The unstable equivariant fixed point index and * the equivariant degree Wac_law Marzantowicz1 Faculty of Mathematics and Computer Science, UAM, Pozna'n, POLAND Carlos Prieto2 Instituto de Matem'aticas, UNAM, 04510 M'exico, D.F., MEXICO Abstract A correspondence between the equivariant degree introduced by Ize, Massab'o, and Vignoli and an unstable version of the equivariant fixed point index defined by the second author and Ulrich is shown. With the help of conormal maps and properties of the unstable index, we prove a sum decomposition formula for the index and consequently also for the degree. As an application, we decompose equivariant homotopy groups as direct sums of smaller groups of fixed orbit types, and we give a geometric interpretation of each summand in terms of conormal maps. 0 Introduction In this paper we shall study the equivariant degree defined by Ize, Massab'o, and Vignoli by comparing it with the equivariant fixed point index defined by Prieto and Ulrich. In the sequel, we shall define an unstable equivariant fixed point index with nice properties, which will be helpful to prove some results about the degree. ______________________________ *2000 Math. Subj. Class.: Primary 54H25; Secondary 55M20, 55M25, 55N91 Keywords and phrases: Topological degree, fixed point index, equivariant cohomo* *logy 1This author was supported by KBN grant No. 2 PO3A 04522, by SRE fellowship 811.5(438)/137, by Fenomec, and by CONACYT grant No. 25427-E. 2This author was partially supported by CONACYT project on Topological Methods * *in Nonlinear Analysis II and CONACYT grant No. 32223-E. 1 In the first two sections, in order to establish notation, we recall the definitions of the equivariant degree ([13]) and of the equivariant fixed point index ([21]). After comparing both concepts in the next section, we use the degree to define in Section 4 an unstable version of the fixed point index as an element of some unstable equivariant homotopy group. Its properties allow to extend the unstable index to G-ENR s. Using this, in Section 5 we prove a sum formula, similar to the already proved formula for the stable index [21, 2.13], that reflects the stratification of a G-ENR in different or* *bit types. This formula in turn provides a corresponding one for the degree, that using different techniques was already obtained by Balanov and Krawcewicz [2]. For doing this, we introduce the notion of conormal map, that in a sense is dual to the notion of normal map used by others. We show that any equivariant map with compact fixed point set is equivariantly homotopic to a conormal map, that is unique up to conormal homotopy. As an application, in section 6 we give a direct sum decomposition of equivariant homotopy groups, and illustrate how our sum formula can be easily used to prove Segal's theorem stating that the G-equivariant 0th stable homotopy group of a point is isomorphic to the Burnside ring of G, namely, ßst0G(*) ~=A(G). 1 The equivariant degree In this section, we shall provide the definition of the equivariant degree, as given in [13]. Let G be a compact Lie group, and let M and N denote G-modules with orthogonal linear actions of G, of dimensions m and n, respectively. Denote by SN , SM the n- and m-dimensional spheres obtained as one-point compactifications of N = Rn and M = Rm , with the corresponding G-actions. Farther below in the paper we use the unit spheres of G-modules M, which we denote by S(M). Note that there is a canonical equivariant homeomorphism between SM and S(M R) that sends the point at 1 2 SM to (0, 1) 2 M x R, where R has trivial G-action. Definition 1.1. For an equivariant map f : V -! M, where and V N is an open G-invariant set such that Z = f-1 (0) is compact, the following is done: 0. Shrinking_V if necessary, we may assume that V is bounded, f is defined in V , and that f-1 (0) V . 2 __ 1. Take R large enough, such that V BR , where BR denotes the open ball centered about the origin with radius R in N. 2. Using the Tietze-Gleason extension theorem, extend f to a map fb: BR -! M. Denote_by Zb the zero-set of fb. bf-1(0) = Zb = Z [ Z0, where Z0 BR - V . __ ___ 3. After taking an open set V 0such that V V 0 BR , V 0\ bZ= Z, using an equivariant version of Urysohn's lemma, construct a G-invariant map ' : BR - ! [0, 1] = I, such that '|BR-V 0= 1 and '|__V= 0. 4. Define F : I x BR - ! R x M by F (t, x) = (2t + 2'(x) - 1, bf(x)) . 5. Since F (t, x) = 0 if and only if t = 1=2 and x 2 Z, then F has no zeroes on the boundary @(I x BR ) SN and therefore, F determines, by restriction, a map F 0: SN - ! R x M - 0 -! SM , where the second map is the usual retraction onto the unit sphere, that obviously coincides equivariantly with SM . By definition, the unstable class degG (f) = [F 0] 2 [SN , SM ]G is the equiv- ariant degree of f. The figure illustrates the construction. Remark 1.2. The excision property of the degree [13, (c), p. 443] guar- antees that the definition above is independent of the equivariant shrinking mentioned in 0. 3 2 The equivariant fixed point index In this section, we recall the definition of the stable equivariant fixed point index, as given in [21], but in a special case. Let G be a compact Lie group. Given an equivariant map ' : V -! K M0, where K, M0 and N0 are G-modules and V K N0 is an open and G-invariant set such that the fixed point set F = Fix(') = {(y, z) 2 V K N0 | '(y, z) = (y, 0) 2 K M0} V is compact, one has an equivariant fixed point index, IG ('), which is an element of the (M -N)-homology group hM-N (*), where hG is some RO (G)-graded equivariant homology theory and M - N 2 RO (G) is the element in the real representation ring of G rep- resented by the (virtual) difference of M = K M0 and N = K N0 (cf. [18]). Definition 2.1. The fixed point index of ' is defined as follows. Consider the diagram j-' (2.2) (V, V -"F`)_____//_(M,MMM- 0) oe (1)|| ß fflffl| (N, N -OFO) ~ | fflj | ff ?Ø| i' (N, N -"Br)` " (2)|| w - fflffl|qt (N, N - 0) where j : V -! M is such that j(y, z) = (y, 0) 2 K M0, (y, z) 2 V K N0. Since F is closed and V is open in N, then (1) is an excision; (2) is a homotopy equivalence in the second term of the pair, thus both induce isomorphisms in homology. Therefore, the dotted arrow i' induces a well-defined homomorphism (i')* : hGj+N(N, N - 0) -! hGj+N(M, M - 0) , where æ 2 RO (G), which, after desuspending by M, determines a homomor- phism IG': hGj(*) -! hGj+N-M (*) 4 and, taking the image of the element 1 2 hG0(*), also an element IG (') = IG'(1) 2 hGN-M (*). Particularly interesting is the case where hG is equivariant stable ho- motopy ßG . Then the index IG (') is a stable element in ßGstN-M(*) = {SN , SM }G = colimK [SN K , SM K ]G , where K varies over a cofinite set of G-modules. Note that this homotopy group can also be considered as the cohomotopy group ßstM-NG(*). 3 Comparison of the degree with the fixed point index Recall Section 1, where given a map f : V - ! M, V N open G-invariant such that Z = f-1 (0) is compact, we defined the degree deg G(f) as the equivariant homotopy class of a map F 0: SN - ! SM . In order to compare the construction of the equivariant degree with the one for the equivariant fixed point index, first, using the linear homeomor- phism D1 = [-1, 1] - ! I, t 7! t+1_2, we change the map F in point 4. of Definition 1.1 to a map G : D1 x BR - ! R x M. Thus G(t, x) = (t + 2'(x), bf(x)) . Then we can extend the map G further to a map eF: R x N -! R x M, say by taking first 8 >>G(t, x) if |t| 1 and |x| R >> < G(_t_, x) if |t| 1 and |x| R eF(t, x) = |t| >>G(t, R_x_) if |t| 1 and |x| R >> |x| : G(_t_ _x_ |t|, R|x|)if |t| 1 and |x| R Then, the zero set eZ= eF -1(0) = {0} x Z and we have indeed a map of pairs eF: (R, R - 0) x (N, N - Z) -! (R x M, R x M - 0). The triangle idxf (3.1) (R, R - 0) x"(V,`V - Z)_____//(R, R3-30) x (M, M - 0) eFhhhhhhhh '|| hhhhhhh fflffl|hhhh (R, R - 0) x (N, N - Z) 5 commutes up to equivariant homotopy of pairs, since if (t, x) 2 R x V , then eF(t, x) = (t, f(x)) if |t| 1 and = (_t_, f(x)), if |t| 1. |t| One has the map of pairs df : (RxN, RxN -BR ) -! (RxM, RxM -0) defined by the following diagram eF (3.2) (R, R - 0) xO(N,ON - Z) ____//_(R, R - 0) x (M, M - 0) | || | || ?Ø| || (R x N, R x N - BR ) ` `d ` `//(R x M, R x M - 0) f Proposition 3.3. The map df : (RxN, RxN -BR ) -! (RxM, RxM -0) induces in homotopy classes the element degG(f) 2 [(R x N, R x N - 0); (R x M, R x M - 0)]G ~= [SN , SM ]G . Proof.Let k* be any graded reduced homotopy functor with a natural exact sequence for pairs of spaces, such as either equivariant homotopy groups ßG* (see [1]), or any equivariant reduced homology theory ehG*. Take the following diagram (3.5) ~= kj(R x N - 0)oo_________kj(SN ) OO | ~=(3)| | ~= | | kj+1((R, R - 0) x (N, N - BR))//_kj(R x (N - BR) [ (R - 0) x N) | _____________________________________________________________|__(1) ____________________________________________________________|| d0f| _______________________________________fflffl|fflffl| | df______________________________kj+1((R,/R/-_0)kxj(N,(NR-xZ))N - (0|x Z)) ___________________________________________ | ____________________________________________________________eF||(Fe|)*| __**____________________________________________________ffl*fflffl|ffl* *|fflffl|~~ kj+1((R, R - 0) x (M, M -_0)__=(2)__//_kj(R x M -o0)o_=_____kj(SM ) , where the horizontal arrows on the left ladder are given by the corresponding connecting homomorphisms, and the two on the right by inclusions. Hor- izontal arrows (1) and (2) are natural isomorphisms, since kj+1(R x N) = kj+1(R x M) = kj(R x N) = kj(R x M) = 0 and the vertical arrow (3) is an isomorphism given by a canonical homotopy equivalence. The curved arrow on the left is the homomorphism df defined above. The two isomorphisms on the right-hand side ladder follow because the inclusion of the unit spheres in R x N - 0, resp. R x M - 0, are equivariant homotopy equivalences, and these spheres are equivariantly homeomorphic to SN and SM , respectively. 6 In the special case kj = ßGN = [SN , -]G , the homomorphism d0fcorre- sponds to a homomorphism [SN , SN ]G - ! [SN , SM ]G , __ that sends [idSN] to degg(f). |__| Given any element [ff] 2 [SN , SM ]G , it induces a homomorphism ff* : ehG*(SN ) -! ehG*(SM ). Corollary 3.6. If 1 2 hG0(*) ~= ehG0(S0) = ehGN(SN ), then deg G(f)*(1) = IG (j - f) 2 ehGN(SM ) ~=ehGN-M(S0) ~=hGN-M (*). In particular, if ehG*is equi* *v- ariant stable homotopy, then deg G(f)*(1) 2 {SN , SM }G is the stabilization of degG (f) 2 [SN , SM ]G , which we call the stable degree. Proof.Diagrams (3.1) and (3.2), put together, give us Diagram (2.2) sus- pended by taking the product with (R, R - 0) on the left, and taking K = 0; therefore, j = 0, and ' = j - f. Then F = Z, i.e., Fix (') = f-1 (0). Hence, taking kj = ehGN, the homomorphism d0fin Diagram (3.5) sends 1 to __ IG (j - f) 2 ehGN(SM ) ~=ehGN-M(S0) ~=hGN-M (*). |__| 4 The unstable fixed point index In this section we redefine the equivariant fixed point index to obtain an un- stable version of it. We shall use the equivariant degree instead of Diagram (2.2) in Section 2, that was used to define the stable index. Definition 4.1. Let M, N and K be G-modules and let V N x K be open and invariant. If ' : V - ! M x K is such that F = Fix(') = {(y, e) 2 V | '(y, e) = (0, e)} is compact, then, if j : V -! M x K is such that j(y, z) = (0, z) and f(y, z) = (j - ')(y, z), define the unstable equivariant fixed point index of ' by IuG(') = degG (f) 2 [SN K , SM K ]G . This group is abelian if dim (NG KG ) > 0 (see [13] or [12]). This unstable index has the following properties which are either direct consequences of the corresponding properties of the equivariant degree, or can be obtained by a slight modification of the corresponding proofs in [21] for the stable index (cf. also [13, (c), (b), (e) p. 443]). 7 (a) Localization. (Corresponding to the excision property of the degree). If W V is open and G-invariant and F W , then IuG(') = IuG('|W ) 2 [SN K , SM K ]G . (b) G-Homotopy. Let 'fi: Vfi-! M x E be such that Ffi= Fix('fi) = {(y, e) 2 Vfi| 'fi(y, e) = (0, e)} is compact for every ø 2 I, then IuG('fi) = IuG('0) 2 [SN K , SM K ]G , ø 2 I . Such a homotopy 'fiwill be called admissible. (c) Additivity. Let ' : V -! M x K, = 1, 2, V N x K open and G-invariant, be such that the fixed point sets F = Fix(f ) are compact and disjoint. By the localization property, one can thus assume that the domains V are also disjoint. If V = V1[V2 and ' : V - ! M xK is such that '|V = ' , then ' has compact fixed point set F = Fix(') = F1 [ F2 and (IuG(')) = (IuG('1)) + (IuG('2)) 2 [SN K+1 , SM K+1 ]G , where : [SN K , SM K ]G -! [SN K+1 , SM K+1 ]G is the suspension homomorphism and, for any L, SL+1 denotes the one-point compacti- fication of the G-module L R. (The additivity holds, thus, already after one suspension). Moreover, the unstable index has a property that the degree does not have. (d) Commutativity. (Corresponding to [21, 1.15]) Let M, N, K, and K0 be G-modules and let U N x K, W K0 be open invariant sets. If ff : U - ! M x K0 and fi : W - ! K are continuous equivariant maps such that the map (1M xfi)ff N x K ff-1(M x W ) __________//_M x K has compact fixed point set F = Fix((1M x fi)ff), then also the map ff(1N xfi) N x K0 (iN x fi)-1(U) __________//_M x K0 8 has compact fixed point set F 0= Fix(ff(1N x fi)). Moreover, both F and F 0are homeomorphic and __ __0 K IuG((1M x fi)ff) = K IuG(ff(1N x fi)) 2 [SN L , SM L ]G , ___ ___0 where L is the smallest G-module, such that K K = L and K0 K = L and denotes the corresponding suspension homomorphism. In particular, if K = K0, one can take L = K = K0 and then one does not need to suspend in order to have the commutativity property. Using this last, as in [21], one can extend the definition of the unstable index to more general situations. To that purpose, let E be a G-euclidean neighborhood retract, or a G- ENR , namely E U K, where U is open and G-invariant, and there is an equivariant retraction r : U - ! E (see [15, 23] for general properties of G-ENRs). Let i : E ,! K be the inclusion. Definition 4.2. Let V N x E be open, invariant and ' : V - ! M x E is such that F = Fix(') = {(y, e) 2 V | '(y, e) = (0, e)} is compact. Then we define the unstable equivariant fixed point index of ' taking e': eV- ! M xK, such that eV= (1N x r)-1(V ) N x K and e'= (1M x i) O ' O (1N x r) and putting IuG(') = IuG('e) = degG (j - e') 2 [SN K , SM K ]G , where, as before, j : eV- ! M x K is such that j(y, z) = (0, z). This general unstable equivariant index for maps (partially) defined on G-ENR s is well defined and has all properties (a)-(d), which the previous case has. Remark 4.3. For the sake of notational simplicity, given a map ', resp. f, partially defined on N x K and image in N x K with compact fixed point set, resp. zero-set, for the unstable index IuG('e) 2 [SN K , SM K ]G , resp. * *the degree deg G(f) 2 [SN K , SM K ]G , one might write instead of the suspen- sions LIuG('e), resp. L degG(f), simply IuG('e) 2 [SN K L , SM K L ]G , res* *p. degG(f) 2 [SN K L , SM K L ]G , since from the term L in the homotopy set one can infere that one is dealing with the L-suspension. Even though the unstable equivariant fixed point index is defined via the equivariant degree, it allows us to extend the definition of the degree to a more general situation, that will be useful later on in Theorem 5.9. 9 Definition 4.4. Given a G-retract E of an open invariant set U in a G- module K with retraction r : U - ! E, such that 0 2 E, and a map f : N x E -! M x E, such that Z = f-1 (0) is compact, one may define degG(f) = IuG(') 2 [SN K , SM K ]G , if ' = j - f(1 x r) : N x U -! M x K, where j : N x U -! M x K is such that j(y, z) = (0, z). 5 Sum decomposition formula In this section we show that the unstable equivariant index decomposes as a sum of elements, each corresponding to one orbit type. This leads to a decomposition of the group [SN , SM ]G into a direct sum, as was shown by Balanov and Krawcewicz [2] using the equivariant degree. Our approach is based on a method used in [21], where the formula was proved for the stable index (see also [20]), and goes back to [23]. This approach is simpler since it does not need any G-transversality as it was the case in [2], and thus it works in more general situations (G-ENR s). We begin by recalling a few notions of compact transformation group theory. Let X be any G-space and H G be a closed subgroup. We use the following notation of [21]: X(H) = {x 2 X | (H) (Gx)} , X(H_)= {x 2 X | (H) ( (Gx)} , X(H) = {x 2 X | (H) = (Gx)} , where (H) (H0) means that some conjugate of H is contained in H0. Therefore, X(H) = X(H) - X(H_)and consists of points of isotropy groups in (H), i.e., of orbit type (G=H). For simplicity we may call the orbit type of these points (H) instead. The set of all orbit types of X, i.e. of conjugacy classes (H) such that X(H) 6= ;, will be denoted by Or (X). Note that for every G-ENR X, the set Or(X) is finite, since by definition, X is an equivariant retract of an open invariant set V M; thus Or (X) Or(V ) Or (M) = Or (S(M)). But Or (S(M)) is finite, because the unit sphere S(M) in M is a smooth, compact G-manifold (cf. [4, IV.1.2]). Next, observe that for a G-space X with a finite set of orbit types there is an ordered indexing (Hj) of Or (X) such that (5.1) (Hj) (Hi) =) j i . 10 Indeed, we may enumerate the minimal elements of Or (X) in an arbitrary way and subtract them from Or (X), then enumerate the minimal elements of the remaining set, and continue this procedure. For such an indexing, we define a filtration of X by [ (5.2) Xi = X(Hj) i j Note that for the difference sets of the filtration (5.2) we have Xi- Xi-1 = X(Hi). If we now take X = E to be a G-ENR , then every Ei is a closed G-ENR subspace of E, because for every H the set E(H) is a closed G-ENR subspace of E (cf. [15, 23]). Now we state the main technical step (cf. [23, II.5.2], see also [21, 2.11]) that we use below, which adapted to our situation reads as follows. Proposition 5.3. Let E be a G-ENR . Consider Rm x E and Rn x E, where Rm and Rn have trivial actions, and let ' : V - ! Rm xE be a G-map with a compact fixed point set F = Fix(') V , V Rn x E. Let moreover D E be a closed G-ENR subspace such that '(V \ (Rn x D)) Rm x D. Then there exists a G-map 'D : V - ! Rm x E, homotopic to ' relative to V D by an admissible homotopy, i.e. a homotopy with a compact fixed point set, of the form 'D = ' O r , __ where r|__U: U - ! D is an equivariant deformation retraction for some open invariant set U D. Proof.By the Localization property of the unstable index, we may restrict ' to a G-numerically open set V with compact closure. Thus V and V D are G-ENRs and so the inclusion V D ,! V is a G-cofibration (see [1, 4.2.13]); hence there exists a G-deformation dfi: V -! V relative to V D such that d-11(V D) is a G-neighborhood of V D (see [1, 4.1.16(b)]). We can make dfistationary_outside of a G-neighborhood U of V D as follows. Take U such that U d-11(V D) (that_is, U is a shrinking of d-11(V D)* *), and take W to be an open G-neighborhood of U in V . Then take oe : V - ! I to be an Urysohn G-function such that oe|__U= 1 and oe|V -W = 0 11 and modulate d by taking (v, ø) 7! dff(v)fi(v) instead. Call this deformation again dfi. Now d0 = idV and dfi|V_-W = idV -W, thus d is now stationary outside of W . We may assume W to be compact and contained in V . The map ' O dfi: V -! Rm x E is a G-homotopy of ' relative_to_ (V D)[(V -W ), and its fixed point set is a closed subset of W xI[Fix (')xI and it is thus compact. Take r = d1. Then the map 'D = ' O r satisfies all __ the requirements of the statement. |__| Proposition 5.3 leads us to the notion of a conormal map, that is dual to the notion of a normal map which was used to study the equivariant degree and was first introduced in [8] for G = S1 (see [9, 2] and the references there* *in for the general case). Definition 5.4. Let E be a G-ENR and _ : V - ! Rm xE be a G-map with compact fixed point set F = Fix(_) V , V Rn x E, where Rm and Rn have trivial actions. We say that _ is conormal if for every (H) 2 Or(E) there exist an open invariant_neighborhood U of V (H_)in V (H)and an equivariant retraction r : U - ! V (H_)such that for the restricted map _(H) = _|V (H)we have __ m (5.5) _(H)|__U= _ O r : U - ! R x E . As a direct consequence of the definition we get. Proposition 5.6. Let _ : V - ! Rm xE be a conormal map and F = Fix(_). Then for every orbit type (H) we have _________ (H ) F \ V(H)\ V __= ; . Moreover, we have IuG(_(H)) = IuG(_(H_)) + IuG(_(H)) 2 [Sn+K+1 , Sm+K+1 ]G , where _(H) = _|V(H). __ Proof.Indeed, for every x 2 U V (H) we have that if _(x) = x, then x 2 V (H_). This shows the first_part of the statement. Take now U and U0 = V (H)- U. By the Additivity and the Localization properties of the unstable index, we have IuG(_(H)) = IuG(_|U ) + IuG(_|U0) = IuG(_|U ) + IuG(_(H)) , 12 because all the fixed points of _|V(H) lie in U0. On the other hand, by the Commutativity property of the index and since _ is conormal, namely of the * * __ form (5.5), IuG(_|U ) = IuG(_|V (H_)) = IuG(_(H_)). * *|__| For any given map, the following theorem states the existence and unique- ness of homotopic conormal maps. Theorem 5.7. Let E be a G-ENR and let ' : V - ! Rm x E be a G-map with a compact fixed point set F = Fix(_) V , V Rn x E, where Rm and Rn have trivial actions. Then we have the following: (a) ' is equivariantly homotopic by an admissible homotopy 'fito a conor- mal map _ = '1 : V -! Rm x E. Moreover, if A V is a closed G-ENR subspace, then this homotopy can be taken relative to A. (b) Furthermore, if '0 and '1 are equivariantly homotopic by an admissi- ble homotopy, and each of them is equivariantly homotopic by an ad- missible homotopy to two conormal maps _0, _1 : V -! Rm x E, respectively, then these two maps are equivariantly homotopic by an admissible conormal homotopy. Note that in the second part of (a), _ is conormal, provided it is conormal on A. Otherwise it is conormal relative to A only. On the other hand, what (b) really states is that any two homotopic conormal maps can be deformed to each other by a conormal homotopy. Proof.By induction over the length of the filtration Ei of E defined in 5.2. For E = E1 the statement is trivial and the wanted conormal map is _1 = '. Let now E = E2 and take D1 = E1 [_A._ We apply Proposition 5.3. Let U1 = U, W1 = W , and d1fi= dfi: U1 :- ! D1 be as in the proof of 5.3. Then _2 = _1 O d11= ' O r1 is a conormal map. Assume now that the result has been proved up to length n - 1 and take E = En. Assume that _n-1 : V -! Rm x E is the already constructed conormal map for En-1 such that _n-1 = _n-2 O dn-11= ' O r1O r2O . .O.rn-1, where r1, r2, . .,.rn-1 are the corresponding local retractions. We now take Dn = En-1 [ A En and apply Proposition_5.3 again. Thus we have Un = U, Wn = W , and dnfi= dfi: Un -! Dn as in the proof of 5.3. Take _n = _n-1 O dn1= ' O r1 O r2 O . .O.rn. In order to see that that _ = _n is a conormal map, note that by itsT construction _ is equivariantly homotopic to ', relative to A and V - ni=1Wi; 13 thus it is homotopic via an admissible homotopy. Suppose that for a given orbit type (H) we have (H) = (Hi+1) in the ordering (5.1). As U we can ___ take Ui\ V (Hi)and as the retraction ri|V (Hi). ri is equivariant and ri(Ui) V (Hi)\ Ei = V (H_i), so that we have completed the proof of (a). To prove (b), it is enough to apply (a) to the following situation. Take E x R instead of E as the given G-ENR ; instead of the map ' take the homotopy 'fibetween '0 and '1, defined on the open set ~V= V x(-", 1+"). Moreover, take the homotopies from '0 to _0 and from '1 to _1. Thus there is a homotopy, that we call 'fibetween the two conormal maps _0 and _1 that can be extended constantly over (-ffl, 0] and [1, 1 + ffl). As the closed subset A we take V x {0} [ V x {1}. Thus (a) provides the desired conormal __ homotopy. |__| We should point out that an analogous statement has been shown by Komiya ([16, Lem. 1]) for m = n = 0 and E a compact, smooth G-manifold. We are in position to prove our main theorem on the decomposition of unstable fixed point index that corresponds to [21, 2.13] for the stable fixed point index. Theorem 5.8. Let ' : V - ! Rm x E, V Rn x E open G-invariant, E a G-ENR , be a G-map with compact fixed point set, and let _ : V - ! Rm x E be a homotopic conormal map by an admissible homotopy. Then X X IuG(')= IuG(_(H))= (IuG('(H)) - IuG('(H_)))2[Sn+K+1 , Sm+K+1 ]G , (H) (H) where the sum runs over (H) 2 Or(V ). Additionally, for every fixed (H0) 2 Or(V ) we have X X IuG('(H0))= IuG(_(H))= (IuG('(H)) - IuG('(H_))) 2[Sn+K+1 , Sm+K+1 ]G , (H) (H) where the sum now runs over (H) 2 Or (V ) such that (H) (H0). This decomposition agrees with the additive structure of [Sn+K+1 , Sm+K+1 ]G , in the sense that every (H)-coordinate of the sum of two elements ', '0is given by the sum of their corresponding coordinates. Proof.We start proving a sum formula for a conormal map. We do it by induction over the filtration (5.2) and the explicit form of a conormal map given in the proof of Theorem 5.7. Suppose that this formula holds for all 14 (Hj), j i. Note that the map _ = ' O r1 . .O.rlpreserves this filtration and _|Ei+1= ' O r1 . .O.ri-1O ri, where ri is the end of_a_G-homotopy defined on Vi+1 relative to Vi such that the restriction ri : Ui -! Vi is a retraction, for some invariant neighborhood Ui of Vi. Repeating the argument of the proof of Corollary 5.6, we get IuG(_|Vi+1) = IuG(_|Vi) + IuG(_|Vi+1-Vi) . But Vi+1 - Vi = V(Hi+1), and consequently IuG(_|Vi+1-Vi) = IuG(_(Hi+1)) - IuG(_(Hi+1_)), by Corollary 5.6. The sum formula is thus proved for a conormal map. By Theorem 5.7, any equivariant map ' : V - ! Rm x E is G-homotopic to a conormal map _. Thus IuG(') = IuG(_), and IuG('(H)) = IuG(_(H)), IuG('(H_)) = IuG(_(H_)). This proves the first sum formula of the statement. The second sum formula follows from the first, when applied to the G-equivariant map '(H0). As to the last assertion of the statement, it follows from the fact that any sum of two fixed point indices can be realized as the fixed point index of one map, by taking a disjoint union. This is always possible in our case, since we are dealing with suspensions by taking the product with R (that has no __ action), using the Additivity property. |__| P We shall call the first equation IuG(') = (H)IuG(_(H)) of Theorem 5.8 the decomposition formula, because it decomposes IuG(f) into a sum of indices (of another map, in general) each of which corresponds to the index on the nonsingular open part of the natural invariantPstratification {E(H)} of E. We shall call the second equation IuG(') = (H)IuG(_(H)) of Theorem 5.8, or the equation of Theorem 5.9 below, the sum formula, because it shows the numerical value of each term of the above mentioned decomposition. We now apply our decomposition and sum formulas of Theorem 5.8 to get similar formulas for the equivariant degree. Since our spaces are not open subsets of a G-module, but only retracts of them, we use here the concept of equivariant degree given in Definition 4.4. If f : V - ! Rm x K is a G-map such that V Rn x K is open and invariant and the zero-set Z = f-1 (0) is compact, then degG (f) = IuG('), where ' = jP- f, j : V - ! Rm x K such that j(y, z) = (0, z). Thus, since IuG(') = (IuG('i) - IuG('i-1)), we have X deg G(f) = IuG('(Hi)) = 15 X X = (IuG('(Hi)) - IuG('(H_i))) = (deg G(f(Hi)) - degG(f(H_i))) and we obtain the desired decomposition formula for the equivariant degree. Thus we have the following. Theorem 5.9. Let f : V - ! Rm x K be a G-map such that V Rn x K is an open invariant set and the zero-set Z = f-1 (0) is compact. Then X deg G(f) = (deg G(f(H)) - degG(f(H_))) 2 [Sn+K+1 , Sm+K+1 ]G , where f(H) = j -'(H), and the sum is taken over all orbit types (H) 2 Or(V ). Moreover, under the same hypotheses as above, for any (fixed) subgroup H0 G, X degG(f(H0)) = (deg G(f(H)) - degG(f(H_))) 2 [Sn+K+1 , Sm+K+1 ]G , where the sum is taken over all orbit types (H) 2 Or (V ) such that (H) (H0). tu Remark 5.10. Using techniques of differential topology, namely the notion of a regular normal map, Balanov and Krawcewicz [2] obtained the decom- position formulaPfor the equivariant degree, that corresponds to the equation IuG(') = (H)IuG(_(H)) in Theorem 5.8 stated as X (5.11) degG(f, V ) = degG (f(H), V ) (H) provided that f is normal. However, they do not have the sum formula of Theorem 5.9, because they, and previous authors, did not have defined degrees in the more general context that we have in Definition 4.4. On the other the hand, we must add that if f is regular normal, by a transversality argument, it follows that in Formula (5.11) ([2, (2.1)]) there are no terms that correspond to (H) such that dim W (H) > n-m, where W (H) = N(H)=H is the Weyl group of H. We could not show it using conormal map techniques. To finish this section we include an algebraic scheme that allows to com- pute the coordinates of the decomposition theorems 5.8, 5.9. Recall that for any poset (X, ), one can define a function i by ( 1 if x y, i(x, y) = 0 otherwise. 16 This produces an ü pper triangular matrix" Z with "entries" Zxy= i(x, y) and 1's along the diagonal. Thus there is (see, for instance. [3, 7.5.2]) another ü pper triangular matrix" M, known as the Moebius matrix of the poset, such that it is an inverse matrix, in the sense that MZ = I and ZM = I, or entrywise, such that X X MxzZzy= ffixz and ZxzMzy= ffixz, z z where ffixzis the Kronecker ffi-function. Call ~(x, y) the entries Mxyof this matrix. ~ is the so-called Moebius function of the poset. Thus, given any two abelian group-valued functions ff, fi : X -! such that X X (5.12) ff(y) = fi(x) , then fi(y) = ~(x, y)ff(x) . x y x y This last is called the Moebius inverse formula. Applying (5.12) to the second sum formula of 5.9, we obtain the following. Theorem 5.13. Under the same hypotheses of the previous results X IuG('(H0)) - IuG('(H_)) = ~((H), (H0)) IuG('(H)) , X degG (f(H0)) - degG(f(H0_)) = ~((H), (H0)) degG (f(H)) , where the sum is taken over the orbit types (H) 2 Or(V ) such that H H0, and ~ is the Moebius function of the poset {(H) | H is a subgroup of G } .tu Remark 5.14. A similar formula using the generalized Moebius function obviously holds also for the fixed point index using the sum formula for the index as in [21, 2.13] instead. Komiya in [16] deals with a similar formula for the classical equivariant fixed point index, that in our terms corresponds to the case m = n = 0, and applies it to an equivariant fixed point problem. Remark 5.15. Making use of a GAP programming package, one may derive the Moebius function ~ for the poset of conjugacy classes of subgroups of G, provided that the group G is included in the library of the package. 17 6 Direct sum decomposition of equivariant homotopy groups To begin this section we show that our decomposition theorem leads to al- ready known decompositions of unstable as well as stable equivariant homo- topy groups graded by integers. Definition 6.1. Given n, m 2 N [ {0}, a G-module K, and an orbit type (H) 2 Or (Rn K) = Or (K), we define the subset [Sn+K , Sm+K ]G,(H) [Sn+K , Sm+K ]G , as the set of elements of the form IuG(_, V ) = IuG(_) 2 [Sn+K , Sm+K ]G , where _ : V -! Rm K is a conormal map with compact fixed point set Fix(_) V(H) and V is an open invariant subset of Rn K. We have the following theorem (cf. [2]). Theorem 6.2. Suppose m > 0 or dim KG > 0. Then for every (H) 2 Or(K), the set [Sn+K+1 , Sm+K+1 ]G,(H) is a subgroup of [Sn+K+1 , Sm+K+1 ]G , and M [Sn+K+1 , Sm+K+1 ]G ~= [Sn+K+1 , Sm+K+1 ]G,(H), (H) where the sum is taken over all (H) 2 Or(K). Moreover [Sn+K+1 , Sm+K+1 ]G,(H)= 0, if dim W (H) > n-m, where W (H) is the Weyl group of H. Proof.The fact that [Sn+K+1 , Sm+K+1 ]G,(H) is a subgroup follows from the decomposition theorem (5.8), since the (H)-coordinate of the sum of two elements is the sum of their corresponding (H)-coordinates. In order to see that it is a decomposition as a direct sum, suppose that ['] 6= 0 lies in [Sn+K+1 , Sm+K+1 ]G,(H1)as well as in [Sn+K+1 , Sm+K+1 ]G,(H2). Then it is of t* *he form IuG(_1, V1), as well as IuG(_2, V2), where _ , = 1, 2, are conormal maps and Fix(_1) V1 (H1), Fix(_2) V2 (H2). Using the Localization property of the index, we may assume that V1 = V2 = V by taking V = V1 [ V2. By Theorem 5.7, _1 and _2 are homotopic by a conormal homotopy. On the other hand it is easy to check that a conormal homotopy does not change the orbit type, i.e. Fix(_1) \ V(H) 6= ; if and only if Fix(_2) \ V(H) 6= ;. Th* *is shows that (H1) = (H2), which completes the proof of the decomposition. 18 Theorem 5.8 shows that every element of the form IuG(', V ) belongs to the above direct sum. We are left with the task of showing that every element in [Sn+K+1 , Sm+K+1 ]G is of the form IuG(', V ). Since (Rm+1 K)G 6= {0}, we can construct an equivariant isotopy on Sm+K+1 that takes any given point x0 2 Sm+K+1 to 1. Consequently, every class [f] 2 [Sn+K+1 , Sm+K+1 ]G has a representative f such that f(1) = 1. Take V = Rm+1 K = Sm+K+1 -{1} and ' = j-f. Since f(1) = 1, 1 is not an accumulation point of zeros of f, thus neither of fixed points of '. Consequently IuG(', V ) = degG (f, V ) = [f]. To show that [Sn+K , Sm+K ]G,(H)= 0 if dim W (h) > n - m, one needs a __ transversality argument (cf. [2]). |__| Remark 6.3. We reproved a theorem about the decomposition of the groups [Sn+K+1 , Sm+K+1 ]G into a direct sum of subgroups [Sn+K+1 , Sm+K+1 ]G,(H). Our interpretation of each element of the latter as an index seems to make the construction of some special elements easier. Note that we need only construct a conormal map on an open invariant set. Moreover, besides the decomposition, we have the sum formula of Theo- rems 5.9 and 5.13, that give the ün meric" values of the coordinates of this decomposition. Of course all the decompositions and sums for the unstable index and degree imply, after stabilizing, the corresponding results in the stable range. To gain confidence on the above results, we established a connection be- tween our decomposition and sum formulas and the Segal theorem that states that the stable cohomotopy group ßst0G(*) is isomorphic to the Burnside ring A(G) of G. This theorem was proved independently by Hauschild [10] (see also [11]), Kosniowski [17], and Rubinsztein [22], with a correction of some gap in the latter by Dancer [5] (see also [12]). Recall that the Burnside ring A(G) of a finite group G is an additively free group with generators given by the orbitsP(G=H), i.e., every element ff 2 A(G) can be uniquely written as ff = (H)k(H)(G=H), k(H) 2 Z. Recall that the unit sphere S(K R) coin- cides with the one-point compactification SK of K and the point (0, 1) in the former corresponds to the point at infinity 1 in the latter. Either of these points is taken as the natural base point. We denote by [S(K R), S(K R)]*G (or [SK , SK ]*G) the set of pointed equivariant homotopy classes. We also set V1 = SK - {1} = K. Suppose that G is finite, K is a complex representation of G, and f : SK - ! SK is an equivariant (pointed) map. We assign to f an element !(f) 19 of A(G) Q by X Iu((j - f)(H), V1(H)) - Iu((j - f)(H_), V1(H_)) (6.4) !(f) = _________________________________________(G=H) , (H) |G=H| where in the numerator of the fraction, we write nonequivariant (unstable) indices, whose difference is an integer, and the sum runs over all (H) 2 Or(V ). Note that !(f) is a well-defined equivariant homotopy invariant, i.e. it depends only on [f]. Furthermore, !(f1 + f2) = !(f1) + !(f2). Proposition 6.5. The element !(f) lies in the Burnside ring A(G); i.e., all coefficients k(H) in (6.4) are integers, and !(f) determines the homotopy class [f] of f. In other words, the mapping [f] 7! !(f) defines a monomorphism from [SK , SK ]G to A(G). Proof.The first statement follows from the fact that Iu((j - f)(H), V1(H)) - Iu((j - f)(H_), V1(H_)) is divisible by |G=H| (cf. [23]), consequently !(f) 2 A(G). Next we remind that an element ff 2 A(G) is uniquely determined by the collection {ØH (ff)} of values of some homomorphisms ØH : A(G) -! Z, (H) 2 Or(G) (cf. [6, 7] for the definitions and properties of ØH ). One can show that for the element !(f) we have ØH (!(f)) = deg(fH ) for every subgroup H G (cf. [23]). On the other hand, by a theorem of tom Dieck it follows that the collection {deg fH }, H 2 Or(G) , determines the homotopy class of f, provided that 0 0 dim KL - dim KL 2 for every two subgroups L _ L G (see [6]). This __ latter condition is satisfied if K is complex. |__| Lemma 6.6. Let _ : V -! K, V K, be an equivariant conormal map such that Fix(_) V(H). Then Iu((_)(H)) - Iu((_)(H_)) IuG(_) = IuG(_(H)) - IuG(_(H_)) = _______________________. |G=H| Consequently, Formula (6.4) is the sum formula of Theorems 5.8, 5.9 if we understand elements of [SK , SK ]G as elements of A(G) by Proposition 6.5. Proof.The statement follows once more by comparing all values of ØL, L 2 __ Or(V ), with deg(j - _)L, both as elements of A(G). |__| 20 Now we show that our sum formula allows us to see any element of A(G) as an index of an equivariant map, which consequently leads to the subsequent result. Proposition 6.7. Let G be a finite group. Let K be the complex regular representation of G or any other complex unitary representation of G that contains all irreducible representations of G as summands. Then the mapping given in 6.5 ! : [SK , SK ]*G= [S(K R), S(K R)]*G- ! A(G) yields an epimorphism. Consequently [SK+1 , SK+1 ]*G~= A(G), and thus also eßst0G(*) ~=A(G). Proof.We apply Lemma 6.6. Since the the sum formulas of Theorems 5.8 and 5.9 are additive with respect to the addition in [SK , SK ]G , at least aft* *er one suspension, it is enough to construct, for a fixed (H), a conormal map _ : V - ! 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