Appendix A Homology of Discrete Groups In this appendix we summarize basic facts about group (co ) homology. We assume a basic familiarity with the homology of topological spaces and homo­ logical algebra. Good references for most of this material are the books of K. Brown [21] and L. Evens [37]. A.1. Basic Concepts A.I.I. The definition. Let G be a group and choose a presentation of G: 1----7R----7F----7G----71 where Rand F are free groups. Construct a CW-complex BG as follows. Take a point x and for each generator of G (i. e., for each element of F) attach a I-cell to x. One now has a bouquet of circles, X(l), with 7T1 (X(1)) = F. Each element of R is a word in the generators of F and hence corresponds to a path , in X(l). For each such element, attach a 2-cell e2 via a map f : ae 2 -+ ,. This yields a space X(2) with 7T1 (X(2)) = F / R = G (each path corresponding to an element of R is now nUllhornotopic). Now, attach a 3-ccll for each generator of 7T2(X(2)) to obtain a space X(3) with 7T1(X(3)) = G and 7T2(X(3)) = O. Continue this process, adding i-cells to obtain a space X(i) at each stage with 7TJ(X(i)) = G and 7Tj(X(i)) = 0 for 1 < j < i. Now define BG = UX(i). Clearly, 7T1 (BG) = G and 7Tj (BG) = 0 for j > 1. This construction is covari­ antly functorial in G and one checks easily that BG is the unique space, up to homotopy equivalence, satisfying 7T1 = G and 7Tj = 0 for j > 1. EXAMPLE A.l.l. If G = Z, then a presentation for G is id o ----7 0 ----7 Z ----7 Z ----7 O. Thus, to build BG, we take a point and attach a I-cell. This gives X(1) = Sl. Since 7Ti(Sl) = 0 for i > 1, the process stops and hence BG = Sl. 150 A. Homology of Discrete Groups EXAMPLE A.1.2. Consider G = Zj2. A presentation is x2 o ---> Z ---> Z ---> Zj2 ---> o. Thus, X(1) = Sl and we attach a 2-cell e2 to Sl via the map 8e 2 = Sl ~ Sl. The resulting space X(2) is the real projective plane lRlP'2. Now, 7r2(lRlP'2) 7r2(S2) = Z and the generator f : S2 --t lRlP'2 is the double cover. Hence, X(3) = lRlP'2 Uf e3 = lRlP'3. Continuing this process, we see that at each stage we attach an i-cell via the double covering map so that X(i) = lRlP'i. Hence, BZj2 = lRlP'oo. DEFINITION A.1.3. The homology of the group G with coefficients in the trivial module A is H.(G, A) = H.(BG, A). The cohomology is defined similarly as The above computations of the homotopy types of BZ and BZj2 give the following homology groups: {z i = 0, 1 o i> 1 z i = 0 {Zj2 i odd o i even. Unfortunately, it is usually impossible to obtain such simple models for BG. Moreover, we would like to be able to compute homology with coefficients in nontrivial G-modules M. We carry this out via the following device. Let ZG be the group ring of Gover Z and let M be a (left) ZG-module. A projective resolution of Mover ZG is an exact sequence of ZG-modules where each Pj is a projective G-module. Such resolutions exist for any M. Now , let G be a group and choose a resolution p. ---> Z of the trivial module Z. If M is a G-module, we define the homology groups of G with coefficients in M to be Hi(G, M) = Hi(P. ®zc M). A.I. Basic Concepts 151 (In the sequel, we shall abbreviate ®ze by ®e.) The cohomology groups are defined similarly: H·(G, M) = H·(Homzc(P., M)). That this is well-defined is a consequence of the following. PROPOSITION A.1.4. Let p. ---+ Z and Q. ---+ Z be two projective resolu­ tions over ZG. Then there is a ZG-linear chain map f. : p. ---+ Q. such that f. is a homotopy equivalence of chain complexes which is unique up to a unique chain homotopy equivalence. D To see that our new definition agrees with the previous definition using BG, consider the (contractible!) universal cover X ---+ BG. The group G acts on X as the group of deck transformations and hence the cellular chain complex C.(X) is a chain complex of G-modules. Moreover, since X is contractible, the augmented complex C.(X) ---+ Z is a (free) resolution of Z over ZG. Each C;(X) is a free G-module with one basis element for each G-orbit of i-cells. This new definition allows us to use ad hoc resolutions to compute homol­ ogy. For example, let G = (t : t n = 1) be the cyclic group of order n with generator t (written multiplicatively). Denote by ~ the endomorphism of ZG given by multiplication by 1 + t + ... + tn-I. Consider the sequence t-I II t-I c ... --> ZG --> ZG --> ZG --> ZG --> Z --> O. This sequence is exact since ~(t - 1) = tn - 1 = O. This gives the following result. PROPOSITION A.1.5. The integral homology of the cyclic group Zin is z i=O Hi(Zln, Z) = {Zin i odd o i even. PROOF. Apply - ®e Z to the above resolution: ... ~ Z !.=..l Z ~ Z !.=..l Z. Since t acts trivially on Z, this complex has the form ···~Z~Z~Z~Z. The homology of this complex is easily computed. D The standard resolution. This resolution is obtained from the "simplex" spanned by G; i.e., we build a space X with vertices the elements of G and simplices the finite subsets of G. This space is clearly contractible. The corre­ sponding free resolution F. = C.(X) is explicitly given as follows. The module Fn is the free abelian group with basis all (n + I)-tuples (gO,g1, ... ,gn)' The 152 A. Homology of Discrete Croups G-action is given by g(gO, ... ,gn) = (ggo, ... ,ggn) and the boundary map a is defined as n a(go, ... ,gn) = 2) -1)i(go, ... ,gi,· .. ,gn). i=O A basis for the free ZG-module Fn consists of those (n + I)-tuples whose first element is 1. Write such a tuple as (1, gl, glg2,· .. , g1g2 ... 9n) and introduce the bar notation [gllg21·· ·Ign] = (l,gl,glg2,.·. ,glg2··· gn). (If n = 0, there is only one such element, denoted [ ].) In terms of this basis, the map a is given as a = L~=o( -1)idi, where 9t[921 .. ·Ign] i = 0 ddgll·· ·Ign] = { [gIl·· ·lgi-llgigi+llgi+21·· ·Ign] 0 < i < n [gIl· . ·Ign-l] z = n. In low dimensions, the bar resolution has the form D 82 8 1 E r 2 ------. F1 ------. ~'71G ------. Z ------. 0, where c(l) = 1, a1([g]) = g[ ]- [ ] = 9 - 1, and a2([glh]) = g[h]- [gh] + [g]. PROPOSITION A.l.6. Let M be a G-module. Then Ho(G, M) is the module Me of coinvariants; that is, Me = M/(gm - m: 9 E G,m EM). PROOF. Apply - ®c M to the standard resolution: ------.8 2 Fl ®c M ------.8 1 Z G ®c M. We have Ho(G, M) = coker(ad. The formula for a1 is given by a1 ([g] ® m) = (g - 1) ® m and under the isomorphism ZG ®c M --> M, we have (g -1) ®m f---> gm - m. 0 REMARK A.l.7. Let M and N be G-modules. Then there is a canonical isomorphism M ®c N ~ (M ® N)c, where G acts diagonally on M ® N. This gives us a second definition of H.(G, M). If F. is a projective resolution of Z, then H.(G, M) = H.((F. ® M)c). PROPOSITION A.l.S. For any group G, H 1 (G,Z) = Gab, the abelianiza­ tion ofG. A.I. Basic Concepts 153 PROOF. Applying - ®c Z to the standard resolution, we obtain ffiZ~ffiZ~Z. [glhJ [gJ The map 81 maps the element [g] ® n to gn - n. But since Z is a trivial G­ module, gn - n = n - n = O. Thus, H1(G,Z) = coker(82 ) and the map 82 is g[h] - [gh] + [g] [h]- [gh] + [g]. Therefore, H1(G,Z) = ffiZ/([g] + [h] = [gh]), [gJ which is Gab, as claimed. o REMARK A.1.9. This also follows from the fact that the first homology group of the space BG is the abelianization of 7r1 (BG) = G. The definition of H. (G) is clearly functorial. Relative Homology. Let G be a group, MaG-module, and H <;;; G a subgroup. We define the relative homology groups H.(G, H; M) as follows. Consider the sequence of complexes 0----7 C.(H) ®H M ----7 C.(G) ®c M ----7 C.(G, H; M) ----70, where C.(G, H; M) = C.(G) ®c M/C.(H) ®H M. We claim that this is exact; that is, we claim that the map C.(H) ®H M ----7 C.(G) ®c M is injective. To see this, note that this map is none other than the inclusion C.(BH; M) ----7 C.(BG; M), where M is the coefficient system on BG given by M. A less topological proof is obtained by noting that the map in question is the composition C.(H) ®H M ----7 C.(G) ®H M ----7 C.(G) ®c M.