COMPUTATION OF THE EQUIVARIANT 1-STEM BY A DECOMPOSITION OF EQUIVARIANT STABLE HOMOTOPY CLASSES WAC_LAW MARZANTOWICZ AND CARLOS PRIETO Abstract. For any compact Lie group G, we give a decompo- sition of the group {X, Y }kGof (unpointed) stable G-homotopy classes as a direct sum of subgroups of fixed orbit types. This is done by interpreting the G-homotopy classes in terms of the gen- eralized fixed point transfer and making use of conormal maps. Finally, we give a full computation of the first equivariant (stable) stem for G, ßG1st= {*, *}-1G. 0. Introduction A description of the homotopy classes, or of the stable homotopy classes of maps between two topological spaces has been a classical question in topology. Particularly, the stable homotopy classes of (pointed) maps between spheres, namely the so-called stable stems, ßst*, have been important objects to study. Historically, via the Brouwer degree theory, the 0-stem was computed, namely ßst0~=Z. The Hopf map and the Pontryagin theorem provided ßst1~=Z2. Nowadays, a lot on this subject is already known and the literature on it amounts to hundreds (maybe thousands) of papers. A variant of the question arises when we assume that a compact Lie group G acts on all spaces involved and that all the maps con- sidered commute with the group action, that is, that the maps are G-equivariant -G-maps for short. Then the corresponding question is to provide a description of the stable G-homotopy classes between G-spaces. Especially, the stable homotopy classes of maps between unit spheres of orthogonal representations pose an important question. ____________ 1991 Mathematics Subject Classification. Primary 54H25; Secondary 55M20, 55M25, 55N91. Key words and phrases. Equivariant stable homotopy groups, equivariant stems, equivariant fixed point index and fixed point transfer. The first 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.. The second author was partially supported by CONACYT project on Topological Methods in Nonlinear Analysis II and PAPIIT-UNAM grant No. IN110902. 1 2 W. MARZANTOWICZ AND C. PRIETO It is quite easy to show that the negative G-stems are zero, that is ßGkst= 0 if k < 0. In 1970, Segal [24 ], stated that for any finite group G, ßG0st~= A(G), where A(G) is the Burnside ring of G. This result was proved by Kosniowski [15 ], and independently by Rubinsztein [22 ] with a gap that was filled later by Dancer [5]. T. tom Dieck [6] proved the same result for a general compact Lie group G, giving a convenient definition of the Burnside ring A(G) for this case. Lately, the groups ßGkst, k > 0, have been studied intensively by people working on nonlinear analysis since they provide very interesting applications to problems on bifurcations with symmetries (see [9]). Ize et al. have made many computations of ßG*stwhen G is abelian ([9, 10, 11, 12, 13, 14]). Balanov and Krawcewicz [1] showed for a general compact Lie group G that there is a direct sum decomposition M ßGkst~= k(H) , (H) where k(H) denotes the subgroup of ßGkstcorresponding to the isotropy type (H), for a subgroup H G; the sum ranks over all (H) such that dim W (H) k. Here W (H) = NH=H is the Weyl group of H. More- over, this splitting is in the unstable range (see [17 ] for an alternative proof of this fact), unlike that given in in [16 , V.9.1]. Following com- putations made in [8], where a construction of the equivariant degree is given, one obtains that if dim W (H) = k, then 1(H) ~=Z or Z2, de- pending on whether W (H) is biorientable or not. On the other hand, in the treatment of ßG1stmade in [1] it was shown that for 1(H) ßG1st, with dim W (H) = 0, there is a short exact sequence (0.1) 0 -! Z2 -! 1(H) -! W (H)ab -! 0 , where W (H)ab denotes the abelianization of the Weyl group. In [2], Balanov, Krawcewicz, and Steinlein, using results of Ize and purely algebraic arguments, proved that this sequence splits when G is abelian. J. Cruickshank [3] has also considered stable equivariant homotopy groups of spheres. One should beware, however, that his concept of equivariant 1-stem differs from that of our first equivariant stem. In this paper we start giving a decomposition of the group of equi- variant stable homotopy classes of maps between two G-spaces X and Y , provided that X has trivial G-action (Theorem 1.8). A similar result was proven by Lewis, Jr., May, and McClure in [16 , V.10.1] un- der other assumptions (they consider more general symmetry but their space X is a finite CW-complex) and using rather different methods. An advantage of our approach is that it gives a short proof showing COMPUTATION OF THE EQUIVARIANT 1-STEM 3 the geometric interpretation of the maps that form a term of this de- composition, even in the unstable range as in [17 ]. In particular, we do not need the Adams and Wirthmüller isomorphisms to define the splitting homomorphism. To carry out the decomposition, we use the equivariant fixed point transfer given by the second author in [19 ], and the fixed point theoretical arguments used in [17 ]. In the second part of the paper we give another geometrical inter- pretation of the kernel in the short exact sequence (0.1) and then show that the sequence always splits (Theorem 2.18). This, together with well-known facts, leads to a complete description of ßG1stfor any com- pact Lie group G in Theorem 2.6. It is worth to point out that this theorem works in the unstable range, provided that the representation fulfills some conditions (see Proposition 2.17). We want to thank W. Krawcewicz, who, after reading the preprint, pointed out a mistake in the proof of Proposition 2.17. 1. The general decomposition formula In this section, we use the generalized fixed point transfer to give a direct sum decomposition of {X, Y }kG. All along the paper, G will denote a compact Lie group. We shall assume that X and Y are metric spaces with a G-action. Definition 1.1. Let V , W , M, and N be finite dimensional real G- modules, namely, orthogonal representations of G, and let æ be the element [M] - [N] 2 RO (G). Then the elements of {X, Y }jG = G- Stab j(X, Y ) are stable homotopy classes represented by equivariant maps of pairs ff : (N x V, N x V - 0) x X -! (M x V, M x V - 0) x Y . Such a map will be stably homotopic to another ff0: (N x V 0, N x V 0- 0) x X -! (M x V 0, M x V 0- 0) x Y , if after taking the product of each map with the identity maps of some (W, W - 0) and (W 0, W 0- 0), respectively, they become G-homotopic, where V x W ~=G V 0x W 0. Denote the class of ff by {ff}. Remark 1.2. Taking the product of X with a pair (L, L - 0) for some orthogonal representation L of G amounts to the same as smashing X+ = X t {*} with the sphere SL that is obtained as the one-point compactification of L (which is G-homeomorphic to the unit sphere S(L R) in the representation L R, with trivial action on the last coordinate). Thus {X, Y }jG~= colimV[SN ^ SV ^ X+ , SM ^ SV ^ Y +]G , where the colimit of pointed G-homotopy classes is taken over a cofinal system of G-representations V . 4 W. MARZANTOWICZ AND C. PRIETO In [19 ] (see also [20 ]) one proves that any {ff} 2 {X, Y }jG can be written as a composite (1.3) {ff} = ' O ø (f) , where ø (f) is the equivariant fixed point transfer of an equivariant fixed point situation f (1.4) N x EQ U _______________//M x E QQQQ uuuu QQQQ uuu p.projEQQQ((QQ zzup.projEu X , where E -! X is a G-ENRX and the fixed point set Fix(f) = {(s, e) 2 U | f(s, e) = (0, e) 2 M x E} lies properly over X, æ = [M] - [N] 2 RO (G). The transfer is a stable map ø (f) : (N x V, N x V - 0) x X -! (M x V, M x V - 0) x U , for some orthogonal representation V , and ' : U -! Y is a nonstable equivariant map (by the localization property of the fixed point trans- fer, U can always be assumed to be a very small open G-neighborhood of the fixed point set Fix(f); see [20 , 4.4]), (the composite is made after suspending ' by taking its product with the identity of (M x V, M x V - 0)). We denote by Or G the set of orbit types of G, that is the set of conjugacy classes (H) of subgroups H G. For any G-ENRX E, where X has trivial G-action, the set of orbit types in E, denoted by OrG (E), is always finite. In what follows, we shall only be concerned with the special case N = Rn, M = Rn+k , k 2 Z, and we shall assume that X is a space with trivial G-action. For the statement of the main result of this section we need the following definitions. The first of them was originally given in [17 , 5.4]. Definition 1.5. Consider the fixed point situation (1.4) above. We say that the map f : U - ! Rn+k x E is conormal if for every orbit type (H) 2 Or G(Rn x E) = Or G(E), there exist an open invariant __ neighborhood V of U(H_)in U(H) and an equivariant retraction r : V -! U(H_)such that for the restricted map f(H) = f|U(H) we have __ n+k f(H)|__V= f O r : V -! R x E . Here U(H) consists of the points in U with isotropy larger than (H) and U(H_)to those with isotropy strictly larger than (H). COMPUTATION OF THE EQUIVARIANT 1-STEM 5 Definition 1.6. For any subgroup H G, we define the subgroup {X, Y }k(H)of {X, Y }kGas the subgroup of those classes {ff} such that {ff} = ' O ø (f), where (a) f is a conormal map, and (b) Fix(f) U(H), where U(H) consists of the points in U with isotropy group conjugate to H. Remark 1.7. The fact that {X, Y }k(H)is a subgroup of {X, Y }kGfollows easily by observing that both properties (a) and (b) are preserved by the sum of two elements {ff} = ' O ø (f), {fi} = _ O ø (g), that, by the additivity property of the fixed point transfer, corresponds to the disjoint union f + g of the fixed point situations (see [21 , 1.17]). The main result in this section is the following. Theorem 1.8. Let X be a space with trivial G-action. Then there is an isomorphism M {X, Y }kG~= {X, Y }k(H). (H) For the proof we need some preliminary results. Consider a fixed point situation as (1.4). First note that it is always possible to pro- vide Or G(E) with an order (Hj), jS= 1, 2, . .,.l such that (Hi) (Hj) implies j i. Define Ei E as i jE(Hj). These G-subspaces de- termine a filtration of E such that Ei- Ei-1 = E(Hi). Let fi = f|Ui : Ui -! Rn+k x Ei, where Ui = U \ (Rn x Ei). Proposition 1.9. For every i = 1, 2, . .,.l there exists an invariant_ neighborhood Vi of Ei-1 in Ei and an equivariant retraction ri : Vi -! Ei-1 that is admissibly homotopic to the identity. Thus fi is admissibly homotopic to f0i-1= fi-1 O (idRn x ri). The proof is similar to those of [17 , 5.3 and 5.7]. tu Proposition 1.10. The following hold: (a) f is equivariantly homotopic by an admissible homotopy ffito a conormal map f0 = f1 : V - ! Rm x E. Moreover, if A U is a closed G-ENR subspace, then this homotopy can be taken relative to A. (b) Furthermore, if f0 and f1 are equivariantly homotopic by an admissible homotopy, and each of them is equivariantly homo- topic by an admissible homotopy to two conormal maps f00, f01: U -! Rm x E, respectively, then these two conormal maps are equivariantly homotopic by an admissible conormal homotopy. The proof is the same as that of [17 , 5.7] (see also [21 , 2.10 and 2.11] or [25 , II.6.8 and III.5.2]). tu 6 W. MARZANTOWICZ AND C. PRIETO We also need a lemma. Lemma 1.11. Let f : U -! Rn+k x E be a fixed point situation over X such that f is a conormal map and take (H) 2 Or G(E). Then there is a neighborhood V of Fix (f|U(H)) such that g = f|V : V - ! Rn+k x E is a conormal map with Fix(g) = Fix(f|U(H)). Denote g by f(H). Consequently, X (1.12) ø (f) = ø (f(H)) . (H)2OrG(E) Proof. Since f is conormal, the set F = Fix(f|U(H)) is separated from all other fixed points. Then there is a neighborhood V of F in U such that Fix(f) \ V = F . Hence g = f|V : V -! Rn+k x E is a conormal map with the desired properties. By the additivity property of the transfer we obtain the decomposition (1.12). We now pass to the proof of Theorem 1.8. Proof. Any {ff} 2 {X, Y }kG can be written as the composite (1.3) ' O ø (f), where ø (f) is the equivariant fixed point transfer of an equi- variant fixed point situation (1.4). By Proposition 1.10 (a), f can bePassumed to be a conormal map, and by Lemma 1.11, ø (f) = (H)2OrG(E)ø (f(H)). Defining {ff(H)} by {ff(H)} = '|U(H) O ø (f(H)), we have immediately X (1.13) {ff} = {ff(H)} , (H)2OrG(E) where {ff(H)} 2 {X, Y }k(H). So, by Proposition 1.10 (b), we may define M : {X, Y }kG- ! {X, Y }k(H) by ({ff}) = (H){ff(H)} . (H) If (H) 6= (K), then {X, Y }k(H)\ {X, Y }k(K)= 0 as easily follows with the same argument used in the proof of [17 , 6.2]. Thus we may also define M X : {X, Y }k(H)-! {X, Y }kG by ( (H){ff(H)}) = {ff(H)} . (H) (H) Then and are inverse isomorphisms. COMPUTATION OF THE EQUIVARIANT 1-STEM 7 2. Computation of the first G-stem In this section, we compute the first equivariant stem for any compact Lie group G. Given any orthogonal representation V of G, SV will denote the one- point compactification of V with the induced G-action (see Remark 1.2). Definition 2.1. We define the kth equivariant stem for a compact Lie group G or briefly the kth G-stem, k = 0, 1, 2, . .,.by ßGkst= colim [SV +k, SV ]G . V where V varies along a cofinal set of orthogonal G-representations and [-, -]G denotes the set of pointed G-homotopy classes of pointed G- maps. Of course, V + k denotes the orthogonal representation V Rk with G acting trivially in the second summand. Remark 2.2. According to Definition 1.1 and to Remark 1.2, we have that the elements of ßGkstare represented by maps of pairs ff : (V x Rk, V x Rk - 0) -! (V, V - 0) , for some orthogonal representation V of G. That is ßGkst= {*, *}-kGin our notation of the previous section. In other words, ßGkstis the kth ho- motopy group of the infinite loop space QG = 1GS1 = colimV V SV , where V SV = Maps G (SV , SV ). Denote by k(H) the subgroup {*, *}-k(H)of ßGkstas in Definition 1.6. Let, moreover, W (H) denote the Weyl group of H G, defined by W (H) = NH=H, where NH G is the normalizer of H in G. In the rest of the paper, we denote W (H) by H or simply by if there is no danger of confusion. For the kth G-stem, one has a stronger form of Theorem 1.8 derived using an equivariant transversality argument in [1, 2.7]; namely M (2.3) ßGkst~= k(H) . (H)2OrG dim k Recall that a compact Lie group is said to be biorientable if it has an orientation invariant under left and right translations (see [1], [8], or [18 ]). From considerations in [8] (see also [18 ]) the following can be proved. Proposition 2.4. Let dim = k. Then ( Z if W (H) is biorientable, k(H) ~= Z2 otherwise. tu 8 W. MARZANTOWICZ AND C. PRIETO Note 2.5. For instance, a compact Lie group is biorientable if it is either finite, abelian, or connected (cf. [8]). The simplest nonbiori- entable group (of dimension 1) is O(2). In what follows, we shall compute the subgroups 1(H) of the first G-stem to obtain the main result of this section, as follows. Theorem 2.6. There is a sum decomposition of the first G-stem M ßG1st= 1(H) . (H)2OrG dimW(H) 1 Here, if dim W (H) = 0, (2.7) 1(H) ~=Z2 W (H)ab, where W (H)ab is the abelianization of W (H), and, if dim W (H) = 1, ( Z W (H) is biorientable, (2.8) 1(H) ~= Z2 if W (H) is not biorientable. In view of Proposition 2.4, we only need to prove Equation (2.7). For doing this, we shall make some general considerations. Assumption 2.9. V denotes a G-module and the elements in ßGkstare represented by maps (V x Rl+k, V x Rl+k- 0) -! (V x Rl, V x Rl- 0), with l k + 3, where G acts trivially on the second factor. Note that the Weyl group of H G acts effectively on V H. We denote by U the representation V HxRl+k of , with the obvious action, and by U0 the representation V H x Rl. Let (P ) be the principal orbit type of the action of on U, and let UP = U - S, where S consists of all points in U with isotropy group type different from (P ) (see [6]). Note 2.10. 1. The set UP is in general disconnected; however, it is connected,0 provided that dim (U -UP ) dim U -2. This holds if dim U dim U -2 for any ( 0) > (P ), and this can always be attained in the stable range. For this, it is enough to replace V by V V . 2. Even being UP connected, it need not be simply connected. By Lefschetz duality, UP will be simply connected if dim (U -UP ) 0 0 dim U -3. This holds if dim U dim U -3 for any ( ) > (P ). For this, it is enough to replace V by V V V . 3. Increasing the size of V further (summing again with itself) we may also assume that UP has an orientation-preserving - action. COMPUTATION OF THE EQUIVARIANT 1-STEM 9 Denote by k fr(UP ) the group of bordism classes of -framed k- submanifolds of UP . One has the following result of Balanov and Krawcewicz. Proposition 2.11. ([1, 3.2]) Let dim k. Then k(H) ~= k fr(UP ). tu To focus on the proof of Equation (2.7), assume in what follows that dim = 0; that is, is a finite group. Note that acts effectively on U, but since is finite, the principal orbit type corresponds to trivial isotropy,_i.e., the action of on UP is in fact free. Let U P denote the quotient space UP = . If [M, j] 2 frk(UP ), then M is a framed -submanifold of UP (and j is a -trivialization_of the normal bundle), and thus__ acts freely on M and M = M= is an oriented submanifold of_U P. Consequently, there is a homomorphism k : k fr(UP ) -! k(U P), where k denotes the usual oriented bor- dism of k-submanifolds. The image of the canonical homomorphism k fr(UP ) -! frk(UP ) lies inside frk(UP ) , where frk(UP ) has the ac- tion of induced by that on UP . By the_Steenrod-Thom_theorem, we know that there is a homomorphism k(U P) -! Hk(U P; Z) that is an isomorphism for k 3 and an epimorphism for k = 4 (see [23 ]). In the case k = 1, that we are concerned with, we thus have an isomoprphism. An essential step in deriving 1(H) when dim = 0 was done in [1, 4.3], where details on the previous comments can be seen; namely we have the following. Theorem 2.12. ker ~=Z2 and thus one has a short exact sequence __ (2.13) 0 -! Z2 -! 1 fr(UP ) -! H1(U P; Z) -! 0 . Moreover, ker consists of those bordism classes of G-framed invariant manifolds [M, j] 2 1 fr(UP ), where M diffS1 and j is an equivariant_ trivialization_of_the normal bundle such that the quotient manifold M = M= U P, M S1, is nullbordant. tu In [2, 2.4], it is shown that if G is abelian, then the sequence (2.13) splits. Their argument is purely algebraic and makes use of the com- putation in [12 ] of 1(H) as a product of p-factors, p prime. We show in what follows that (2.13) always splits. Note 2.14. There is an isomorphism (2.15) * fr( x) = * fr( ) ~= fr*(*) , that is a consequence of the following well-known fact (see [6]). Namely, there is a bijection [SV ^ X, SV ^ Y ^ + ] ~= [SV ^ X, SV ^ Y ], that provides the isomorphism (2.15), since the homology theory * fris 10 W. MARZANTOWICZ AND C. PRIETO equivalent to the theory ß* st. In particular, 1 fr( x) ~= fr1(*) ~=Z2 . Lemma 2.16. Take x 2 UP . If ix : ~= x ,! Up is the inclusion and i* = ix *: 1 fr( ) -! 1 fr(UP ) is the induced homomorphism, then ker = im(i*) . __ Proof. Recall first that d = dim UP = dim U P 3 (see Assumption 2.9), and assume that we have a metric on UP that isF -invariant and take " > 0 sufficiently small, that ß-1 (D"(__x)) = fl2 flD"(x) x D"(x),_where D" denotes the corresponding_d-balls of radius ", and let M be the boundary @D2"=2(__x) U P of a 2-disk of radius "=2 contained ___ in D"(__x). Hence M _is_diffeomorphic_to S1. Let __j0,___j1:_ (M ) -_!_M x Rd-1 be trivializations of the_normal bundle of M such that [M , __j0] = 0 2 fr1(D"(__x)) and [M , __j1] 6= 0 2 fr1(D"(__x)). Let jx : D"(__x)) -! D"(x))_be the inverseFdiffeomorphism to that induced by ß, and call Mx = jx(M ). Define M = fl2flMx ___ UP . M is homeomorphic to x M . ___ Note that (Mx) D"(x) is diffeomorphic to (M ) via the map (m, v) 7! (ß(m), Dß(m)v). On the other hand, flMx flD"(x) = D"(flx) and (flMx) = fl( (Mx)) , since fl induces a diffeomorphism, because it is a linear orthogonal map. Consequently, the tubular neighborhood G (M) = fl( (Mx)) , fl2 and thus we can define an equivariant trivialization ji : (M) - ! M x U0, i = 0, 1, by ji(flm, flv) = (flm, fl__ji(ß(m), Dß(m)v)) for m 2 Mx and v 2 m (Mx). Observe that ji is equivariant, since for ~ 2 we have ji(~(flm, flv))= ji((~fl)m, (~fl)v) = ((~fl)m, (~fl)(__ji(ß(m), Dß(m)v))) = (~(flm), ~(fl__ji(ß(m), Dß(m)v))) = ~ji(flm, flv) . COMPUTATION OF THE EQUIVARIANT 1-STEM 11 Hence we get_that_[M, j0], [M, j1] 2 1 fr( xD"(x)) 1 fr(UP ). Con- sequently, M = M= is nullbordant, thus implying that [M, j0], [M, j1] 2 ker . By construction, they lie in im (i*) = im (iD"* ), where iD" : x D" ,! UP , and obviously, [M, j1] 6= 0 in 1 fr(UP ). Proposition 2.17. If is finite, UP is connected and the action of preserves the orientation, then __ 1 fr(UP ) ~=Z2 H1(U P; Z) . If, moreover, UP is simply connected, then 1(H) ~=Z2 ab. Proof. First observe that for any manifold U with a free action of a finite group , and its projection ß : U -! U= onto its orbit space, there is a well-defined transfer homomorphism ß! : fr*(U= ) -! * fr(U) given as follows. o Every manifold M U= that represents an element in fr*(U= ) is mapped by ß! to the -invariant submanifold fM = ß-1 M U. o Observe that the tangent-bundle morphism (ß, Dß) : ø (U) -! ø (U= ) provides an isomorphism ø (U) ~= ß!(ø (U= )), where here ß! denotes the transfer on bundles (in KO-theory). In particular, we have ø (U)|fM ~= ß!(ø (U= ))|M = ß!(ø (M)) ß!( (M)) ~=ø (fM ) (fM ), where stands for the normal bun- dle, as before. o Note that for any bundle ,, ß!(,) is always a -bundle. More- over, if , is trivial, then ß!(,) is -trivial. Besides, if j = {ei(x)}ki=1, ei(x) : Ex(,) - ! V , is a frame for the trivializa- tion of ,, then ej= {eei(x) = ei(ß(x))}ki=1is a frame for the - trivialization of ß!(,). In the special case of the tangent bundle ø (U) and its subbundles, the -trivialization is obtained from the equality ß O fl = ß, for any fl 2 . Thus Dß O Dfl = Dß. o Consequently, an element [M, j] 2 fr*(U= ) determines an el- ement [fM , ej] 2 * fr(U), since a bordism between two such manifolds lifts in the same way. o Note that ß! : fr*( x= ) = fr*(*) - ! * fr( x) = * fr( ) is the inverse of the canonical isomorphism given above in Note 2.14. 12 W. MARZANTOWICZ AND C. PRIETO Consider now the following diagram that commutes by the very defini- tion of the homomorphisms. j* __ __ 0 _____//_ fr1(*)____//_ fr1(U_P)__//_H1(U P; Z)____//_0 i!|| |i!| |1| fflffl| fflffl| fflffl|_ 0 _____// 1 fr(_)i*_// 1 fr(UP )___//_H1(U P; Z)___//_0 , where the horizontal homomorphisms on the left-hand side are in- duced by inclusions, is the homomorphism given by Balanov and Krawcewicz described above, and__ is the forgetful_homomorphism mapping to oriented bordism 1(U P) = H1(U P; Z) also mentioned above. By the commutativity of the diagram and the five lemma, the homomorphism ß! in the middle is an isomorphism. __ The homomorphism j* on the_top row splits by s* : fr1(U P; Z) -! fr1(*) induced by s : UP - ! *. This gives us the splitting ho- momorphism oe : 1 fr(UP ) - ! 1 fr( ), that is explicitly given by oe = ß! O s* O ß!-1. Therefore, we have the following. Theorem 2.18. The short exact sequence qffq]`a // __ // 0 ____//_ 1 fr( x)i*_//_ 1 fr(UP_)___H1(U P; Z) _____0 . splits. tu Combining 2.11 and 2.12 with the previous theorem, we obtain our main theorem 2.6. 3. Some examples of the first G-stem To finish the paper, we discuss briefly some examples of Theorem 2.6. Examples 3.1. 1. Let G = 1 be the trivial group. Then there is only one H G and W (H) = G=H = 1 has dimension 0. Thus ßst1= Z2. 2. Historically, the first case of ßG1stdescribed was for G = S1, when 1st M ßS1 ~= Z2 Z , H S1 and was given by Dylawerski [7]. COMPUTATION OF THE EQUIVARIANT 1-STEM 13 3. Let G be a finite group. Then for every H G, dim W (H) = 0. Thus M (3.2) ßG1st~= (Z2 W (H)ab) . (H)2Or(G) If G is abelian, then W (H) = G=H and thus M (3.3) ßG1st~= (Z2 G=H) . H G Particular special cases are G = Zp, where p is prime. Then ßZpst1~=Z2 Z2 Zp . We leave to the reader the verification that the description (3.3) agrees with the decomposition in terms of prime factors of G given by Ize and Vignoli [12 , 13, 14]. 4. Let G be either O(2) or SO (3). Then G has infinitely many conjugacy classes of closed subgroups H such that W (H) is finite (see [4, IV.(4.10) Ex.9]). Thus ßG1sthas infinitely many Z2-summands and for each of them also a W (H)ab-summand (see [2] for further details on the case G = SO (3)). 5. Let G = O(k). Then G has infinitely many finite conjugacy classes of subgroups H generated by reflections such that W (H) is finite (see [4, V.(2.18) Ex.6]). Thus, as in the previous ex- ample, ßO(k)1sthas infinitely many Z2-summands and for each of them also a W (H)ab-summand. Remark 3.4. Examples 3. and 4. above show that in general, for non- abelian compact Lie groups G, ßG1stis an infinite -and quite complicated- group. 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Ulrich, Fixed Point Theory of Parametrized Equivariant Maps, Lecture Notes in Math. 1343, Springer-Verlag, Berlin, Heidelberg, 1988 COMPUTATION OF THE EQUIVARIANT 1-STEM 15 Faculty of Mathematics and Computer Science, UAM, Pozna'n, POLAND E-mail address: marzan@main.amu.edu.pl Instituto de Matem'aticas, UNAM, 04510 M'exico, D.F., MEXICO E-mail address: cprieto@math.unam.mx