II.I Chapter II. Equivariant Obstruction Theory In this chapter we shall assume that the reader is reasonably familiar with obstruction theory on CW-complexes. We shall attempt to strike a reasonable balance between giving no details on the one hand and developing the theory from scratch on the other by making use of the results, without proof, of the classical theory. 1. The obst!uction coczcle In this section n > I will be an integer, fixed throughout the discussion. Let K be a G-complex and L a G-subcomplex. Let Y be a G-space. We shall assume, for simplicity, that the set yH of stationary points of H on Y is non-empty, arcwise connected and n-simple for each subgroup H C G, (We note here that the theory could be generalized to relative CW-complexes (K,L) with no trouble.) Assume that we are given an equivariant map 9: KnUL § Y. Let o be an (n+l)-ce11 of K and let fo: Sn § Kn be a characteristic map for a (note that the characteristic maps may be chosen equivariantly). The subgroup G leaves K(o), and hence Im f , stationary. 0 O It follows that Go leaves Im(~ofo) stationary. That is, G __f ~ n) = y a. G Thus ~~ defines an element c~(o)~ ~n(Y o), and clearly c~ (o) = 0 if o is in L. But, with ~n(u defined as in example (3) of Chap. I, w this defines a cochain c~ E cn+l(l,L;~n(Y)), II .2 -1 Gg o gGog Now c~ (go) is represented by ~Ofgo: Sn § Y = Y and ~ = ~~176 = go,of o so that c~ (g~) = g#(c~ (a)). This means that c~ is an equivariant cochain (by the defintion of ~n (Y))' that iS c~c~+l(x,L'',~n(Y)). It is called the obstruction cochain. (1.1) Proposition. 6c~ = O. Proof. Let t be an (n+2)-cell and consider the compu- tation of (6c~)(T). To calculate this, one "pushes" the coeffi- G cients to those on t; that is to ~ (y t), and calculates the n classical coboundary. But c~ restricted to K(z) and with G coefficients pushed to ~ n (y T ) is just the obstruction cochain, in the classical sense, to extending ~IK n n K(T) to Kn+l~ K(z). Thus (6c~)(t) = 0 is a fact from the classical theory. (1.2) Proposition. c~ = 0 s ~ can be extended eaui- variantly to K n+l UL. Proof. If c cp(O) = 0 then clearly we may extend ~ to G KnL) LDo in such a way that 7(o) c Y o. Define, for geG and xGo, -1 G gGog G cpCgx) = g ~,(x)e gCY o) = y = y go If gx = g'x then g, = gh for some he G so that g' ~Cx) = g ~C x) U (since ~(x) e YG~ which shows that this definition is valid. The proof is completed by taking an (n+l)-cell from each orbit of G on the (n+1)-cells and following the procedure above. II.3 Now suppose that ~ and O are equivariant maps Kn UL § Y and let F: (Kn-lu L)xI § Y he an equivariant homotopy between Ix n-lU L and O[K n-IUL. Define an equivariant map ?#FO: (KxI) n U(LzI) § Y by (~o# FO)(x,O) = ~(x) FO)(x,1) = ~(x) f FO)(x,t) = F(x,t), If ~r = OIK n-I UL and F is the constant homotopy # will denote #F" The deformation cochain d n ~F,O r (Y)) is defined by d~,F,@(o) = C~#F@(Oxl) o It is clear that (1.3) 6d ~,F,O = c 9 = c~ . If #F = #' that is if F is constant, then we put d 0 = d F, O. (1.4) Proposition. Let 9~: KntJL § Y be equivariant and let d SCG(K,L;~n(Y)).n Then there is an equivariant map O: KnUL § Y, coinciding with cf on Kn-Iu L, such that d~, 0 = d. Proof. Let ~ be an n-cell of K, not in L, and choose a characteristic map f : (Bn,S n-l) § (Kn,K n-l) for a. Let o G jn = Bnx(o) usn-lxI C Bnxl and define u jn § y o by u = ~(fo(x)). As shown in non-equivariant obstruction theory, u may G be extended to a map u ~(Bn• § y o representing the element G (or any element) d(o)~ ~ (Y o). It is clear that such extensions n may be chosen equivariantly, since d is an equivariant cochain. II .4 Now 0 can be defined by gIK n-1UL ffi I Kn-1 UL and, for an n-cell o and xa o, O(x) ffi T'(fS 1 (x) ,1). It is clear that d~, 0 = d. The cocycle c~ ~ -Gc n+l (K,L;~n(Y)) represents a cohomology class [c~]~ H n+l(K,L;~n(y)G ) which depends, by (1,3), only on the equivariant homotopy class of ~IK n-1U L. Moreover, if [c~ ] ffi 0, then by (1.4) q[K n-lu L extends to 0: Kn UL + Y such that c 0 = 0 (choose d with 6d = -c~). Hence, by (1.2), we have the following result: (1.5) Theorem. Let ~: Kn UL § Y be equivariant. Then ~[ K n-I UL can be extended to an equivariant ma r Kn+lu L § Y iff [c~] - o. Remark. Suppose that ~,0: K § Y are equivariant and that F: (Kn-Iu L)xI § Y is an equivariant homotopy between the restrictions of ~ and 0 to Kn-Iu L. As above we obtain an equivariant map ~#FO= (Kn-l• U Q § u where Q ffi (LxI) U (K• Then the obstruction to extending ~#F 0 to (KnxI) U q is ^n§ C~#F@ ~c G (KxI,LxIU Kx~l;~n(Y)) ~ This group is isomorphic to Cn(K,L;~n(Y))G and this isomorphism takes C~#FO into d~,F, @ (now a cocycle). II .5 2. Primary obstructions At various points in this section we shall make one or more of the following assumptions: (I) yH is r-simple, non-empty and arcwise connected for all r and H c G (e.g. ~0 (Y) = 0 = ~1 (Y))" (2) H~+I(K,L;~ (Y)) = 0 for all r < n. r (3) HG(K,L;~r(Y)) = 0 for all r < n. r-I (4) H G (K,L;~ (Y)) = 0 for all r < n. r Numbers appearing in each statement indicate which of these assumptions are used. The results in this section are all easy applications of w to the study of extensions of equivariant maps and homotopies. The proofs will be omitted since they offer no difficulties. Suppose first that we are given an equivariant map f: L § Y. (2. I) Lemma. (1,2) There exists an equivariant exten- sion f of f to K n UL. n (2.2) Lemma. (I,3) If fn and gn are equivariant exten- sions of f to K nUL then [Cfn ] = [c ]. gn (Hint : Use (2.1) to find a homotopy fn-i ~ gn-I relative to L.) (2.3) Definition. (1,2,3) Let y n+l (f) e-GNn+l (K'L;~n(Y)) be the (unique) cohomology class [cf ] for any equivariant n+l n extension fn of f to K nUL. y (f) is called the primary ob- struction to. extendin# f and is an invariant of the equivariant homotopy class of f. II .6 (2.4) Proposition. If k: K' § K is cellular and equi- variant then ~n+l(f-k) = k *(Tn§ (f)) when this is defined. (This is also true without cellularity but we have not yet defined k in the general case.) (2.5) Theorem (Extension). (1,2,3) If we also have that -GHr§ ffi 0 for n r dim(K-L) then an equivariant map f: L § Y has an equivariant extension to K iff 7n+l(f) ffi 0. Now suppose that we are given two equivariant maps f,g: K § Y such that f[L ffi g[L. These induce an equivariant map f#g: Q § Y where Q = (KX~I) U (LXI). There is a natural isomorphism (2.6) ~: HGn(K,L;~n(Y)) ~-~ NG'n+l(lxI,Q'~n(Y)), (induced by the obvious isomorphism on the cochain level). We define, under conditions (1,5,4): C2.7) n(f,g) = ~-l(Tn+l(f#g)) and note that n n n C2.8) (f,g) + ~ (g,h) = ~ (f,h) and (2.9) ~n(fok,gok) ffi k (n(f,g)) (where k: (K',L') § (K,L) is cellular and equivariant) when this is defined. An application of (2.5) to this situation yields: II .7 (2.10) Theorem (Homotop~v~. (1,S,4) If we also have that H Gr(K,L;~r(Y)) = 0 for n < r < dim(K-L) and if f,g: K § Y are equivariant with f[L = g[L, then f and g are equivariantly homo- topic (relative to L) iff n(f,g) = 0. A standard argument now proves the following result: (2.11) Theorem (Classification). Assume that (1) holds and also that ~ ~ r-1 H (K,L;&r(Y)) = 0 = H G (K,L;~r(Y)) fox" r < n r+l ~Pl (K,L;~r(u 0 H G (K,L;~r(Y)) for r > n, Let f: K § Y be an equivariant map.