=== PAGE 110 === The Idea of Homology 101 generated by this 3 cellC, and we define a boundary homomorphism @3:C3!C2 sendingCtoA−Bsince the cycle A−Bshould be viewed as the boundary of C in the same way that the 1 dimensional cycle a−bis the boundary of A. Now we have a sequence of three boundary homomorphisms C3@3------------!C2@2------------!C1@1------------!C0and the quotientH2„X4…ƒKer@2=Im@3has become trivial. Also H3„X4…ƒKer@3ƒ0. The groupH1„X4…is the same as H1„X3…, namelyZZ, so this is the only nontrivial homology group of X4. It is clear what the general pattern of the examples is. For a cell complex Xone has chain groups Cn„X…which are free abelian groups with basis the ncells ofX, and there are boundary homomorphisms @n:Cn„X…!Cn−1„X…, in terms of which one defines the homology group Hn„X…ƒKer@n=Im@n‚1. The major difficulty is how to define @nin general. For nƒ1 this is easy: The boundary of an oriented edge is the vertex at its head minus the vertex at its tail. The next case nƒ2 is also not hard, at least for cells attached along cycles that are simply loops of edges, forthen the boundary of the cell is this cycle of edges, with the appropriate signs takingorientations into account. But for larger n, matters become more complicated. Even if one restricts attention to cell complexes formed from polyhedral cells with niceattaching maps, there is still the matter of orientations to sort out. The best solution to this problem seems to be to adopt an indirect approach. Arbitrary polyhedra can always be subdivided into special polyhedra called simplices(the triangle and the tetrahedron are the 2 dimensional and 3 dimensional instances) so there is no loss of generality, though initially there is some loss of efficiency, inrestricting attention entirely to simplices. For simplices there is no difficulty in defin-ing boundary maps or in handling orientations. So one obtains a homology theory,called simplicial homology, for cell complexes built from simplices. Still, this is arather restricted class of spaces, and the theory itself has a certain rigidity that makesit awkward to work with. The way around these obstacles is to step back from the geometry of spaces decomposed into simplices and to consider instead something which at first glanceseems wildly more complicated, the collection of all possible continuous maps ofsimplices into a given space X. These maps generate tremendously large chain groups C n„X…, but the quotients Hn„X…ƒKer@n=Im@n‚1, called singular homology groups, turn out to be much smaller, at least for reasonably nice spaces X. In particular, for spaces like those in the four examples above, the singular homology groups coincidewith the homology groups we computed from the cellular chains. And as we shallsee later in this chapter, singular homology allows one to define these nice cellularhomology groups for all cell complexes, and in particular to solve the problem ofdefining the boundary maps for cellular chains. === PAGE 111 === 102 Chapter 2 Homology The most important homology theory in algebraic topology, and the one we shall be studying almost exclusively, is called singular homology. Since the technical appa-ratus of singular homology is somewhat complicated, we will first introduce a moreprimitive version called simplicial homology in order to see how some of the apparatusworks in a simpler setting before beginning the general theory. The natural domain of definition for simplicial homology is a class of spaces we callÑ complexes, which are a mild generalization of the more classical notion of a simplicial complex. Historically, the modern definition of singular homology wasfirst given in [Eilenberg 1944], and Ñ complexes were introduced soon thereafter in [Eilenberg-Zilber 1950] where they were called semisimplicial complexes. Within afew years this term came to be applied to what Eilenberg and Zilber called completesemisimplicial complexes, and later there was yet another shift in terminology asthe latter objects came to be called simplicial sets. In theory this frees up the termsemisimplicial complex to have its original meaning, but to avoid potential confusionit seems best to introduce a new name, and the term Ñ complex has at least the virtue of brevity. ∆–Complexes The torus, the projective plane, and the Klein bottle can each be obtained from a square by identifying opposite edges in the way indicated by the arrows in the follow-ing figures: bb a ac vv v vU L bb aac wv w vU L bb aac vv v vU LRP2T : :¡¡K:¡¡ ¡¡¡ Cutting a square along a diagonal produces two triangles, so each of these surfaces can also be built from two triangles by identifying their edges in pairs. In similarfashion a polygon with any number of sides can be cut alongdiagonals into triangles, so in fact all closed surfaces can beconstructed from triangles by identifying edges. Thus we have bb aac cd da single building block, the triangle, from which all surfaces can be constructed. Using only triangles we could also construct alarge class of 2 dimensional spaces that are not surfaces in the strict sense, by allowing more than two edges to be identifiedtogether at a time. === PAGE 112 === Simplicial and Singular Homology Section 2.1 103 The idea of a Ñcomplex is to generalize constructions like these to any number of dimensions. The ndimensional analog of the triangle is the nsimplex . This is the smallest convex set in a Euclidean spaceR mcontainingn‚1 pointsv0;;vn that do not lie in a hyperplane of dimen- sion less than n, where by a hyperplane we mean the set of solutions of a system v v0v0v0 1v2v1 v1v2v3 v0 of linear equations. An equivalent condi- tion would be that the difference vectorsv 1−v0;;vn−v0are linearly independent. The points viare the vertices of the simplex, and the simplex itself is denoted †v0;;vn‡. For exam- ple, there is the standard nsimplex v v01v2 Ñnƒ „t0;;tn…2Rn‚1jjjjP itiƒ1 andti0 for alli/bracerightbig whose vertices are the unit vectors along the coordinate axes. For purposes of homology it will be important to keep track of the order of the vertices of a simplex, so ‘ nsimplex’ will really mean ‘ nsimplex with an ordering of its vertices.’ A by-product of ordering the vertices of a simplex †v0;;vn‡is that this determines orientations of the edges †vi;vj‡according to increasing sub- scripts, as shown in the two preceding figures. Specifying the ordering of the verticesalso determines a canonical linear homeomorphism from the standard n simplex Ñnonto any other nsimplex†v0;;vn‡, preserving the order of vertices, namely, „t0;;tn…,P itivi. The coefficients tiare the barycentric coordinates of the pointP itiviin†v0;;vn‡. If we delete one of the n‚1 vertices of an nsimplex†v0;;vn‡, then the remainingnvertices span an „n−1…simplex, called a face of†v0;;vn‡.W e adopt the following convention: The vertices of a face, or of any subsimplex spanned by a subset of the vertices, will always be ordered according to their order in the larger simplex. The union of all the faces of Ñnis the boundary ofÑn, written@Ñn. The open simplexÑnisÑn−@Ñn, the interior of Ñn. A∆complex structure on a space Xis a collection of maps  :Ñn!X, withn depending on the index , such that: (i) The restriction  jjÑnis injective, and each point of Xis in the image of exactly one such restriction  jjÑn. (ii) Each restriction of  to a face of Ñnis one of the maps  :Ñn−1!X. Here we are identifying the face of ÑnwithÑn−1by the canonical linear homeomorphism between them that preserves the ordering of the vertices. (iii) A setAXis open iff−1 „A…is open inÑnfor each . === PAGE 113 === 104 Chapter 2 Homology Among other things, this last condition rules out trivialities like regarding all the points ofXas individual vertices. The earlier decompositions of the torus, projective plane, and Klein bottle into two triangles, three edges, and one or two vertices defineÑ complex structures with a total of six  ’s for the torus and Klein bottle and seven for the projective plane. The orientations on the edges in the pictures are compatiblewith a unique ordering of the vertices of each simplex, and these orderings determinethe maps . A consequence of (iii) is that Xcan be built as a quotient space of a collection of disjoint simplices Ñn , one for each  :Ñn!X, the quotient space obtained by identifying each face of a Ñn with theÑn−1 corresponding to the restriction  of  to the face in question, as in condition (ii). One can think of building the quotient space inductively, starting with a discrete set of vertices, then attaching edges tothese to produce a graph, then attaching 2 simplices to the graph, and so on. From this viewpoint we see that the data specifying a Ñcomplex can be described purely combinatorially as collections of nsimplicesÑn for eachntogether with functions associating to each face of each nsimplexÑn an„n−1…simplexÑn−1 . More generally, Ñcomplexes can be built from collections of disjoint simplices by identifying various subsimplices spanned by subsets of the vertices, where the iden-tifications are performed using the canonical linear homeomorphisms that preservethe orderings of the vertices. The earlier Ñ complex structures on a torus, projective plane, or Klein bottle can be obtained in this way, by identifying pairs of edges oftwo 2 simplices. If one starts with a single 2 simplex and identifies all three edges to a single edge, preserving the orientations given by the ordering of the vertices,this produces a Ñ complex known as the ‘dunce cap.’ By contrast, if the three edges of a 2 simplex are identified preserving a cyclic orientation of the three edges, as in the first figure at the right, this does not produce a Ñcomplex structure, although if the 2 simplex is subdivided into three smaller 2 simplices about a central vertex, then one does obtain a Ñcomplex structure on the quotient space. Thinking of a ÑcomplexXas a quotient space of a collection of disjoint sim- plices, it is not hard to see that Xmust be a Hausdorff space. Condition (iii) then implies that each restriction  jjÑnis a homeomorphism onto its image, which is thus an open simplex in X. It follows from Proposition A.2 in the Appendix that these open simplices  „Ñn…are the cellsen of a CW complex structure on Xwith the ’s as characteristic maps. We will not need this fact at present, however. Simplicial Homology Our goal now is to define the simplicial homology groups of a ÑcomplexX. Let Ñn„X…be the free abelian group with basis the open nsimplicesen ofX. Elements === PAGE 114 === Simplicial and Singular Homology Section 2.1 105 ofÑn„X…, callednchains , can be written as finite formal sumsP n en with co- efficientsn 2Z. Equivalently, we could writeP n  where :Ñn!Xis the characteristic map of en , with image the closure of en as described above. Such a sumP n  can be thought of as a finite collection, or ‘chain,’ of nsimplices inX with integer multiplicities, the coefficients n . As one can see in the next figure, the boundary of the nsimplex†v0;;vn‡con- sists of the various „n−1…dimensional simplices †v0;;bvi;;vn‡, where the ‘hat’ symbolboverviindicates that this vertex is deleted from the sequence v0;;vn. In terms of chains, we might then wish to say that the boundary of †v0;;vn‡is the „n−1…chain formed by the sum of the faces †v0;;bvi;;vn‡. However, it turns out to be better to insert certain signs and instead let the boundary of †v0;;vn‡beP i„−1…i†v0;;bvi;;vn‡. Heuristically, the signs are inserted to take orientations into account, so that all the faces of a simplex are coherently oriented, as indicated inthe following figure: v0v1v2v3v v0 1v2v0v1+-@†v0;v1‡ƒ†v1‡−†v0‡ @†v0;v1;v2‡ƒ†v1;v2‡−†v0;v2‡‚†v0;v1‡ @†v0;v1;v2;v3‡ƒ†v1;v2;v3‡−†v0;v2;v3‡ ‚†v0;v1;v3‡−†v0;v1;v2‡ In the last case, the orientations of the two hidden faces are also counterclockwise when viewed from outside the 3 simplex. With this geometry in mind we define for a general ÑcomplexXaboundary homomorphism @n:Ñn„X…!Ñn−1„X…by specifying its values on basis elements: @n„ …ƒX i„−1…i jj†v0;;bvi;;vn‡ Note that the right side of this equation does indeed lie in Ñn−1„X…since each restric- tion jj†v0;;bvi;;vn‡is the characteristic map of an „n−1…simplex ofX. Lemma 2.1. The composition Ñn„X…@n------------!Ñn−1„X…@n−1------------------!Ñn−2„X…is zero. Proof: We have@n„…ƒP i„−1…ijj†v0;;bvi;;vn‡, and hence @n−1@n„…ƒX ji„−1…i„−1…j−1jj†v0;;bvi;;bvj;;vn‡ === PAGE 115 === 106 Chapter 2 Homology The latter two summations cancel since after switching iandjin the second sum, it becomes the negative of the first. tu The algebraic situation we have now is a sequence of homomorphisms of abelian groups -!Cn‚1@n‚1-----------------!Cn@n------------!Cn−1-!-!C1@1------------!C0@0------------!0 with@n@n‚1ƒ0 for eachn. Such a sequence is called a chain complex . Note that we have extended the sequence by a 0 at the right end, with @0ƒ0. From@n@n‚1ƒ0 it follows that Im @n‚1Ker@n, where Im and Ker denote image and kernel. So we can define the nthhomology group of the chain complex to be the quotient group HnƒKer@n=Im@n‚1. Elements of Ker @nare called cycles and elements of Im @n‚1 areboundaries . Elements of Hnare cosets of Im @n‚1, called homology classes . Two cycles representing the same homology class are said to be homologous . This means their difference is a boundary. Returning to the case that CnƒÑn„X…, the homology group Ker @n=Im@n‚1will be denotedHÑ n„X…and called the nthsimplicial homology group ofX. Example 2.2 .XƒS1, with one vertex vand one edge e. ThenÑ0„S1… ve andÑ1„S1…are bothZand the boundary map @1is zero since@eƒv−v. The groups Ñn„S1…are 0 forn2 since there are no simplices in these dimensions. Hence HÑ n„S1… Z fornƒ0;1 0 forn2 This is an illustration of the general fact that if the boundary maps in a chain complex are all zero, then the homology groups of the complex are isomorphic to the chaingroups themselves. Example 2.3 .XƒT, the torus with the Ñcomplex structure pictured earlier, having one vertex, three edges a,b, andc, and two 2 simplicesUandL. As in the previous example,@1ƒ0s oHÑ 0„T…Z. Since@2Uƒa‚b−cƒ@2Landfa;b;a‚b−cgis a basis for Ñ1„T…, it follows that HÑ 1„T…ZZwith basis the homology classes †a‡ and†b‡. Since there are no 3 simplices,HÑ 2„T…is equal to Ker @2, which is infinite cyclic generated by U−Lsince@„pU‚qL…ƒ„p‚q…„a‚b−c…ƒ0 only ifpƒ−q. Thus HÑ n„T…8 < :ZZ fornƒ1 Z fornƒ0;2 0 forn3 Example 2.4 .XƒRP2, as pictured earlier, with two vertices vandw, three edges a,b, andc, and two 2 simplicesUandL. Then Im@1is generated by w−v,s o HÑ 0„X…Zwith either vertex as a generator. Since @2Uƒ−a‚b‚cand@2Lƒa−b‚c, we see that@2is injective, so HÑ 2„X…ƒ0. Further, Ker @1ZZwith basisa−band c, and Im@2is an index-two subgroup of Ker @1since we can choose canda−b‚c === PAGE 116 === Simplicial and Singular Homology Section 2.1 107 as a basis for Ker @1anda−b‚cand 2cƒ„a−b‚c…‚„−a‚b‚c…as a basis for Im@2. ThusHÑ 1„X…Z2. Example 2.5 . We can obtain a Ñcomplex structure on Snby taking two copies of Ñn and identifying their boundaries via the identity map. Labeling these two nsimplices UandL, then it is obvious that Ker @nis infinite cyclic generated by U−L. Thus HÑ n„Sn…Zfor thisÑcomplex structure on Sn. Computing the other homology groups would be more difficult. Many similar examples could be worked out without much trouble, such as the other closed orientable and nonorientable surfaces. However, the calculations do tendto increase in complexity before long, particularly for higher-dimensional complexes. Some obvious general questions arise: Are the groups H Ñ n„X… independent of the choice of Ñcomplex structure on X? In other words, if two Ñcomplexes are homeomorphic, do they have isomorphic homology groups? More generally, do theyhave isomorphic homology groups if they are merely homotopy equivalent? To answersuch questions and to develop a general theory it is best to leave the rather rigidsimplicial realm and introduce the singular homology groups. These have the addedadvantage that they are defined for all spaces, not just Ñ complexes. At the end of this section, after some theory has been developed, we will show that simplicial andsingular homology groups coincide for Ñ complexes. Traditionally, simplicial homology is defined for simplicial complexes , which are theÑcomplexes whose simplices are uniquely determined by their vertices. This amounts to saying that each nsimplex hasn‚1 distinct vertices, and that no other nsimplex has this same set of vertices. Thus a simplicial complex can be described combinatorially as a set X0of vertices together with sets Xnofnsimplices, which are„n‚1…element subsets of X0. The only requirement is that each „k‚1…element subset of the vertices of an nsimplex inXnis aksimplex, inXk. From this combi- natorial data a ÑcomplexXcan be constructed, once we choose a partial ordering of the vertices X0that restricts to a linear ordering on the vertices of each simplex inXn. For example, we could just choose a linear ordering of all the vertices. This might perhaps involve invoking the Axiom of Choice for large vertex sets. An exercise at the end of this section is to show that every Ñcomplex can be subdivided to be a simplicial complex. In particular, every Ñcomplex is then homeo- morphic to a simplicial complex. Compared with simplicial complexes, Ñcomplexes have the advantage of simpler computations since fewer simplices are required. For example, to put a simplicialcomplex structure on the torus one needs at least 14 triangles, 21 edges, and 7 vertices,and forRP 2one needs at least 10 triangles, 15 edges, and 6 vertices. This would slow down calculations considerably! === PAGE 117 === 108 Chapter 2 Homology Singular Homology Asingularnsimplex in a spaceXis by definition just a map :Ñn!X. The word ‘singular’ is used here to express the idea that need not be a nice embedding but can have ‘singularities’ where its image does not look at all like a simplex. All thatis required is that be continuous. Let C n„X…be the free abelian group with basis the set of singular nsimplices inX. Elements of Cn„X…, callednchains , or more precisely singular nchains, are finite formal sumsP iniiforni2Zandi:Ñn!X. A boundary map @n:Cn„X…!Cn−1„X…is defined by the same formula as before: @n„…ƒX i„−1…ijj†v0;;bvi;;vn‡ Implicit in this formula is the canonical identification of †v0;;bvi;;vn‡with Ñn−1, preserving the ordering of vertices, so that jj†v0;;bvi;;vn‡is regarded as a mapÑn−1!X, that is, a singular „n−1…simplex. Often we write the boundary map @nfromCn„X…toCn−1„X…simply as@when this does not lead to serious ambiguities. The proof of Lemma 2.1 applies equally wellto singular simplices, showing that @ n@n‚1ƒ0 or more concisely @2ƒ0, so we can define the singular homology group Hn„X…ƒKer@n=Im@n‚1. It is evident from the definition that homeomorphic spaces have isomorphic sin- gular homology groups Hn, in contrast with the situation for HÑ n. On the other hand, since the groups Cn„X…are so large, the number of singular nsimplices inXusually being uncountable, it is not at all clear that for a ÑcomplexXwith finitely many sim- plices,Hn„X…should be finitely generated for all n, or thatHn„X…should be zero fornlarger than the dimension of X— two properties that are trivial for HÑ n„X…. Though singular homology looks so much more general than simplicial homology, it can actually be regarded as a special case of simplicial homology by means of thefollowing construction. For an arbitrary space X, define the singular complex S„X… to be theÑ complex with one nsimplexÑn for each singular nsimplex:Ñn!X, withÑn attached in the obvious way to the „n−1…simplices ofS„X… that are the restrictions of to the various „n−1…simplices in@Ñn. It is clear from the defini- tions thatHÑ n/parenleftbig S„X… is identical with Hn„X…for alln, and in this sense the singular homology group Hn„X…is a special case of a simplicial homology group. One can regardS„X… as aÑcomplex model for X, although it is usually an extremely large object compared to X. Cycles in singular homology are defined algebraically, but they can be given a somewhat more geometric interpretation in terms of maps from finite Ñcomplexes. To see this, note first that a singular nchaincan always be written in the formP i"iiwith"iƒ1, allowing repetitions of the singular nsimplicesi. Given such annchainƒP i"ii, when we compute @as a sum of singular „n−1…simplices with signs1, there may be some canceling pairs consisting of two identical singu- lar„n−1…simplices with opposite signs. Choosing a maximal collection of such === PAGE 118 === Simplicial and Singular Homology Section 2.1 109 canceling pairs, construct an ndimensional ÑcomplexKfrom a disjoint union of nsimplicesÑn i, one for each i, by identifying the pairs of „n−1…dimensional faces corresponding to the chosen canceling pairs. The i’s then induce a map K!X.I f is a cycle, all the „n−1…simplices ofKcome from canceling pairs, hence are faces of exactly two nsimplices ofK. ThusKis a manifold, locally homeomorphic toRn, except at a subcomplex of dimension at most n−2. All thensimplices of Kcan be coherently oriented by taking the signs of the i’s into account, so Kis actually an oriented manifold away from its nonmanifold points. A closer inspectionshows thatK is also a manifold near points in the interiors of „n−2…simplices, so the nonmanifold points of Kin fact have dimension at most n−3. However, near the interiors of „n−3…simplices it can very well happen that Kis not a manifold. In particular, elements of H1„X…are represented by collections of oriented loops inX, and elements of H2„X…are represented by maps of closed oriented surfaces intoX. With a bit more work it can be shown that an oriented 1 cycle‘ S1 !Xis zero inH1„X…iff it extends to a map of an oriented surface into X, and there is an analogous statement for 2 cycles. In the early days of homology theory it may have been believed, or at least hoped, that this close connection with manifolds continuedin all higher dimensions, but this has turned out not to be the case. There is a sortof homology theory built from manifolds, called bordism , but it is quite a bit more complicated than the homology theory we are studying here. After these preliminary remarks let us begin to see what can be proved about singular homology. Proposition 2.6. Corresponding to the decomposition of a space Xinto its path- componentsX there is an isomorphism of Hn„X…with the direct sumL Hn„X …. Proof: Since a singular simplex always has path-connected image, Cn„X…splits as the direct sum of its subgroups Cn„X …. The boundary maps @npreserve this direct sum decomposition, taking Cn„X …toCn−1„X …, so Ker@nand Im@n‚1split similarly as direct sums, hence the homology groups also split, Hn„X…L Hn„X ….tu Proposition 2.7. IfXis nonempty and path-connected, then H0„X…Z. Hence for any spaceX,H0„X…is a direct sum of Z’s, one for each path-component of X. Proof: By definition, H0„X…ƒC0„X…= Im@1since@0ƒ0. Define a homomorphism ":C0„X…!Zby"/parenleftbigP inii ƒP ini. This is obviously surjective if Xis nonempty. The claim is that Ker "ƒIm@1ifXis path-connected, and hence "induces an iso- morphismH0„X…Z. To verify the claim, observe first that Im @1Ker"since for a singular 1 simplex :Ñ1!Xwe have"@1„…ƒ"/parenleftbig jj†v1‡−jj†v0‡ ƒ1−1ƒ0. For the reverse inclusion Ker "Im@1, suppose"/parenleftbigP inii ƒ0, soP iniƒ0. Thei’s are singular 0simplices, which are simply points of X. Choose a path i:I!Xfrom a basepoint === PAGE 119 === 110 Chapter 2 Homology x0toi„v0…and let0be the singular 0 simplex with image x0. We can view i as a singular 1 simplex, a map i:†v0;v1‡!X, and then we have @iƒi−0. Hence@/parenleftbigP inii ƒP inii−P ini0ƒP iniisinceP iniƒ0. ThusP iniiis a boundary, which shows that Ker "Im@1. tu Proposition 2.8. IfXis a point, then Hn„X…ƒ0forn>0andH0„X…Z. Proof: In this case there is a unique singular nsimplexnfor eachn, and@„n…ƒP i„−1…in−1, a sum ofn‚1 terms, which is therefore 0 for nodd andn−1forn even,n”0. Thus we have the chain complex -!Z------------!Z0------------!Z------------!Z0------------!Z-!0 with boundary maps alternately isomorphisms and trivial maps, except at the last Z. The homology groups of this complex are trivial except for H0Z. tu It is often very convenient to have a slightly modified version of homology for which a point has trivial homology groups in all dimensions, including zero. This isdone by defining the reduced homology groups eH n„X…to be the homology groups of the augmented chain complex -!C2„X…@2------------!C1„X…@1------------!C0„X…"------------!Z-!0 where"/parenleftbigP inii ƒP inias in the proof of Proposition 2.7. Here we had better requireXto be nonempty, to avoid having a nontrivial homology group in dimension −1. Since"@1ƒ0,"vanishes on Im @1and hence induces a map H0„X…!Zwith kerneleH0„X…,s oH0„X…eH0„X…Z. ObviouslyHn„X…eHn„X…forn>0. Formally, one can think of the extra Zin the augmented chain complex as gener- ated by the unique map †;‡!Xwhere†;‡is the empty simplex, with no vertices. The augmentation map "is then the usual boundary map since @†v0‡ƒ†bv0‡ƒ†;‡. Readers who know about the fundamental group 1„X… may wish to make a detour here to look at x2.A where it is shown that H1„X…is the abelianization of 1„X…wheneverXis path-connected. This result will not be needed elsewhere in the chapter, however. Homotopy Invariance The first substantial result we will prove about singular homology is that ho- motopy equivalent spaces have isomorphic homology groups. This will be done byshowing that a map f:X !Yinduces a homomorphism f:Hn„X…!Hn„Y…for each n, and thatfis an isomorphism if fis a homotopy equivalence. For a mapf:X!Y, an induced homomorphism f]:Cn„X…!Cn„Y…is defined by composing each singular nsimplex:Ñn!Xwithfto get a singular nsimplex === PAGE 120 === Simplicial and Singular Homology Section 2.1 111 f]„…ƒf:Ñn!Y, then extending f]linearly viaf]/parenleftbigP inii ƒP inif]„i…ƒP inifi. The mapsf]:Cn„X…!Cn„Y…satisfyf]@ƒ@f]since f]@„…ƒf]/parenleftbigP i„−1…ijj†v0;;bvi;;vn‡ ƒP i„−1…ifjj†v0;;bvi;;vn‡ƒ@f]„… Thus we have a diagram¡¡¡¡¡!¡¡¡¡¡!¡¡¡¡!¡¡¡¡¡!¡¡¡¡¡!¡¡¡¡! ]f]f]f- XCn 1()+¡¡¡¡¡!XCn 1() XCn()@@... ... ¡¡¡¡!¡¡¡¡¡!¡¡¡¡¡!¡¡¡¡! - YCn 1()+ YCn 1() YCn()@@... ... such that in each square the composition f]@equals the composition @f]. A diagram of maps with the property that any two compositions of maps starting at one point inthe diagram and ending at another are equal is called a commutative diagram . In the present case commutativity of the diagram is equivalent to the commutativity relationf ]@ƒ@f], but commutative diagrams can contain commutative triangles, pentagons, etc., as well as commutative squares. The fact that the maps f]:Cn„X…!Cn„Y…satisfyf]@ƒ@f]is also expressed by saying that the f]’s define a chain map from the singular chain complex of X to that ofY. The relation f]@ƒ@f]implies thatf]takes cycles to cycles since @ ƒ0 implies@„f] …ƒf]„@ …ƒ0. Also,f]takes boundaries to boundaries sincef]„@ …ƒ@„f] …. Hencef]induces a homomorphism f:Hn„X…!Hn„Y….A n algebraic statement of what we have just proved is: Proposition 2.9. A chain map between chain complexes induces homomorphisms between the homology groups of the two complexes. tu Two basic properties of induced homomorphisms which are important in spite of being rather trivial are: (i)„fg…ƒfgfor a composed mapping Xg-----!Yf-----!Z. This follows from associativity of compositions Ñn-----!Xg-----!Yf-----!Z. (ii)11ƒ11 where 11 denotes the identity map of a space or a group. Less trivially, we have: Theorem 2.10. If two mapsf;g:X!Yare homotopic, then they induce the same homomorphism fƒg:Hn„X…!Hn„Y…. In view of the formal properties „fg…ƒfgand 11ƒ11, this immediately implies: Corollary 2.11. The mapsf:Hn„X…!Hn„Y…induced by a homotopy equivalence f:X!Yare isomorphisms for all n. tu For example, if Xis contractible then eHn„X…ƒ0 for alln. === PAGE 121 === 112 Chapter 2 Homology Proof of 2.10 : The essential ingredient is a procedure for subdividing the product ÑnIinto„n‚1…simplices. The figure shows the cases nƒ1;2. InÑnI, letÑnf0gƒ †v0;;vn‡andÑnf1gƒ†w0;;wn‡, whereviand wihave the same image under the projection ÑnI!Ñn. Thensimplex†v0;;vi;wi‚1;;wn‡is the graph ofvw w v0 v01 v1v2 1 w1w20 w0the linear function ’i:Ñn!Idefined in barycentric co- ordinates by ’i„t0;;tn…ƒti‚1‚‚tnsince the vertices of this simplex †v0;;vi;wi‚1;;wn‡are on the graph of’iand the simplex projects homeomorphi- cally ontoÑnunder the projection ÑnI!Ñn. The graph of’ilies below the graph of ’i−1since’i’i−1, and the region between these two graphs is the simplex†v 0;;vi;wi;;wn‡, a true„n‚1…simplex since wi is not on the graph of ’iand hence is not in the nsimplex†v0;;vi;wi‚1;;wn‡. From the string of inequalities 0 ƒ’n’n−1’0’−1ƒ1 we deduce that ÑnIis the union of the „n‚1…simplices†v0;;vi;wi;;wn‡, each intersecting the next in an nsimplex face. Given a homotopy F:XI!Yfromftog, we can define prism operators P:Cn„X…!Cn‚1„Y…by P„…ƒX i„−1…iF„11…jj†v0;;vi;wi;;wn‡ for:Ñn!X, whereF„11…is the composition ÑnI!XI!Y. We will show that these prism operators satisfy the basic relation @Pƒg]−f]−P@ Geometrically, the left side of this equation represents the boundary of the prism, and the three terms on the right side represent the top Ñnf1g, the bottom Ñnf0g, and the sides@ÑnIof the prism. To prove the relation we calculate @P„…ƒX ji„−1…i„−1…jF„11…jj†v0;;bvj;;vi;wi;;wn‡ ‚X ji„−1…i„−1…j‚1F„11…jj†v0;;vi;wi;;cwj;;wn‡ The terms with iƒjin the two sums cancel except for F„11…jj†bv0;w0;;wn‡, which isgƒg]„…, and−F„11…jj†v0;;vn;cwn‡, which is−fƒ−f]„…. The terms with i”jare exactly−P@„… since P@„…ƒX ij„−1…i−1„−1…jF„11…jj†v0;;bvj;;vi;wi;;wn‡ === PAGE 122 === Simplicial and Singular Homology Section 2.1 113 Now we can finish the proof of the theorem. If 2Cn„X…is a cycle, then we haveg]„ …−f]„ …ƒ@P„ …‚P@„ …ƒ@P„ … since@ ƒ0. Thusg]„ …−f]„ …is a boundary, so g]„ …andf]„ …determine the same homology class, which means thatgequalsfon the homology class of . tu The relationship @P‚P@ƒg]−f]is expressed by saying Pis achain homotopy between the chain maps f]andg]. We have just shown: Proposition 2.12. Chain-homotopic chain maps induce the same homomorphism on homology. tu There are also induced homomorphisms f:eHn„X…!eHn„Y…for reduced homol- ogy groups since f]"ƒ"f]. The properties of induced homomorphisms we proved above hold equally well in the setting of reduced homology, with the same proofs. Exact Sequences and Excision It would be nice if there was always a simple relationship between the homology groups of a space X, a subspaceA, and the quotient space X=A. For then one could hope to understand the homology groups of spaces such as CW complexes that can bebuilt inductively from successively more complicated subspaces. Perhaps the simplestpossible relationship would be if H n„X… containedHn„A…as a subgroup and the quotient group Hn„X…=Hn„A…was isomorphic to Hn„X=A… . While this does hold in some cases, if it held in general then homology theory would collapse totally sinceevery spaceXcan be embedded as a subspace of a space with trivial homology groups, namely the cone CXƒ„XI…=„Xf0g…, which is contractible. It turns out that this overly simple model does not have to be modified too much to get a relationship that is valid in fair generality. The novel feature of the actualrelationship is that it involves the groups H n„X…,Hn„A…, andHn„X=A… for all values ofnsimultaneously. In practice this is not as bad as it might sound, and in addition it has the pleasant side effect of sometimes allowing higher-dimensional homologygroups to be computed in terms of lower-dimensional groups, which may already beknown by induction for example. In order to formulate the relationship we are looking for, we need an algebraic definition which is central to algebraic topology. A sequence of homomorphisms  ---------!An‚1 n‚1-------------------------!An n------------------!An−1---------! is said to be exact if Ker nƒIm n‚1for eachn. The inclusions Im n‚1Ker n are equivalent to n n‚1ƒ0, so the sequence is a chain complex, and the opposite inclusions Ker nIm n‚1say that the homology groups of this chain complex are trivial. === PAGE 123 === 114 Chapter 2 Homology A number of basic algebraic concepts can be expressed in terms of exact se- quences, for example: (i) 0-!A -----!Bis exact iff Ker ƒ0, i.e., is injective. (ii)A -----!B-!0 is exact iff Im ƒB, i.e., is surjective. (iii) 0-!A -----!B-!0 is exact iff is an isomorphism, by (i) and (ii). (iv) 0-!A -----!B -----!C-!0 is exact iff is injective, is surjective, and Ker ƒ Im ,s o induces an isomorphism CB=Im . This can be written CB=A if we think of as an inclusion of Aas a subgroup of B. An exact sequence 0!A!B!C!0 as in (iv) is called a short exact sequence . Exact sequences provide the right tool to relate the homology groups of a space, a subspace, and the associated quotient space: Theorem 2.13. IfXis a space and Ais a nonempty closed subspace that is a defor- mation retract of some neighborhood in X, then there is an exact sequence -----!eHn„A…i------------!eHn„X…j------------!eHn„X=A…@-----!eHn−1„A…i------------!eHn−1„X…-----! -----!eH0„X=A…-----!0 whereiis the inclusion A>Xandjis the quotient map X!X=A. The map@will be constructed in the course of the proof. The idea is that an elementx2eHn„X=A… can be represented by a chain inXwith@ a cycle inA whose homology class is @x2eHn−1„A…. Pairs of spaces „X;A… satisfying the hypothesis of the theorem will be called good pairs . For example, if Xis a CW complex and Ais a nonempty subcomplex, then„X;A… is a good pair by Proposition A.5 in the Appendix. Corollary 2.14. eHn„Sn…ZandeHi„Sn…ƒ0fori”n. Proof: Forn>0 take„X;A…ƒ„Dn;Sn−1…soX=AƒSn. The terms eHi„Dn…in the long exact sequence for this pair are zero since Dnis contractible. Exactness of the sequence then implies that the maps eHi„Sn…@-----!eHi−1„Sn−1…are isomorphisms for i>0 and that eH0„Sn…ƒ0. The result now follows by induction on n, starting with the case ofS0where the result holds by Propositions 2.6 and 2.8. tu As an application of this calculation we have the following classical theorem of Brouwer, the 2 dimensional case of which was proved in x1.1. Corollary 2.15. @Dnis not a retract of Dn. Hence every map f:Dn!Dnhas a fixed point. Proof:I fr:Dn!@Dnis a retraction, then riƒ11 fori:@Dn!Dnthe inclusion map. The composition eHn−1„@Dn…i------------!eHn−1„Dn…r------------!eHn−1„@Dn…is then the identity map === PAGE 124 === Simplicial and Singular Homology Section 2.1 115 oneHn−1„@Dn…Z. Butiandrare both 0 since eHn−1„Dn…ƒ0, and we have a contradiction. The statement about fixed points follows as in Theorem 1.9. tu The derivation of the exact sequence of homology groups for a good pair „X;A… will be rather a long story. We will in fact derive a more general exact sequence whichholds for arbitrary pairs „X;A… , but with the homology groups of the quotient space X=A replaced by relative homology groups , denotedH n„X;A… . These turn out to be quite useful for many other purposes as well. Relative Homology Groups It sometimes happens that by ignoring a certain amount of data or structure one obtains a simpler, more flexible theory which, almost paradoxically, can give resultsnot readily obtainable in the original setting. A familiar instance of this is arithmeticmodn, where one ignores multiples of n. Relative homology is another example. In this case what one ignores is all singular chains in a subspace of the given space. Relative homology groups are defined in the following way. Given a space Xand a subspaceAX, letC n„X;A… be the quotient group Cn„X…=Cn„A…. Thus chains in Aare trivial inCn„X;A… . Since the boundary map @:Cn„X…!Cn−1„X…takesCn„A… toCn−1„A…, it induces a quotient boundary map @:Cn„X;A…!Cn−1„X;A… . Lettingn vary, we have a sequence of boundary maps -!Cn„X;A…@------------!Cn−1„X;A…-! The relation@2ƒ0 holds for these boundary maps since it holds before passing to quotient groups. So we have a chain complex, and the homology groups Ker @=Im@ of this chain complex are by definition the relative homology groups Hn„X;A… .B y considering the definition of the relative boundary map we see: Elements ofHn„X;A… are represented by relative cycles :nchains 2Cn„X… such that@ 2Cn−1„A…. A relative cycle is trivial inHn„X;A… iff it is a relative boundary : ƒ@ ‚γ for some 2Cn‚1„X…andγ2Cn„A…. These properties make precise the intuitive idea that Hn„X;A… is ‘homology of X moduloA.’ The quotient Cn„X…=Cn„A…could also be viewed as a subgroup of Cn„X…, the subgroup with basis the singular nsimplices:Ñn!Xwhose image is not con- tained inA. However, the boundary map does not take this subgroup of Cn„X…to the corresponding subgroup of Cn−1„X…, so it is usually better to regard Cn„X;A… as a quotient rather than a subgroup of Cn„X…. Our goal now is to show that the relative homology groups Hn„X;A… for any pair „X;A… fit into a long exact sequence -!Hn„A…-!Hn„X…-!Hn„X;A…-!Hn−1„A…-!Hn−1„X…-! -!H0„X;A…-!0 === PAGE 125 === 116 Chapter 2 Homology This will be entirely a matter of algebra. To start the process, consider the diagram¡¡¡¡¡!¡¡¡¡¡!¡¡¡¡¡¡!¡¡¡¡¡¡!¡¡¡¡¡¡!¡¡¡¡¡¡!i j i jAAC 0 ()¡¡¡¡¡!XC 0 n n n () XC() @@ @, ¡¡¡¡¡¡!¡¡¡¡¡¡!¡¡¡¡¡¡!¡¡¡¡¡¡! - AAC 0 () XC 0 n 1 -n 1 -n 1 () XC() , whereiis inclusion and jis the quotient map. The diagram is commutative by the def- inition of the boundary maps. Letting nvary, and drawing these short exact sequences vertically rather than horizontally, wehave a large commutative diagram ofthe form shown at the right, where thecolumns are exact and the rows arechain complexes which we denote A,¡¡¡¡¡!¡¡¡¡¡!¡¡¡¡¡¡¡!¡¡¡¡¡¡¡!¡¡¡¡¡¡¡!¡¡¡¡¡¡¡! jj j- Bn 1+¡¡¡¡¡!Bn 1 Bn@@... ... ¡¡¡¡¡!¡¡¡¡¡!¡¡¡¡¡¡¡!¡¡¡¡¡¡¡!¡¡¡¡¡¡¡!¡¡¡¡¡¡¡!- Cn 1+¡¡¡¡¡! ¡¡¡¡¡!¡¡¡¡¡!¡¡¡¡¡! Cn 1 Cn@@... ... ¡¡¡¡¡!¡¡¡¡¡!¡¡¡¡¡¡¡!¡¡¡¡¡¡¡!¡¡¡¡¡¡¡!¡¡¡¡¡¡¡! ii i- An 1+¡¡¡¡¡!An 1 A00 0 0 0 0n@@... ... B, andC. Such a diagram is called a short exact sequence of chain com-plexes . We will show that when we pass to homology groups, this shortexact sequence of chain complexes stretches out into a long exact sequence of homol-ogy groups  -!Hn„A…i------------!Hn„B…j------------!Hn„C…@-----!Hn−1„A…i------------!Hn−1„B…-! whereHn„A…denotes the homology group Ker @=Im@atAnin the chain complex A, andHn„B…andHn„C…are defined similarly. The commutativity of the squares in the short exact sequence of chain complexes means thatiandjare chain maps. These therefore induce maps iandjon homology. To define the boundary map @:Hn„C…!Hn−1„A…, letc2Cnbe a cycle. Sincejis onto,cƒj„b… for someb2Bn. The element @b2Bn−1 is in Kerjsincej„@b…ƒ@j„b…ƒ@cƒ0. So@bƒi„a… for some a2An−1since KerjƒImi. Note that@aƒ0 sincei„@a…ƒ @i„a…ƒ@@bƒ0 andiis injective. We define @:Hn„C…!Hn−1„A… by sending the homology class of cto the homology class of a, @†c‡ƒ†a‡. This is well-defined since:CnBBnA -n 1-n 1 ¡¡! ¡¡¡! ¡¡¡!¡¡¡!a bb c@ @-¡¡¡!-¡¡¡!- i j The elementais uniquely determined by @bsinceiis injective. A different choice b0forbwould havej„b0…ƒj„b…,s ob0−bis in KerjƒImi. Thusb0−bƒi„a0…for somea0, henceb0ƒb‚i„a0…. The effect of replacing b byb‚i„a0…is to changeato the homologous element a‚@a0sincei„a‚@a0…ƒ i„a…‚i„@a0…ƒ@b‚@i„a0…ƒ@„b‚i„a0……. A different choice of cwithin its homology class would have the form c‚@c0. Sincec0ƒj„b0…for someb0, we then have c‚@c0ƒc‚@j„b0…ƒc‚j„@b0…ƒ j„b‚@b0…,s obis replaced by b‚@b0, which leaves @band therefore also a unchanged. === PAGE 126 === Simplicial and Singular Homology Section 2.1 117 The map@:Hn„C…!Hn−1„A…is a homomorphism since if @†c1‡ƒ†a1‡and@†c2‡ƒ †a2‡via elementsb1andb2as above, then j„b1‚b2…ƒj„b1…‚j„b2…ƒc1‚c2and i„a1‚a2…ƒi„a1…‚i„a2…ƒ@b1‚@b2ƒ@„b1‚b2…,s o@„†c1‡‚†c2‡…ƒ†a1‡‚†a2‡. Theorem 2.16. The sequence of homology groups -!Hn„A…i------------!Hn„B…j------------!Hn„C…@-----!Hn−1„A…i------------!Hn−1„B…-! is exact. Proof: There are six things to verify: ImiKerj. This is immediate since jiƒ0 impliesjiƒ0. ImjKer@. We have@jƒ0 since in this case @bƒ0 in the definition of @. Im@Keri. Herei@ƒ0 sincei@takes†c‡to†@b‡ƒ0. KerjImi. A homology class in Ker jis represented by a cycle b2Bnwith j„b… a boundary, so j„b…ƒ@c0for somec02Cn‚1. Sincejis surjective,c0ƒj„b0… for someb02Bn‚1. We havej„b−@b0…ƒj„b…−j„@b0…ƒj„b…−@j„b0…ƒ0 since @j„b0…ƒ@c0ƒj„b….S ob−@b0ƒi„a… for somea2An. Thisais a cycle since i„@a…ƒ@i„a…ƒ@„b−@b0…ƒ@bƒ0 andiis injective. Thus i†a‡ƒ†b−@b0‡ƒ†b‡, showing that imaps onto Ker j. Ker@Imj. In the notation used in the definition of @,i fcrepresents a homology class in Ker@, thenaƒ@a0for somea02An. The element b−i„a0…is a cycle since@„b−i„a0……ƒ@b−@i„a0…ƒ@b−i„@a0…ƒ@b−i„a…ƒ0. Andj„b−i„a0……ƒ j„b…−ji„a0…ƒj„b…ƒc,s ojmaps†b−i„a0…‡to†c‡. KeriIm@. Given a cycle a2An−1such thati„a…ƒ@bfor someb2Bn, then j„b… is a cycle since @j„b…ƒj„@b…ƒji„a…ƒ0, and@takes†j„b…‡ to†a‡.tu This theorem represents the beginnings of the subject of homological algebra. The method of proof is sometimes called diagram chasing . Returning to topology, the preceding algebraic theorem yields a long exact se- quence of homology groups: -!Hn„A…i------------!Hn„X…j------------!Hn„X;A…@-----!Hn−1„A…i------------!Hn−1„X…-! -!H0„X;A…-!0 The boundary map @:Hn„X;A…!Hn−1„A…has a very simple description: If a class † ‡2Hn„X;A… is represented by a relative cycle , then@† ‡ is the class of the cycle@ inHn−1„A…. This is immediate from the algebraic definition of the boundary homomorphism in the long exact sequence of homology groups associated to a shortexact sequence of chain complexes. This long exact sequence makes precise the idea that the groups H n„X;A… mea- sure the difference between the groups Hn„X…andHn„A…. In particular, exactness === PAGE 127 === 118 Chapter 2 Homology implies that if Hn„X;A…ƒ0 for alln, then the inclusion A>Xinduces isomorphisms Hn„A…Hn„X…for alln, by the remark (iii) following the definition of exactness. The converse is also true according to an exercise at the end of this section. There is a completely analogous long exact sequence of reduced homology groups for a pair„X;A… withA”;. This comes from applying the preceding algebraic ma- chinery to the short exact sequence of chain complexes formed by the short exact se-quences 0 !Cn„A…!Cn„X…!Cn„X;A…!0 in nonnegative dimensions, augmented by the short exact sequence 0 -!Z11-----!Z-!0-!0 in dimension −1. In particular this means that eHn„X;A… is the same as Hn„X;A… for alln, whenA”;. Example 2.17 . In the long exact sequence of reduced homology groups for the pair „Dn;@Dn…, the mapsHi„Dn;@Dn…@-----!eHi−1„Sn−1…are isomorphisms for all i>0 since the remaining terms eHi„Dn…are zero for all i. Thus we obtain the calculation Hi„Dn;@Dn… Zforiƒn 0 otherwise Example 2.18 . Applying the long exact sequence of reduced homology groups to a pair„X;x0…withx02Xyields isomorphisms Hn„X;x0…eHn„X…for allnsince eHn„x0…ƒ0 for alln. There are induced homomorphisms for relative homology just as there are in the nonrelative, or ‘absolute,’ case. A map f:X!Ywithf„A…B, or more concisely f:„X;A…!„Y;B… , induces homomorphisms f]:Cn„X;A…!Cn„Y;B… since the chain mapf]:Cn„X…!Cn„Y…takesCn„A…toCn„B…, so we get a well-defined map on quo- tients,f]:Cn„X;A…!Cn„Y;B… . The relation f]@ƒ@f]holds for relative chains since it holds for absolute chains. By Proposition 2.9 we then have induced homomorphismsf :Hn„X;A…!Hn„Y;B… . Proposition 2.19. If two mapsf;g:„X;A…!„Y;B… are homotopic through maps of pairs„X;A…!„Y;B… , thenfƒg:Hn„X;A…!Hn„Y;B… . Proof: The prism operator Pfrom the proof of Theorem 2.10 takes Cn„A…toCn‚1„B…, hence induces a relative prism operator P:Cn„X;A…!Cn‚1„Y;B… . Since we are just passing to quotient groups, the formula @P‚P@ƒg]−f]remains valid. Thus the mapsf]andg]on relative chain groups are chain homotopic, and hence they induce the same homomorphism on relative homology groups. tu An easy generalization of the long exact sequence of a pair „X;A… is the long exact sequence of a triple „X;A;B… , whereBAX: -!Hn„A;B…-!Hn„X;B…-!Hn„X;A…-!Hn−1„A;B…-! This is the long exact sequence of homology groups associated to the short exact sequence of chain complexes formed by the short exact sequences 0-!Cn„A;B…-!Cn„X;B…-!Cn„X;A…-!0 === PAGE 128 === Simplicial and Singular Homology Section 2.1 119 For example, taking Bto be a point, the long exact sequence of the triple „X;A;B… becomes the long exact sequence of reduced homology for the pair „X;A… . Excision A fundamental property of relative homology groups is given by the following Excision Theorem , describing when the relative groups Hn„X;A… are unaffected by deleting, or excising, a subset ZA. Theorem 2.20. Given subspaces ZAXsuch that the closure of Zis contained in the interior of A, then the inclusion „X−Z;A−Z…>„X;A… induces isomor- phismsHn„X−Z;A−Z…!Hn„X;A… for alln. Equivalently, for subspaces A;BX whose interiors cover X, the inclusion „B;A\B…>„X;A… induces isomorphisms Hn„B;A\B…!Hn„X;A… for alln. AZ XThe translation between the two versions is obtained by settingBƒX−ZandZƒX−B. ThenA\BƒA−Zand the condition clZintAis equivalent to XƒintA[intBsince X−intBƒclZ. The proof of the excision theorem will involve a rather lengthy technical detour involving a construction known as barycentric subdivision, which allows homologygroups to be computed using small singular simplices. In a metric space ‘smallness’can be defined in terms of diameters, but for general spaces it will be defined in termsof covers. For a spaceX, let UƒfUjgbe a collection of subspaces of Xwhose interiors form an open cover of X, and letCU n„X…be the subgroup of Cn„X…consisting of chainsP iniisuch that each ihas image contained in some set in the cover U. The boundary map @:Cn„X…!Cn−1„X…takesCU n„X…toCU n−1„X…, so the groups CU n„X… form a chain complex. We denote the homology groups of this chain complex byH U n„X…. Proposition 2.21. The inclusion :CU n„X…>Cn„X… is a chain homotopy equiva- lence, that is, there is a chain map :Cn„X…!CU n„X…such thatandare chain homotopic to the identity. Hence induces isomorphisms HU n„X…Hn„X…for alln. Proof: The barycentric subdivision process will be performed at four levels, beginning with the most geometric and becoming increasingly algebraic. (1)Barycentric Subdivision of Simplices . The points of a simplex †v0;;vn‡are the linear combinationsP itiviwithP itiƒ1 andti0 for eachi. The barycenter or ‘center of gravity’ of the simplex †v0;;vn‡is the pointbƒP itiviwhose barycen- tric coordinates tiare all equal, namely tiƒ1=„n‚1…for eachi. The barycentric subdivision of†v0;;vn‡is the decomposition of †v0;;vn‡into thensimplices †b;w0;;wn−1‡where, inductively, †w0;;wn−1‡is an„n−1…simplex in the === PAGE 129 === 120 Chapter 2 Homology barycentric subdivision of a face †v0;;bvi;;vn‡. The induction starts with the casenƒ0 when the barycentric subdivision of †v0‡is defined to be just †v0‡itself. The next two cases nƒ1;2 and part of the case nƒ3 are shown in the figure. It follows from theinductive definition that the ver- v v0vb bb0 1v2v1 v0v1v2 v3 tices of simplices in the barycen- tric subdivision of †v0;;vn‡ are exactly the barycenters of allthek dimensional faces †vi0;;vik‡of†v0;;vn‡for 0kn. Whenkƒ0 this gives the original vertices visince the barycenter of a 0 simplex is itself. The barycen- ter of†vi0;;vik‡has barycentric coordinates tiƒ1=„k‚1…foriƒi0;;ikand tiƒ0 otherwise. Thensimplices of the barycentric subdivision of Ñn, together with all their faces, do in fact form a Ñcomplex structure on Ñn, indeed a simplicial complex structure, though we shall not need to know this in what follows. A fact we will need is that the diameter of each simplex of the barycentric subdivi- sion of†v0;;vn‡is at mostn=„n‚1…times the diameter of †v0;;vn‡. Here the diameter of a simplex is by definition the maximum distance between any two of itspoints, and we are using the metric from the ambient Euclidean space R mcontaining †v0;;vn‡. The diameter of a simplex equals the maximum distance between any of its vertices because the distance between two points vandP itiviof†v0;;vn‡ satisfies the inequality v−P itivi ƒ P iti„v−vi… P itijv−vijP itimaxjv−vijƒmaxjv−vij To obtain the bound n=„n‚1…on the ratio of diameters, we therefore need to verify that the distance between any two vertices wjandwkof a simplex†w0;;wn‡of the barycentric subdivision of †v0;;vn‡is at mostn=„n‚1…times the diameter of †v0;;vn‡. If neitherwinorwjis the barycenter bof†v0;;vn‡, then these two points lie in a proper face of †v0;;vn‡and we are done by induction on n.S ow e may supposewj, say, is the barycenter b, and then by the previous displayed inequal- ity we may take wkto be a vertex vi. Letbibe the barycenter of †v0;;bvi;;vn‡, with all barycentric coordinates equal to 1 =nexcept fortiƒ0. Then we have bƒ1 n‚1vi‚n n‚1bi. Thebb vii sum of the two coefficients is 1, so blies on the line segment†vi;bi‡fromvitobi, and the distance from btoviisn=„n‚1…times the length of †vi;bi‡. Hence the distance from btoviis bounded byn=„n‚1…times the diameter of †v0;;vn‡. The significance of the factor n=„n‚1…is that by repeated barycentric subdivision we can produce simplices of arbitrarily small diameter since/parenleftbig n=„n‚1…rapproaches === PAGE 130 === Simplicial and Singular Homology Section 2.1 121 0a srgoes to infinity. It is important that the bound n=„n‚1…does not depend on the shape of the simplex since repeated barycentric subdivision produces simplicesof many different shapes. (2)Barycentric Subdivision of Linear Chains . The main part of the proof will be to construct a subdivision operator S:Cn„X…!Cn„X…and show this is chain homotopic to the identity map. First we will construct Sand the chain homotopy in a more restricted linear setting. For a convex set Yin some Euclidean space, the linear maps Ñn!Ygenerate a subgroup of Cn„Y…that we denote LCn„Y…, the linear chains . The boundary map @:Cn„Y…!Cn−1„Y…takesLCn„Y…toLCn−1„Y…, so the linear chains form a subcom- plex of the singular chain complex of Y. We can uniquely designate a linear map :Ñn!Yby†w0;;wn‡wherewiis the image under of theithvertex ofÑn. To avoid having to make exceptions for 0 simplices it will be convenient to augment the complexLC„Y… by settingLC−1„Y…ƒZgenerated by the empty simplex †;‡, with@†w0‡ƒ†;‡for all 0 simplices†w0‡. Each pointb2Ydetermines a homomorphism b:LCn„Y…!LCn‚1„Y…defined on basis elements by b„†w0;;wn‡…ƒ†b;w0;;wn‡. Geometrically, the homo- morphismbcan be regarded as a cone operator, sending a linear chain to the cone having the linear chain as the base of the cone and the point bas the tip of the cone. Applying the usual formula for @, we obtain the relation @b„†w0;;wn‡…ƒ †w0;;wn‡−b„@†w0;;wn‡…. By linearity it follows that @b„ …ƒ −b„@ … for all 2LCn„Y…. This expresses algebraically the geometric fact that the boundary of a cone consists of its base together with the cone on the boundary of its base. Therelation@b„ …ƒ −b„@ … can be rewritten as @b‚b@ƒ 11,sobis a chain homotopy between the identity map and the zero map on the augmented chain complex LC„Y… . Now we define a subdivision homomorphism S:LCn„Y…!LCn„Y…by induction onn. Let:Ñn!Ybe a generator of LCn„Y… and letbbe the image of the barycenter of Ñnunder. Then the inductive formula for SisS„…ƒb„S@… whereb:LCn−1„Y…!LCn„Y…is the cone operator defined in the preceding para- graph. The induction starts with S„†;‡…ƒ†;‡,s oSis the identity on LC−1„Y…. It is also the identity on LC0„Y…, since when nƒ0 the formula for Sbecomes S„†w0‡…ƒw0„S@†w0‡…ƒw0„S„†;‡……ƒw0„†;‡…ƒ†w0‡. Whenis an embed- ding, with image a genuine nsimplex†w0;;wn‡, thenS„… is the sum of the nsimplices in the barycentric subdivision of †w0;;wn‡, with certain signs that could be computed explicitly. This is apparent by comparing the inductive definitionofSwith the inductive definition of the barycentric subdivision of a simplex. Let us check that the maps Ssatisfy@SƒS@, and hence give a chain map from the chain complex LC„Y… to itself. Since Sƒ 11o nLC0„Y…andLC−1„Y…, we certainly have@SƒS@onLC0„Y…. The result for larger nis given by the following calculation, in which we omit some parentheses to unclutter the formulas: === PAGE 131 === 122 Chapter 2 Homology @Sƒ@/parenleftbig b„S@… ƒS@−b„@S@… since@b‚b@ƒ11 ƒS@−b„S@@… by induction on n ƒS@ since@@ƒ0 We next build a chain homotopy T:LCn„Y…!LCn‚1„Y…betweenSand the iden- tity, fitting into a diagram¡¡¡¡¡!¡¡¡¡¡!¡¡¡¡¡¡¡!¡¡¡¡¡¡¡!¡¡¡¡¡¡¡!¡¡¡¡¡¡¡!¡¡¡¡¡¡¡!¡¡¡¡¡¡¡! ST ¡¡¡¡¡¡¡!T ¡¡¡¡¡¡¡!T 0SS- YLC2() YLC1() YLC0() YLC 01() ¡¡¡¡¡!S¡¡¡¡¡!... ¡¡¡¡¡¡¡!¡¡¡¡¡¡¡!¡¡¡¡¡¡¡!¡¡¡¡¡¡¡!¡¡¡¡¡¡¡! - YLC2() YLC1() YLC0() YLC 01() ...11 11 We defineTonLCn„Y…inductively by setting Tƒ0 fornƒ−1 and letting Tƒ b„−T@… forn0. The geometric motivation for this formula is an inductively defined subdivision of ÑnIobtained by joining all simplices in Ñnf0g[@ÑnI to the barycenter of Ñnf1g, as indicated in the figure in the case nƒ2. WhatT actually does is take the image of this sub-division under the projection Ñ nI!Ñn. The chain homotopy formula @T‚T@ƒ11−Sis trivial onLC−1„Y…whereTƒ0 andSƒ11. Verifying the formula on LCn„Y…withn0 is done by the calculation @Tƒ@/parenleftbig b„−T@… ƒ−T@−b/parenleftbig @„−T@… since@bƒ11−b@ ƒ−T@−b„S@‚T@@… by induction on n ƒ−T@−S since@@ƒ0 andSƒb„S@… Now we are done with inductive arguments and we can discard the group LC−1„Y… which was used only as a convenience. The relation @T‚T@ƒ11−Sstill holds without LC−1„Y…sinceTwas zero onLC−1„Y…. (3)Barycentric Subdivision of General Chains . DefineS:Cn„X…!Cn„X…by setting Sƒ]SÑnfor a singular nsimplex:Ñn!X. SinceSÑnis the sum of the nsimplices in the barycentric subdivision of Ñn, with certain signs, Sis the corre- sponding signed sum of the restrictions of to thensimplices of the barycentric subdivision of Ñn. The operator Sis a chain map since @Sƒ@]SÑnƒ]@SÑnƒ]S@Ñn ƒ]S/parenleftbigP i„−1…iÑn i whereÑn iis theithface ofÑn ƒP i„−1…i]SÑn i ƒP i„−1…iS„jjÑn i… ƒS/parenleftbigP i„−1…ijjÑn i ƒS„@… === PAGE 132 === Simplicial and Singular Homology Section 2.1 123 In similar fashion we define T:Cn„X…!Cn‚1„X…byTƒ]TÑn, and this gives a chain homotopy between Sand the identity, since the formula @T‚T@ƒ11−Sholds by the calculation @Tƒ@]TÑnƒ]@TÑnƒ]„Ñn−SÑn−T@Ñn…ƒ−S−]T@Ñn ƒ−S−T„@… where the last equality follows just as in the previous displayed calculation, with S replaced byT. (4)Iterated Barycentric Subdivision . A chain homotopy between 11 and the iterate Sm is given by the operator DmƒP 0i0 such that every set of diameter less than "lies in some set of the cover; such a number exists by an elementary compactness argument.) We cannot expect the samenumbermto work for all ’s, so let us define m„… to be the smallest msuch that S mis inCU n„X…. Suppose we define D:Cn„X…!Cn‚1„X…byDƒDm„…. To see whether Dis a chain homotopy, we manipulate the chain homotopy equation @Dm„…‚Dm„…@ƒ−Sm„… into an equation whose left side is @D‚D@ by moving the second term on the left side to the other side of the equation and adding D@ to both sides: @D‚D@ƒ− Sm„…‚Dm„…„@…−D„@… If we define„… to be the expression in brackets in this last equation, then this equation has the form „…@ D  ‚D@ƒ−„… We claim that „…2CU n„X…. This is obvious for the term Sm„…. For the remaining partDm„…„@…−D„@… , note first that if jdenotes the restriction of to thejth face ofÑn, thenm„j…m„… , so every term TSi„j…inD„@… will be a term in Dm„…„@… . ThusDm„…„@…−D„@… is a sum of terms TSi„j…withim„j…, and these terms lie in CU n„X…sinceTtakesCU n−1„X…toCU n„X… We can thus regard the equation „…as defining:Cn„X…!CU n„X…. For varying nthese’s form a chain map since „…implies@„…ƒ@−@D@„…ƒ„@… . === PAGE 133 === 124 Chapter 2 Homology The equation „…says that@D‚D@ƒ11−for:CU n„X…>Cn„X…the inclusion. Furthermore, ƒ11 sinceDis identically zero on CU n„X…,a sm„…ƒ0i fis in CU n„X…, hence the summation defining Dis empty. Thus we have shown that is a chain homotopy inverse for . tu Proof of the Excision Theorem : We prove the second version, involving a decom- positionXƒA[B. For the cover UƒfA;Bgwe introduce the suggestive notation Cn„A‚B…forCU n„X…, the sums of chains in Aand chains in B. At the end of the preceding proof we had formulas @D‚D@ƒ11−andƒ11. All the maps ap- pearing in these formulas take chains in Ato chains inA, so they induce quotient maps when we factor out chains in A. These quotient maps automatically satisfy the same two formulas, so the inclusion Cn„A‚B…=Cn„A…>Cn„X…=Cn„A…induces an isomorphism on homology. The map Cn„B…=Cn„A\B…!Cn„A‚B…=Cn„A…induced by inclusion is obviously an isomorphism since both quotient groups are free withbasis the singular n simplices inBthat do not lie in A. Hence we obtain the desired isomorphism Hn„B;A\B…Hn„X;A… induced by inclusion. tu All that remains in the proof of Theorem 2.13 is to replace relative homology groups with absolute homology groups. This is achieved by the following result. Proposition 2.22. For good pairs „X;A… , the quotient map q:„X;A…!„X=A;A=A… induces isomorphisms q:Hn„X;A…!Hn„X=A;A=A…eHn„X=A… for alln. Proof: LetVbe a neighborhood of AinXthat deformation retracts onto A.W e have a commutative diagram A ¡¡¡¡¡!XHn(),¡¡¡¡¡¡¡! ¡¡¡¡¡¡¡! q⁄V ¡¡¡¡¡!XHn(), q⁄V ¡¡¡¡¡!XAHn() , q⁄-A- AX/AA/Hn() , AX/ AX/AA/AA/ AV/ AV/ Hn() ,¡¡¡¡¡! ¡¡¡¡¡! Hn() ,-- The upper left horizontal map is an isomorphism since in the long exact sequence of the triple„X;V;A… the groupsHn„V;A… are zero for all n, because a deformation retraction ofVontoAgives a homotopy equivalence of pairs „V;A…’„A;A… , and Hn„A;A…ƒ0. The deformation retraction of VontoAinduces a deformation retrac- tion ofV=A ontoA=A, so the same argument shows that the lower left horizontal map is an isomorphism as well. The other two horizontal maps are isomorphismsdirectly from excision. The right-hand vertical map q is an isomorphism since q restricts to a homeomorphism on the complement of A. From the commutativity of the diagram it follows that the left-hand qis an isomorphism. tu This proposition shows that relative homology can be expressed as reduced abso- lute homology in the case of good pairs „X;A… , but in fact there is a way of doing this for arbitrary pairs. Consider the space X[CAwhereCAis the cone„AI…=„Af0g… === PAGE 134 === Simplicial and Singular Homology Section 2.1 125 whose baseAf1gwe identify with AX. Using terminologyp CA XAintroduced in Chapter 0, X[CAcan also be described as the map- ping cone of the inclusion A>X. The assertion is that Hn„X;A… is isomorphic to eHn„X[CA… for allnvia the sequence of iso- morphisms eHn„X[CA…Hn„X[CA;CA…Hn„X[CA−fpg;CA−fpg…Hn„X;A… wherep2CAis the tip of the cone. The first isomorphism comes from the exact sequence of the pair, using the fact that CAis contractible. The second isomorphism is excision, and the third isomorphism comes from the deformation retraction ofCA−fpgontoA. Here is an application of the preceding proposition: Example 2.23 . Let us find explicit cycles representing generators of the infinite cyclic groups Hn„Dn;@Dn…andeHn„Sn…. Replacing„Dn;@Dn…by the equivalent pair „Ñn;@Ñn…, we will show by induction on nthat the identity map in:Ñn!Ñn, viewed as a singularnsimplex, is a cycle generating Hn„Ñn;@Ñn…. That it is a cycle is clear since we are considering relative homology. When nƒ0 it certainly represents a generator. For the induction step, let ÓÑnbe the union of all but one of the „n−1…dimensional faces of Ñn. Then we claim there are isomorphisms Hn„Ñn;@Ñn…------------!Hn−1„@Ñn;Ӆ -------------Hn−1„Ñn−1;@Ñn−1… The first isomorphism is a boundary map in the long exact sequence of the triple „Ñn;@Ñn;Ӆ, whose third terms Hi„Ñn;Ӆare zero since Ñndeformation retracts ontoÓ, hence„Ñn;Ӆ’„Ó;Ӆ. The second isomorphism comes from the preceding proposition since we are dealing with good pairs and the inclusion Ñn−1>@Ñnas the face not contained in Óinduces a homeomorphism of quotients Ñn−1=@Ñn−1 @Ñn=Ó. The induction step then follows since the cycle inis sent under the first isomorphism to the cycle @inwhich equalsin−1inCn−1„@Ñn;Ӆ. To find a cycle generating eHn„Sn…let us regardSnas twonsimplicesÑn 1and Ñn 2with their boundaries identified in the obvious way, preserving the ordering of vertices. The difference Ñn 1−Ñn 2, viewed as a singular nchain, is then a cycle, and we claim it represents a generator of eHn„Sn…, assumingn>0 so that the latter group is infinite cyclic. To see this, consider the isomorphisms eHn„Sn…------------!Hn„Sn;Ñn 2… -------------Hn„Ñn 1;@Ñn 1… where the first isomorphism comes from the long exact sequence of the pair „Sn;Ñn 2… and the second isomorphism is justified by passing to quotients as before. Underthese isomorphisms the cycle Ñ n 1−Ñn 2in the first group corresponds to the cycle Ñn 1 in the third group, which represents a generator of this group as we have seen, so Ñn 1−Ñn 2represents a generator of eHn„Sn…. === PAGE 135 === 126 Chapter 2 Homology The preceding proposition implies that the excision property holds also for sub- complexes of CW complexes: Corollary 2.24. If the CW complex Xis the union of subcomplexes AandB, then the inclusion„B;A\B…>„X;A… induces isomorphisms Hn„B;A\B…!Hn„X;A… for alln. Proof: Since CW pairs are good, Proposition 2.22 allows us to pass to the quotient spacesB=„A\B…andX=A which are homeomorphic, assuming we are not in the trivial caseA\Bƒ;. tu Here is another application of the preceding proposition: Corollary 2.25. For a wedge sumW X , the inclusions i :X >W X induce an iso- morphismL i :L eHn„X …!eHn„W X …, provided that the wedge sum is formed at basepoints x 2X such that the pairs „X ;x …are good. Proof: Since reduced homology is the same as homology relative to a basepoint, this follows from the proposition by taking „X;A…ƒ„‘ X ;‘ fx g…. tu Here is an application of the machinery we have developed, a classical result of Brouwer from around 1910 known as ‘invariance of dimension,’ which says in partic-ular thatR mis not homeomorphic to Rnifm”n. Theorem 2.26. If nonempty open sets URmandVRnare homeomorphic, thenmƒn. Proof: Forx2Uwe haveHk„U;U−fxg…Hk„Rm;Rm−fxg…by excision. From the long exact sequence for the pair „Rm;Rm−fxg…we getHk„Rm;Rm−fxg… eHk−1„Rm−fxg…. SinceRm−fxgdeformation retracts onto a sphere Sm−1, we con- clude thatHk„U;U−fxg…isZforkƒmand 0 otherwise. By the same reasoning, Hk„V;V−fyg…isZforkƒnand 0 otherwise. Since a homeomorphism h:U!V induces isomorphisms Hk„U;U−fxg…!Hk„V;V−fh„x…g…for allk, we must have mƒn. tu Generalizing the idea of this proof, the local homology groups of a spaceXat a pointx2Xare defined to be the groups Hn„X;X−fxg…. For any open neigh- borhoodUofx, excision gives isomorphisms Hn„X;X−fxg…Hn„U;U−fxg…, so these groups depend only on the local topology of Xnearx. A homeomorphism f:X!Ymust induce isomorphisms Hn„X;X−fxg…Hn„Y;Y−ff„x…g…for allx andn, so these local homology groups can be used to tell when spaces are not locally homeomorphic at certain points, as in the preceding proof. The exercises give somefurther examples of this. === PAGE 136 === Simplicial and Singular Homology Section 2.1 127 Naturality The exact sequences we have been constructing have an extra property that will become important later at key points in many arguments, though at first glance thisproperty may seem just an idle technicality, not very interesting. We shall discuss theproperty now rather than interrupting later arguments to check it when it is needed,but the reader may prefer to postpone a careful reading of this discussion. The property is called naturality . For example, to say that the long exact sequence of a pair is natural means that for a map f:„X;A… !„Y;B… , the diagram f-n 1 -n 1@ @... ...... ... ¡¡¡¡¡!¡¡¡¡¡!¡¡¡¡¡!¡¡¡¡¡!¡¡¡¡¡!¡ ¡¡¡¡!ijAAH()¡¡¡¡¡!¡¡¡¡¡!XHn n n () XH() , ¡¡¡¡¡!¡¡¡¡¡!¡¡¡¡¡!¡ ¡¡¡¡!¡¡¡¡¡! ¡¡¡¡¡!ijBBH() YHn n nAH() BH() () YH() ,⁄⁄⁄ ⁄ ⁄f⁄ f⁄ f⁄ is commutative. Commutativity of the squares involving iandjfollows from the obvious commutativity of the corresponding squares of chain groups, with Cnin place ofHn. For the other square, when we defined induced homomorphisms we saw that f]@ƒ@f]at the chain level. Then for a class † ‡2Hn„X;A… represented by a relative cycle , we havef@† ‡ƒf†@ ‡ƒ†f]@ ‡ƒ†@f] ‡ƒ@†f] ‡ƒ@f† ‡. Alternatively, we could appeal to the general algebraic fact that the long exact sequence of homology groups associated to a short exact sequence of chain complexesis natural: For a commutative diagram of short exact sequences of chain complexes ¡¡¡¡¡¡¡!¡¡¡¡¡¡¡!¡¡¡¡¡¡¡!¡¡¡¡¡¡¡! ¡¡¡¡¡¡¡!¡¡¡¡¡¡¡!¡¡¡¡¡¡¡! ¡¡¡¡¡¡¡!¡¡¡¡¡¡¡!¡¡¡¡¡¡¡!- Bn 1+ Bn 1 Bn... ... ¡¡¡¡¡!¡¡¡¡¡!¡¡¡¡¡¡¡! - Cn 1+¡¡¡¡¡!Cn 1 Cn... ...¡¡¡¡¡¡¡! ii i- An 1+ An 1 A00 0 0 0 0n... ... ¡¡¡¡¡!¡¡¡¡¡!¡¡¡¡¡¡¡! ¡¡¡¡¡¡¡! ¡¡¡¡¡¡¡! ¡ ¡¡¡¡¡¡! jj j- Bn 1+¡¡¡¡¡!Bn 1 Bn... ... ¡¡¡¡¡!¡¡¡¡¡!¡¡¡¡¡¡¡! ¡¡¡¡¡¡¡! ¡¡¡¡¡¡¡! ¡ ¡¡¡¡¡¡!- Cn 1+¡¡¡¡¡! ¡¡¡¡¡!¡¡¡¡¡!¡¡¡¡¡! Cn 1 Cn... ... ¡¡¡¡¡!¡¡¡¡¡!¡¡¡¡¡¡¡! ¡¡¡¡¡¡¡! ¡¡¡¡¡¡¡! ¡ ¡¡¡¡¡¡! i i i- An 1+¡¡¡¡¡!An 1 A00 0 0 0 0n... ... 0 00 00 00 0 00 0 00 0 0fi fl °fi fl °fi fl ° @@@ @@@ @@ @ @@ @ ¡¡¡¡¡!¡¡¡¡¡!¡¡¡¡¡! ¡¡¡¡¡!¡¡¡¡¡!¡¡¡¡¡! ¡¡¡¡¡!¡¡¡¡¡!¡¡¡¡¡! ¡¡¡¡¡¡¡! ¡¡¡¡¡¡¡! ¡ ¡¡¡¡¡¡! ¡¡¡¡¡¡¡! ¡¡¡¡¡¡¡! ¡ ¡¡¡¡¡¡!¡¡¡¡¡¡¡! ¡¡¡¡¡¡¡! ¡ ¡¡¡¡¡¡! jj j the induced diagram -n 1 -n 1@ @... ...... ... ¡¡¡¡¡!¡¡¡¡¡!¡¡¡¡¡!¡¡¡¡¡!¡¡¡¡¡!¡ ¡¡¡¡!ijAH()¡¡¡¡¡!¡¡¡¡¡!CHn n n () BH() ¡¡¡¡¡!¡¡¡¡¡!¡¡¡¡¡!¡ ¡¡¡¡!¡¡¡¡¡! ¡¡¡¡¡!ijAH() CHn n nAH() AH() () BH()⁄⁄ ⁄ ⁄ ⁄⁄⁄0 000 00fi⁄fi fl ° is commutative. Commutativity of the first two squares is obvious since iƒi0 implies iƒi0  andγjƒj0 impliesγjƒj0  . For the third square, recall that the map @:Hn„C…!Hn−1„A…was defined by @†c‡ƒ†a‡wherecƒj„b… and i„a…ƒ@b. Then@†γ„c…‡ƒ† „a…‡ sinceγ„c…ƒγj„b…ƒj0„ „b…… andi0„ „a……ƒ i„a…ƒ @„b…ƒ@ „b… . Hence@γ†c‡ƒ †a‡ƒ @†c‡. === PAGE 137 === 128 Chapter 2 Homology This algebraic fact also implies naturality of the long exact sequence of a triple and the long exact sequence of reduced homology of a pair. Finally, there is the naturality of the long exact sequence in Theorem 2.13, that is, commutativity of the diagram Y/X/ f-n 1 -n 1@ @... ...... ... ¡¡¡¡¡!¡¡¡¡¡!¡¡¡¡¡!¡¡¡¡¡!¡¡¡¡¡!¡ ¡¡¡¡!i qAAH()¡¡¡¡¡!¡¡¡¡¡!Hn n n () XH() ¡¡¡¡¡!¡¡¡¡¡!¡¡¡¡¡!¡ ¡¡¡¡!¡¡¡¡¡! ¡¡¡¡¡!i qBBH() Hn n nAH() BH() () YH()⁄⁄⁄ ⁄ ⁄f⁄ f⁄ f⁄»» » » »»» »¡ whereiandqdenote inclusions and quotient maps, and f:X=A!Y=B is induced byf. The first two squares commute since fiƒifandfqƒqf. The third square expands into -n 1 -n 1@ @ ¡¡¡¡¡!¡¡¡¡¡! ¡¡¡¡¡¡!j q¡¡¡¡¡!¡¡¡¡¡! ¡¡¡¡¡! ¡¡¡¡¡¡!¡¡¡¡¡! ¡¡¡¡¡!j qAH() BH()⁄ ⁄ ⁄ ⁄f⁄ f⁄ f⁄» »Y/X/A ¡¡¡¡¡!Hn() BHn()f⁄» »¡¡AXHn(), BYHn(),AX/AA/Hn() , BY/BB/Hn() ,… …… … We have already shown commutativity of the first and third squares, and the second square commutes since fqƒqf. The Equivalence of Simplicial and Singular Homology We can use the preceding results to show that the simplicial and singular homol- ogy groups of Ñcomplexes are always isomorphic. For the proof it will be convenient to consider the relative case as well, so let Xbe aÑcomplex with AXa sub- complex. Thus Ais theÑcomplex formed by any union of simplices of X. Relative groupsHÑ n„X;A… can be defined in the same way as for singular homology, via relative chainsÑn„X;A…ƒÑn„X…=Ñn„A…, and this yields a long exact sequence of simplicial homology groups for the pair „X;A… by the same algebraic argument as for singular homology. There is a canonical homomorphism HÑ n„X;A…!Hn„X;A… induced by the chain map Ñn„X;A…!Cn„X;A… sending each nsimplex ofXto its characteristic map:Ñn!X. The possibility Aƒ; is not excluded, in which case the relative groups reduce to absolute groups. Theorem 2.27. The homomorphisms HÑ n„X;A…!Hn„X;A… are isomorphisms for allnand allÑcomplex pairs „X;A… . Proof: First we do the case that Xis finite-dimensional and Ais empty. For Xk thekskeleton ofX, consisting of all simplices of dimension kor less, we have a commutative diagram of exact sequences: H()n¡¡¡¡!¡¡¡¡!¡¡¡¡!¡¡¡¡! XH()nk -XXH(),nk 1k -Xk 1H()-Xk 1H()n 1+-XX,k 1k -n 1 H()n¡¡¡¡!¡¡¡¡!¡¡¡¡!¡¡¡¡! XH()nk -XXH(),nk 1k -Xk 1H()-Xk 1H()n 1+-XX,k 1k -n 1 ¡¡¡!¡¡¡!¡¡¡!¡¡¡!¡¡¡!†† † † † === PAGE 138 === Simplicial and Singular Homology Section 2.1 129 Let us first show that the first and fourth vertical maps are isomorphisms for all n. The simplicial chain group Ñn„Xk;Xk−1…is zero forn”k, and is free abelian with basis theksimplices ofXwhennƒk. HenceHÑ n„Xk;Xk−1…has exactly the same description. The corresponding singular homology groups Hn„Xk;Xk−1…can be com- puted by considering the map Ø:‘ „Ñk ;@Ñk …!„Xk;Xk−1…formed by the character- istic mapsÑk!Xfor all theksimplices ofX. SinceØinduces a homeomorphism of quotient spaces‘ Ñk =‘ @Ñk Xk=Xk−1, it induces isomorphisms on all singu- lar homology groups. Thus Hn„Xk;Xk−1…is zero forn”k, while fornƒkthis group is free abelian with basis represented by the relative cycles given by the char-acteristic maps of all the k simplices ofX, in view of the fact that Hk„Ñk;@Ñk…is generated by the identity map Ñk!Ñk, as we showed in Example 2.23. Therefore the mapHÑ k„Xk;Xk−1…!Hk„Xk;Xk−1…is an isomorphism. By induction on kwe may assume the second and fifth vertical maps in the pre- ceding diagram are isomorphisms as well. The following frequently quoted basic alge-braic lemma will then imply that the middle vertical map is an isomorphism, finishingthe proof when Xis finite-dimensional and Aƒ;. TheFive-Lemma. In a commutative diagram of abelian groups as at the right, if the two rowsA fifl° ¡¡¡¡!¡¡¡¡!¡¡¡¡!¡¡¡¡!B¡¡¡¡!C¡¡¡¡!¡¡¡¡! AB¡¡¡¡!C¡¡¡¡!¡¡¡¡!i ij j00000–"¡¡¡¡!DE¡¡¡¡! DE¡¡¡¡!‘ ‘0k k000are exact and , ,, and"are isomorphisms, thenγis an isomorphism also. Proof: It suffices to show: (a)γis surjective if andare surjective and "is injective. (b)γis injective if andare injective and is surjective. The proofs of these two statements are straightforward diagram chasing. There is really no choice about how the argument can proceed, and it would be a good exercisefor the reader to close the book now and reconstruct the proofs without looking. To prove (a), start with an element c 02C0. Thenk0„c0…ƒ„d… for somed2D sinceis surjective. Since "is injective and "‘„d…ƒ‘0„d…ƒ‘0k0„c0…ƒ0, we deduce that‘„d…ƒ0, hencedƒk„c… for somec2Cby exactness of the upper row. The difference c0−γ„c… maps to 0 under k0sincek0„c0…−k0γ„c…ƒk0„c…−k„c…ƒ k0„c0…−„d…ƒ0. Thereforec0−γ„c…ƒj0„b0…for someb02B0by exactness. Since is surjective,b0ƒ „b… for someb2B, and thenγ„c‚j„b……ƒγ„c…‚γj„b…ƒ γ„c…‚j0 „b…ƒγ„c…‚j0„b0…ƒc0, showing that γis surjective. To prove (b), suppose that γ„c…ƒ0. Sinceis injective,k„c…ƒk0γ„c…ƒ0 impliesk„c…ƒ0, socƒj„b… for someb2B. The element „b… satisfiesj0 „b…ƒ γj„b…ƒγ„c…ƒ0,so „b…ƒi0„a0…for somea02A0. Since is surjective,a0ƒ „a… for somea2A. Since is injective, „i„a…−b…ƒ i„a…− „b…ƒi0 „a…− „b…ƒ i0„a0…− „b…ƒ0 impliesi„a…−bƒ0. Thusbƒi„a…, and hencecƒj„b…ƒji„a…ƒ0 sincejiƒ0. This shows γhas trivial kernel. tu === PAGE 139 === 130 Chapter 2 Homology Returning to the proof of the theorem, we next consider the case that Xis infinite- dimensional, where we will use the following fact: A compact set in Xcan meet only finitely many open simplices of X, that is, simplices with their proper faces deleted. This is a general fact about CW complexes proved in the Appendix, but here is adirect proof for Ñ complexes. If a compact set Cintersected infinitely many open simplices, it would contain an infinite sequence of points xieach lying in a different open simplex. Then the sets UiƒX−S j”ifxjg, which are open since their preimages under the characteristic maps of all the simplices are clearly open, form an open coverofCwith no finite subcover. This can be applied to show the map H Ñ n„X…!Hn„X…is surjective. Represent a given element of Hn„X…by a singularncyclez. This is a linear combination of finitely many singular simplices with compact images, meeting only finitely many open sim-plices ofX, hence contained in X kfor somek. We have shown that HÑ n„Xk…!Hn„Xk… is an isomorphism, in particular surjective, so zis homologous in Xk(hence inX)t o a simplicial cycle. This gives surjectivity. Injectivity is similar: If a simplicial ncycle zis the boundary of a singular chain in X, this chain has compact image and hence must lie in some Xk,s ozrepresents an element of the kernel of HÑ n„Xk…!Hn„Xk…. But we know this map is injective, so zis a simplicial boundary in Xk, and therefore inX. It remains to do the case of arbitrary XwithA”;, but this follows from the absolute case by applying the five-lemma to the canonical map from the long exactsequence of simplicial homology groups for the pair „X;A… to the corresponding long exact sequence of singular homology groups. tu We can deduce from this theorem that H n„X…is finitely generated whenever X is aÑcomplex with finitely many nsimplices, since in this case the simplicial chain groupÑn„X…is finitely generated, hence also its subgroup of cycles and therefore also the latter group’s quotient HÑ n„X…. If we writeHn„X…as the direct sum of cyclic groups, then the number of Zsummands is known traditionally as the nthBetti number ofX, and integers specifying the orders of the finite cyclic summands are called torsion coefficients . It is a curious historical fact that homology was not thought of originally as a sequence of groups, but rather as Betti numbers and torsion coefficients. One canafter all compute Betti numbers and torsion coefficients from the simplicial boundarymaps without actually mentioning homology groups. This computational viewpoint,with homology being numbers rather than groups, prevailed from when Poincar ´e first started serious work on homology around 1900, up until the 1920s when the moreabstract viewpoint of groups entered the picture. During this period ‘homology’ meantprimarily ‘simplicial homology,’ and it was another 20 years before the shift to singularhomology was complete, with the final definition of singular homology emerging only === PAGE 140 === Simplicial and Singular Homology Section 2.1 131 in a 1944 paper of Eilenberg, after contributions from quite a few others, particularly Alexander and Lefschetz. Within the next few years the rest of the basic structureof homology theory as we have presented it fell into place, and the first definitivetreatment appeared in the classic book [Eilenberg & Steenrod 1952]. Exercises 1. What familiar space is the quotient Ñcomplex of a 2 simplex†v0;v1;v2‡obtained by identifying the edges †v0;v1‡and†v1;v2‡, preserving the ordering of vertices? 2. Show that the Ñcomplex obtained from Ñ3by performing the edge identifications †v0;v1‡†v1;v3‡and†v0;v2‡†v2;v3‡deformation retracts onto a Klein bottle. Find other pairs of identifications of edges that produce Ñcomplexes deformation retracting onto a torus, a 2 sphere, and RP2. 3. Construct a Ñcomplex structure on RPnas a quotient of a Ñcomplex structure onSnhaving vertices the two vectors of length 1 along each coordinate axis in Rn‚1. 4. Compute the simplicial homology groups of the triangular parachute obtained from Ñ2by identifying its three vertices to a single point. 5. Compute the simplicial homology groups of the Klein bottle using the Ñcomplex structure described at the beginning of this section. 6. Compute the simplicial homology groups of the Ñcomplex obtained from n‚1 2simplicesÑ2 0;;Ñ2 nby identifying all three edges of Ñ2 0to a single edge, and for i>0 identifying the edges †v0;v1‡and†v1;v2‡ofÑ2 ito a single edge and the edge †v0;v2‡to the edge†v0;v1‡ofÑ2 i−1. 7. Find a way of identifying pairs of faces of Ñ3to produce a Ñcomplex structure onS3having a single 3 simplex, and compute the simplicial homology groups of this Ñcomplex. 8. Construct a 3 dimensional ÑcomplexXfromntetrahe- draT1;;Tnby the following two steps. First arrange the tetrahedra in a cyclic pattern as in the figure, so that each Ti shares a common vertical face with its two neighbors Ti−1 andTi‚1, subscripts being taken mod n. Then identify the bottom face of Tiwith the top face of Ti‚1for eachi. Show the simplicial homology groups ofXin dimensions 0, 1, 2, 3 are Z,Zn,0 ,Z, respectively. [The space Xis an example of a lens space ; see Example 2.43 for the general case.] 9. Compute the homology groups of the ÑcomplexXobtained from Ñnby identi- fying all faces of the same dimension. Thus Xhas a singleksimplex for each kn. 10. (a) Show the quotient space of a finite collection of disjoint 2 simplices obtained by identifying pairs of edges is always a surface, locally homeomorphic to R2. (b) Show the edges can always be oriented so as to define a Ñcomplex structure on the quotient surface. [This is more difficult.] === PAGE 141 === 132 Chapter 2 Homology 11. Show that if Ais a retract of Xthen the map Hn„A…!Hn„X…induced by the inclusionAXis injective. 12. Show that chain homotopy of chain maps is an equivalence relation. 13. Verify that f’gimpliesfƒgfor induced homomorphisms of reduced homology groups. 14. Determine whether there exists a short exact sequence 0 !Z4!Z8Z2!Z4!0. More generally, determine which abelian groups Afit into a short exact sequence 0!Zpm!A!Zpn!0 withpprime. What about the case of short exact sequences 0!Z!A!Zn!0? 15. For an exact sequence A!B!C!D!Eshow thatCƒ0 iff the map A!B is surjective and D!Eis injective. Hence for a pair of spaces „X;A… , the inclusion A>Xinduces isomorphisms on all homology groups iff Hn„X;A…ƒ0 for alln. 16. (a) Show that H0„X;A…ƒ0 iffAmeets each path-component of X. (b) Show thatH1„X;A…ƒ0 iffH1„A…!H1„X…is surjective and each path-component ofXcontains at most one path-component of A. 17. (a) Compute the homology groups Hn„X;A… whenXisS2orS1S1andAis a finite set of points in X. (b) Compute the groups Hn„X;A… andHn„X;B… forX a closed orientable surface of genus two with AandB AB the circles shown. [What are X=A andX=B?] 18. Show that for the subspace QR, the relative homology group H1„R;Q…is free abelian and find a basis. 19. Compute the homology groups of the subspace of IIconsisting of the four boundary edges plus all points in the interior whose first coordinate is rational. 20. Show that eHn„X…eHn‚1„SX… for alln, whereSXis the suspension of X. More generally, thinking of SXas the union of two cones CXwith their bases identified, compute the reduced homology groups of the union of nconesCXwith their bases identified. 21. Making the preceding problem more concrete, construct explicit chain maps s:Cn„X…!Cn‚1„SX… inducing isomorphisms eHn„X…!eHn‚1„SX… . 22. Prove by induction on dimension the following facts about the homology of a finite-dimensional CW complex X, using the observation that Xn=Xn−1is a wedge sum ofnspheres: (a) IfXhas dimension nthenHi„X…ƒ0 fori>n andHn„X…is free. (b)Hn„X…is free with basis in bijective correspondence with the ncells if there are no cells of dimension n−1o rn‚1. (c) IfXhaskn cells, thenHn„X…is generated by at most kelements.