AA stable approach to theequivariant Hopf theorem
Markus SzymikMarch 2007
Let G be a finite group. For semi-free G -manifolds which are oriented in thesense of Waner [20], the homotopy classes of G -equivariant maps into a G -sphere are described in terms of their degrees, and the degrees occurringare characterised in terms of congruences. This is first shown to be a sta-ble problem, and then solved using methods of equivariant stable homotopytheory with respect to a semi-free G -universe. Introduction
All manifolds considered will be smooth, closed, and connected. If M is an ori-ented n -manifold, any continuous map from M to an n -dimensional sphere has adegree, which is an integer. The Hopf theorem [11] says that the degree characte-rises the homotopy class of such a map, and that every integer occurs as the degreeof such a map. Note that the degree map from homotopy classes to the integersfactors through the group { M , S n } of stable homotopy classes of maps from M to S n . The Hopf theorem follows from the facts that both the stabilisation andthe stable degree map are isomorphisms. It is this approach to that result which a r X i v : . [ m a t h . A T ] F e b ill be generalised here to the equivariant category: it will first be shown that theproblem, which at first sight seems to be unstable, is in fact stable, and then it willbe solved using stable techniques.The Hopf theorem has been generalised to equivariant contexts by a numberof people, motivated by the problem to describe maps between representationspheres or homotopy representations. See Section 8.4. in [4], [10], Section II.4in [5], or [3], [2], [7], and the references therein for recent contributions. Allof these work unstably with a fairly elaborate obstruction machinery. Further-more, they require the top-dimensional cohomology of the fixed point spaces tobe cyclic, neglecting situations in which the fixed point space is not connected.The point of view taken here avoids elaborate notions of degree (and correspond-ing Lefschetz and Euler numbers) as in [6], [19], [16], [12], and the referencestherein, to name just a few. For the present purpose, the degree of a map is itsstable homotopy class, and calculations of stable homotopy groups – which aremuch more accessible than their unstable relatives – process this into numericalinformation.Throughout the paper, let G be a finite group. There will be occasions to assumethat its order is a prime, but only for illustrative purposes. Let W be a real G -representation. Recall that a G -manifold M is called W -dimensional , if for everypoint x in M the tangential representation T x M of the stabiliser G x is isomorphicto the restriction of W from G to G x . In particular, the components of the fixedpoint space M G = ∏ α M G α are manifolds of equal dimension. In addition, a W -dimensional manifold M is called oriented if W is oriented and there is a compatible choice of isomor-phisms T x M ∼ = W which are orientation preserving. In particular, the componentsof the fixed point space inherit an orientation. (See [20] and the references thereinfor details.) 2 xample 1. If M → N is a cyclic branched covering of complex manifolds, therepresentation W is a direct sum of trivial summands and a complex line on whichthe cyclic group acts in the natural way.The following example has been considered and applied in [18]. Note that con-trary to the previous example, the fixed point space is not connected here. Example 2.
For a prime order group G , and a complex G -representation V ,let M = C P ( V ) be the associated projective space. The fixed point space is the dis-joint union of the projective spaces of the isotypical summands, hence is equidi-mensional if V is a multiple of the regular G -representation. In that case, alltangential representations are isomorphic, as complex G -representations, to thecomplement of a trivial line in V . This we may take as our W .In this writing, we will be concerned with oriented W -dimensional semi-free G -manifolds M , and prove an equivariant Hopf theorem which gives a descriptionof the set of G -homotopy classes of G -maps from M to S W in the generic casewhen the dimension and codimension of the fixed point space are at least 2. (SeeSection 2 for free actions; trivial actions can be dealt with as in Example 1 below.)By orientability, the codimension of the fixed point space is always at least two ifthe action is non-trivial.While the Hopf problem looks like an unstable problem, its turns out to be sta-ble. Equivariant stable homotopy theory is more complicated than ordinary stablehomotopy theory, as there is a stable homotopy category for each G - universe ,which is an infinite-dimensional G -representation which contains the trivial rep-resentation, and contains each of its subrepresentation with infinite multiplic-ity. (Standard references for equivariant stable homotopy theory are [1], [15], [17],and [8].) The full-flavoured equivariant stable homotopy category correspondsto a complete G -universe, which contains every irreducible G -representation. A semi-free universe is obtained from a complete G -universe by restriction to thesemi-free subrepresentations. The stable homotopy category depends only onthe G -isomorphism class of the universe, and we will write { X , Y } G sf for the mor-3hism groups from X to Y in an equivariant stable homotopy category correspond-ing to a semi-free G -universe. As the stable categories requires based objects,we will let M + denote the G -space which is M with a disjoint G -fixed basepointadded. By adjunction, there is a canonical bijection between the set [ M + , S W ] G ofbased G -homotopy classes and the set of (unbased) G -homotopy classes of mapsfrom M to S W . The proof of the following result is in Section 1. It justifies workingstably afterwards. Theorem 1.
Let W be a G-representation, and let M be a W -dimensional semi-free G-manifold such that the dimension and codimension of the fixed point spaceare at least . Then the stabilisation map [ M + , S W ] G −→ { M + , S W } G sf is bijective. In order to study the stable groups { M + , S W } G sf , the standard approach is to asso-ciate to a stable G -map f the set of all the degrees of the fixed point maps f H forthe various subgroups H of G . In the semi-free case, the only subgroups relevantare 1 and G , so that the fixed point maps can be organised into the ghost map { M + , S W } G sf −→ { M + , S W } ⊕ { M G + , S W G } (1)which sends a stable G -map f to the pair ( f , f G ) . A priori, the first factor of theghost map lies in the G -invariants of { M + , S W } , but the present assumptions onthe G -actions on M and W ensure that G acts trivially on this group. Using theordinary Hopf theorem, the image of an f under (1) can be interpreted in terms ofdegrees: an integer x and a sequence ( y α ) of integers, indexed by the componentsof M G . (See Section 8 for details.) One may show – using standard splitting andlocalisation techniques – that the ghost map (1) is an isomorphism away from theorder of G , but this will also be shown directly in the course of this investigation.It means that the kernel and cokernel are finite abelian groups. The followingsummarises Propositions 7 and 8 of the main part.4 heorem 2. Let M be an oriented W -dimensional G-manifold. The source andthe target of the ghost map (1) are free abelian. Their rank is one larger than thenumber of components of M G . The map is injective and the cokernel is cyclic oforder | G | . In other words, stable G -maps are determined by their degrees, and the numbersthat occur satisfy a relation. The following result, which is proved in Section 8,says which. Theorem 3.
The image of the ghost map (1) is given by the subgroup of integers xand sequences ( y α ) for which the congruencex ≡ ∑ α y α mod | G | is satisfied. The chosen approach to the two theorems separates the homotopy theory from thegeometry. Let me illustrate this by explaining how the well-known description ofthe Burnside rings of prime order groups can be deduced using these methods.
Example 3.
Let G be a prime order group, and let M be a point. Then M + and S W are both 0-dimensional spheres. While the G -actions on M is trivial,of course, it will no longer be after stabilisation with a suitable oriented G -representation. Let us assume that this has been done without changing the nota-tion. The group { S , S } G maps via the forgetful map to the group { S , S } = Z .On the other hand, it maps to the group { ( S ) G , ( S ) G } = Z via the fixed pointmap. The product { S , S } G −→ Z ⊕ Z of the two maps is the ghost map. It is injective, and the image has index | G | . Theimage is not yet determined by this. There are still | G | + S is G -equivariant. It is mapped to the pair ( , ) .5his determines the image, which consists of those pairs ( x , y ) in Z ⊕ Z whichsatisfy the relation x ≡ y mod | G | .The homotopy theory for Theorem 2 is done in Sections 3 to 7. The method ofproof is to deal with the G -trivial space M G and with the G -space M / M G , which isfree away from the base point, separately. The latter uses a result from Section 2which sometimes allows for a comparison of the group of G -maps out of a free G -space with the group of ordinary maps out of the quotient in the case when G does not act trivially on the target. In the final Section 8, the geometry of the situationis used to construct some maps whose images under the ghost map are easy todetermine, leading to Theorem 3. Acknowledgments
I would like to thank a referee of an earlier version for her or his stimulatingreport. Part of this work has been supported by the SFB 343 at the University ofBielefeld.
In this section, we will prove the stability of the Hopf problem, and the proof givenwill also show that the semi-stable universe is the appropriate context in which towork.Let us recall the classical situation. If M is an oriented n -manifold, by Freuden-thal’s suspension theorem, the stabilisation map [ M + , S n ] −→ { M + , S n } is bijective if n (cid:62)
2. In the equivariant situation, we may use the equivariantextension of Freudenthal’s theorem due to H. Hauschild, see [9] (or [1] for an6xposition in English). This implies that the suspension map [ X , Y ] G −→ [ Σ V X , Σ V Y ] G is bijective for finite based G -CW-complexes X and Y , if two conditions are sat-isfied: the connectivity of Y H has to be at least dim ( X H ) / H of G such that V H (cid:54) =
0, and the connectivity of Y K has to be at least dim ( X H ) + K < H of G such that V H (cid:54) = V K . This will now be used to proveTheorem 1. Proof.
The first condition corresponds to Freudenthal’s condition in the non-equivariant case. It is therefore satisfied for all H – and for all V – in our situ-ation X = M + and Y = S W as long as the dimension of M G is at least 2.The other condition refers to a genuinely equivariant phenomenon. Suppose firstthat K = H is a non-trivial subgroup of G . Then M H = M G since M is semi-free, and the condition is satisfied for these K and H – and for all V – if and onlyif the codimension of M G in M is at least 2.If K and H are non-trivial subgroups of G such that V H (cid:54) = V K , the condi-tion dim ( M G ) (cid:54) dim ( W G ) − V H = V G = V K if V is semi-free, so this doesnot occur, and the proof is finished.There is a change-of-universe natural transformation from { X , Y } G sf to the corre-sponding group with respect to a complete G -universe, see [13]. However, thepreceding proof shows that one should not expect it to be bijective. It is clearlybijective if the order of G is a prime, since semi-free universes are complete forsuch G . In a similar vein, the proof above can easily be adapted to show that thestabilisation map from [ M + , S W ] G to the corresponding group of stable G -mapswith respect to a complete G -universe is always bijective if M satisfies a suitablegap hypothesis: The fixed point space M G has to be at least 2-dimensional, andthe codimension of M H in M K has to be at least 2 for all subgroups K < H of G .7 A comparison result
Let G be a finite group. A based G -space is called free if the group G acts freelyon the complement of the base point (which is fixed by G ). Let F be a finite freebased G -CW-complex. Since G is finite, this also yields an ordinary CW-structureon F . And it gives an ordinary CW-structure on the quotient Q = F / G .If Y is a G -space, the group { F , Y } G sf is not obviously related to { Q , Y } sincethe G -action on Y need not be trivial. If Y were a trivial G -space, therewould be a tautological map from { Q , Y } to { Q , Y } G sf . One could then use asemi-free version of (5.3) from [1], which says that in this case the composi-tion { Q , Y } → { Q , Y } G sf → { F , Y } G sf with the map induced by the projection from F to Q would be an isomorphism. However, since the action on Y is non-trivial, thearrow on the left is not defined. Nevertheless, there sometimes is a way to com-pare { F , Y } G sf with { Q , Y } . This will be explained now.By mapping the CW-filtrations into Y , one gets three spectral sequences, whichconverge to { F , Y } G sf , { F , Y } and { Q , Y } , respectively.For an integer s , let I ( s ) denote the (finite) set of s -dimensional G -cells in F . Thefiltration gives rise to a spectral sequence E s , t = (cid:8) (cid:95) I ( s ) G + , Σ t Y (cid:9) G sf = ⇒ { F , Σ s + t Y } G sf . (2)In order to compute the groups on the E -page, the differentials on the E -pageneed to be discussed. The t -th row is obtained by applying the functor { ? , Σ t Y } G sf to the G -cellular complex ∗ ←− (cid:95) I ( ) G + ←− (cid:95) I ( ) G + ←− · · · ←− (cid:95) I ( n ) G + ←− ∗ of F . Note that (cid:8) (cid:95) I ( s ) G + , Σ t Y (cid:9) G sf ∼ = (cid:77) I ( s ) (cid:8) G + , Σ t Y (cid:9) G sf , (3)8nd that the groups { G + , Σ t Y } G sf are right modules over { G + , G + } G sf via compo-sition. Because of (3), the differentials can be identified with matrices, and theentries are given by right multiplication with elements of { G + , G + } G sf . Therefore,let me pause to discuss this action in more detail.Sending an element g of G to the G -map G + → G + which sends an element x to xg induces an isomorphism of the group ring Z G with { G + , G + } G sf which reverses theorder of the multiplication. Therefore, { G + , Σ t Y } G sf is a left Z G -module. As such,it is isomorphic with the left Z G -module { S , Σ t Y } , the G -action being inducedby the action on Y . (The adjunction isomorphism (5.1) in [1] is G -linear.) Thisfinishes the digression on the Z G -module structures.One can now try to compare the spectral sequence (2) to the one E s , t = (cid:8) (cid:95) I ( s ) S , Σ t Y (cid:9) = ⇒ { Q , Y } s + t (4)obtained by the induced CW-filtration of Q . (This is of course the Atiyah-Hirze-bruch spectral sequence for Y -cohomology.) As mentioned above, there is noreasonable map between the targets in sight, and I cannot offer a map of spectralsequences. However, note that the groups on the E -pages are always isomor-phic: { G + , Σ t Y } G sf ∼ = { S , Σ t Y } , again by the adjunction isomorphism (5.1) in [1].As for the differentials, the following is true. Proposition 1. If { S , Σ t Y } is a trivial Z G-module, the t-rows on the E -terms ofthe spectral sequences (2) and (4) are isomorphic as complexes. In particular, thegroups on the t-rows of the E -pages are isomorphic.Proof. The differentials on the E -page of the spectral sequence (4) are obtainedby applying the functors { ? , Σ t Y } to the cellular complex ∗ ←− (cid:95) I ( ) S ←− (cid:95) I ( ) S ←− · · · ←− (cid:95) I ( n ) S ←− ∗ Q . As for (2), they can be thought of as matrices. This time, the entries areobtained from the entries of those in (2) by passage to quotients, i.e. by applyingthe map ε : { G + , G + } G sf → { S , S } , g (cid:55)→
1. This means that the diagram { (cid:87) I ( s ) G + , Σ t Y } G sf (cid:47) (cid:47) { (cid:87) I ( s + ) G + , Σ t Y } G sf { (cid:87) I ( s ) S , Σ t Y } (cid:47) (cid:47) ∼ = (cid:79) (cid:79) { (cid:87) I ( s + ) S , Σ t Y } ∼ = (cid:79) (cid:79) (obtained from the isomorphisms above and the differentials) is only commutativeif the elements in Z G which appear in the matrix of the top arrow act via ε . Whileat first sight it seems that one needs to know the details of the G -CW-structure toproceed, this is not the case: if { S , Σ t Y } is a trivial Z G -module, the condition isfulfilled for all elements of Z G .In nice situations, this result implies that { F , Y } G sf and { Q , Y } are in fact isomor-phic. One of these situations will be encountered in the following section. Let W be a orientable real G -representation which in non-trivial and semi-free. The results of the previous section will now be applied to the free G -space F = S W / S W G and Y = S W . Let n and n G be the real dimensions of S W and S W G , respectively. Note that the number n − n G is positive and even. Proposition 2.
The three groups { S W / S W G , S W } , { ( S W / S W G ) / G , S W } , and { S W / S W G , S W } G sf are all free abelian on one generator. roof. As for the group { S W / S W G , S W } : Since the G -representation W is non-trivial, one knows that the quotient S W / S W G is non-equivariantly equivalent toa wedge S n ∨ S n G + . The map from the group { S W / S W G , S W } to { S W , S W } = Z induced by the collapse map is an isomorphism. Note that { S W / S W G , S W } is iso-morphic to H n ( S W / S W G ; Z ) via the Hurewicz map.As for { ( S W / S W G ) / G , S W } : Also via the Hurewicz map, this group is isomorphicto the group H n (( S W / S W G ) / G ; Z ) . Therefore, one can use the spectral sequence E s , t = H s ( G ; H t ( S W / S W G ; Z )) = ⇒ H s + t (( S W / S W G ) / G ; Z ) (5)for that. From the stable homotopy type of S W / S W G it follows that the only non-trivial groups on the E -page are in two rows: t = n G + t = n . Each ofthese contains the cohomology of the group G with coefficients in the trivial G -module Z .Thus there is only one page on which non-trivial differentials may occur. Sincethe dimension of ( S W / S W G ) / G is at most n = dim R ( W ) , all differentials betweennon-trivial groups must be non-trivial. This determines the E ∞ -page. There are noextension problems. It follows that { ( S W / S W G ) / G , S W } ∼ = Z . But, notice the edgehomomorphism of the spectral sequence, which is induced by the quotient mapfrom S W / S W G to ( S W / S W G ) / G . It can immediately be read off that the generatorof H n (( S W / S W G ) / G ; Z ) is mapped to | G | times the generator of H n ( S W / S W G ; Z ) .Of course, this just reminds us that the quotient map has degree | G | .As for { S W / S W G , S W } G sf , Proposition 1 from the previous section can be used. Theassumption on the action is satisfied for Y = S W and all t since the G -action on W preserves the orientation. We can thus deduce some of the groups on the E -pageof the spectral sequence (2) from the previously calculated groups on the E -page of the spectral sequence (4): the groups H s (( S W / S W G ) / G ; π t ( S W )) vanishif s > n or t >
0, and hence the only pair ( s , t ) with s + t = n and E s , t (cid:54) = ( s , t ) = ( n , ) . The corresponding group H n (( S W / S W G ) / G ; Z ) has beenshown to be isomorphic to Z . There are no non-trivial differentials into and out ofit, so it survives to E ∞ . There are no extension problems.11ow that the structure of those three groups is known, it is desirable to know themaps between them. The proof of the preceding proposition shows that the map { ( S W / S W G ) / G , S W } → { S W / S W G , S W } is injective with cyclic cokernel of order | G | . The same holds for the forgetfulmap: Proposition 3.
The forgetful map { S W / S W G , S W } G sf −→ { S W / S W G , S W } is injective and has a cyclic cokernel of order | G | .Proof. Let us contemplate the following diagram. { S W G , S W } G sf { S W , S W } G sf (cid:15) (cid:15) (cid:111) (cid:111) { S W / S W G , S W } G sf (cid:15) (cid:15) (cid:111) (cid:111) { S W , S W } { S W / S W G , S W } ∼ = (cid:111) (cid:111) The horizontal maps are induced by the obvious cofibre sequence. Therefore, thetop row is exact. The vertical maps are the forgetful maps and make the diagramcommute. Now if one picks a map in { S W / S W G , S W } G sf , its image in { S W , S W } G sf hasdegree zero when restricted to the fixed point spheres. It follows that the degreeof the image itself is a multiple of | G | by following proposition. Proposition 4.
Let f : S W → S W be G-equivariant for some finite group G. If thedegree of the restriction of f to the fixed sphere is zero, then the degree of f itselfis divisible by the order of G.Proof. We will use Borel cohomology b ∗ ( X ) = H ∗ ( EG + ∧ G X ; Z / | G | ) with co-efficients in Z / | G | . The b ∗ -modules b ∗ ( S W G ) and b ∗ ( S W ) are free of rank 1 by12he suspension theorem and the Thom isomorphism, respectively. The inclu-sion S W G ⊆ S W induces an inclusion b ∗ ( S W ) ⊆ b ∗ ( S W G ) since the quotient S W / S W G is free and therefore has b ∗ -torsion Borel cohomology. By hypothesis, the map f induces to zero map on b ∗ ( S W G ) , and so it has to on b ∗ ( S W ) . Let W be an orientable real G -representation which is non-trivial and semi-free,as before. The results of the previous section will now be generalised from S W tooriented W -dimensional G -manifolds M . Proposition 5.
Source and target of the forgetful map { M / M G , S W } G sf −→ { M / M G , S W } are free abelian on one generator. The map is injective with a cyclic cokernel oforder | G | .Proof. Choose an orientation preserving G -embedding of W onto a neighbour-hood of a fixed point. Let A be the complement of the image. The collapse mapfrom M to M / A ∼ = S W and the forgetful maps induce a commutative diagram { M / M G , S W } G sf (cid:15) (cid:15) { S W / S W G , S W } G sf (cid:15) (cid:15) (cid:111) (cid:111) { M / M G , S W } { S W / S W G , S W } . (cid:111) (cid:111) By Proposition 3 it is sufficient to show that the two horizontal maps are isomor-phisms. The bottom row is isomorphic to { M + , S W } ←− { S W , S W } , M / M G into S W / S W G is A / A G which is G -free and cohomologically at most ( n − ) -dimensional. It follows that { A / A G , S W } G sf is trivial. It does not follow, however,that { Σ ( A / A G ) , S W } G sf is trivial, too. But injectivity of the top arrow follows frominjectivity of the right arrow, see Proposition 3, and injectivity of the bottom arrow,which has already been proven. Let T be a trivial G -space, and let M be an oriented W -dimensional G -manifoldas before. There is a map from { T , S W G } to { T , S W G } G sf , using the fact that anymap between trivial G -spaces is a G -map, and there is a map from { T , S W G } G sf to { T , S W } G sf induced by the inclusion of S W G into S W which is a G -map. Thecomposition { T , S W G } −→ { T , S W } G sf (6)has a retraction, namely the fixed point map. The splitting theorem implies thata complement for the image of the group { T , S W G } in the group { T , S W } G sf is iso-morphic to { T , EG + ∧ G S W } . (See [14], Section 2, for the version for incompleteuniverses needed here.) One can use that to prove the following. Proposition 6.
The group { M G + , S W } G sf is isomorphic to the { M G + , S W G } , which isfree abelian. The rank is the number of components of M G .Proof. Since S W is ( n − ) -connected, so is EG + ∧ G S W . The dimension of M G + is smaller than n . It follows that the group { M G + , EG + ∧ G S W } is zero. Thus, themap (6) is an isomorphism in the case T = M G + . This and the ordinary Hopf the-orem imply that { M G + , S W } G sf ∼ = { M G + , S W G } , which is free abelian of the indicatedrank. 14 All points
Let M be an oriented W -dimensional G -manifold as before. We are now in theposition to use the information gathered on the free points and on the fixed pointsin order to determine the structure of the source { M + , S W } G sf of the ghost map. Proposition 7.
The group { M + , S W } G sf is free abelian of rank one more than thenumber of components of M G .Proof. The starting point for the calculation of { M + , S W } G sf is the cofibre sequence M G + −→ M + −→ M / M G . Mapping this into S W , we get a long exact sequence · · · ←− { M G + , S W } G sf ←− { M + , S W } G sf ←− { M / M G , S W } G sf ←− · · · . The group in the middle is to be computed. On the left side, since the G -space M / M G is free and of smaller dimension than Σ S W , the next group on theleft { Σ − ( M / M G ) , S W } G sf , which is isomorphic to { M / M G , Σ S W } G sf , is trivial. Onthe right side, the diagram { M / M G , S W } G sfforget (cid:15) (cid:15) { Σ M G + , S W } G sfforget (cid:15) (cid:15) (cid:111) (cid:111) { M / M G , S W } { Σ M G + , S W } (cid:111) (cid:111) shows that the top map from { Σ M G + , S W } G sf to { M / M G , S W } G sf is zero: the left mapis injective by Proposition 5, and the bottom right group is trivial by the dimensionand connectivity of the spaces involved.To sum up, there is a short exact sequence0 ←− { M G + , S W } G sf ←− { M + , S W } G sf ←− { M / M G , S W } G sf ←− .
15y Proposition 6 again, { M G + , S W } G sf ∼ = { M G + , S W G } is free abelian of the indicatedrank. Thus, the short exact sequence must be splittable. Let M be an oriented W -dimensional G -manifold as before. Now that the structureof the source and the target of the ghost map are known, it is time to study the mapitself. Proposition 8.
The ghost map (1) is injective with a cyclic cokernel of order | G | .Proof. One may compare the short exact sequence used in the proof of Proposi-tion 7 to the short exact sequence0 ← { M G + , S W G } ← { M + , S W } ⊕ { M G + , S W G } ← { M / M G , S W } ← , which is built by using the isomorphism between the group { M / M G , S W } and thegroup { M + , S W } discussed before and the identity on { M G + , S W G } . The two shortexact sequences yield the rows in the diagram { M G + , S W } G sf (cid:15) (cid:15) { M + , S W } G sf (cid:15) (cid:15) (cid:111) (cid:111) { M / M G , S W } G sf (cid:15) (cid:15) (cid:111) (cid:111) { M G + , S W G } { M + , S W } ⊕ { M G + , S W G } (cid:111) (cid:111) { M / M G , S W } , (cid:111) (cid:111) which will now be used to compare both of them. The vertical arrow on theright is the forgetful map. This map was shown to be injective with a cycliccokernel of order | G | in Proposition 3. The vertical map in the middle is the ghostmap: it sends a G -map f to the pair ( f , f G ) . The vertical arrow on the left is theisomorphism which sends f to f G as discussed above. The diagram commutes,and the snake lemma implies the result.16rom what has already been shown, it is by now established that the image ofthe group { M + , S W } G sf under the ghost map is a subgroup of index | G | in thedirect sum { M + , S W } ⊕ { M G + , S W G } . This subgroup contains ( | G | , ) and projectsonto { M G + , S W G } . But, this does not determine the image. Additional informationfrom the geometric situation seems to be required. This will be supplied for in thefollowing final section. Let M be an oriented W -dimensional G -manifold as before. In this final section,the group { M + , S W } , which is free abelian on one generator by the ordinary Hopftheorem, has a distinguished generator, namely the one that preserves the orienta-tions. The elements of this group can hence be thought of as integers. Similarly,the fixed point space of M decomposes into components: M G = ∏ α M G α . The dimension of any of the components M G α agrees with that of S W G . Thus thegroup { ( M G α ) + , S W G } has a distinguished generator, too. Collecting these together,the restriction of an element in { M + , S W } G sf to the fixed point space gives a familyof integers, one for each α . Using these identifications, the ghost map sends anequivariant map to a pair ( x , y ) consisting of an integer x and a family y = ( y α ) ofintegers.With these preparations, we may now prove Theorem 3 from the introduction.The proof works as in Example 3 from the introduction. Proof.
Let us fix an α . For any point in M G α , a neighbourhood in M is G -homeomorphic to the G -representation W . Collapsing the complement yields a G -map f α from M + to S W . Note that this map is the chosen generator of { M + , S W } ,17nd the restriction to ( M G α ) + is the chosen generator of the group { ( M G α ) + , S W } .On the other hand, for β (cid:54) = α , the collapse map sends the subspace ( M G β ) + toa point. Thus, the corresponding element in { ( M G β ) + , S W } is zero. Thus if 1 α denotes the characteristic function of α , the element of { M + , S W } G sf which is rep-resented by f α is mapped to ( , α ) .Now for each α there has been produced a G -map f α in { M + , S W } G sf which theghost map sends to a pair ( x , y ) = ( , α ) satisfying the relation in the theo-rem. The subgroup of all pairs satisfying that relation is the unique subgroupof index | G | which contains the pairs ( , α ) for all α . References [1] J.F. Adams, Prerequisites (on equivariant stable homotopy) for Carlsson’slecture, Algebraic Topology (Aarhus 1982), Lecture Notes in Mathematics1051, Springer, Berlin, 1984, 483-532.[2] Z. Balanov, Equivariant Hopf theorem, Nonlinear Anal. 30 (1997) 3463–3474.[3] Z. Balanov and A. Kushkuley, On the problem of equivariant homotopicclassification, Arch. Math. 65 (1995) 546–552.[4] T. tom Dieck, Transformation groups and representation theory, LectureNotes in Mathematics 766, Springer, Berlin, 1979.[5] T. tom Dieck, Transformation groups, de Gruyter Studies in Mathematics 8,Walter de Gruyter, Berlin, 1987.[6] A. Dold, Lectures on algebraic topology, Springer-Verlag, New York-Berlin,1972.[7] D.L. Ferrario, On the equivariant Hopf theorem, Topology 42 (2003) 447-465. 188] J.P.C. Greenlees and J.P. May, Equivariant stable homotopy theory, in:Handbook of algebraic topology, 277-323, North-Holland, Amsterdam,1997.[9] H. Hauschild, ¨Aquivariante Homotopie. I. Arch. Math. 29 (1977) 158–165.[10] H. Hauschild, Zerspaltung ¨aquivarianter Homotopiemengen, Math. Ann.230 (1977) 279–292.[11] H. Hopf, Abbildungsklassengruppen n -dimensionaler Mannigfaltigkeiten,Math. Ann. 96 (1926) 209-224.[12] J. Ize, Equivariant degree, in: Handbook of topological fixed point theory,301–337, Springer, Dordrecht, 2005.[13] L.G. Lewis, Jr., Change of universe functors in equivariant stable homotopytheory, Fund. Math. 148 (1995), 117–158.[14] L.G. Lewis, Jr., Splitting theorems for certain equivariant spectra, Mem.Amer. Math. Soc. 144 (2000).[15] L.G. Lewis, J.P. May and M. Steinberger, Equivariant stable homotopy the-ory, with contributions by J.E. McClure, Lecture Notes in Mathematics1213, Springer, Berlin, 1986.[16] W. L ¨uck, The equivariant degree, in: Algebraic topology and transformationgroups (G ¨ottingen, 1987) 123–166, Lecture Notes in Math. 1361, Springer,Berlin, 1988.[17] J.P. May, Equivariant homotopy and cohomology theory, with contributionsby M. Cole, G. Comeza˜na, S. Costenoble, A. D. Elmendorf, J. P. C. Green-lees, L. G. Lewis, R. J. Piacenza, G. Triantafillou, and S. Waner, dedicatedto the memory of Robert J. Piacenza, Regional Conference Series in Mathe-matics 91, American Mathematical Society, Providence, 1996.1918] M. Szymik, Bauer-Furuta invariants and Galois symmetries. Q. J. Math. 63(2012) 1033-1054.[19] H. Ulrich, Fixed point theory of parametrized equivariant maps. LectureNotes in Mathematics, 1343. Springer-Verlag, Berlin, 1988.[20] S. Waner, Equivariant RO ( G ))