boundary operator of the generalized MV-sequence becomes, in the special cases X D X0 D X1 D X01 and A D A0 D A1 D A01 X01, the same as in the previously discussed algebraic derivation of the MV-sequences. 10.7. Mayer–Vietoris Sequences 269 (10.7.8) Example. Let i 1, x 7!.x; y0/, resp. x 7!.x0; y/. i W Hn.S n/ ˚ Hn.S n/! Hn.S n S n/ is an isomorphism (n 1/ Then hi 1 ; i 2 /. We ﬁx a generator z 2 Hn.S n/ and write zj D i j with inverse.pr1 z. Then.z1;
z2/ is a Z-basis of Hn.S n S n/. Let ˛ D.˛1; ˛2 be a map with bi-degree.a; b/ of ˛1 and bi-degree.c; d / of ˛2. Then ˛.z1/ D az1 C cz2 and ˛.z2/ D bz1 C dz2. ; pr2 Construct a space, a.2n C 1/-manifold, X by identifying in DnC1 S n C DnC1 S n the point.x; y/ 2 S n S n in the ﬁrst summand with ˛.x; y/ in the second summand via a homeomorphism ˛ D.˛1; ˛2/ of S n S n as above. The two summands are embedded as X1 and X2 into X. We use the MV-sequence of.XI X1; X2/ to determine the integral homology of X. Let us consider a portion of this sequence Hn.S n S n/ j! Hn.DnC1 S n/ ˚ Hn.DnC1 S n/! Hn.X/: We use the Z-basis.z1; z2/ as above. The inclusion S n! DnC1 S n, x 7!.0; x/ give us as image of z the generators u1; u2 in the summands Hn.DnC1 S n/. The image of the basis elements under j is seen to be j.z1/ D cu2, j.z2/ D u1 C du2 (we do not use the minus sign for the second summand). We conclude for c 6D 0 that Hn.X/ is the cyclic group of order jcj; the other homology groups of X are in this case H0.X/ Š Z Š H2nC1.X/ and Hj.X/ D 0 for j 6D 0; n; 2n C 1. We Þ leave the case c D 0 to the reader. Problems 1. Let Rn be the union of two open sets U and V. (i) If U and V are path connected, then U \ V is
path connected. (ii) Suppose two of the sets 0.U /; 0.V /; 0.U \ V / are ﬁnite, then the third set is also ﬁnite and the relation j0.U \ V /j.j 0.U /j C j0.V /j/ C j0.U [ V /j D0 holds. (iii) Suppose x; y 2 U \ V can be connected by a path in U and in V. Then they can be connected by a path in U \ V. Can you prove these assertions without the use of homology directly from the deﬁnition of path components? 2. Let the real projective plane P be presented as the union of a Möbius band M and a disk D, glued together along the common boundary S 1. Determine the groups and homomorphisms in the MV-sequence of.P I M; D/. Do the same for the Klein bottle.KI M; M /. (Singular homology with arbitrary coefﬁcients.) 3. Let.X1; : : : ; Xn/ be an open covering of X and.Y1; : : : ; Yn/ be an open covering of Y. Let f W X! Y be a map such that f.Xi / Yi. Suppose that the restriction a2A Xa! T a2A Ya of f induces a homology isomorphism for each ; 6D A f1; : : : ; ng. Then f T 270 Chapter 10. Homology induces a homology isomorphism. 4. Suppose AB A and AB B are closed coﬁbrations. Then p W N.A; B/! A [ B is an h-equivalence. 5. The boundary operators in (10.7.2) and (10.7.3) which result when we interchange the roles of A and B differ from the original ones by 1. (Apply the Hexagon Lemma to the two boundary operators.) 6. The triad.X @; X 1 [ A I / is always excisive. (Excision of X 0 [ A Œ0; 1=2Œ and h-equivalence.) 7. Verify the assertions about the morphisms in the sequence (10.7.7). 10.8 Colimits The additivity axiom for a homology theory
expresses a certain compatibility of homology and colimits (namely sums). We show that this axiom has consequences for other colimits. Let.X; f/ be a sequence X1 : : : of continuous maps f j. Recall that a colimit (a direct limit) of this sequence consists of a space X and continuous maps j k W Xk! X with the following universal property: X3 X2 f 1 f 2 f 3 (1) j kC1f k D j k. (2) If ak W Xk! Y is a family of maps such that akC1f k D ak, then there exists a unique map a W X! Y such that aj k D ak. (This deﬁnition can be used in any category.) Let us write colim.X; f/ D colim.Xk/ for the colimit. In the case that the f k W Xk XkC1 are inclusions, we can take as colimit the union X D i Xi together with the colimit topology: U X open if and only if U \ Xn open in Xn for each n. S Colimits are in general not suitable for the purpose of homotopy theory, one has to weaken the universal property “up to homotopy”. We will construct a so-called homotopy colimit. Colimits of sequences allow a special and simpler treatment than general colimits. A model of a homotopy colimit in the case of sequences is the i Xi Œi; i C 1 the point.xi ; i C 1/ with.f i.xi /; i C 1/ telescope. We identify in for xi 2 Xi. Denote the result by ` T D T.X; f/ D hocolim.X; f/: We have injections j k W Xk! T, x 7!.x; k/ and a homotopy k W j kC1f k'j k, a linear homotopy in Xk Œk; k C 1. Thus the telescope T consists of the mapping cylinders of the maps f i glued together. The data deﬁne the homotopy colimit of the sequence. 10.8. Colimits 271 Given maps ak W Xk! Y and homotopies hk W Xk �
�k; k C 1! Y from akC1f k to ak. Then there exists a map a W T! Y such that j ka D ak, and the composition of the canonical map Xk Œk; k C 1! T with a is hk. We have subspaces Tk T, the image of k1 iD1 Xi Œi; i C 1 C Xk fkg. The canonical inclusion k W Xk! Tk is a homotopy equivalence (compare the analogous situation for a mapping cylinder). In homology we have the equality j k f k W hn.Xk/! hn.XkC1/! hn.T /. By the universal property of the colimit of groups we therefore obtain a homomorphism D j kC1 ` W colim hn.Xk/! hn.T.X; f//: (10.8.1) Theorem. In an additive homology theory is an isomorphism. Proof. We recall an algebraic construction of the colimit A1 abelian groups. Consider a1 A2 a2 : : : of L k1 Ak! L k1 Ak;.xk/ 7!.xkC1 ak.xk//: The cokernel is the colimit, together with the canonical maps (inclusion of the j -th k Ak! colimAk. We therefore summand composed with the projection) Aj! need a computation of hn.T / which has this form. We cover T by the subspaces L A D T X S i1 X2i f2i C 1 2 g; B D T X S i1 X2i1 f2i 1 2 g: (10.8.2) Lemma. The inclusions ` i1 X2i f2ig!A; ` i1 X2i1 f2i 1g!B; ` i1 Xi fig!A \ B are h-equivalences, and.A; B/ is a numerable covering of T. Because of this lemma we have a Mayer–Vietoris sequence hn.A \ B/ L Š hn.Xj / ˛ The map ˛ has the form hn.A/ ˚ hn.B/ hn.T / L j 0.2/ h
n.Xj / ˚ Š L j 1.2/ hn.Xj /: ˛.x2i / D.x2i ; f 2i.x2i // and ˛.x2iC1/ D.f 2iC1.x2iC1/; x2iC1/ for xj 2 hn.Xj /. We see that ˛ is injective; therefore we can obtain hn.T / as cokernel of ˛. The automorphism.xi / 7!..1/i xi / transforms ˛ into the map which was used in the algebraic deﬁnition of the colimit. 272 Chapter 10. Homology For applications we have to ﬁnd conditions under which the homotopy colimit is h-equivalent to the colimit. We consider the case that the f k W Xk! XkC1 are S inclusions, and denote the colimit by X D k Xk. We change the deﬁnition of the telescope slightly and consider it now as the subspace S T D k Xk Œk; k C 1 X Œ0; 1Œ : The topology of T may be different, but the proof of (10.8.1) works equally well in this case. The projection onto the ﬁrst factor yields p W T! X. (10.8.3) Example. Let W X! Œ1; 1Œ be a function such that s D.id; / W X! X Œ1; 1Œ has an image in T. Then s W X! T is a section of p. The composition sp is homotopic to the identity by the homotopy..x; u/; t/ 7!.x;.1 t/u C t.x//. This is a homotopy over X, hence p is shrinkable. The property required by amounts to.x/ < i ) x 2 Xi1 for each i. Let.Ui j i 2 N/ be a numerable covering of X with locally ﬁnite numeration 1 iD1 ii. Þ.i / (see the chapter on partitions of unity). Set Xk D Then.x/ < i implies x 2 Xi1. S
k iD1 Ui and D P (10.8.4) Proposition. Suppose the inclusions Xk XkC1 are coﬁbrations. Then T X Œ1; 1Œ is a deformation retract. Proof. Since Xk X is a coﬁbration, there exists by (5.1.3) a homotopy hk t W X Œ1; 1Œ! X Œ1; 1Œ rel Yk D Xk Œ1; 1Œ [X Œk C 1; 1Œ from the identity to a retraction X Œ0; 1Œ! Yk. The retraction Rl acts as the identity on Xk Œ1; 1Œ for l > k, and therefore the inﬁnite composition Rj D ı Rj C2 ı Rj C1 ı Rj is a well-deﬁned continuous map. From hj we obtain a homotopy Rj'Rj C1 relative to Yj. We can concatenate these homotopies and obtain a homotopy from the retraction R1 to the identity relative to T. Problems 1. Let T be a subring. Find a system of homomorphisms Z! Z! such that the colimit is T. 2. Let S n! S n! S n! be a sequence of maps where each map has degree two. Let X be the homotopy colimit. Show that n.X/ Š ZŒ1=2, the ring of rational numbers with denominators a power of two. What system of maps between S n would yield a homotopy colimit Y such that n.Y / Š Q? 3. Let X be a CW-complex and T the telescope of the skeleton ﬁltration. Then the inclusion T X Œ1; 1Œ induces isomorphisms of homotopy groups and is therefore a homotopy equivalence. One can also apply (10.8.4) in this case. 10.9. Suspension 273 10.9 Suspension Recall the homological suspension isomorphism 10.2.5. We use abbreviations of the type I A D IA, @I A D @IA, 0 X D 0A. We set kn.A; B/ D hn..
I; @I /.A; B//. The k./ are the data for a new homology theory. The boundary operator of this homology theory is deﬁned for a triple.A; B; C / by Q@ W hnC1.IA; @IA [ IB/ @! hn.@IA [ B; @IA [ IC / Š hn.IB; @IB [ IC /: In order to work with this deﬁnition we use (10.9.1) Lemma. For each triple.A; B; C / the triad.@IA [ IBI IB; @IA [ IC / is excisive. Proof. The inclusion induces an isomorphism hn.@IA; @IB/! hn.@IA [ IC; @IB [ IC /; excise 1 2 C and use a homotopy equivalence. If we use this also for B D C we conclude that hn.@IA [ IC; @IB [ IC /! hn.@IA [ IB; IB/ is an isomorphism. The exact sequence of the triple.IA; @IA [ IB; @IA/ is transformed with the isomorphism kn.B/ D hn.IB; @IB/ Š hn.IB [ @IA; @IA/ into the exact sequence of.A; B/ for the k-groups. Let U B A and xU B ı. The excision isomorphism for the k-theory claims that hn.IA X U /; I.B X U / [ @I.A X U //! hn.IA; IB [ @IA/ is an isomorphism. This is a consequence of I U D I xU.IB [ @IA/ı and the usual excision isomorphism. The isomorphisms.A;B/ W hn.A; B/! knC1.A; B/ deﬁned in (10.2.5) form a natural transformation. The next proposition says that they are natural transformations of homology theories of degree 1. (10.9.2) Proposition. For each triple.A; B; C / the diagram hnC1.A; B/ @ hn.B; C /.A;B/.B;C / knC2.A; B/ Q@ kn
C1.B; C / is commutative. 274 Chapter 10. Homology Proof. By naturality it sufﬁces to consider the case C D ;. The Hexagon Lemma shows ˛ D ˇ for the maps ˛ W hnC1.IA; IB [ @IA/! hn.IB [ @IA; @IA/ Š hn.IB [ 0A; 0A [ 1B/! hn1.0A [ 1B; 0A/; ˇ W hnC1.IA; IB [ @IA/! hn.IB [ @IA; IB [ 0A/ Š hn.@IA; 0A [ 1B/! hn1.0A [ 1B; 0A/; and the center group hn.IB [ @IA; 1B [ 0A/. Let j be the isomorphism j W hn1.1B [ 0A; 0A/ Š hn1.1B/ Š hn1.B/: By diagram chasing one veriﬁes.B/j˛ D Q@ and jˇ.A;B/ D @. (10.9.3) Lemma. Let.A; B; C / be a triple. Then we have an isomorphism h1; 0; i W hn.A; B/˚hn.A; B/˚hn.IB; @IB [IC /! hn.@IA[IB; @IB [IC /: Here is induced by the inclusion, and by a 7!.; a/. Proof. This is a consequence of (10.9.1) and (10.7.1). (10.9.4) Proposition. For each triple.A; B; C / the diagram hn.A; B/ ˛.A;B/ hnC1.IA; @IA [ IB/ @ ˇ Š hn.@IA [ IB; @IB [ IC / hn.A; B/ ˚ hn..A; B/ ˚ hn.IB; @IB [ IC / is commutative; here ˛.x/ D.x; x;.B;C /@x/, and the isomorphism ˇ is taken
from (10.9.3). Proof. The assertion about the third component of ˛ follows from (10.9.2). The other components require a little diagram chasing. For the veriﬁcation it is helpful to use the inverse isomorphism of ˇ given by the procedure of (10.7.1). The minus sign in the second component is due to the fact that the suspension isomorphism changes the sign if we interchange the roles of 0; 1, see 10.2.5. Chapter 11 Homological Algebra In this chapter we collect a number of algebraic deﬁnitions and results which are used in homology theory. Reading of this chapter is absolutely essential, but it only serves practical purposes and is not really designed for independent study. “Homological Algebra” is also the name of a mathematical ﬁeld – and the reader may wish to look into the appropriate textbooks. The main topics are diagrams and exact sequences, chain complexes, derived functors, universal coefﬁcients and Künneth theorems. We point out that one can imitate a lot of homotopy theory in the realm of chain complexes. It may be helpful to compare this somewhat simpler theory with the geometric homotopy theory. 11.1 Diagrams Let R be a commutative ring and denote by R- MOD the category of left Rmodules and R-linear maps. (The category ABEL of abelian groups can be identiﬁed with Z- MOD.) Recall that a sequence of R-modules and R-linear maps ai! Ai1! is exact at Ai if Im.aiC1/ D Ker.ai / and! AiC1 exact if it is exact at each Ai. The language of exact sequences is a convenient way to talk about a variety of algebraic situations. aiC1! Ai (1) 0! A a! B exact, a injective. (We also use A B for an injective homomorphism.) (2) B b! C! 0 exact, b surjective. (We also use B C for a surjective homomorphism.) b a (3 exact, a is injective, b is surjective, b induces an isomorphism of the cokernel of a with C. (4) Let A.1/! B.2/! C.3/! D
be exact. Then the following are equivalent:.1/ surjective,.2/ zero,.3/ injective: An exact sequence of the form (3) is called a short exact sequence. It sometimes happens that in a longer exact sequence every third morphism has the property (1) (or (2), (3)). Then the sequence can be decomposed into short exact sequences. Note that the exact homotopy sequences and the exact homology sequences have “period” 3. L A family.Mj j j 2 J / of modules has a direct sum Q j 2J Mj, the sum j 2J Mj, the product in the category in the category R- MOD, and a product 276 Chapter 11. Homological Algebra Q R- MOD. The underlying set of the product is the Cartesian product of the underlying sets, consisting of all families.xj 2 Mj j j 2 J /; the R-module structure is given by pointwise addition and scalar multiplication. The canonical projection pk W j Mj! Mk onto the k-th factor is part of the categorical product strucj Mj is a submodule of the product and consists of all families ture. The sum.xj / where all but a ﬁnite number of xj are zero. We have the canonical injection L ik W Mk! j Mj, deﬁned by pkik D id and pkil D 0 for k 6D l. If fj W Mj! N is a family of R-linear maps, then L L hfj i W Mj! N denotes the morphism determined by hfj i ıi k D fk. If gj W L! Mj is a family of R-linear maps, then Q.gj / W L! j Mj denotes the morphism determined by pk ı.gj / D gk. S Let.Aj j j 2 J / be a family of submodules of M. Then P j Aj denotes the j Aj. We say that M is the internal direct sum of the Aj submodule generated by if the map L j Aj! M;.aj / 7! P j aj L is an isomorphism. In that case we also write M D j Aj. A submodule A M is called a direct summand of M if there exists a complement of A, i
.e., a submodule B such that M is the internal direct sum of A and B. We assume known the structure theory of ﬁnitely generated abelian groups. An element of ﬁnite order in an abelian group A is called a torsion element. The torsion elements form a subgroup, the torsion subgroup T.A/. The torsion subgroup of A=T.A/ is trivial. If the group is ﬁnitely generated, then the torsion subgroup has a complement F, and F is a free abelian group. The cardinality of a basis of F is called the rank of A. A ﬁnitely generated torsion group is the direct sum of cyclic groups of prime power order Z=.pk/, and the number of factors isomorphic to Z=.pk/ is uniquely determined by the group. A similar structure theory exists for ﬁnitely generated modules over principal ideal domains. A linear map p W M! M with the property p ı p D p is called a projection operator on M. (11.1.1) Splitting Lemma. Let 0! E sequence of modules. Then the following assertions are equivalent:! G! 0 be a short exact f! F g (1) The image of f is a direct summand of F. (2) There exists a homomorphism r W F! E such that rf D id. (3) There exists a homomorphism s W G! F such that gs D id. If.1/–.3/ holds, we say the sequence splits. We then call r and s splittings, r a retraction of f, s a section of g. In case.2/ f r is a projection operator, hence we 11.1. Diagrams 277 have F D Im.f r/ ˚ Ker.f r/ D Im.f / ˚ Ker.r/. In case.3/ sg is a projection operator, hence we have F D Im.sg/ ˚ Ker.sg/ D Im.s/ ˚ Ker.g/. (11.1.2) Sum Lemma. Suppose given a commutative diagram in R- MODW B1 a1 A1 j2 $$$$$$ i1 C $$$$$$ j1 i2 B2 a
2 A2. Assume jkik D 0 for k D 1; 2..1/ If the ak are isomorphisms and.i2; j2/ is exact, then hi1; i2 i W A1 ˚ A2! C and.j2; j1/ W C! B2 ˚ B1 are isomorphisms and.i1; j1/ is exact..2/ If hi1; i2 i is an isomorphism and.i2; j2/ is short exact, then a1 is an isomorphism (j1; a2 are not needed). If.j2; j1/ is an isomorphism and.i1; j1/ is short exact, then a1 is an isomorphism (i2; a2 are not needed). Proof. (1) The hypothesis implies.j2; j1/ ı hi1; i2 i Da 1 ˚ a2. We show that hi1; i2 i is surjective. Given c 2 C we have j2.c i1a1 1 j2.c// D 0, by commutativity. Hence there exists by exactness x2 2 A2 such that c i1a1 1 j2.c/ D i2.x2/, i.e., c is contained in the image of hi1; i2 i. Let j1.c/ D 0 and write c D i1x1Ci2x2. Then 0 D j1.c/ D j1i1x1Cj1i2x2 D j1i2x2 D a2.x2/, hence x2 D 0 and c 2 Im.i1/. (2) Exercise. (11.1.3) Hexagon Lemma. Given a commutative diagram of abelian groups. G0 1 k1 G1 l1 j1 ) &&&&&& %%%%%% i1 h1 l2 j2 %%%%%% &&&&&& ) i2 h2 i0 j0 G0 G G0 0 G0 2 k2 G2 Suppose that k1; k2 are isomorphisms,.i1; j2/ exact,.i2; j1/ exact, and j0i0 D 0. Then h1k1 1 l1 D h2k1 2 l2. ) * *
) 278 Chapter 11. Homological Algebra Proof. The part i; j; k satisﬁes the hypothesis of the Sum Lemma (11.1.2). Given x 2 G0 there exist xj 2 Gj such that i0x D i1x1 C i2x2. We compute 0 D j0i0x D j0i1x1 C j0i2x2 D h1x1 C h2x2; l1x D j1i0x D j1i1x C j1i2x D j1i1x D k1x1; hence x1 D k1 1 l1x and similarly x2 D k1 2 l2x. (11.1.4) Five Lemma. Given a commutative diagram of groups and homomorphisms with exact rows: ˛ ˛0 A a A0 D0 E e E0. (1) a surjective, b; d injective ) c injective. (Here the E-part of the diagram is not needed.) (2) b; d surjective, e injective ) c surjective. (Here the A-part of the diagram is not needed.) (3) a surjective, b; d bijective, e injective ) c bijective. Proof. For another proof see (11.2.7). We give here a direct proof by the “method” called diagram chasing. One refers to diagram chasing whenever the proof (chasing elements through the diagram) does not really require a mathematical idea, only careful patience. (1) Let c.w/ D 0. Then 0c.w/ D d.w/ D 0, and injectivity of d shows.w/ D 0. By exactness, ˇ.v/ D w for some v. Since ˇ0b.v/ D cˇ.v/ D 0, we have ˛0.u0/ D b.v/ for some u0, by exactness, and a.u/ D u0 by surjectivity of a. By injectivity of b and commutativity we see ˛.u/ D v and hence by exactness w D ˇ.v/ D 0. (2) Given w0 2 C 0. Choose x such that d.x/ D 0
.c0/. By exactness, commutativity, and injectivity of e, we see ı.x/ D 0 and hence.w/ D x for some w. By commutativity, c.w/ and w0 have the same image under 0. Hence w0 D c.w/ˇ0.v0/ for some v0. Then c.w ˇ.v// D c.w/cˇ.v/ D c.w/ˇ0b.v/ D c.w/ ˇ0.v0/ D w0, i.e., w0 is contained in the image of c. (3) A consequence of (1) and (2). Problems 11.2. Exact Sequences 279 1. Let p be a projection operator on M. Then 1 p is a projection operator. The equalities Im.1 p/ D Ker.p/ and Ker.1 p/ D Im.p/ hold. Moreover M D Im.p/ ˚ Im.1 p/. The submodule A of M is a direct summand if and only if there exists a projection operator with image A. 2. Let.Aj j j 2 J / be a ﬁnite family of modules. Suppose given linear maps i k W Ak! A and pl W A! Al such that pki k D id and pki l D 0 for k 6D l (we write pki l D ıkl in this j i j pj is a projection operator. Hence case). Then.pk/ ı h i k i Did and h i k i ı.p k/ D the following are equivalent: (1) h i k i is an isomorphism. (2).pk/ is an isomorphism. (3) P P j i j pj D id. 3. Let p be a prime number. Determine the number of subgroups of Z=.pk/ ˚ Z=.pl /. 4. Consider the group Z=.6/˚Z. Determine the subgroups of index 2; 3; 4; 5; 6. Determine the number of complements of the torsion subgroup. 5. Let A be a ﬁnitely generated abelian group. Then A ˝Z Q is a Q-vector space. Show that its dimension is
the rank of A. 6. Let Mj, j 2 J be submodules of M. The following assertions are equivalent: P j Mj is the direct sum of the Mj. P (1) (2) For each i 2 J, Mi \ (3) Suppose P j;j ¤i Mj D f0g: j xj D 0, xj 2 Mj, almost all xj D 0, then xj D 0 for each j 2 J. 11.2 Exact Sequences We start with a commutative diagram of modules. a 0 a A ˛ A0 It yields two derived diagrams (Ke D kernel, Ko D cokernel, Im D Image). Ke.˛/ a Ke.ˇ/ b Ke./.1/ D.2/ Ke.a0˛/ a Ke.ˇ/ b Ke./ \ Im.b/ a0 Ko.ˇ/ b0 Ko./ Ko.˛/.3/ A0 Im.˛/ C Ke.a0/ a0 D B 0 Im.ˇ/ b0.4/ C 0 Im.b/ 280 Chapter 11. Homological Algebra The morphisms named a; b; a0; b0 are induced by the original morphisms with the same name by applying them to representatives. (1) and (2) are inclusions, (3) and (4) are quotients. (11.2.1) Proposition. Let.a; b/ and.a0; b0/ be exact. Then a connecting morphism ı W Ke./ \ Im.b/! A0 Im.˛/ C Ke.a0/ is deﬁned by the correspondence.a0/1ˇb1. Proof. For z 2 Ke./ \ Im.b/ there exists y 2 B such that b.y/ D z; since z 2 Ke./ and b D b0ˇ we have ˇ.y/ 2 Ke.b0/; since Ke.b0/ Im.a0/, there exists x0 2 A0 such that a0.x0/ D ˇ.y/. We set ı.z/ D x0 and show that this assignment is well-deﬁned. If Qy 2 B, b. Qy/ D z, then b.y Qy/ D
0; since Ke.b/ Im.a/, there exists x 2 A such that a.x/ D y Qy. We have ˇ.y/ ˇ. Qy/ D a0˛.x/, because of a0˛ D ˇa, and with a0. Qx0/ D ˇ. Qy/ we obtain a0.x0 Qx0 ˛.x// D 0, i.e., x0 Qx0 mod Im.˛/ C Ke.a0/. We add further hypotheses to the original diagram and list the consequences for the derived diagrams. We leave the veriﬁcation of (11.2.2), (11.2.3), (11.2.4), (11.2.5) to the reader. 11.2.2 If a0 is injective, then (1) and (3) are bijective. If b is surjective, then (2) Þ and (4) are bijective. 11.2.3 Let.a0; b0/ be exact. Then A0 Im.˛/ C Ke.a0/ a0! B 0 Im.ˇ/ b0! C 0 Im.b/ is exact. If, moreover, b is surjective, then (4) is bijective and therefore Ko.˛/ a0! Ko.ˇ/ b0! Ko./ is exact. If b0 is surjective, then the derived b0 is surjective too. Þ 11.2.4 Let.a; b/ be exact. Then Ke.a0˛/ a! Ke.ˇ/ b! Ke./ \ Im.b/ is exact. If, moreover, a0 is injective, then (1) is bijective and therefore Ke.˛/ a! Ke.ˇ/ b! Ke./ is exact. If a is injective, then the derived a is injective too. 11.2. Exact Sequences 281 Þ 11.2.5 Let.a; b/ and.a0; b0/ be exact. Then, as we have seen, ı is deﬁned. Under these assumptions the bottom lines of the derived diagrams together with ı yield Þ an exact sequence. (See the special case (
11.2.6).) (11.2.6) Kernel–Cokernel Lemma. If in the original diagram a0 is injective and b surjective, then.1/;.2/;.3/, and.4/ are bijective and the kernel-cokernel-sequence Ke.˛/ a! Ke.ˇ/ b! Ke./ ı! Ko.˛/ a0! Ko.ˇ/ b0! Ko./ is exact. Proof. We show the exactness at places involving ı; the other cases have already been dealt with. The relations ıb D 0 and a0ı D 0 hold by construction. If the class of x0 is contained in the kernel of a0, then there exists y such that a0.x0/ D ˇ.y/. Hence z D b.y/ 2 Ke./ by commutativity, and ı.z/ D x0. Suppose z 2 Ke./ is contained in the kernel of ı. Then there exists y such that z D b.y/; ˇ.y/ D a0.x0/ and ˇ.z/ D ˛.x/ 2 Im.˛/. Then b.y a.x// D z and ˇ.y a.x// D ˇ.y/ ˇa.x/ D ˇ.y/ a0˛.x/ D ˇ.y/ a0.x0/ D 0. Hence y a.x/ is a pre-image of z. We now relate the Kernel–Cokernel Lemma to the Five Lemma (11.2.7); see also (11.1.4). Given a commutative ﬁve-term diagram of modules and homomorphisms with exact rows. ˛ ˛0 A a A0 D0 E e E0 We have three derived diagrams. 0 0 Ke.ı/ Qd Ke.ı0/ D d D0 ı ı 0 E e E0, ˛ A a A0 ˛0 B b B 0 Ko.˛/ Qb Ko.˛0/ 0 0, 0 0 Ko.˛/ ˇ C Qb Ko.˛0/ ˇ 0 c C 0 0 Ke.ı/ Qd
Ke.ı0/ 0 0. 282 Chapter 11. Homological Algebra The rows of the ﬁrst two diagrams are exact for trivial reasons. The exactness of the rows of the original diagram implies that the third diagram has exact rows. From the considerations so far we obtain the exact sequences Ke.e/ \ Im.ı/! Ko. Qd /! Ko.d /; Ke.b/! Ke. Qb/! A0=.Im.a/ C Ke.˛0//; 0! Ke. Qb/! Ke.c/! Ke. Qd /! Ko. Qb/! Ko.c/! Ko. Qd /! 0: This yields: (11.2.7) Five Lemma. Given a ﬁve-term diagram as above. Then the following holds: (1) Ke. Qb/ D 0; Ke. Qd / D 0 ) Ke.c/ D 0: (2) Ko. Qb/ D 0; Ko. Qd / D 0 ) Ko.c/ D 0: (3) Ke.b/ D 0; A0=.Im.a/ C Ke.˛0// D 0 ) Ke. Qb/ D 0: (4) Ke.e/ \ Im.ı/ D 0; Ko.d / D 0 ) Ko. Qd / D 0: (5) a surjective, b; d injective ) c injective. (Here the E-part of the diagram is not needed.) (6) b; d surjective, e injective ) c surjective. (Here the A-part of the diagram is not needed.) (7) a surjective, b; d bijective, e injective ) c bijective. Problems 1. Given homomorphisms f W A! B and g W B! C between R-modules. Then there is a natural exact sequence 0! Ke.f /! Ke.gf /! Ke.g/! Ko.f /! Ko.gf /! Ko.g/! 0: The connection to the previous considerations: The commutative diagram with exact rows 0 0.1;f / A ˚ B h f;1 i gf ˚1.g;1/ C ˚ B h 1; can be viewed as an exact sequence of chain complexes
/ D Im.@nC1 W CnC1! Cn/; Hn D Hn.C/ D Zn=Bn: We call Cn (Zn, Bn) the module of n-chains (n-cycles, n-boundaries) and Hn the n-th homology module of the chain complex. (The boundary relation @@ D 0 implies Bn Zn, and therefore Hn is deﬁned.) Two n-chains whose difference is a boundary are said to be homologous. Often, in particular in the case R D Z, we talk about homology groups. Let C D.Cn; cn/ and D D.Dn; dn/ be chain complexes. A chain map f W C! D is a sequence of homomorphisms fn W Cn! Dn which satisfy the commutation rules dn ı fn D fn1 ı cn. A chain map induces (by restriction and passage to the factor groups) homomorphisms of the cycles, boundaries, and homology groups Zn.f/ W Zn.C/! Zn.D/; Bn.f/ W Bn.C/! Bn.D/; f D Hn.f/ W Hn.C/! Hn.D/: 284 Chapter 11. Homological Algebra A (short) exact sequence of chain complexes 00! 0 fn! Cn consists of chain maps f and g such that 0! C 0 n for each n. gn! C 00 n! 0 is exact We certainly have the induced morphisms Hn.f / and Hn.g/. Moreover, there exists a connecting morphism @n W Hn.C 00/! Hn1.C 0/, also called boundary operator, which is induced by the correspondence f 1 n1 ı dn ı g1 n. gn Cn C 00 n 3 z00 z0 2 C 0 n1 fn1 dn Cn1 (11.3.1) Lemma. For a cycle z00 2 C 00 n with pre-image z under gn the relation gn1dnz D d 00 n z00 D 0 and exactness shows that there exists z0 with n gnz D d 00 dn.z/ D fn1.z0/. The assignment z00 7! z0 induces a well-deﬁned homomorph
ism @n W Hn.C 00/! Hn1.C 0/. Proof. The relation fn2d 0 n1z0 D dn1fn1z0 D dn1dnz D 0 and the injectivity of fn2 show that z0 is a cycle. If we choose another pre-image z C fnw0 of z00, then we have to replace z0 by z0 C d 0 nw0, so that the homology class of z0 is welldeﬁned. Finally, if we change z00 by a boundary, we can replace z by the addition of a boundary and hence dnz does not change. (11.3.2) Proposition. The sequence! Hn.C 0/ f! Hn.C / g! Hn.C 00/ @n! Hn1.C 0/! is exact. Proof. The boundary operator dn W Cn! Cn1 induces a homomorphism dn W Kn D Cn=Bn! Zn1; and its kernel and cokernel are Hn and Hn1. By (11.2.3) and (11.2.4) the rows of the next diagram are exact. K0 n d 0 n Z0 n1 fn gn Kn fn1 dn Zn1 gn1 K00 n d 00 n Z00 n1 0 0 The associated sequence (11.2.6) H 0 is the exact homology sequence. n! Hn! H 00 n @! H 0 n1! Hn1! H 00 n1 11.4. Cochain complexes 285 Let f; g W C D.Cn; cn/! D D.Dn; dn/ be chain maps. A chain homotopy s from f to g is a sequence sn W Cn! DnC1 of homomorphisms which satisfy dnC1 ı sn C sn1 ı cn D gn fn: (This deﬁnition has two explanations; ﬁrstly, one can deﬁne “chain homotopy” in analogy to the topological deﬁnition by using the chain complex analogue of the unit interval; secondly, it codiﬁes the boundary relation of a geometric homotopy.) We call f and g homotopic or chain homotopic, if there
exists a chain homotopy s from f to g, in symbols s W f'g. “Chain homotopic” is an equivalence relation on the set of chain maps C! D; the data s W f'g and t W g'h imply.sn C tn/ W f'h. This relation is also compatible with composition; if and t W g'g0 W D! E, then.gnC1sn/ W gf'gf 0 and.tnfn/ W gf'g0f. We call f W C! D a chain equivalence, if there exists a chain map g W D! C and chain homotopies fg'id and gf'id. (11.3.3) Proposition. Chain homotopic maps induce the same morphisms between the homology groups. Proof. Let x 2 Cn be a cycle. The homotopy relation gn.x/ fn.x/ D dnC1sn.x/ shows that fn.x/ and gn.x/ are homologous. 11.4 Cochain complexes Let C D.Cn; @n/ be a chain complex of R-modules. Let G be another R-module. We apply the functor HomR.; R/ to C and obtain a chain complex C D.C n; ın/ of R-modules with C n D HomR.Cn; R/ and the R-linear map ın W C n D HomR.Cn; R/! HomR.CnC1; R/ D C nC1 deﬁned by ın.'/ D.1/nC1' ı @nC1 for'2 Hom.Cn; R/. For the choice of this sign see 11.7.4. The reader will ﬁnd different choices of signs in the literature. Other choices will not effect the cohomology functors. But there seems to be an agreement that our choice is the best one when it comes to products. Now some “co” terminology. A cochain complex C D.C n; ın/ is a Zgraded module.C n j n 2 Z/ together with homomorphisms ın W C n! C nC1, called coboundary operators or differentials1, such that ınC1ın D
> 0 and each model Bn;j. (11.5.1) Theorem. Let F be a free and G be an acyclic functor. For each natural transformation x' W H0 ı F0! H0 ı G0 there exists a natural transformation'W F! G which induces x'. Any two natural transformations'and with this property are naturally chain homotopic ([57]). Proof. We specify a natural transformation '0 by the condition that '.b0;j / represents the homology class x'Œb0;j. Let now 'i W Fi! Gi be natural transfori 'i D 'i1d F mations.0 i < n/ such that d G for 0 < i < n. Consider the i elements 'n1d F n bn;j 2 Gn1.Bn;j /. For n D 1 this element represents 0 in H0, by the construction of '0. For n > 1 we see from the induction hypothesis that d G n bn;j D 0. Since G is acyclic we ﬁnd gn;j 2 Gn.Bn;j / such that d G n bn;j. We specify a natural transformation'by the conditions '.bn;j / D gn;j. This transformation then satisﬁes n 'n D 'n1d F d G n. This ﬁnishes the induction step. n bn;j D 'n2d F n gn;j D 'n1d F n1'n1d F n1d F Let now'and be given. Then 0.b0;j / '0.b0;j / D d G 1 c0;j for some c0;j, since 0.b0;j / and '0.b0;j / represent the same homology class. We deﬁne the transformation s0 W F0! G1 by the condition s0.b0;j / D c0;j. Suppose now that sn W Fn! GnC1 are given such that d G i D i 'i for 0 i < n iC1si C si1d F (and s1 D 0). We compute with the induction hypothesis 11.6. Chain Equivalences 287 n. n 'n sn1d F d G n / n 'n1d
F D n1d F n. n1d F n 'n1d F n sn2d F n1/d F n D 0: Thus, by acyclicity, we can choose cn;j 2 GnC1.Bn;j / such that nC1cn;j D. n 'n sn1d F d G n /.bn;j /: We now specify a natural transformation sn W Fn! GnC1 by sn.bn;j / D cn;j. It then has the required property d G nC1sn D n 'n sn1d F n. Problems 1. Let F0 F1 be a chain complex of free R-modules Fi and D0 D1 an exact sequence of R-modules. A chain map.' W Fi! Di j i 2 N0/ induces a homomorphism H0.'/ W H0.F/! H0.D/. Given a homomorphism ˛ W H0.F/! H0.D0/ there exists up to chain homotopy a unique chain map.'i / such that H0.'/ D ˛. This can be obtained as a special case of (11.5.1). The reader should now study the notion of a projective module (one deﬁnition is: direct summand of a free module) and then show that a similar result holds if the Fi are only assumed to be projective. An exact sequence of the form 0 A P0 P1 with projective modules Pi is called a projective resolution of the module A. The result stated at the beginning says that projective resolutions are unique up to chain equivalence. (Fundamental Lemma of homological algebra). Each module has a free resolution. 11.6 Chain Equivalences A chain map which induces an isomorphism of homology groups is under certain circumstances a chain equivalence. This is one of the results of this section. The notion of a chain homotopy can be used to develop a homotopy theory of chain complexes in analogy to the topological homotopy theory. We have the null complex; the chain groups are zero in each dimension. A chain complex is called contractible if it is chain equivalent to the null complex, or equivalently, if the identity is chain homotopic to the null map. A chain complex is said to be acyclic if its homology
groups are zero. Let f W.K; d K/!.L; d L/ be a chain map. We construct a new chain complex Cf, the mapping cone of f, by.Cf /n D Ln ˚ Kn1; d Cf.y; x/ D.d Ly C f x; d Kx/: This can also be written in matrix form d L 0 d K y x 7! f : y x 288 Chapter 11. Homological Algebra The suspension †K of K is deﬁned by.†K/n D Kn1 and d †K D d K. The canonical injection and projection yield an exact sequence of chain complexes 0! L! Cf! †K! 0. Associated is an exact sequence (11.3.2), and the boundary morphism @ W HnC1.†K/! Hn.L/ equals Hn.f /, if we use the canonical identiﬁcations HnC1.†K/ Š Hn.K/. The next result shows a typical difference between the topological and the algebraic homotopy theory. (11.6.1) Theorem. Let Cf be contractible. Then f is a chain equivalence. Proof. The inclusion W L! Cf, y 7!.y; 0/ is null homotopic, since Cf is contractible. Let s W'0 be a null homotopy. We write s.y/ D..y/; g.y// 2 L ˚ K (without notation for the dimensions). The condition @s C s@ D then reads.@y C fgy C @y; @gy C g@y/ D.y; 0/; i.e., @g D g@ and @ C @ D id fg. Hence g is a chain map, and because of the -relation, a right homotopy inverse of f. The projection W Cf! †K is likewise null homotopic. Let t W'0 be a null homotopy. We write t.y; x/ D h.y/ C.x/. The equality @t C t@ D then means @hy C h@y @x C @x C hf x D x; hence @h D h@ and @ C @ D hf id. Therefore h is a chain map and a left homot
opy inverse of f. (11.6.2) Proposition. Let K be acyclic and suppose that Zn Kn is always a direct summand. Then K is contractible. @ Proof. We have the exact sequence 0! Zn! Kn! Bn1! 0, and since K is acyclic we conclude Zn D Bn. Moreover there exists tn1 W Bn1! Kn with @tn1 D id, since Zn is a direct summand of Kn. We therefore have a direct decomposition Kn D Bn ˚ tn1Bn1. We deﬁne s W Kn! KnC1 by sjBn D tn and snjtn1Bn1 D 0. With these deﬁnitions one veriﬁes separately on Bn as well as on tn1Bn1 that @s C s@ is the identity, i.e., s is a null homotopy of the identity. (11.6.3) Theorem. Let f W K! L be a chain map between chain complexes which consist of free modules over a principal ideal domain R. If f induces isomorphisms f W H.K/ Š H.L/, then f is a chain equivalence. Proof. The exact homology sequence and the hypothesis imply that Cf is acyclic. A submodule of a free R-module is free. Hence the boundary groups of the complex Cf are free, and therefore the exact sequence 0! Zn! Cfn! Bn1! 0 splits. Now we apply (11.6.1) and (11.6.2), in order to see that f is a chain equivalence. 11.7. Linear Algebra of Chain Complexes 289 In the topological applications we often have to work with large chain complexes. In some situations it is useful to replace them by smaller chain equivalent complexes. A graded R-module A D.An/ is said to be of ﬁnite type if the modules An are ﬁnitely generated R-modules. (11.6.4) Proposition. Let R be a principal ideal domain. Let C D.Cn/ be a chain complex of free R-modules such that its homology groups are ﬁnitely generated. Then there exists a free chain complex D of ﬁnite type which is chain equivalent
to C. Proof. Let Fn be a ﬁnitely generated submodule of Zn.C / which is mapped onto Hn.C / under the quotient map Zn.C /! Hn.C /, and denote by Gn the kernel of the epimorphism Fn! Hn.C /. Deﬁne a chain complex D D.Dn; dn/ by Dn D Fn ˚ Gn1 and dn.x; y/ D.y; 0/. Then D is a free chain complex of ﬁnite type and Hn.D/ D Fn=Gn Š Hn.C /. Since Gn is a free submodule of Bn.C / we can choose for each n a homomorphism 'n W Gn! CnC1 such that cnC1'n.y/ D y for each y 2 Gn. Deﬁne n W Dn D Fn ˚ Gn1! Cn,.x; y/ 7! x C 'n1.y/. One veriﬁes that D. n/ is a chain map which induces an isomorphism of homology groups. By (11.6.3), is a chain equivalence. 11.7 Linear Algebra of Chain Complexes We work in the category R- MOD for a commutative ring R. 11.7.1 Graded modules. Let A D.An/ and B D.Bn/ be Z-graded left Rmodules over a commutative ring R. The tensor product A ˝ B is the module and g W B! B 0 with are morphisms of some degree, then their tensor product f ˝ g is deﬁned by pCqDn Ap ˝R Bq as entry in degree n. If f W A! A0 L.f ˝ g/.a ˝ b/ D.1/jgjjajf.a/ ˝ g.b/: Here jaj denotes the degree of a. The formula for the tensor product obeys the (heuristic) “graded sign rule”: Whenever entities of degree x and y are interchanged, then the sign.1/xy appears. The tensor product of objects and of morphisms is associative and compatible with composition (in the graded sense).
f ˝ g/ ı.f 0 ˝ g0/ D.1/jgjjf 0jff 0 ˝ gg0 (sign rule). This composition is associative, as it should be. When we use the degree as upper index (e.g., in cohomology), then the agreement Ak D Ak is sometimes Þ suitable. 11.7.2 Graded algebras. A Z-graded R-algebra A is a Z-graded R-module.An j n 2 Z/ together with a family of R-linear maps Ai ˝R Aj! AiCj ; x ˝ y 7! x y: 290 Chapter 11. Homological Algebra The algebra is associative, if always x.y z/ D.x y/ z holds, and commutative, if always x y D.1/jxjjyjy x holds (sign rule). A unit element 1 2 A0 of the algebra satisﬁes 1 x D x D x 1. Let M D.M n/ be a Z-graded R-module. A family Ai ˝ M j! M iCj ; a ˝ x 7! a x of R-linear maps is the structure of an A-module on M, provided the associativity a.b x/ D.a b/ x holds for a; b 2 A and x 2 M. If A has a unit element, then the module is unital, provided 1 x D x always holds. Let A and B be Zgraded algebras. Their tensor product A ˝ B is the tensor product of the underlying graded modules.A ˝ B/n D iCj Dn Ai ˝ B j together with the multiplication.a ˝ b/.a0 ˝ b0/ D.1/jbjja0jaa0 ˝ bb0 (sign rule). If A and B are associative, then A ˝ B is associative. If both have a unit element 1, then 1 ˝ 1 is a unit element for the tensor product. If both algebras are commutative, then their tensor product is commutative. The tensor product of graded algebras is an associative functor.Þ L 11.7.3 Tensor
product of chain complexes. Let.A; dA/ and.B; dB / be chain complexes. Then the graded module A ˝ B is a chain complex with boundary operator d D dA ˝ 1 C 1 ˝ dB. Here we have to take the sign rule into account, i.e., d.a ˝ b/ D dAa ˝ b C.1/jaja ˝ dB b: One veriﬁes dd D 0, using this sign rule. Passage to homology induces Hp.A/ ˝ Hq.B/! HpCq.A ˝ B/; Œa ˝ Œb 7! Œa ˝ b: The tensor product of chain complexes is associative. Þ 11.7.4 Dual chain complex. We regard the ground ring R as a trivial chain complex with R in degree 0 and zero modules otherwise. Let.An; @/ be a chain complex. We deﬁne the dual graded R-module by A n D HomR.An; R/. We require a boundary operator ı W A n1 on the dual module such that the evaluation " ˝ An! R;'˝ a 7! '.a/; " D 0 otherwise, becomes a chain map. This condition, ".' ˝ a/ D 0 and 11.7.3 yield for'˝ a 2 A n1 ˝ An 0 D d ".' ˝ a/ D ".d.' ˝ a/ D ".ı' ˝ a C.1/j'j' ˝ @a D.ı'/.a/ C.1/j'j'.@a/; i.e., we have to deﬁne ı' D.1/j'jC1' ı @. Þ 11.7. Linear Algebra of Chain Complexes 291 11.7.5 Hom-Complex. For graded modules A and B we let Hom.A; B/ be a2Z Hom.Aa; BaCn/ as component in degree n. On this Homthe module with module we use the boundary operator Q d.fi / D.@ ı fi /..1/nfi ı @/ for.fi W Ai! BiCn/, i.
e., the a-component pra.df / 2 Hom.Aa; BaCn1/ for f D.fa/ 2 Hom.A; B/n is deﬁned to be pra.df / D @ ı fa.1/nga1 ı @: One veriﬁes dd D 0. This deﬁnition generalizes our convention about the dual Þ module. 11.7.6 Canonical maps. The following canonical maps from linear algebra are chain maps. (1) The composition Hom.B; C / ˝ Hom.A; B/! Hom.A; B/;.fi / ˝.gj / 7!.flCjgj ı gl /: (2) The adjunction ˆ W Hom.A˝B; C /! Hom.A; Hom.B; C //; ˆ.fi /.x/.y/ D fjxjCjyj.x ˝y/: (3) The tautological map W Hom.C; C 0/ ˝ Hom.D; D0/! Hom.C ˝ D; C 0 ˝ D0/ with.f ˝ g/.x ˝ y/ D.1/jgjjxjf.x/ ˝ g.y/ (sign rule). (4) The trace map W A ˝ B! Hom.A; B/,.' ˝ b/.a/ D.1/jajjbj'.a/b. Þ Problems 1. Tensor product is compatible with chain homotopy. Let be a chain homotopy. Then s ˝ id W f ˝ id'g ˝ id W C ˝ D! C 0 ˝ D is a chain homotopy. 2. A chain complex model of the unit interval is the chain complex I with two non-zero groups I1 Š R with basis e, I0 Š R ˚ R with basis e0; e1 and boundary operator d.e/ D e1 e0 (in the topological context: the cellular chain complex of the unit interval). We use this model to deﬁne chain homotopies with the cylinder I ˝ C. Note Cn ˚ Cn
˚ Cn1 Š.I ˝ C /n;.x1; x0; y/ 7! e1 ˝ x1 C e0 ˝ x0 C e ˝ y: A chain map h W I ˝ C! D consists, via these isomorphisms, of homomorphisms ht n W Cn! Dn and sn W Cn! DnC1. The ht are chain maps (t D 0; 1) and dsn.y/ D h1 n.y/ sn1cy, i.e., s W h1 n.y/ h0 is a chain homotopy in our previous deﬁnition. 3. Imitate the topological deﬁnition of the mapping cone and deﬁne the mapping cone of a chain map f W C! D as a quotient of I ˝C ˚D. The n-th chain group is then canonically isomorphic to Cn1 ˚ Dn and the resulting boundary operator is the one we deﬁned in the section on chain equivalences. Consider also the mapping cylinder from this view-point.'h0 292 Chapter 11. Homological Algebra 11.8 The Functors Tor and Ext Let R be a principal ideal ring. We work in the category R- MOD; this comprises the category of abelian groups (Z-modules). An exact sequence 0! F1! F0! A! 0 with free modules F1; F0 is a free resolution of A. Since submodules of free modules are free, it sufﬁces to require that F0 is free. Let F.A/ denote the free R-module generated by the set A. Denote the basis element of F.A/ which belongs to a 2 A by Œa. We have a surjective homomorphism p W F.A/! A, P naa. Let K.A/ denote its kernel. The exact sequence naŒa 7! P 0! K.A/ i! F.A/ p! A! 0 will be called the standard resolution of A. We take the tensor product (over R) of this sequence with a module G, denote the kernel of i ˝ 1 by TorR.A; G/ D Tor.A; G/ and call
it the torsion product of A; G. We now derive some elementary properties of torsion products. We show that Tor.A; G/ can be determined from any free resolution, and we make Tor.; / into a functor in two variables. In the next lemma we compare free resolutions. (11.8.1) Lemma. Given a homomorphism f W A! A0 and free resolution F and F 0 of A and A0, there exists a commutative diagram 0 0 F1 i F0 p A 0 F f1 F 0 1 f0 F 0 0 s i 0 p0 f A0 0 F 0 (without s). If. xf1; xf0/ is another choice of homomorphisms making the diagram commutative, then there exists a homomorphism s W F0! F 0 0 D i 0s and f1 f 0 1 with f0 f 0 1 D si. k 2 F 0 Proof. Let.xk/ be a basis of F0. Choose x0 Deﬁne f0 by f0.xk/ D x0 exactness of F 0 a unique f1 such that f0i D i 0f1. Since p0.f0 f 0 0, the elements.f0 f 0.f0 f 0 i 0.f1 f 0 k/ D fp.xk/. k. Then fp D p0f0. Since p0f0i D 0, there exists by 0/ D fp fp D 0/.xk/ are contained in the kernel of p0. Hence we have o/.xk/ D i 0.yk/ for suitable yk. We deﬁne s by s.xk/ D yk. From 1/ D f0i f 0 1 D si. 0i D i 0si and the injectivity of i 0 we conclude f1 f 0 0 such that p0.x0 We take the tensor product ˝G of the diagram in (11.8.1). The homomorphism f1 ˝ 1 induces a homomorphism Ker.i ˝ 1/! Ker.i 0 ˝ 1/ and f1 f 0 1 D si shows that this homomorphism does not depend on the choice of.f1; f0/. Let us denote this homomorphism by T.f I F
; F 0/. If g W A0! A00 is given and F 00 a 11.8. The Functors Tor and Ext 293 free resolution of A00, then T.gI F 0; F 00/ ı T.f I F ; F 0/ D T.gf I F ; F 00/. This implies that an isomorphism f induces an isomorphism T.f I F ; F 0/. In particular each free resolution yields a unique isomorphism Ker.i ˝ 1/ Š Tor.AI G/, if we compare F with the standard resolution. The standard resolution is functorial in A. This fact is used to make Tor.; G/ into a functor. It is clear that a homomorphism G! G0 induces a homomorphisms Tor.A; G/! Tor.A; G0/. Hence Tor is also a functor in the variable G (and the two functor structures commute). If we view 0! F1! F0! 0 in (11.8.1) as a chain complex, then.f1; f0/ is a chain map and s yields a chain homotopy between.f1; f0/ and.f 0 1; f 0 0/. (11.8.2) Proposition. Elementary properties of torsion groups in the category of abelian groups are: (1) Let A be a free abelian group. Then Tor.A; G/ D 0. (2) Tor.Z=n; G/ Š fg 2 G j ng D 0g G. (3) If G is torsion free, then Tor.Z=n; G/ D 0. (4) Tor.Z=m; Z=n/ Š Z=d with d the greatest common divisor of m; n. (5) A direct sum decomposition A Š A1˚A2 induces a direct sum decomposition Tor.A; G/ Š Tor.A1; G/ ˚ Tor.A2; G/. Proof. (1 is a free resolution. (2) Use the free resolution 0! Z n! Z! Z=n! 0. (3) and (4) are consequences of (2). In order to verify (5), use the direct sum of free resolutions. We can also work with a resolution of the other variable. Let Q1
Q0 B be a free resolution and deﬁne Tor0.A; B/ D Ker.A ˝ Q1! A ˝ Q0/. (11.8.3) Proposition. There exists a canonical isomorphism Tor.A; B/ Š Tor0.A; B/: Proof. Let P1! P0! A be a free resolution. From the resolutions of A and B we obtain a commutative diagram: Tor.A; B/ P1 ˝ Q1 P1 ˝ Q0 P1 ˝ B ˛ ˇ P0 ˝ Q1 P0 ˝ Q0 P0 ˝ B Tor0.A; B/ A ˝ Q1 A ˝ Q0 A ˝ B. 294 Chapter 11. Homological Algebra The Kernel–Cokernel Lemma (11.2.6) yields an isomorphism ı of Tor.A; B/ D Ker./ with the submodule Tor0.A; B/ of Coker ˛. Interchanging the tensor factors yields an isomorphism Tor.B; A/ Š Tor0.A; B/. We combine this with (11.8.3) and see that the isomorphisms (11.8.2) also hold if we interchange the variables. It is now no longer necessary to use the notation Tor0. The functor Ext is deﬁned in analogy to the functor Tor, the tensor product is replaced by the Hom-functor. Let R be a principal ideal domain and 0! K.A/ p! A! 0 the standard free resolution of A as above. We apply the functor HomR.; B/ to this sequence. The cokernel of i W Hom.F.A/; B/! Hom.K.A/; B/ is deﬁned to be ExtR.A; B/ D Ext.A; B/. We show that Ext.A; B/ can be determined from any free resolution. We start with a diagram as in (11.8.1) and obtain a well-deﬁned homomorphism Coker.Hom.i; B//! Coker.Hom.i 0; B//; in particular we obtain an isomorphism Ext.A; B/ Š Coker.
Hom.i; B//. i! F.A/ (11.8.4) Proposition. Elementary properties of Ext in the category of abelian groups are: (1) Ext.A; B/ D 0 for a free abelian group A. (2) Ext.Z=n; B/ Š B=nB. (3) Ext.Z=n; B/ D 0 for B D Q; Q=Z; R. (4) Ext.Z=m; Z=n/ Š Z=.m; n/. (5) Ext.A1 ˚ A2; B/ Š Ext.A1; B/ ˚ Ext.A2; B/. The foregoing develops what we need in this text. We should at least mention the general case. Let 0 C P0 P1 be a projective resolution of the R-module C and let A be another R-module. We apply Hom.; A/ to the chain complex P and obtain a cochain complex Hom.P; A/; its i-th cohomology group (i 1) is denoted Exti R.C; A/. Since projective resolutions are unique up to chain equivalence, the Exti R-groups are unique up to isomorphism. For principal ideal domains only Ext1 occurs, since we have resolution of length 1. The notation Ext has its origin in the notion of extensions of modules. An exact sequence 0! A! Bn1!! B1! B0! C! 0 is called an n-fold extension of A by C. One can obtain Extn R.C; A/ as certain congruence classes of n-fold extension of A by C, see [120, Chapter III]. Write E E0 if there exists a commutative diagram Bn1 B 0 n1 B0 B 0 0 A D A 0 0. The congruence relation is generated by. 11.9. Universal Coefﬁcients 295 Problems 1. Suppose Tor.A; Z=p/ D 0 for each prime p. Then the abelian group A is torsion free. 2. The kernel of A! A ˝Z Q, a 7! a ˝ 1 is the torsion subgroup of A. 3. Does there exist a non-trivial abelian group A such that A ˝ F D
0 for each ﬁeld F? 11.9 Universal Coefﬁcients We still work in R- MOD for a principal ideal domain R. Let C D.Cn; cn/ be a chain complex of modules. Then C ˝ G D.Cn ˝ G; cn ˝ 1/ is again a chain complex. (11.9.1) Proposition (Universal Coefﬁcients). Let C be a chain complex of free modules. Then there exists an exact sequence 0! Hq.C / ˝ G ˛! Hq.C ˝ G/ ˇ! Tor.Hq1.C /; G/! 0: The sequence is natural in C and G and splits. The homomorphism ˛ sends Œz ˝ g for a cycle z to the homology class Œz ˝ g. cn! Bn1! 0 is exact; Bn1 is a submodule Proof. The sequence 0! Zn! Cn of Cn1 and hence free. Therefore the sequence splits and the induced sequence 0! Zn ˝ G! Cn ˝ G! Bn1 ˝ G! 0 is again a split exact sequence. We consider the totality of these sequences as an exact sequence of chain complexes, the Z- and the B-complex have trivial boundary operator. Associated to this short exact sequence of chain complexes is a long exact homology sequence of the form Bn ˝ G i˝1 Zn ˝ G Hn.C ˝ G/ Bn1 ˝ G i˝1 Zn1 ˝ G: One veriﬁes that the boundary operator (11.3.1) of the homology sequence is i ˝ 1, where i W Bn Zn. The sequence Bn ˝ G! Zn ˝ G! Hn ˝ G! 0 is exact, hence the cokernel of i ˝ 1 is Hn.C / ˝ G, and the resulting map Hn.C / ˝ G! Hn.C ˝ G/ is ˛. The kernel of Bn1 ˝ G! Zn1 ˝ G is Tor.Hn1.C /; G/, because
0! Bn1! Zn1! Hn1.C /! 0 is a free resolution. Let r W Cn! Zn be a splitting of Zn Cn. Then Zn.C ˝ G/ Cn ˝ G r˝1! Zn ˝ G! Hn.C / ˝ G maps Bn.C ˝G/ to zero and induces W Hn.C ˝G/! Hn.C /˝G with ˛ D id, i.e., a splitting of the universal coefﬁcient sequence. 296 Chapter 11. Homological Algebra Let again C D.Cn; cn/ be a chain complex with free R-modules Cn. We obtain the cochain complex with cochain groups Hom.Cn; G/ and cohomology groups H n.C I G/. (11.9.2) Proposition (Universal Coefﬁcients). There exists an exact sequence 0! Ext.Hn1.C /; G/! H n.C I G/ ˛! Hom.Hn.C /; G/! 0: The map ˛ sends the cohomology class of the cocycle'W Cn! G to the homomorphism Hn.C /! G, Œc 7! '.c/. The sequence is natural with respect to chain maps (variable C ) and module homomorphisms (variable G). The sequence splits, and the splitting is natural in G but not in C. Proof. Again we start with the split exact sequence 0! Zn! Cn! Bn1! 0 and the induced exact sequence 0 Hom.Zn; G/ Hom.Cn; G/ Hom.Bn1; G/ 0: We consider the totality of these sequences as an exact sequence of cochain complexes, the Z- and the B-complex have trivial coboundary operator. Associated to this short exact sequence of cochain complexes is a long exact cohomology sequence of the form Hom.Bn; G/ d n Hom.Zn; G/ H n.C I G/ Hom.Bn1; G/ which induces a short exact sequence 0 Ker d n ˛ H n.C I G/ Coker d n1 0:.4/ We need: (11.9.3) Lem
ma. The formal coboundary operator d n (without the additional sign introduced earlier!) is the homomorphism induced by i W Bn! Zn. Proof. Let'W Zn! G be given. Then d n.'/ is obtained as follows: Extend'to Q' W Cn! G. Apply ı and ﬁnd a pre-image of ı. Q'/ D Q'cnC1 in Hom.Bn; G/. One veriﬁes that 'i is a pre-image. From the exact sequence 0! Bn! Zn! Hn.C /! 0 we obtain the exact sequence Hom.Bn; G/ i Hom.Zn; G/ Hom.Hn.C /; G/ 0: We use it to identify the Ker i with Hom.Hn.C /; G/. One veriﬁes that ˛ is as claimed in the statement (11.9.2). From the free presentation and the deﬁnition 11.9. Universal Coefﬁcients 297 of Ext we thus obtain the exact sequence of the theorem. The naturality of this sequence is a consequence of the construction. It remains to verify the splitting. We choose a splitting r W Cn! Zn of the inclusion Zn Cn. Now consider the diagram 0 0 Zn.Hom.C; G// Hom.Cn; G/ ı Hom.CnC1; G/ Hom.Hn.C /; G/ r Hom.Zn; G/ i Hom.Bn; G/. If'2 Ker i, then r.'/ D'ı r 2 Ker ı. The splitting is induced by Ker i! Zn.Hom.C; G//,'7! 't. Without going into the deﬁnition of Ext we see from the discussion: (11.9.4) Proposition. Suppose Hn1.C / is a free R-module. Then the homomorphism ˛ W H n.C I G/! Hom.Hn.C /; G/ in (11.9.2) is an isomorphism. Proof. The sequence 0! Bn1! Zn1! Hn1! 0 splits and therefore the cokernel of d n1 is zero. Given a cochain complex C D.C
q; ıq/ we can view it as a chain complex C D.Cq; @q/ by a shift of indices: We set Cq D C q and we deﬁne @q W Cq! Cq1 as ıq W C q! C qC1. We can now rewrite (11.9.1): (11.9.5) Proposition. Let C be a cochain complex of free R-modules. Then we have a split exact sequence 0! H q.C / ˝ G! H q.C ˝ G/! Tor.H qC1.C /; G/! 0: Let now C be a chain complex of free modules. We apply (11.9.5) to the dual cochain complex with C q D Hom.Cq; R/ and cohomology groups H q.C I R/. (11.9.6) Proposition. Let C be a free chain complex and G be a module such that either H.C / is of ﬁnite type or G is ﬁnitely generated. Then there exists a natural exact sequence 0! H p.C / ˝ G! H q.C I G/! Tor.H qC1.C /; G/! 0 and this sequence splits. Proof. If G is ﬁnitely generated we have a canonical isomorphism of the form Hom.C; R/ ˝ G Š Hom.C; G/; we use this isomorphism in (11.9.5). If H.C / is of ﬁnite type we replace C by a chain equivalent complex C 0 of ﬁnite type (see (11.6.4)). In that case we have again a canonical isomorphism Hom.C 0; R/ ˝ G Š Hom.C 0; G/. We apply now (11.9.5) toC 0. 298 Chapter 11. Homological Algebra (11.9.7) Proposition. Let f W C! D be a chain map between complexes of free abelian groups. Suppose that for each ﬁeld F the map f ˝ F induces isomorphisms of homology groups. Then f is a chain equivalence. Proof. Let C.f / denote the mapping cone of f. The hypothesis implies that H.
C.f / ˝ F / D 0. We use the universal coefﬁcient sequence. It implies that Tor.H.C.f //; Z=p/ D 0 for each prime p. Hence H.C.f // is torsion-free. From H.C.f // ˝ Q we conclude that H.C.f // is a torsion group. Hence H.C.f // D 0. Now we use (11.6.3). 11.10 The Künneth Formula Let C and D be chain complexes of R-modules over a principal ideal domain R. We have the tensor product chain complex C ˝R D and the associated homomorphism ˛ W Hi.C / ˝R Hj.D/! HiCj.C ˝R D/; Œx ˝ Œy 7! Œx ˝ y: We use the notation for TorR. The next theorem and its proof generalizes the universal coefﬁcient formula (11.9.1). (11.10.1) Theorem (Künneth Formula). Suppose C consists of free R-modules. Then there exists an exact sequence 0! L iCj Dn Hi.C /˝RHj.D/! Hn.C ˝RD/! L iCj Dn1 Hi.C /Hj.D/! 0: If also D is a free complex, then the sequence splits. Proof. We consider the graded modules Z.C / and B.C / of cycles and boundaries as chain complexes with trivial boundary. Since Z.C / is free, we have the equalities (canonical isomorphisms).Z.C / ˝ Z.D//n D Ker.1 ˝ @ W.Z.C / ˝ D/n!.Z.C / ˝ D/n1/ and.Z.C / ˝ B.D//n D Im.1 ˝ @ W.Z.C / ˝ D/nC1!.Z.C / ˝ D/n/; and they imply H.Z.C / ˝ D/ Š Z.C / ˝ H.D/ (homology commutes with the tensor product by
a free module). In a similar manner we obtain an isomorphism H.B.C /˝D/ Š B.C /˝H.D/. We form the tensor product of the free resolution of chain complexes 0! B.C / i! Z.C /! H.C /! 0 11.10. The Künneth Formula 299 with H.D/. We obtain the following exact sequence, referred to as./, with injective morphism (1) and surjective morphism (2) H.C / H.D/.1/ B.C / ˝ H.D/ i˝1 Z.C / ˝ H.D/.2/ H.C / ˝ H.D/ Š Š H.B.C / ˝ D/.i˝1/ H.Z.C / ˝ D/. Let us use the notation.AŒ1/n D An1 for a graded object A. We tensor the exact sequence of chain complexes 0! Z.C /! C! B.C /Œ1! 0 with D and obtain an exact sequence 0! Z.B.C / ˝ D/Œ1! 0: Its exact homology sequence has the form : : :! H.B.C / ˝ D/.1/! H.Z.C / ˝ D/! H.C ˝ D/! H.B.C / ˝ D/Œ1.1/! H.Z.C / ˝ D/Œ1! : One veriﬁes that (1) is the map.i ˝ 1/. Hence we obtain the exact sequence 0! Coker.i/! H.C ˝ D/! Ker.i/Œ1! 0 which yields, together with the sequence./, the exact sequence of the theorem. Choose retractions r W Cn! Zn.C / and s W Dn! Zn.D/. Then.C ˝ D/n! H.C / ˝ H.D/, c ˝ d 7! Œr.c/ ˝ Œs.d / sends the boundaries of.C ˝ D/n to zero and induces a retraction W Hn.
C ˝ D/!.H.C / ˝ H.D//n of ˛. As in the case of the universal coefﬁcient theorem we can rewrite (11.10.1) in terms of cochain complexes. Under suitable ﬁniteness conditions we can then apply the result to the dual complex of a chain complex and obtain: (11.10.2) Theorem (Künneth Formula). Let C and D be free chain complexes such that H.C / or H.D/ is of ﬁnite type. Then there exists a functorial exact sequence 0! L iCj Dn H i.C / ˝ H j.D/! H n.C ˝ D/! and this sequence splits. L iCj DnC1 H i.C / H j.D/! 0 Chapter 12 Cellular Homology In this chapter we ﬁnally show that ordinary homology theory is determined on the category of cell complexes by the axioms of Eilenberg and Steenrod. From the axioms one constructs the cellular chain complex of a CW-complex. This chain complex depends on the skeletal ﬁltration, and the boundary operators of the chain complex are determined by the so-called incidence numbers; these are mapping degrees derived from the attaching maps. The main theorem then says that the algebraic homology groups of the cellular chain complex are isomorphic to the homology groups of the homology theory (if it satisﬁes the dimension axiom). From this fact one obtains immediately qualitative results and explicit computations of homology groups. Thus if X has k.n/ n-cells, then Hn.XI Z/ is a subquotient of the free abelian group of rank k.n/. A ﬁnite cell complex has ﬁnitely generated homology groups. We deduce that the combinatorial Euler characteristic is a homotopy invariant that can be computed from the homology groups. In the case of a simplicial complex we show that singular homology is isomorphic to the classical combinatorial simplicial homology. In this context, simplicial homology is a special case of cellular homology. 12.1 Cellular Chain Complexes Let h be an additive homology theory. Let X be obtained from A by attaching
/!.X; A/. The characteristic map of the n-cells via.ˆ; '/ W cell e is denoted by.ˆe; 'e/. The index e distinguishes different copies. e2E.Dn e ; S n1 e ` (12.1.1) Proposition. The induced map ˆn D hˆe i W L e h.Dn e ; S n1 e /! h.X; A/ is an isomorphism. e.Dn Proof. By (10.4.6), ˆ W h Now apply the additivity isomorphism and compose it with ˆ. ` e ; S n1 L e / e h.Dn! h.X; A/ is an isomorphism. e ; S n1 e e ; S n1 e / Š h e.Dn ` The isomorphism inverse to ˆn is obtained as follows. Given z 2 hk.X; A/. We use the inclusion pe W.X; A/.X; X X e/ and the relative homeomorphism ˆe W.Dn; S n1/!.X; X X e/. Let ze 2 hk.Dn / denote the image of z under e ; S n1 e z 2 hk.X; A/ pe! hk.X; X X e/ ˆe hk.Dn e ; S n1 e / 3 ze: 12.1. Cellular Chain Complexes 301 Then z 7!.ze j e 2 E/ is inverse to ˆn. Let X be a CW-complex. The boundary operator @ W hkC1.X nC1; X n/! hk.X n; X n1/ of the triple.X nC1; X n; X n1/ is transformed via the isomorphisms (12.1.1) into a matrix of linear maps m.e; f / W hkC1.DnC1 f ; S n f /! hk.Dn e ; S n1 e / for each pair.f; e/ of an.n C 1/-cell f and an n-cell e (as always in linear algebra). Let e;f be the composition 'f! X n qe! X n=.X n X e/ ˆ
e Dn=S n1: S n f If we compose e;f with an h-equivalence n W Dn=S n1! S n, then ne;f has as a self-map of S n a degree d.e; f /. We call d.e; f / the incidence number of the pair.f; e/ of cells. The case n D 0 is special, so let us consider it separately. Note that D0=S 1 is the point D0 D f0g together with a disjoint base point fg. Let 0 be given by 0.0/ D C1 and 0./ D 1. We have two0-cells 'f.˙1/ D e˙ (they could coincide). With these conventions d.f; e˙/ D ˙1. In the following considerations we use different notation @, @0, @00 for the bound- ary operators. (12.1.2) Proposition. The diagram hkC1.DnC1 f ; S n f / m.e;f / hk.Dn e ; S n1 e / @00 Qhk.S n f / p e;f Qhk.Dn e =S n1 e / is commutative. Proof. Consider the diagram hkC1.X nC1; X n/ ˆf hkC1.DnC1; S n/ @0 @00 Qhk.X n/ 'f Qhk.S n/ j hk.X n; X n1/ pe hk.X n; X n X e/ p Qhk.X n=X n1/ p Qhk.X n=X n X e/ ˆe ˆe hk.Dn; S n1/ p Qhk.Dn=S n1/: Given x 2 hkC1.DnC1; S n/. Then pm.e; f /x is, by deﬁnition of m.e; f /, the image of x in Qhk.Dn=S n1/. Now use the commutativity of the diagram. 302 Chapter 12. Cellular Homology (12.1.3) Corollary. Let W hk.Dn; S n1/! h
kC1.DnC1; S n/ be a suspension isomorphism. Then m.e; f / ı is the multiplication by d.e; f /, provided the relation @00 ı D n ı p holds. We now write the isomorphism (12.1.1) in a different form. We use an iterated suspension isomorphism n W hkn! hk.Dn; S n1/ in each summand. Let Cn.X/ denote the free abelian group on the n-cells of.X; A/. Elements in Cn.X/ ˝Z hkn will be written as ﬁnite formal sums e e ˝ ue where ue 2 hkn; the elements in Cn.X/ ˝ hkn are called cellular n-chains with coefﬁcients in hkn. We thus have constructed an isomorphism P n W Cn.X/ ˝Z hkn! hk.X; A/; e ˆe The matrix of incidence numbers provides us with the Z-linear map e e ˝ ue 7! n.ue/: P P M.n/ W CnC1.X/! Cn.X/; f 7! P e d.e; f /e: The sum is ﬁnite: d.e; f / can only be non-zero if the image of 'f intersects e (property (W3) of a Whitehead complex). From the preceding discussion we obtain: (12.1.4) Proposition. Suppose and are chosen such that the relation (12.1.3) holds. Then the diagram hkC1.X nC1; X n/ @ hk.X n; X n1/ nC1 n CnC1.X/ ˝ hkn M.n/˝id Cn.X/ ˝ hkn is commutative. The composition of the boundary operators (belonging to the appropriate triples) hmC1.X nC1; X n/ @! hm.X n; X n1/ @! hm1.X n1; X n2/ is zero, because the part hm.X n/! hm.X n; X n1/! hm
1.X n1/ of the exact sequence of the pair.X n; X n1/ is “contained” in this composition. We set hn;k.X/ D hnCk.X n; X n1/. Thus the groups.hn;k.X/ j n 2 Z/ together with the boundary operators just considered form a chain complex h;k.X/. (12.1.5) Proposition. The product M.n 1/M.n/ of two adjacent incidence matrices is zero. The cellular chain groups Cn.X/ together with the homomorphisms M.n/ W C.n/! C.n 1/ form a chain complex C.X/. This chain complex is called the cellular chain complex of X. Proof. The relation M.n 1/M.n/ D 0 follows from (12.1.4) applied to the chain complex H;0.X/ obtained from singular homology with coefﬁcients in Z. 12.1. Cellular Chain Complexes 303 The cellular chain complex has its algebraically deﬁned homology groups. In the next section we prove that in the case of an ordinary homology theory the algebraic homology groups of the cellular chain complex are naturally isomorphic to the homology groups of the theory. We should point out that the algebraic homology groups of the chain complexes h;k.X/ only depend on the space and the coefﬁcients of the homology theory, so are essentially independent of the theory. Nevertheless, they can be used to obtain further information about general homology theories – this is the topic of the so-called spectral sequences [130]. The deﬁnition of incidence numbers uses characteristic maps and a homotopy equivalence. These data are not part of the structure of a CW-complex so that the incidence numbers are not completely determined by the CW-complex. The choice of a characteristic map determines, as one says, an orientation of the cell. If ˆ; ‰ W.Dn; S n1/!.X n; X n1/ are two characteristic maps of a cell e, then ‰1ˆ W Dn=S n1! X n=X n X e Dn=S n1 is a homeomorphism and hence has degree ˙1. One concludes that the incidence numbers are deﬁ
ned up to sign by the CW-complex. (12.1.6) Proposition. A cellular map f W X! Y induces a chain map with components f W hm.X n; X n1/! hm.Y n; Y n1/. Homotopic cellular maps induce chain homotopic maps. Proof. The ﬁrst assertion is clear. Let f; g W X! Y be cellular maps and let'W X I! Y; f'g be a homotopy between them. By the cellular approximation theorem we can assume that'is cellular, i.e., '..X I /n/ Y n. Note that.X I /n D X n @I [ X n1 I. We deﬁne a chain homotopy as the composition sn W hm.X n; X n1/! hmC1..X n; X n1/.I; @I // '! hmC1.Y nC1; Y n/: In order to verify the relation @sn D g f sn1@ we apply (10.9.4) to.A; B; C / D.X n; X n1; X n2/ and compose with '. (12.1.7) Proposition. Let W k./! l./ be a natural transformation between additive homology theories such that induces isomorphisms of the coefﬁcient groups W kn.P / Š ln.P /, n 2 Z, P a point. Then is an isomorphism k.X/! l.X/ for each CW-complex X. Proof. Since is compatible with the suspension isomorphism we see from (12.1.1) that W k.X n; X n1/ Š l.X n; X n1/. Now one uses the exact homology sequences and the Five Lemma to prove by induction on n that is an isomorphism for n-dimensional complexes. For the general case one uses (10.8.1). 304 Chapter 12. Cellular Homology Problems p 1 kxk2x; 2kxk2 1/ induces a homeomorphism n. Let be the 1. The map x 7!.2 suspension isomorphism (10.2.5). Then commutativity holds in (12.1.3). For the proof show! Dn=S n
1 n that S n! S n=Dn! S n, with the projection r which deletes the last coordinate, has degree 1. 2. Prove M.n 1/M.n/ D 0 without using homology by homotopy theoretic methods. C Š Dn =S n1 r 12.2 Cellular Homology equals Homology Let H./ D H.I G/ be an ordinary additive homology theory with coefﬁcients in G (not necessarily singular homology). The cellular chain complex C.X/ D C.XI G/ of a CW-complex X with respect to this theory has its algebraically deﬁned homology groups. It is a remarkable and important fact that these algebraic homology groups are naturally isomorphic to the homology groups of the space X. This result says that the homology groups are computable from the combinatorial data (the incidence matrices) of the cellular complex. (12.2.1) Theorem. The n-th homology group of the cellular chain complex C.X/ is naturally isomorphic to Hn.X/. Proof. We show that the isomorphism is induced by the correspondence Hn.X n; X n1/ Hn.X n/! Hn.X/: We divide the proof into several steps. (1) A basic input is Hk.X n; X n1/ D 0 for k 6D 0; this follows from our determination of the cellular chain groups in (12.1.1) and the dimension axiom. (2) Hk.X n/ D 0 for k > n. Proof by induction on n. The result is clear for X 0 by the dimension axiom. Let k > n C 1. We have the exact sequence Hk.X n/! Hk.X nC1/! Hk.X nC1; X n/. The ﬁrst group is zero by induction, the third by (1). (3) Since Hn1.X n2/ D 0, the map Hn1.X n1/! Hn1.X n1; X n2/ is injective. Hence the cycle group Zn of the cellular chain complex is the kernel of @ W Hn.X n; X n1/! Hn1.X n1/. (4) The exact sequence 0!
Hn.X n/! Hn.X n; X n1/! Hn1.X n1/ induces an isomorphism.b/ W Hn.X n/ Š Zn. (5) Hk.X; X n/ D 0 for k n. One shows by induction on t that the groups Hk.X nCt ; X n/ are zero for t 0 and k n. We know that for an additive theory the canonical map colimt Hk.X nCt ; X n/! Hk.X; X n/ is an isomorphism (see (10.8.1) and (10.8.4)). For singular homology one can also use that a singular chain has compact support and that a compact subset of X is contained in some skeleton X m. 12.2. Cellular Homology equals Homology 305 (6) The map Hn.X nC1/! Hn.X/ is an isomorphism. This follows from the exact sequence of the pair.X; X nC1/ and (5). (7) The diagram HnC1.X nC1; X n/ @ Zn Zn=Bn D.b/ Š.a/ Š HnC1.X nC1; X n/ @ Hn.X n/ Hn.X nC1/ +’’’’’’’’’’’’ Š 0 0 Hn.X/ shows us that we have an induced isomorphism.a/ (Five Lemma). (12.2.2) Corollary. Suppose X has a ﬁnite number of n-cells; then Hn.XI Z/ is a ﬁnitely generated abelian group. Let X be n-dimensional; then Hk.XI G/ D 0 for k > n. (12.2.3) Example (Real projective space). The diagram S i1 Di p RP i1 ˆ RP i 1 kxk2 is a pushout. The incidence map with with attaching map ˆ W x 7! Œx; the homomorphism i1 of Problem 1 in the previous section is computed to be S i1! S i1,.y; t/ 7!.2ty;
C t. We can again orient the 2-cells such that d2 is multiplication by 1 C t. If we continue in this manner, we see that dk D 1t for k odd, and dk D 1Ct for k even. The homology module Hn.S n/ D Z."n/ carries the t-action "n D.1/nC1, the degree of the antipodal map. One can, of course, determine the boundary operator by a computation of degrees. We leave this as an exercise. Similar results hold Þ for S 1. Let X and Y be CW-complexes. The product inherits a cell decomposition. The cross product induces an isomorphism L kClDn Hk.X k; X k1/ ˝ Hl.Y l ; Y l1/! Hn..X Y /n;.X Y /n1/: With a careful choice of cell orientations these isomorphisms combine to an isomorphism C.X/ ˝ C.Y / Š C.X Y / of cellular chain complexes. 12.3 Simplicial Complexes We describe the classical combinatorial deﬁnition of homology groups of polyhedra. These groups are isomorphic to the singular groups for this class of spaces. The combinatorial homology groups of a ﬁnite polyhedron are ﬁnitely generated abelian groups and they are zero above the dimension of the polyhedron. This ﬁnite generation is not at all clear from the deﬁnition of the singular groups. 12.3. Simplicial Complexes 307 Recall that a simplicial complex K D.E; S/ consists of a set E of vertices and a set S of ﬁnite subsets of E. A set s 2 S with q C 1 elements is called a q-simplex of K. We require the following axioms: (1) A one-point subset of E is a simplex in S. (2) s 2 S and ; 6D t s imply t 2 S. An ordering of a p-simplex is a bijection f0; 1; : : : ; pg!s. An ordering of K is a partial order on E which induces a total ordering on each simplex. We write s D hv0; : : : ; vp
i, if the vertices of s satisfy v0 < v1 < < vp in the given partial ordering. Let Cp.K/ denote the free abelian group with basis the set of p-simplices. Its elements are called the simplicial p-chains of K. Now ﬁx an ordering of K and deﬁne a boundary operator @ W Cp.K/! Cp1.K/; hv0; : : : ; vp i 7! P p iD0.1/i hv0; : : : ; bvi ; : : : ; vp i: The symbol bvi means that this vi is to be omitted from the string of vertices. The boundary relation @@ holds (we set Cp.K/ D 0 for p 1). We denote the p-th homology group of this chain complex by Hp.K/. This is the classical combinatorial homology group. A simplicial complex K has a geometric realization jKj. An ordered simplex s D hv0; : : : ; vp i has an associated singular simplex ˆs W p! jKj;.t0; : : : ; tp/ 7! P tj vj : We extend s 7! ˆs by linearity to a homomorphism p W Cp.K/! Sp.jKj/. The boundary operators are arranged so that D.p/ is a chain map. (12.3.1) Theorem. induces isomorphisms Rp W Hp.K/ Š Hp.jKj/. Proof. We write X D jKj. Let S.p/ be the set of p-simplices. The characteristic maps ˆs W.p; @p/!.X p; X p1/ yield an isomorphism (12.1.1), L ˆp W s2S.p/ Hp.p s ; @p s /! Hp.X p; X p1/: The identity of p represents a generator p of Hp.p; @p/. Let xs be its image. Then.xs j s 2 S.p// is a Z-basis of Hs.X p; X p1/. If we express under ˆs x 2 Hp.X p; X p1
/ in terms of this basis, x D s nsxs, then ns is determined by the image nsp of x under P x 2 Hp.X p; X p1/! Hp.X p; X p X es/ ˆs Hp.p; @p/ 3 nsp: Here es is the open simplex which belongs to s. Let s.i/ denote the i-th face of p and xs.i/ 2 Hp1.X p1; X p2/ the corresponding basis element. We claim P p iD0.1/i xs.i/. It is clear for geometric reasons that the expression of @xs @xs D 308 Chapter 12. Cellular Homology in terms of the basis.xt j t 2 S.p 1// can have a non-zero coefﬁcient only for the xs.i/. The coefﬁcient of xs.i/ is seen from the commutative diagram Hp.p; @p/ ˆs Hp.X p; X p1/ @ @ Hp1.@p; @p X s.i/ı/ ˆs Hp1.X p1; X p2/.d p i / Hp1.p1; @p1/ ˆs.i/ H p1.X p1; X p1 X es.i//: Note ˆsd p i D ˆs.i/. The left column sends p to.1/i p1. We have constructed so far an isomorphism of C.K/ with the cellular chain complex C.jKj/ of jKj. Let Pp W Hp.C.K//! Hp.C.jKj// be the induced isomorphism. Let Qp W Hp.C.jKj//! Hp.jKj/ be the isomorphism in the proof of (12.2.1). Tracing through the deﬁnitions one veriﬁes Rp D QpPp. Hence Rp is the composition of two isomorphisms. An interesting consequence of (12.3.1) is that W C.K/! S.jKj/ is a chain equivalence. Thus, for a ﬁnite complex K, the singular complex
/ D 0 holds. For the zero-module M we have.M / D 0 since there exists an exact sequence 0! M! M! M! 0. We consider only categories which contain with a module also its submodules and its quotient modules as well as all exact sequences between its objects. Let C W 0! Ck @k! Ck1!! C1 @1! C0 @0! 0 be a chain complex in this category. Then its homology groups Hi.C/ are also contained in this category. (12.4.1) Proposition. Let be an additive invariant for M. Then for each chain complex C in M as above the following equality holds: P k iD0.1/i.Ci / D P k iD0.1/i.Hi.C//: Proof. Induction on the length k of C. We set Hi D Hi.C/, Bi D Im @iC1, Zi D Ker @i. For k D 1 there exist, by deﬁnition of homology groups, exact sequences 0! B0! C0! H0! 0; 0! H1! C1! B0! 0: We apply the additivity (1) to both sequences and thereby obtain.H0/.H1/ D.C0/.C1/. For the induction step we consider the sequences C 0 W 0! Ck1!! C0! 0; 0! Hk! Ck! Bk1! 0; 0! Bk1! Zk1! Hk1! 0I the last two are exact and the ﬁrst one is a chain complex. The homology groups of the chain complex are, for k 2, Hi.C 0 / D Hi.C/; 0 i k 2; Hk1.C 0 / D Zk1: We apply the induction hypothesis to C 0 the desired result by eliminating.Bk1/ and.Zk1/. and (1) to the other sequences. We obtain 310 Chapter 12. Cellular Homology The relation of the combinatorial Euler characteristic to homology groups goes back to Henri Poincaré [150], [152]. The i-th Betti number, named after Enrico Betti [20], bi.X/ of X is the rank of Hi.XI Z/, i.e., the cardinality of a
basis of its free abelian part, or equivalently, the dimension of the Q-vector space Hi.XI Z/ ˝ Q Š Hi.XI Q/. The result of Poincaré says: (12.4.2) Theorem. For each ﬁnite CW-complex X the combinatorial Euler characteristic equals the homological Euler characteristic i0.1/i bi.X/. P Proof. For ﬁnitely generated abelian groups A 7! rank A is an additive invariant. We apply (12.4.1) to the cellular chain complex C.X/ of X and observe that rank Ci.X/ D fi.X/. If is an additive invariant for M and C a chain complex of ﬁnite length in i0.1/i.Hi.C// the Euler i0.1/i.Ci / D P P M, then we call.C/ D characteristic of C with respect to. (12.4.3) Proposition. Let 0 H 0 0 H0 H 00 0 H 0 1 H1 H 00 1 H 0 2 H2 be an exact sequence of modules in M which consists eventually of zero-modules. i0.1/i.Hi / and similarly for H 0 and H 00. Then Let.H/ D P /.H/ C.H 00 Proof. Apply (12.4.1) to the given exact sequence, considered as chain complex, and order the terms according to H, H 0, and H 00. / D 0:.H 0 One can deﬁne the Euler characteristic by homological methods for spaces which are not necessarily ﬁnite CW-complexes. There are several possibilities depending on the homology theory being used. Let R be a principal ideal domain. We call.X; A/ of ﬁnite R-type if the groups Hi.X; AI R/ are ﬁnitely generated R-modules and only ﬁnitely many of them are non-zero. In that case we have the associated homological Euler characteristic P.X; AI R/ D i0.1/i rankRHi.X; AI R/: (12.4.4) Proposition. If.X; A/ is of ﬁnite Z-type, then it is of �
�nite R-type and the equality.X; AI Z/ D.X; AI R/ holds. Proof. If.X; A/ is of ﬁnite Z-type, then the singular complex S.X; A/ is chain equivalent to a chain complex D of ﬁnitely generated free abelian groups with only ﬁnitely many of the Dn non-zero (see (11.6.4)). Therefore P P.X; AI R/ D D i.1/i rankRHi.D ˝ R/ D i.1/i rankZ.Di / D.X; AI Z/; P i.1/i rankR.Di ˝ R/ by (12.4.1), and some elementary algebra. 12.5. Euler Characteristic of Surfaces 311 Proposition (12.4.3) has the following consequence. Suppose two of the spaces A, X,.X; A/ are of ﬁnite R-type. Then the third is of ﬁnite R-type and the additivity relation.AI R/ C.X; AI R/ D.XI R/ holds. Let A0; A1 be subspaces of X with MV-sequence, then.A0I R/ C.A1I R/ D.A0 [ A1I R/ C.A0 \ A1I R/ provided the spaces involved are of ﬁnite R-type. Similarly in the relative case. Let.X; A/ and.Y; B/ be of ﬁnite R-type. Then the Künneth formula is used to show that the product is of ﬁnite R-type and the product formula.X; AI R/.Y; BI R/ D..X; A/.Y; B/I R/ holds. These relations should be clear for ﬁnite CW-complex and the combinatorial Euler characteristic by counting cells. For the more general case of Lefschetz invariants and ﬁxed point indices see [51], [52], [109], [116]. 12.5 Euler Characteristic of Surfaces We report about the classical classiﬁcation of surfaces and relate this to the Euler characteristic. For details of the combinatorial or differentiable class
iﬁcation see e.g., [167], [80], [123]. See also the chapter about manifolds. Let F1 and F2 be connected surfaces. The connected sum F1#F2 of these surfaces is obtained as follows. Let Dj Fj be homeomorphic to the disk D2 with boundary Sj. In the topological sum F1 X Dı 2 we identify x 2 S1 with '.x/ 2 S2 via a homeomorphism'W S1! S2. The additivity of the Euler characteristic is used to show 1 C F2 X Dı.F1/ 1 C.F2/ 1 D.F1#F2/; i.e., the assignment F 7!.F / 2 is additive with respect to the connected sum. Let mF denote the m-fold connected sum of F with itself. We have the standard surfaces sphere S 2, torus T, and projective plane P. The Euler characteristics are.S 2/ D 2;.mT / D 2 2m;.nP / D 2 n: If F is a compact surface with k boundary components, then we can attach k disks D2 along the components in order to obtain a closed surface F. By additivity.F / D.F / C k. Connected surfaces Fj with the same number of boundary components are homeomorphic if and only if the associated surfaces F j are homeomorphic. 312 Chapter 12. Cellular Homology (12.5.1) Theorem. A closed connected surface is homeomorphic to exactly one of the surfaces S 2, mT with m 1, nP with n 1. The nP are the non-orientable surfaces. The homeomorphism type of a closed orientable surface is determined by the orientation behaviour and the Euler characteristic. The homeomorphism type of a compact connected surface with boundary is determined by the orientation behaviour, the Euler characteristic and the number of boundary components. The sphere has genus 0, mT has genus m and nP has genus n. 12.5.2 Platonic solids. A convex polyhedron is called regular if each vertex is the end point of the same number of edges, say m, and each 2-dimensional face has the same number of boundary edges, say n. If E is the number of vertices, K the number of edges and F the number of 2-faces, then mE D 2
K and nF D 2K. We insert this into the Euler relation E C F D K C 2, divide by 2K, and obtain : We have m 3, n 3. The equation has only the solutions which are displayed in the next table. E F m n K solid 4 4 3 3 6 tetrahedron 8 6 3 12 4 octahedron 6 8 4 12 3 cube 5 30 dodecahedron 20 12 3 5 12 20 3 30 icosahedron 12.5.3 Lines in the projective plane. Let G1; : : : ; Gn be lines in the projective plane P. We consider the resulting cells decomposition of P. Let tr be the number of points which are incident with r lines. We have the Euler characteristic relation f0 f1 C f2 D 1 where fi is the number of i-cells. Thus f0 D t2 C t3 C. From an r-fold intersection point there start 2r edges. The sum over the vertices yields f1 D 2t2 C 3t3 C 4t4 C. Let pn denote the numbers of n-gons, then sps. We insert these relations into the Euler characteristic ps; f2 D relation and obtain P 2f1 D P P P r2.3 r/tr C s2.3 s/ps D 3f0 f1 C 3f2 2f1 D 3: We now assume that not all lines are incident with a single point; then we do not have 2-gons. From 2f1 3f2 and then f1 3.f0 1/ we conclude P t2 3 C r4.r 3/tr : Thus there always exist at least three double points. 12.5. Euler Characteristic of Surfaces 313 Þ (12.5.4) Proposition. Let X be a Hausdorff space and p W X! Y a local homeomorphism onto a connected space. Then p is a covering with ﬁnitely many leaves if and only if p is proper. Proof. (1) Suppose p is proper. For n 2 N let Yn D fy 2 Y j n jp1.y/jg. We show that Yn is open and closed. Since Y is connected, either Yn D ; or Yn D Y. The inclusion Yn YnC1 shows that there is a largest n such
that Yn D Y and YnC1 D ;. Hence the ﬁbres have the cardinality n. Let p1.y/ D fx1; : : : ; xmg. Since X is a Hausdorff space and p a local homeomorphism, there exist open pairwise disjoint sets Ui 3 xi which are mapped homeomorphically under p onto the same open set V 3 y. Hence each ﬁbre p1.z/, z 2 V has at least cardinality n. If y 2 Yn, then m n and hence z 2 Yn, i.e., V Yn. This shows that Yn is open. Let p1.y/ D fx1; : : : xt g, t < n, i.e., y 62 Yn. Let again the Ui 3 xi be open with homeomorphic image V 2 Y. Since p is closed, being a proper map, the set C D.X X.U1 [ [ Ut // is closed in Y. This set does not contain y. Hence Y X C D W is an open neighbourhood of y and p1.W / D p1.Y / X p1p.X X.U1 [ [ Ut // U1 [ [ Ut : This shows jp1.z/j t for each z 2 W, and the complement of Yn is seen to be open. We now know that all ﬁbres of p have the same cardinality, and since p is a local homeomorphism it must be a covering. (2) Suppose conversely that p is an n-fold covering. We have to show that p is closed. A projection pr W B F! B with a ﬁnite discrete set F as ﬁbre is closed. Now we use (1.5.4). A continuous map p W X! Y between surfaces is called a ramiﬁed covering if for each x 2 X there exist centered charts.U; '; U 0/ about x and.V; ; V 0/ about y D p.x/ with p.U / V such that '1 W '.U / D U 0! C; z 7! zn with n 2 N. We call n 1 the ramiﬁcation index of p at x. In the case that n D 1 we say that p is unram
iﬁed at x and for n > 1 we call x a ramiﬁcation point. (12.5.5) Proposition. Let p W X! Y be a ramiﬁed covering between compact connected surfaces. Let V be the image under p of the ramiﬁcation points. Then p W X X p1.p/! Y X V is a covering with ﬁnitely many leaves. 314 Chapter 12. Cellular Homology Proof. For each y 2 Y the set p1.y/ X is closed and hence compact. The pre-images p1.y/ in a ramiﬁed covering are always discrete, hence ﬁnite. The set V is also discrete and hence ﬁnite. The map p is, as a continuous map between compact Hausdorff spaces, closed. Thus we have shown that the map in question is proper. Now we use (12.5.4). (12.5.6) Proposition (Riemann–Hurwitz). Let p W X! Y be a ramiﬁed covering between compact connected surfaces. Let P1; : : : ; Pr 2 X be the ramiﬁcation points with ramiﬁcation index v.Pj /. Let n be the cardinality of the general ﬁbre. Then for the Euler characteristics the relation P.X/ D n.Y / r j D1 v.Pj / holds. Proof. Let Q1; : : : ; Qs be the images of the ramiﬁcation points. Choose pairwise disjoint neighbourhoods D1; : : : ; Ds Y where Dj is homeomorphic to a disk. Then S p W X0 D X X s j D1 p1.Dı j /! Y X j D1 Dı j D Y0 is an n-fold covering (see (12.5.4)). We use the relation.X0/ D n.Y0/ for n-fold coverings. If C is a ﬁnite set in a surface X, then.X X C / D.X/ jC j. Thus we see S P.X/ s j D1 jp1.Qj /j Dn..Y / s/: Moreover P s j D1 jp1.Qj /j D
ns P r iD1 v.Pi /; P n.j / since for p1.Qj / D fP.j /g the relation tD1.v.P j t / C 1/ D n holds. An interesting application of the Riemann–Hurwitz formula concerns actions of ﬁnite groups on surfaces. Let F be a compact connected orientable surface and G F! F an effective orientation preserving action. We assume that this action has the following properties: (1) The isotropy group Gx of each point x 2 F is cyclic. (2) There exist about each point x a centered chart'W U! R2 such that U is Gx-invariant and'transforms the Gx-action on U into a representation on R2, i.e., a suitable generator of Gx acts on R2 as rotation about an angle 2=jGxj. In this case the orbit map p W F! F=G is a ramiﬁed covering, F=G is orientable, and the ramiﬁcation points are the points with non-trivial isotropy group. One can show that each orientation preserving action has the properties (1) and (2). Examples are actions of a ﬁnite group G SO.3/ on S 2 by matrix multiplication and of a ﬁnite group G GL2.Z/ on the torus T D R2=Z2 by matrix multiplication. 12.5. Euler Characteristic of Surfaces 315 The ramiﬁed coverings which arise as orbit maps from an action are of a more special type. If x 2 F is a ramiﬁcation point, then so is each point in p1px, and these points have the same ramiﬁcation index, since points in the same orbit have conjugate isotropy groups. Let C1; : : : ; Cr be the orbits with non-trivial isotropy group and let nj denote the order of the isotropy group of x 2 Ci ; hence jCi jni D jGj. The Riemann–Hurwitz formula yields in this case: 12.5.7 Riemann–Hurwitz formula for group actions..F / D jGj.F=G/ P r j D1.1 1=nj / : In the
case of a free action r D 0 and there is no sum. Þ 12.5.8 Actions on spheres. Let F D S 2 and jGj 2. Since.S 2/ D 2 we see that.F=G/ 0 is not compatible with 12.5.7, hence.F=G/ D 2 and the orbit space is again a sphere. We also see that r 3 and r D 0; 1 are not possible. For r D 2 we have 2=jGj D1=n 1 C 1=n2, 2 D jC1 C jC2. Hence there are two ﬁxed points (example: rotation about an axis). For r D 3 one veriﬁes that 1 C 2 jGj D 1 n1 C 1 n2 C 1 n3 has the solutions (for n1 n2 n3) displayed in the next table. n1 jGj=2 3 4 5 n2 n3 2 2 2 3 2 3 2 3 jGj jGj 12 24 60 Examples are the standard actions of subgroups of SO.3/, namely D2n (dihedral), A4 (tetrahedral), S4 (octahedral), A5 (icosahedral). Up to homeomorphism there Þ are no other actions. 12.5.9 Action on the torus. Let F D T D S 1 S 1 be the torus,.F / D 0. The Riemann–Hurwitz formula shows that for r 1 we must have.F=G/ D 2. The cases r 5 and r D 1; 2 are impossible. For r D 4 we must have n1 D n2 D n3 D n4 D 2 and G D Z=2. For r D 3 the solutions of 12.5.7 are displayed in a table. n1 n2 n3 3 3 3 2 6 3 4 4 2 316 Chapter 12. Cellular Homology Consider the matrices in SL2.Z The cyclic groups generated by A; A2; A3 realize cases 2 and 1 of the table and the Þ case r D 4 above. The matrix B realizes case 3 of the table. Problems 1. Let G act effectively on a closed orientable surface F of genus 2 preserving the orientation. Then jGj divides 48 or 10. There exist groups of orders 48 and 10 which act on a surface of genus 2
. The group of order 48 has a central subgroup C of order 2 and G=C is the octahedral group S4 acting on the sphere F=C. Study the solutions of 12.5.7 and determine the groups which can act on F. Use covering space theory and work towards a topological classiﬁcation of the actions. 2. The nicest models of surfaces are of course Riemann surfaces. Here we assume known the construction of a compact Riemann surface from a polynomial equation in two variables. The equation y2 D f.x/ with 2g C2 branch points deﬁnes a surface of genus g. Such curves are called hyper-elliptic (g 2). It is known that all surfaces of genus 2 are hyper-elliptic. A hyper-elliptic surface always has the hyper-elliptic involution I.x; y/ D.x; y/. Here are some examples. Let us write e.a/ D exp.2 ia/. (1) y2 D x.x2 1/.x2 4/ has a Z=4-action generated by A.x; y/ D.x; e.1=4/y/. Note A2 D I. (2) y2 D.x3 1/=.x3 8/ has a Z=3-action generated by B.x; y/ D.e.1=3/x; y/. Since B commutes with I, we obtain an action of Z=6. (3) y2 D.x3 1/=.x3 C 1/ has a Z=6-action generated by C.x; y/ D.e.1=6/x; 1=y/. Since C commutes with I, we obtain an action of Z=6 ˚ Z=2. It has an action of Z=4 generated by D.x; y/ D.1=x; e.1=4/y/. Note D2 D I. The actions C and D do not commute, in fact CD D DC 5. Thus we obtain an action of a group F which is an extension 1! Z=2! F! D12! 1 where Z=2 is generated by I and D12 denotes the dihedral group of order 12. (4) y2 D x.x4 1/ has the following automorphisms (
see also [121, p. 94]) G.x; y/ D.e.1=4/x; e.1=8/y/; G8 D id; G4 D I; H.x; y/ D.1=x; e.1=4/y=x3/; H 4 D id; H 2 D I; K.x; y/ D..x i/=.x C i/; 2 2e.1=8/y=.x C i//; K3 D I: p The elements G; H; K generate a group of order 48. If we quotient out the central hyperelliptic involution we obtain the octahedral group of order 24 acting on the sphere. Thus there also exists an action of a group of order 24 such that the quotient by the hyper-elliptic involution is the tetrahedral group (and not the dihedral group D12, as in (3)). (5) y2 D x5 1 has an action of Z=5 generated by J.x; y/ D.e.1=5/x; y/. It commutes with I and gives an action of Z=10. 12.5. Euler Characteristic of Surfaces 317 3. By an analysis of 12.5.7 one can show that, for an effective action of G on a closed orientable surface of genus g 2, the inequality jGj 84.g 1/ holds. There exists a group of order 168 which acts on a surface of genus 3 [63, p. 242]. Chapter 13 Partitions of Unity in Homotopy Theory Partitions of unity and numerable coverings of a space are useful tools in order to obtain global results from local data. (A related concept is that of a paracompact space.) We present some notions about partitions of unity in the context of point-set topology. Then we use them to show that, roughly, local homotopy equivalences are global ones and a map is a ﬁbration if it is locally a ﬁbration (see the precise statements in (13.3.1) and (13.4.1)). We apply the results to prove a theorem of Dold about ﬁbrewise homotopy equivalences (see (13.3.4)). Conceptually, partitions of unity are used to relate the homotopy colim

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