Equivariant Lefschetz and Fuller indices via topological intersection theory

Topology and its Applications 180 (2015) 16–43

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Equivariant Lefschetz and Fuller indices via topological intersection theory Philipp Wruck 1 University of Sheffield, Hicks Building, Sheffield, S3 7RH, United Kingdom

a r t i c l e

i n f o

Article history: Received 17 March 2014 Received in revised form 31 October 2014 Accepted 1 November 2014 Available online xxxx MSC: 55N91 55M20 Keywords: Equivariant Equivariant Equivariant Equivariant

a b s t r a c t For a compact Lie group G, we use G-equivariant Poincaré duality for ordinary RO(G)-graded homology to define an equivariant intersection product, the dual of the equivariant cup product. Using this, we give a homological construction of the equivariant Lefschetz number and a simple proof of the equivariant Lefschetz fixed point theorem. We relate this invariant to existing notions of equivariant Lefschetz numbers and give an explicit computational formula. Using similar techniques, we also construct an equivariant Fuller index with values in the rationalized Burnside ring. © 2014 Elsevier B.V. All rights reserved.

intersection theory Lefschetz number Fuller index fixed point theory

0. Introduction There have been various approaches in the literature to obtain equivariant Lefschetz numbers, both for finite groups and general compact Lie groups. Lewis, May and Steinberger define an equivariant trace for any compact Lie group action in [16], which specializes to a Lefschetz invariant. An invariant for discrete groups has been defined by Lück and Rosenberg in [17]. All these approaches have in common that they are related to group orbits of fixed points. There are similar invariants related to fixed group orbits, treated in e.g. [2,8,6] or [15]. An axiomatic approach to both variants of equivariant Lefschetz numbers has been developed by the author in [22] as a generalization of [13]. In this paper, we give a new definition of a G-equivariant Lefschetz number LG for any compact Lie group G. The basic idea is to use ordinary equivariant homology, cohomology and the respective dual theories as constructed by Costenoble and Waner in [5]. We do not need the full generality of their approach,

1

E-mail address: [email protected]. The author was supported by research grant WR 180/1-1 of the German Research Foundation (DFG).

http://dx.doi.org/10.1016/j.topol.2014.11.001 0166-8641/© 2014 Elsevier B.V. All rights reserved.

P. Wruck / Topology and its Applications 180 (2015) 16–43

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which is rooted in parametrized homotopy theory. We may restrict to spaces over a point and obtain ordinary RO(G)-graded theories which satisfy equivariant Poincaré duality, and so we can define an equivariant intersection product as the dual of the equivariant cup product. The investigation of this topological equivariant intersection product has not appeared in the literature before. The advantages of our approach are threefold. Firstly, it allows for a direct geometric interpretation of the Lefschetz invariant which is rather obscure with the existing definitions. Secondly, it enables an explicit computation of the invariant in terms of local fixed point data, which has been virtually impossible in the case of compact Lie groups of positive dimension. And thirdly, the approach can be generalized to define other equivariant homotopy invariants with a similar geometric background. The paper is organized as follows. In the next section, we recall the main properties of ordinary equivariant homology, cohomology and the respective dual theories. In particular, we describe products and pairings for these theories, review the definition and relation of Thom classes and fundamental classes and end with the equivariant Poincaré duality theorem. The main reference for this material is [16], which treats all this in some generality. A more specialized account, focussing on homology rather than on spectra, can be found in [5]. For an overview, we also refer to [18]. We should also mention that this work has profited greatly from the author’s study of [4], and the presentation is often similar to the one in that source. Section 2 is concerned with the relation of Thom classes and fundamental classes to several constructions, namely restriction to subgroups and fixed points, taking products, and inclusion of submanifolds. The latter two relations are most crucial for the theory and, to our knowledge, have not been investigated so far. Restricting to the special case of orientable V -manifolds allows the theory to be developed along the lines of the non-equivariant constructions in Section 3, yielding an element in the Burnside ring A(G) of G. Standard constructions from fixed point theory allow us to drop most auxiliary assumptions on the manifold, and we can prove as our first main result an equivariant Lefschetz theorem. Recall that for a closed subgroup H of G, we have the fixed point homomorphism ηH : A(G) → A(W (H)), W (H) being the Weyl group of H. Theorem 3.4. Let M be a G-manifold with finite orbit type, f : M → M a G-map with compact fixed point set. Then if f has no fixed point of orbit type (K) with (H) ≤ (K), we have ηH (LG (f )) = 0. We proceed to prove a generalization of this theorem, which calculates the equivariant Lefschetz number in terms of local fixed point data. This result is similar to Theorem 2.1 of [17]. To state it, we introduce some basic concepts. A proper map f : V → V of a G-representation to itself induces a map SV → SV of the one-point compactifications. We denote the stable equivariant homotopy class of this induced map by Deg G (f ). We have an induction map tG H : A(H) → A(G) for subgroups H of G. Our second main result then is the following. Theorem 3.5. Let M be a G-manifold of finite orbit type and f : M → M a G-map with finitely many G-orbits of fixed points Gx1 , . . . , Gxn . We assume that Nxi f has no eigenvalue of unit modulus for i = 1, . . . , n, where Nx f is the component of Tx f normal to the orbit Gx. Then the equivariant Lefschetz number is given as LG (f ) =

n 

  tG Gx Deg Gx (id −Nxi f ) . i

i

i=1

From this theorem it also follows immediately that this equivariant Lefschetz number satisfies all the properties one would expect of such an invariant, like normalization and additivity, compare [20]. We add a short Section 4 to establish relations of our invariant with existing notions. We will show that in case G is finite, our definition agrees with Definition (4.1) of [17] of the “equivariant Lefschetz class with

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P. Wruck / Topology and its Applications 180 (2015) 16–43

values in the Burnside ring”. For general compact Lie groups it follows from the uniqueness of an equivariant Lefschetz number in the sense of [22] that our invariant agrees with the equivariant Lewis–May–Steinberger Lefschetz number. Thus, our approach opens a new viewpoint on these classical invariants and makes them calculable in a rather straightforward manner. We finish the paper by exploiting another major advantage of our approach in Section 5, namely the possibility to define an equivariant homotopy invariant related to flows instead of maps. This is an equivariant version of the Fuller index as constructed in [12] or [11]. The rough idea is to assign to an orbit of fixed points of a flow the equivariant Lefschetz number of a return map related to that flow. Our machinery allows us to do this homologically, obtaining an equivariant index which is related to the index constructed in [23] in the same way as the homological Fuller index of [11] is related to the ordinary index of Fuller, see [12]. Theorem 5.1. There exists an equivariant Fuller index FG for a flow ϕ with respect to an isolated set C of periodic points. It is a G-homotopy invariant with the following properties. i) It takes values in the rationalized Burnside ring A(G) ⊗ Q. ii) If C consists of finitely many periodic orbits γ1 , . . . , γn and ϕi is the flow ϕ, restricted to an isolating neighbourhood of the orbit γi , then

FG (ϕ) =

n 

FG (ϕi ).

i=1 1 iii) If ϕ has a single periodic orbit of multiplicity m, then FG (ϕ) = LG (P m ) ⊗ m ∈ A(G) ⊗ Q, where P is an equivariant Poincaré map for the orbit, considered with multiplicity one. iv) If ηH (FG (ϕ)) = 0, then ϕ has a periodic orbit of orbit type (K) with (H) ≤ (K).

Such an invariant has not appeared in the literature apart from [23], where the author used techniques from the theory of dynamical systems to construct the index. We will show that FG actually equals the dynamical index of [23]. The author would like to thank John Greenlees for many interesting discussions on the subject. 1. Preliminaries We have to establish some conventions at the beginning. Throughout the paper, unless stated otherwise, G will be a compact Lie group. Subgroups of G are always assumed to be closed. A G-space X is a pointed topological space X with a left G-action G × X → X, (g, x) → gx. The base point of X is fixed by G. A G-manifold is assumed to be smooth with a smooth G-action and is not assumed to be pointed. For homological considerations, we will work with pointed G-spaces throughout and we will add a disjoint base point to unbased spaces Y , denoting the result by Y+ . We will use several standard constructions from equivariant topology, all of which can be found in [1]. Most notable is the twisted product X×H Y of a right unpointed H-space X with a left unpointed H-space Y . This is defined to be the quotient space (X × Y )/H, where H acts as h(x, y) = (xh−1 , hy). The twisted product becomes a left unpointed G-space provided X carries a left G-action such that g(xh) = (gx)h for all g ∈ G, h ∈ H, x ∈ X. Then g[x, y] = [gx, y] is a well defined action of G, where [x, y] denotes the class of (x, y) in X ×H Y . We also have the pointed analogue X ∧H Y , which is obtained from X ∧ Y by identifying [xh, y] with [x, hy]. As before, this carries a left G-action, provided X carries a left G-action with the aforementioned compatibility assumption. In almost all cases, X will be the G-space G, acting by left and right translations on itself.

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In general, when we have to denote a class of an element x under a standard quotient map, such as G → G/H, X × Y → X ∧ Y and the like, we will use the notation [x] for that class, as long as no confusion is probable. The remainder of this chapter consists of an overview of constructions and results of [5] and [16]. It is well-known, see e.g. [18], that there exist ordinary RO(G)-graded equivariant homology and cohomology theories and their respective duals with coefficients general Mackey functors. In this paper we are only concerned with the special contravariant Burnside coefficient system. Recall that the Burnside ring of a compact Lie group G is the free abelian group on orbit types G/H for closed subgroups H ⊆ G such that N (H)/H = W (H), the Weyl group of H, is finite. The Burnside coefficient system is given by AG/G (G/H) = {G/H+ , G/G+ }G , where here and in the following, for G-spaces X, Y we denote by {X, Y }G the abelian group of stable equivariant homotopy classes of stable G-maps X → Y . The name stems from the fact that AG/G (G/H) is isomorphic to A(H), a fundamental property proven in [21]. The ordinary theories are represented by equivariant ring spectra and we will sometimes work on the represented level. For a construction of the representing spectra, see e.g. [14]. We write H for the spectrum representing ordinary cohomology. The homology represented by H is called “dual ordinary homology”. We note that in case G is infinite, this is different from ordinary homology. We suppress the coefficient functor from the notation and write ∗ ∗ HG (X) = HG (X; AG/G ),

H∗G (X) = H∗G (X; AG/G ).

The curly H denotes equivariant dual ordinary homology. The coefficient ring of these theories is the Burnside ring A(G). Since we will need it repeatedly, we mention the Wirthmüller isomorphism [16, Chapter II.6]. For K ⊆ G a subgroup, α = V − W ∈ RO(G) and a K-spectrum X, this is a natural isomorphism α HG (G+ ∧K X) ∼ = HK (X). α|K

If L denotes the tangential space at [e] in G/K, then there is a natural dual Wirthmüller isomorphism    G K Hα G+ ∧K S−L ∧ X ∼ (X). = Hα|K These isomorphisms allow us to define restriction maps with respect to subgroups. We recall from [16] that the Spanier–Whitehead dual of an orbit G/K + is given by the spectrum G+ ∧K S−L . This, together with the Wirthmüller isomorphisms, yields maps α|K

α HG (X) → HK (X),

G K Hα (X) → Hα|K (X).

There are also restrictions with respect to fixed sets for normal subgroups K ⊂ G. For the actual definition, we refer to [4], Definition 1.5.6, or the general theory of [16]. In any case we obtain natural maps  K α αK HG (X) → HG/K X ,

G/K 

G Hα (X) → HαK

 XK .

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If K is any closed subgroup, we define restriction to fixed sets by the composition with restriction to the α αK K normalizer N (K) of K, followed by the restriction above. Thus, we obtain maps HG (X) → HW (K) (X ), and similarly for the dual theories. There are several products and pairings, which we summarize in the following statement. The precise definitions are straightforward, see e.g. [16, Chapter III.3], using the representing ring spectra. Proposition 1.1. Let X, Y, Z be pointed G-spaces, α, β ∈ RO(G) and let d : Z → X ∧ Y be a G-map. There is a cohomology cross product β α+β α × : HG (X) ⊗ HG (Y ) → HG (X ∧ Y )

This yields a cup product β α+β α ∪d : HG (X) ⊗ HG (Y ) → HG (Z),

by following up the cross products with the map induced by d in cohomology. Furthermore, there is a pairing α G

− , −d : HG (Y ) ⊗ HβG (Z) → Hβ−α (X).

The products and pairing are natural in the expected way: Let X  , Y  , Z  be G-spaces and g : X → X  , ∗ ∗ h : Y → Y  , f : Z → Z  be G-maps. Then for x ∈ HG (X  ), y  ∈ HG (Y  ) we have       g ∗ x × h∗ y  = (g ∧ h)∗ x × y  . Consequently, if the diagram d

Z f

X ∧Y g∧h



d

Z

X ∧ Y 

commutes, then       g ∗ x ∪d h∗ y  = f ∗ x ∪d y  . Furthermore, if z ∈ H∗G (Z), then 



y  , f∗ (z)

d

    = g∗ h∗ y  , z d .

In addition, the products and pairings respect restriction to subgroups and fixed spaces. If the map d is a diagonal d : X → X ∧ X, or closely related to such a diagonal, we also use the notation ξ ∩d x for ξ, xd and speak of the cap product. Occasionally we will need associativity of the cup product. Proposition 1.2. Let A, B, C, X, Y , Z be G-spaces and let d1 : A → X ∧ Y , d2 : B → A ∧ Z, d3 : C → Y ∧ Z and d4 : B → X ∧ C be G-maps such that the diagram

P. Wruck / Topology and its Applications 180 (2015) 16–43

d2

B

A∧Z d1 ∧id

d4

X ∧C

21

id ∧d3

X ∧Y ∧Z

commutes. Then β γ α HG (X) ⊗ HG (Y ) ⊗ HG (Z)

∪d1 ⊗id

α+β γ HG (A) ⊗ HG (Z) ∪d2

id ⊗∪d3 β+γ α HG (X) ⊗ HG (C)

∪d4

α+β+γ HG (B)

commutes. To make use of the equivariant duality theory for manifolds as in [5], we have to restrict our attention to a special case of G-manifolds, namely, manifolds modelled with respect to a single representation V . This is due to the fact that we want to avoid twisted gradings of our theories. We can generalize to arbitrary G-manifolds later, so it doesn’t really matter, but it may be interesting to carry out the constructions using the general twisted theory of [5], which establishes duality for arbitrary G-manifolds. The bundles and manifolds we will be looking at are defined as follows. Definition 1.3. Let p : E → B be a G-vector bundle. p is said to be a V -vector bundle, if every fibre p−1 (x) over a point x is Gx -isomorphic to the restriction of a G-representation V . A G-manifold M is said to be a V -manifold, if its tangential bundle is a V -vector bundle. An alternative and more conceptual description of V -vector bundles is that a V -vector bundle is a map p : E → B such that every point x ∈ B has a neighbourhood U such that p−1 (U ) is Gx -bundle isomorphic to U × V . A V -manifold then is a G-manifold M such that every point x ∈ M is contained in a chart Gx -isomorphic to V . The notions of Thom classes and fundamental classes for V -bundles and V -manifolds are straightforward generalizations of the non-equivariant situation. Definition 1.4. Let p : E → B be a V -vector bundle and M be a V -manifold. V i) Let T (p) be the Thom space of p. A Thom class for p is a class τ ∈ HG (T (p)) such that for every G-map ϕ : G/H → B, the image of τ under the maps

    ∗  V V HG T (p) → HG T ϕ (p)   V ∼ G+ ∧H SV = HG  V V ∼ S = HH ∼ = A(H) is a generator (i.e. a unit). ii) If M is compact, a fundamental class for M is a class O ∈ HVG (M+ ) such that the following holds. For any point x ∈ M , let νx be a normal bundle of the embedding Gx → M and let T (νx ) be the Thom space of this embedding. Let Lx be the tangential space of Gx at x and ψ : M+ → T (νx ) be the Pontryagin–Thom map. Then the image of O under the maps

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P. Wruck / Topology and its Applications 180 (2015) 16–43

  ψ∗ HVG (M+ ) −→ HVG T (νx )   ∼ = HVG G+ ∧H SV −Lx   ∼ = HVH SV ∼ = A(H) is a generator. iii) If M is compact with boundary, a fundamental class for M is a class O ∈ HVG (M/∂M ) such that for any point x ∈ M \ ∂M with normal bundle νx , Thom space T (νx ), Lx and ψ : M/∂M → T (νx ) as above, the image of O in   ψ∗ HVG (M/∂M ) −→ HVG T (νx )   ∼ = HVG G+ ∧H SV −Lx   ∼ = HVH SV ∼ = A(H) is a generator. iv) If N ⊆ M is a compact W -submanifold of the compact V -manifold M with a fundamental class, we M G define the fundamental class ON of N in M to be the image iM N ∗ (ON ) ∈ HW (M+ ) of the fundamental G M class ON ∈ HW (N+ ) under the inclusion iN : N → M . Existence of fundamental classes follows from the equivariant Thom isomorphism theorem [5, Theorem C]. More precisely, a fundamental class exists if and only if a Thom class for the normal bundle of an embedding into a G-representation exists. This in turn is the case if and only if the bundle is G-orientable in the sense of [7]. This last assertion is essentially the following equivariant Thom isomorphism theorem. Theorem 1.5. ([5], Theorem C) Let p : E → B be a V -vector bundle. p is G-orientable if and only if it has V an equivariant Thom class τ ∈ HG (T (p)). In that case, the map   α+V α HG (B) → HG T (p) ,

a → a ∪d τ

is an isomorphism, where d : T (p) → B+ ∧ T (p) is the Thom diagonal. The connection between Thom classes and fundamental classes is given by the following result. It follows directly from the identification of duals of manifolds as in [16, Chapter III.5]. Proposition 1.6. Let M be a compact V -manifold with (possibly empty) boundary and let M → W be an embedding into an orthogonal G-representation W . Let ν be the normal bundle of that embedding. The following sequence consists of isomorphisms   α+W −V HG (T ν) ∼ = T ν, Σ α+W −V H G   ∼ = Σ W D(M/∂M ), Σ α+W −V H G   ∼ = SV −α , M/∂M ∧ H G ∼ = HVG−α (M/∂M ).

P. Wruck / Topology and its Applications 180 (2015) 16–43

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W −V In addition, taking α = 0, a class τ ∈ HG (T ν) is a Thom class for ν if and only if its image O ∈ HVG (M/ ∂M ) is a fundamental class for M .

Using the preceding proposition together with the Thom isomorphism theorem and some more calculations of duals from [16, Chapter III.5], one can prove the following equivariant Poincaré duality theorem. It can be seen as a specialization of the more general discussion in [16, Chapter III.6]. Theorem 1.7. Let M be a compact orientable V -manifold with possibly empty boundary. Then M has a fundamental class, and the following pairings induce isomorphisms, the equivariant Poincaré duality isomorphisms, given by capping with the fundamental class of M . i) With d : M/∂M → M+ ∧ (M/∂M ) induced by the diagonal of M+ , ∩

d α HG (M+ ) ⊗ HVG (M/∂M ) −→ HVG−α (M/∂M )

induces the Poincaré isomorphism ∼ =

α HG (M+ ) −→ HVG−α (M/∂M ).

ii) With d : M/∂M → (M/∂M ) ∧ M+ induced by the diagonal of M+ , ∩

d α HG (M/∂M ) ⊗ HVG (M/∂M ) −→ HVG−α (M+ )

induces the Poincaré isomorphism ∼ =

α HG (M/∂M ) −→ HVG−α (M+ ).

2. Restriction, products and submanifolds In this section we investigate the behaviour of fundamental classes and Thom classes with respect to products and restriction homomorphisms. We also derive a relation between Thom classes of submanifolds which will be important when we prove the equivariant Lefschetz fixed point theorem. We begin with restriction properties. With the obvious adaptions, the result is also true for manifolds with boundary. While a proof for the following result for Thom classes instead of fundamental classes is given in [4], the remaining results in this section seem not to appear in the literature so far. Proposition 2.1. Let M be a V -manifold. Then the following are equivalent for a class O ∈ HVG (M+ ): i) ii) iii) iv)

O is a fundamental class for M . O|H is a fundamental class for M as an H-manifold for every subgroup H of G. OH is a fundamental class for M H as a W (H)-manifold for every subgroup H of G. OH |e is a fundamental class for M H as an e-manifold for every subgroup H of G.

Proof. i) =⇒ ii): A fundamental class O is characterized by the property that for every G-embedding ϕ : G/ H → M , O maps to a generator of HVG (T νϕ ), where νϕ is the Thom space of a normal bundle associated with the embedding.

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P. Wruck / Topology and its Applications 180 (2015) 16–43

So let ϕ : H/K → M be an H-map and let Φ : G/K → M be its unique extension to a G-map. We can write V = L ⊕ N = L1 ⊕ L2 ⊕ N, where L = T[e] G/K, L1 = T[e] H/K and L2 = T[e] G/H. Then the Thom space of the embedding ϕ is H-homeomorphic to H+ ∧K SL2 ⊕N and the Thom space of the embedding Φ is G-homeomorphic to G+ ∧K SN . We have to investigate the diagram HVG (M+ )

ψ∗G

HVG (T νΦ ) ∼ =

  HVG G+ ∧H H+ ∧K SV −L G wH

G rH

  HVH H+ ∧K SV −L1 ∼ =

HVH (M+ )

ψ∗H

HVH (T νϕ ),

where ψ G and ψ H are the Pontryagin–Thom maps of the embeddings Φ and ϕ, respectively. The map down the right is essentially the Wirthmüller isomorphism. If we could show that the diagram commutes, the claim would follow. For this, it suffices to show that the dual of the diagram commutes. We embed M into a G-representation W with normal bundle ν and use the facts that   D(T νϕ ) ∼ = Σ −W T ϕ∗ ν ,

D(M+ ) ∼ = Σ −W T ν,

  D(T νΦ ) ∼ = Σ −W T Φ∗ ν

and D(ψ G ) = Σ −W T Φ, D(ψ H ) = Σ −W T ϕ. Here, νΦ , νφ are normal bundles of the embeddings ϕ and Ψ into M . These isomorphisms, and the identification of the duals of the Pontryagin–Thom maps, are established [16, Chapter III.5], in particular Proposition III.5.5. The dual diagram therefore has the form W −V HG (T ν)

T Φ∗

  ∗  W −V HG T Φ ν ∼ =

  W −V HG G+ ∧H H+ ∧K SW −V G rH

G wH

  W −V HH H+ ∧K SW −V ∼ = W −V HH (T ν)

∗

Tϕ

  W −V HH T (ϕ∗ ν) .

G But now it is obvious that this diagram commutes, because the restriction rH is actual restriction of G-maps G to H-maps, wH is the usual adjunction and T Φ = G+ ∧H T ϕ.

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ii) =⇒ i): This is trivial. ii) =⇒ iii): We can assume that H is normal in G, otherwise we restrict to N (H) first. Let K ⊆ G be a subgroup containing H and ϕ : G/K → M an embedding. If G+ ∧K SN is the Thom space of that embedding, since H is normal, we have  H H ∼ G+ ∧K SN = G/H + ∧K/H SN . Now consider the commutative diagram G/H 

HVG (M+ )

HV H G/H 

HV H

G/H + ∧K/H SN

∼ =

A(K)



wG/K

K/H  V H 

HV H

σ

  H0K S0

H

G/H

G wK

  HVK SV



ψH

ψ

  HVG G+ ∧K SN

H M+

S

σ K/H  0 

H0

S

∼ =

A(K/H),

where the horizontal maps are restrictions and the vertical maps are Pontryagin–Thom maps, Wirthmüller isomorphisms and suspensions, respectively. The bottom horizontal map sends a generator [K/L] to [(K/L)H ]. This map is a ring homomorphism, so it sends units to units. Thus, if we start with the fundamental class in the upper left, we end up with a unit in the lower left by definition, and this maps to a unit in the lower right. Therefore, the restriction of the fundamental class maps to a unit via the vertical map on the right, which characterizes it as a fundamental class. iii) =⇒ iv): This follows from the equivalence of i) and ii). iv) =⇒ i): The fundamental class is determined by the property that it maps to a unit under       HVG (M+ ) → HVG G+ ∧H SN ∼ = HVH SV ∼ = A(H) ∼ = colimW SW , SW H . L

L

An element in colimW {SW , SW }H is a unit if and only if its restriction to the colimit of the sets {SW , SW } is a unit for every subgroup L of H. From this, the claim follows immediately. 2 In the non-equivariant case it follows easily from the Kuenneth theorem that the homology cross product of two fundamental classes is a fundamental class for the product manifold. The situation is more difficult in the equivariant case and we take a more direct approach to prove this fact. The essential ingredient is to show that the cohomology cross product of two Thom classes is a Thom class for the product bundle. Proposition 2.2. Let p : E → B be an orientable V -vector bundle, q : E  → B  be an orientable W -vector V W bundle, with Thom classes τ ∈ HG (T (p)), τ  ∈ HG (T (q)), respectively. Then the product bundle p × q :   E × E → B × B is an orientable V ⊕ W -bundle and a Thom class is given by the image of τ ⊗ τ  under     ×     V +W V +W V W HG T (p) ⊗ HG T (q) −→ HG T (p) ∧ T (q) ∼ T (p × q) . = HG

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P. Wruck / Topology and its Applications 180 (2015) 16–43

Proof. We show that the stated class is a Thom class for p × q. So let ϕ = (ϕ1 , ϕ2 ) : G/H → B × B  be a G-map. Consider the diagram     V W T (p) ⊗ HG T (q) HG

×

  V +W T (p × q) HG

    V W HG T (ϕ∗1 p) ⊗ HG T (ϕ∗2 q)

×

  V +W HG T (ϕ∗1 p × ϕ∗2 q)

∼ =

∼ =

 V   W  V W HG Σ G/H + ⊗ HG Σ G/H +

×

 V +W  V +W HG Σ G/H + ∧ G/H +

∼ = 0 0 HG (G/H + ) ⊗ HG (G/H + )

∼ = ×

0 HG (G/H + ∧ G/H + ) Δ∗

∼ =

0 HG (G/H + ) ∼ =

A(H) ⊗ A(H)

μ

A(H).

The top square commutes due to multiplicativity of Thom spaces, and the second square commutes since we just use the standard identification of Thom spaces over group orbits. The third square commutes due to naturality of the product with respect to suspension, the last square is a restatement of the fact that the cup product generalizes the ring multiplication in A(H). This follows directly from the definition of the product and the fact that the Burnside ring product is induced by smash product of representatives. If we start with the product of two Thom classes in the upper left, these are mapped to a tensor product of units in the lower left by definition of the Thom class. So they are mapped to a unit in the lower right. But the map down the right column is just the composition     V +W V +W HG T (p × q) → HG (T ϕ∗ (p × q)  V +W  V +W ∼ Σ G/H + = HG 0 ∼ (G/H + ) = HG ∼ = A(H),

since ϕ1 × ϕ2 ◦ Δ = (ϕ1 , ϕ2 ) = ϕ. Thus, the potential Thom class maps to a generator under restriction via ϕ, which characterizes it as a Thom class. 2 Our intended result for fundamental classes follows immediately. Corollary 2.3. Let M, N be orientable V, V  -manifolds, respectively. Then M × N is an orientable V ⊕ V  -manifold and the product of the fundamental classes is a fundamental class for M × M . Proof. Let O, O be fundamental classes for M, N . Let ν be a normal bundle for an embedding M → W W −V into a G-representation, similarly ν  for an embedding N → W  . We have Thom classes τ ∈ HG (T ν),   W −V   τ ∈ HG (T ν ) such that in the commutative diagram

P. Wruck / Topology and its Applications 180 (2015) 16–43

W −V W HG (T ν) ⊗ HG



−V 

(T ν  )

×

W +W HG



−V −V 

∼ =



T (ν × ν  )

∼ =

0 0 HG (M+ ) ⊗ HG (N+ ) ∼ =



27

×

  0 HG (M × N )+ ∼ =

P ⊗P 

P×



 HVG+V  (M × N )+ ,

HVG (M+ ) ⊗ HVG (N+ )

the element O ⊗ O in the lower left maps to τ ⊗ τ  in the upper left. τ ⊗ τ  maps to a Thom class of the normal bundle of the embedding M × N → W × W  , and this Thom class maps to a class in the lower right which, by Proposition 2.2, is a fundamental class. 2 We want to investigate the behaviour of Thom classes of submanifolds in several situations. For our first result, we need a variation of the Thom diagonal. It is readily constructed by realizing Thom spaces via certain tubular neighbourhoods. Let P ⊆ M ⊆ W be equivariant embeddings of invariant submanifolds. We find an invariant tubular neighbourhood UPW of P in W , an invariant tubular neighbourhood UPM ⊆ UPW of P in M and an invariant W W tubular neighbourhood UM of M in W . Let s : UM → M be the tubular retraction. Then we can choose W W the neighbourhoods in a way that, if z ∈ ∂UP , then either z ∈ / UM , or s(z) ∈ / UPM . For example, we can choose UPW such that it contains s−1 (UPM ). With this choice, the map W d : T νPW → T νPM ∧ T νM ,



x → s(x), x ,

is well defined. Proposition 2.4. Let M be an orientable V -manifold, P ⊆ M an orientable Z-submanifold and M → W an embedding of M into a G-representation W . Let νPM be a normal bundle for the embedding P → M , W νM a normal bundle for the embedding M → W and let νPW be the induced normal bundle of the embedding P → W . Let  M  W V −Z W −V W τPM ∈ HG T νP , τM ∈ HG T νM W −Z W W be Thom classes for the normal bundles νPM , νM , respectively. Then τPM ∪d τM ∈ HG (T νPW ) is a Thom W class for the bundle νP , where the cup product is taken with respect to the map

W d : T νPW → T νPM ∧ T νM , x → s(x), x

defined above. ∗ W W M Proof. Let ϕ : G/H → P be a G-embedding. ϕ induces maps T ϕW P : T (ϕ (νP )) → T νP , T ϕP : ∗ W W T (ϕ∗ (νPM )) → T νPM and T ϕW M : T (ϕ (νM )) → T νM . These fit together, after identification of the Thom spaces over orbits, in the diagram d

T νPW

W T ϕM P ∧T ϕM

T ϕW P

G+ ∧H SW −Z

W T νPM ∧ T νM

d

G+ ∧H SV −Z ∧ G+ ∧H SW −V ,

where d is induced by the diagonal map on G+ and an identification of the spheres.

28

P. Wruck / Topology and its Applications 180 (2015) 16–43

This implies commutativity of the top square in  M  W V −Z W −V HG T νP ⊗ HG T νM

 W W −Z HG T νP

∪d

    V −Z W −V G+ ∧H SV −Z ⊗ HG G+ ∧H SW −V HG  V −Z   W −V  V −Z W −V HH S ⊗ HH S

  W −Z G+ ∧H SW −Z HG

∪d

 W −Z  W −Z HH S

×

μ

A(H) ⊗ A(H)

A(H).

The rest of the diagram commutes due to naturality of the Wirthmüller isomorphism and the fact that the cross product is compatible with suspension. W Starting with the product τPM ⊗ τM in the upper left, this maps to a product of units in A(H) ⊗ A(H) W = τPW is a Thom class. 2 and thus to a unit in the lower right, implying that τPM ∪d τM Next, assume that P ⊆ M is an invariant G-orientable submanifold of the G-orientable V -manifold M and moreover that P is a W -manifold for some G-subrepresentation W ⊆ V . Let U be an invariant tubular neighbourhood of P . We can regard this as a V − W -bundle νPM : U → P , and this bundle has a Thom V −W class τ living in HG (T νPM ). Now we have the Pontryagin–Thom map M+ → T νPM , and we denote the image of τ under the induced map  M V −W V −W HG T νP → HG (M+ ) by τPM and call it the Thom class of P in M . Proposition 2.5. Let M be an orientable V -manifold and P, Q be orientable ZP - and ZQ -submanifolds, respectively. Assume that P ∩ Q is a Z-manifold with V − ZP = ZQ − Z ∈ RO(G) and such that ∗ M  iP = νPQ∩Q P ∩Q νP ∗

M M M is a normal bundle for the inclusion iQ P ∩Q : P ∩ Q → Q. Then if τP is a Thom class for P in M , iQ (τP ) is a Thom class for P ∩ Q in Q.

Proof. Firstly, let ϕ : G/H → P ∩ Q be any G-map. iP P ∩Q ◦ ϕ : G/H → P is a G-map making the diagram  M V −ZP HG T νP

T ϕ∗ T iP P ∩Q

∗

  ∗ P ∗  M  V −ZP HG T ϕ iP ∩Q νP ∼ =

∗ T (iP P ∩Q )

  P ∗  M   Z −Z  V −ZP ∼ HG T iP ∩Q νP T νPQ∩Q = HG Q

T ϕ∗

Z −Z 

HG Q

   T ϕ∗ νPQ∩Q

commutative. This shows that the image of the Thom class of νPM is a Thom class for νPQ∩Q .

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29

Secondly, let U ⊆ M be an invariant tubular neighbourhood for P in M such that νPM : U → P is the bundle projection of the normal bundle. By assumption, the pullback of this bundle via iP P ∩Q is a normal bundle for P ∩ Q in Q, which implies that the diagram T (νPQ∩Q )

Q+

  T iP P ∩Q

iM Q

  T νPM

M+

commutes. It follows that in cohomology,   Q  V −ZP HG T νP ∩Q

V −ZP HG (Q+ )

∗ T (iP P ∩Q )

iM Q

  M  V −ZP T νP HG

∗

V −ZP HG (M+ )

commutes. Starting with the Thom class in the lower left, we have τPM in the lower right, which maps up ∗ M Q Q to iM Q (τP ). In the upper left we have the Thom class for νP ∩Q , which by definition maps to τP ∩Q under ∗ Q M the upper horizontal map. Thus, the identity iM Q (τP ) = τP ∩Q is proven. 2 It should be noted that the above result holds in particular when P ∩ Q is empty. In that case, it shows ∗ M V −ZP that iM (Q+ ). Q (τP ) = 0 ∈ HG Our last result will relate Thom classes of submanifolds to fundamental classes of submanifolds. This will help us to define the equivariant Lefschetz number in a more general setting then just compact orientable V -manifolds. It will also be of use in the proof of the equivariant Lefschetz fixed point Theorem 3.4. Proposition 2.6. Let P ⊆ M be a Z-submanifold of the V -manifold M . Let τPM be a Thom class of P in M . Let OPM be the image of its corresponding fundamental class OP under the inclusion P → M . Then τPM ∩ OM = OPM i.e. OPM is the equivariant Poincaré dual of τPM . Proof. Again we have to fix the various tubular neighbourhoods first. P embeds into W via the composition of the embeddings of P in M and of M in W . We find invariant tubular neighbourhoods U of M in W and U  of P in M , together with retractions s : U → M , r : U  → P . We can arrange this such that s−1 (U  ) = U  is a neighbourhood of P in W contained in U and r ◦ s : U  → P is the tubular retraction. W We obtain a Pontryagin–Thom map for the inclusion U  → U , i.e. a map ψ : T νM → T νPW . Using the fact that the map dual to the inclusion is the Pontryagin–Thom map, which follows from [16, Lemma III.3.7], the duality diagram coming from Proposition 1.6 becomes G HZ (P+ )

i∗

∼ =

 W W −Z HG T νP

G HZ (M+ ) ∼ =

ψ∗

 W W −Z HG T νM ,

30

P. Wruck / Topology and its Applications 180 (2015) 16–43

We now start with the bottom row and complete it to the following diagram.  W W −Z HG T νP

 W W −Z HG T νM ∪tW P

∪tW P 0 HG (P+ )

∪tW M ∪tM P

∪tM P

 M V −Z T νP HG

V −Z HG (M+ )

The cup products involved come from different diagonals. The upper left triangle commutes due to commutativity of the diagram P+ ∧ T νPW

W T νM ψ

id ∧ id

T νPW

P+ ∧ T νPW .

The upper horizontal map is the modified Thom diagonal, with the ψ in the second component and the retraction onto P in the first. The lower horizontal map is the actual Thom diagonal. Naturality of the cup product implies the commutativity of the first triangle. The lower left triangle commutes by the same reasoning, using the diagram M+

P+ ∧ T νPM id ∧ id

T νPM

P+ ∧ T νPM .

0 Thus, we see that an element x ∈ HG (P+ ) maps to x ∪ tW P in the upper right. The various diagonals involved fit together in an associativity diagram as in Proposition 1.2. That proposition therefore implies M W that mapping x to x ∪ tM P in the lower right and then with the Thom isomorphism to (x ∪ tP ) ∪ tM gives W W x ∪ (tM P ∪ tM ). By Proposition 2.4, this is equal to x ∪ tP , so the whole diagram commutes. The two maps on the left are both Thom isomorphisms, so they compose to give an isomorphism which maps the Thom classes to one another. Pasting the two diagrams together yields

G HZ (P+ )

i∗

∼ =

 W W −Z HG T νP ∼ =

 M V −Z HG T νP

G HZ (M+ ) ∼ =

ψ∗

 W W −Z HG T νM ∼ = V −Z HG (M+ ),

and the lower horizontal map is induced by the Pontryagin–Thom map of P ⊆ M . The composition up the right is the cap product with the fundamental class, which follows from Proposition 1.6 and the definition

P. Wruck / Topology and its Applications 180 (2015) 16–43

31

of the cap product. It therefore follows that starting with the fundamental class of P in the upper left, we end up with the Thom class of the embedding P → M in the lower left, and by definition, we end up with τPM in the lower right. Then going up gives τPM ∩ OM , and this is equal, by commutativity of the diagram, to i∗ (OP ) = OPM . 2 3. The equivariant Lefschetz number and its properties We are now prepared to define our central objects, namely the equivariant intersection product and the equivariant Lefschetz number, and to derive their fundamental properties. The intersection product is simply the Poincaré dual of the cup product. ∗ (M+ ) → HVG−∗ (M+ ) be the Definition 3.1. Let M be an orientable compact V -manifold and let P : HG Poincaré duality isomorphism. The intersection product on M is defined as G G • : Hα (M+ ) ⊗ HβG (M+ ) → Hα+β−V (M+ ),

  x ⊗ y → P P −1 (y) ∪ P −1 (x) .

The definition of the equivariant Lefschetz number on a compact orientable V -manifold is now straight forward. Definition 3.2. Let M be a compact orientable V -manifold and f : M → M a G-map. Let OΔ ∈ HVG ((M × M )+ ) be the fundamental class of the diagonal submanifold of M × M in M × M , and let OΓ ∈ HVG ((M × M )+ ) be the fundamental class of the graph manifold Γ = {(x, f (x)) | x ∈ M } ⊆ M × M . The equivariant Lefschetz number of f is defined to be the intersection product of OΔ and OΓ , followed by the evaluation map ε : H0G ((M × M )+ ) → H0G (S0 ) ∼ = A(G): LG (f ) = ε(OΔ • OΓ ). To generalize this definition to a more general type of manifold, we first rephrase it a bit. Consider the diagram M+

Δ

(id,f )

(M × M )+

(M × M )+ ((id,f ),(id,f ))

Δ×

(M × M × M × M )+ ,

where Δ× is the diagonal embedding of the product M × M . Using naturality of the cap product, we can α conclude that for classes x ∈ HG ((M × M )+ ) and y ∈ HβG (M+ ), we have   (id, f )∗ (id, f )∗ (x) ∩ y = x ∩ (id, f )∗ (y). We apply this to the intersection product OΔ • OΓ , and use the fact that the dual of OΔ is given by the M ×M element τΔ = ψ ∗ (τ ), where τ is the Thom class of the embedding Δ ⊆ M × M and ψ : (M × M )+ → M ×M is the associated Pontryagin–Thom map. Similarly, the dual of OΓ is an element τΓM ×M , associated T νΔ to the graph embedding Γ → M × M . These are both consequences of Proposition 2.6. We obtain  M ×M  OΔ • OΓ = τΔ ∪ τΓM ×M ∩ OM ×M   M ×M = τΔ ∩ τΓM ×M ∩ OM ×M M ×M = τΔ ∩ (id, f )∗ (OM )   M ×M   = (id, f )∗ (id, f )∗ τΔ ∩ OM .

P. Wruck / Topology and its Applications 180 (2015) 16–43

32

Since the diagram

M

(id,f )

M ×M

∗ commutes, it follows for the equivariant Lefschetz number that   M ×M   LG (f ) = ε (id, f )∗ τΔ ∩ OM . Writing this in diagrammatic form, we have that the equivariant Lefschetz number equals the image of the Thom class τ of the diagonal embedding of M into M × M under the sequence of maps  M ×M  ψ∗   (id,f )∗ V ∩OM ε V V HG T νΔ −→ HG (M × M )+ −→ HG (M+ ) −→ H0G (M+ ) −→ A(G), where the first map is induced by the Pontryagin–Thom map. Now if M is a non-compact orientable V -manifold and f has compact fixed point set, let N be an invariant open neighbourhood of Fix(f ), such that N is an invariant manifold with boundary. As an open subset of M ×M an orientable V -manifold, N is an orientable V -manifold. Let T νΔ be the Thom space of the diagonal embedding of M into M × M . We can realize this Thom space as an invariant neighbourhood of Δ such that the image of the boundary of N under (id, f ) does not lie in that neighbourhood. Then (id, f ) induces M ×M a map (id, f ) : N /∂N → T νΔ , and we define the equivariant Lefschetz number of f to be the image of the Thom class under  M ×M  (id,f )∗ V ∼ ε = V HG −→ HG (N /∂N ) −→ H0G (N+ ) −→ A(G) T νΔ In the general case where M is not necessarily orientable or a V -manifold, we embed M into a G-representation V with invariant tubular neighbourhood U . As an open subset of V , U is a non-compact orientable V -manifold. We define a map f0 : U → U by f0 = i ◦ f ◦ r, where r : U → M is the tubular retraction, i : M → U the embedding. If the fixed point set of f0 (equal to that of f ) is compact, we define LG (f ) = LG (f0 ), the right hand term having been defined previously. Summarizing, we have defined an equivariant Lefschetz number for G-manifolds M with finite orbit type (those are embeddable into finite dimensional G-representations, see [21]), and G-maps f : M → M with compact fixed point set. There is one final generalization we should mention, because it is needed in a later comparison result. If X is a G-ENR (see e.g. [19]), and f : X → X a self map, we can embed X as a retract into an open neighbourhood U of some G-representation. Again denoting by r and i the retraction and the inclusion, respectively, we have the map f0 = i ◦ f ◦ r : U → U , whose equivariant Lefschetz number has been defined previously. It is well-known that every finite G-CW complex is a G-ENR, so we have in fact defined an equivariant Lefschetz number for every finite G-CW complex. We will not use this most general version in the remainder of the paper and stick to manifolds throughout, with the exception of the comparison with the Lewis–May–Steinberger invariant.

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We should address the question of independence of the equivariant Lefschetz number of all the choices made in the various definitions. However, this is straightforward and does not differ from the non-equivariant situation, which can be found in [20], so we omit a rigorous proof. The general idea is that, given two embeddings into V and W , we obtain a diagonal embedding into V ⊕ W , and the Lefschetz numbers defined via the embedding into V and into W both equal the Lefschetz number defined via the diagonal embedding. Having defined the equivariant Lefschetz number, we turn to derive some basic properties. We begin by showing that it behaves well with respect to restriction to fixed points. Proposition 3.3. Let M be a G-manifold with finite orbit type and let H ⊆ G be a subgroup. Let ηH : A(G) → A(W (H)) be the fixed point homomorphism. Then     ηH LG (f ) = LW (H) f H . Proof. This follows in the case where M is a compact orientable V -manifold from the commutative diagram

W (H) 

HV H

H M+

    HVG (M × M )+ ⊗ HVG (M × M )+

Δ∗ ⊗(id,f )∗

HVG (M+ ) 

H ∗ Δ∗ H ⊗(id,f )

W (H) 

HV H

MH × MH

  +

W (H) 

⊗ HV H

MH × MH

  , +

where the vertical maps are restriction maps. The fundamental class OM of M maps to OΔ ⊗ OΓf under the upper horizontal map. Since OM restricts to OM H by Proposition 2.1, the restriction of OΔ ⊗ OΓf is OΔH ⊗ OΓf H , where ΔH denotes the diagonal of M H . From this, the formula for the Lefschetz numbers follows immediately. The case for more general M follows similarly by writing down the defining diagrams of the Lefschetz number, applying restriction and using the fact that Thom classes restrict to Thom classes, compare the remarks before Proposition 2.1. 2 The remainder of this section is devoted to the proofs of Theorems 3.4 and 3.5 and to the comparison of our invariant with existing notions. We start to give the desired geometric proof of the equivariant Lefschetz fixed point theorem. Theorem 3.4. Let M be a G-manifold with finite orbit type, f : M → M a G-map with compact fixed point set. Then if f has no fixed point of orbit type (K) with (H) ≤ (K), we have ηH (LG (f )) = 0. Proof. It suffices to prove the theorem in the case where M is a compact orientable V -manifold. Furthermore, we can assume that H = e, in the general case, we just replace G by W (H) and use Proposition 3.3. So we have to show that if f has no fixed points, LG (f ) = 0. If f has no fixed points, the intersection of the diagonal Δ ⊆ M × M with the graph Γ of f is empty. Both Δ and Γ are orientable V -manifolds, fundamental classes are given by the images of the fundamental class of M via the canonical homeomorphisms between M and each of these manifolds. Using the terminology of Definition 3.2, we claim that OΔ • OΓ = 0. Indeed, let τΔ , τΓ be the classes dual to OΔ and OΓ , i.e. τΔ ∩ OM ×M = OΔ ,

τΓ ∩ OM ×M = OΓ .

P. Wruck / Topology and its Applications 180 (2015) 16–43

34

Then by Proposition 2.6, τΔ and τΓ are just images of the Thom classes of normal bundles to the respective M ×M embeddings of the manifolds Δ and Γ under the Pontryagin–Thom maps. Let iM : M → M × M be the diagonal embedding. We can calculate   OΔ • OΓ = P P −1 (OΓ ) ∪ P −1 (OΓ ) = (τΓ ∪ τΔ ) ∩ OM ×M = τΓ ∩ (τΔ ∩ OM ×M )  M ×M  = τΓ ∩ iM ∗ (OM )  M ×M ∗  M ×M = iM (τΓ ) ∩ OM ∗ iM = 0, ∗

M ×M since iM (τΓ ) = 0 by Proposition 2.5.

2

We now want to give an explicit formula for the equivariant Lefschetz number which allows for computations to be carried out. The result also helps to relate our Lefschetz number to similar invariants and to prove several more properties. We denote the element of A(G) corresponding to the one-point compactification of a proper map f : V → V by Deg G (f ). For a subgroup H of G, tG H : A(H) → A(G) is the usual transfer sending an H-space Y to the G-space G ×H Y . We want to prove the following theorem. Theorem 3.5. Let M be a G-manifold of finite orbit type and f : M → M a G-map with finitely many G-orbits of fixed points Gx1 , . . . , Gxn . Assume that Nxi f has no eigenvalue of unit modulus for i = 1, . . . , n, where Nx f is the component of Tx f normal to the orbit Gx. Then the equivariant Lefschetz number LG (f ) is given as LG (f ) =

n 

  tG Gx Deg Gx (id −Nxi f ) . i

i

i=1

The proof will be given via two lemmas. Before we proceed, we make the following generic assumptions and simplifications. i) If f : M → M is a G-map, we can embed M into a G-representation V . In particular, every G-orbit in M embeds into V . Then the Lefschetz number of f equals the Lefschetz number of the map U → U , x → i ◦ f ◦ r, where U is a tubular neighbourhood of M in V , r : U → M the tubular retraction and i : M → U the embedding. Thus, we can assume that M is a V -manifold such that every G-orbit in M embeds into V . ii) The equivariant Lefschetz number is determined via the diagram  M ×M  (id,f )∗ V   ∼ = V HG T νΔ −→ HG (N /∂N ) −→ H0G (N+ ) → H0G S0 , where we have used the same notation as after Definition 3.2. From this it is clear that the Lefschetz number is determined by local data. Realizing the Thom space of the diagonal embedding as an arbitrary small neighbourhood of the diagonal, N will be an arbitrarily small neighbourhood of the fixed point set of f . So in order to understand the equivariant Lefschetz number, we need to understand Lefschetz numbers of maps f : G ×H W → G ×H W , where G ×H W is the tubular neighbourhood in V of an embedding of the orbit G/H and G ×H 0 is the unique fixed orbit of f .

P. Wruck / Topology and its Applications 180 (2015) 16–43

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iii) We can furthermore assume that f is smooth and the fixed orbit is non-degenerate, see e.g. [9] or [23]. This means that the derivative of f at [e, 0] in direction normal to the orbit has no eigenvalue of unit modulus. This assumption implies that there are no points in a neighbourhood of G ×H 0 with f ([g, w]) = [h, w] except for the ones with w = 0. iv) By Krupa’s normal decomposition lemma, see e.g. [10], Lemma 6.2, f is of the form G ×H W → G ×H W,



[g, w] → gγ(w), n(w) ,

where γ : W → G is H-equivariant with respect to the conjugation action on G and n : W → W is an H-map. v) Let A ⊆ G/H be an H-invariant neighbourhood of [e], H-isomorphic to L = T[e] G/H. By Lemma 3.10.2 of [9], we can assume that A is N (H)-invariant and there exists an N (H)-equivariant section σ : A → G, i.e. σ([hg]) = hσ([g])h−1 for h ∈ N (H). The set U=



  

   [g, w], gσ g  , w + w  g ∈ G, g  ∈ A, w ∈ W, w ∈ B1 (W )

is the total space of a normal bundle of the diagonal embedding of G ×H W for any ball B1 (W ) around 0 in W . We can choose B1 (W ) such that we can find another ball B2 (W ) around 0 in W with ([g, w], f ([g, w])) ∈ / U for w ∈ S2 (W ) = ∂B2 (W ). This is possible since G ×H 0 is the only fixed orbit of f . By our non-degeneracy assumption, we can arrange the neighbourhoods in a way such that 



 [g, w], gγ(tw), n(w) ∈ /U

for w ∈ S2 (W ) and t ∈ [0, 1]. This shows that the map (id, f ) : B2 (W )/S2 (W ) → U/∂U ⎧ w ∈ S2 (W ) ⎨∗ w → ∗ ([g, w], f ([g, w])) ∈ /U ⎩ ([g, w], f ([g, w])) else is equivariantly homotopic to the map

B2 (W )/S2 (W ) → U/∂U ,

⎧ ⎨∗ w → ∗ ⎩ ([g, w], [g, n(w)])

w ∈ S2 (W ) ([g, w], [g, n(w)]) ∈ /U else.

So we can assume that (id, f ) already possesses this form. We note in particular that the homotopy only involves the group coordinate, so the derivative of n at 0 equals the derivative of f at [e, 0] in normal direction to the group orbit. Summarizing the assumptions and simplifications, we are left with the task to compute the Lefschetz number of a map f : G ×H W → G ×H W, [g, w] → [g, n(w)] of a tubular neighbourhood of an embedding of G/H into a G-representation V , where T0 n has no eigenvalue of unit modulus.

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36

Lemma 3.6. In the situation above, we have   LG (f ) = tG H LH (n) . Proof. We will continue to use the local data assembled in v) above. The equivariant Lefschetz number is defined as the image of the Thom class under   ∼      (id,f )∗ = V V HG (U/∂U ) −→ HG G+ ∧H SW −→ H0G G ×H B2 (W ) + → H0G S0 , 2 the middle isomorphism being Poincaré duality. We note that the Thom space U/∂U can be identified with G+ ∧H B1 (W )+ ∧ SL ∧ SW via 

 

  

[g, w], gσ g  , w + w → g, w, ζ g  , w ,

where ζ : A → L is an H-homeomorphism satisfying ζ([e]) = 0. The special form of (id, f ) then has the property that the diagram

G+ ∧H SW

(id,f )

G+ ∧H B1 (W )+ ∧ SL ∧ SW Σ L (id,˜ n)

i

G+ ∧H SL ∧ SW commutes, where i is the inclusion and (id, n ˜ ) denotes the map   [g, w] → [g, w], n(w) − w .

G+ ∧H SW → G+ ∧H B1 (W )+ ∧ SW , Thus, we obtain a diagram

  (id,f )∗ V G+ ∧H B1 (W )+ ∧ SL ∧ SW HG

  V G+ ∧H SW HG i∗

Σ L (id,˜ n)∗

  V G+ ∧H SL ∧ SW HG

∼ =

∼ =

 Σ V B1 (W )+ ∧ SL ∧ SW HH

L

(id,n)

∼ =

  W HH B1 (W )+ ∧ SW

∗

 L  V S ∧ SW HH ∼ =

(id,n)

∗

 W W HH S .

We will see below in the proof of Lemma 3.7 that B1 (W )+ ∧ SW is a model for the Thom space of the diagonal embedding of B1 (W ) and the corresponding map (id, n) indeed makes the diagram commutative.

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We complete this diagram by inserting Poincaré duality isomorphisms on the right. We obtain, with some suppressed suspension isomorphisms,   V G+ ∧H SW HG

  p∗ HVG G/H + ∧ SV

P

i∗

  HVG SV

j∗

  V G+ ∧H SW ∧ SL HG

  HVG G+ ∧H SW

P

w

?

w

 W  V HH S ∧ SL

  HVH SV

P

∼ =

∼ =

 W W HH S

 W H HW S ,

P

where j : G+ ∧H SW → G/H + ∧ SV is the map [g, w] → ([g], gw). The two diagrams together imply that the G-equivariant Lefschetz number of f equals the image of the H-equivariant Lefschetz number of n under the map p∗ ◦ j∗ ◦ w, where w is the Wirthmüller isomorphism in the middle right above. Now it follows from the definition of the transfer t : SV → G/H + ∧ SV , [16, Definition II.6.15], and the construction of the Wirthmüller isomorphism, [16, Definition II.6.1], that the diagram  V V   V  w S ,S ∧ H H S , G+ ∧H SW ∧ H G ∼ =

(j∧id)∗



G/H + ∧ SV , SV ∧ H





SV , G/H + ∧ SV ∧ H

G

 G

t∗



S ,S ∧ H V

V

(p∧id)∗

 G

commutes. The vertical map on the left induces the induction map tG H : A(H) → A(G). Consequently, the map labelled with a question mark in the diagram above induces the induction map tG H . This finishes the proof. 2 It remains to calculate the H-equivariant Lefschetz number of the map n. This is an H-map of an H-representation with 0 as a unique fixed point. We emphasize that under our assumptions, id −T0 n is invertible. Lemma 3.7. Let W be an H-representation and n : W → W an equivariant map with 0 as a unique fixed point and id −T0 n invertible. Then LH (n) = Deg H (id −T0 n). Proof. We find two balls B1 = B1 (W ), B2 = B2 (W ) in W around 0 such that We take    U = (w, w) + (z, −z)  w ∈ W, z ∈ B2

w−n(w) 2

∈ / B2 for w ∈ S1 = ∂B1 .

as a tubular neighbourhood of Δ(W ) in W × W with the obvious tubular retraction. The Thom space W ×W T νΔ is the space U /∂U . The map (id, n) defined in the definition of the equivariant Lefschetz number is induced by

P. Wruck / Topology and its Applications 180 (2015) 16–43

38



 N :w→  w, n(w) =



w + n(w) w + n(w) , 2 2



 +

w − n(w) n(w) − w , 2 2

 ∈ U.

In detail,  (id, n) : B1 /S1 → U /∂U ,

(id, n)(w) =

w−n(w) 2

∗ N (w)

∈ / B2

else.

The projection p2 : U /∂U → B2 /S2 ,

(w, w) + (z, −z) → z

is an H-homotopy equivalence, since W is H-contractible. Therefore, the defining diagram for the equivariant Lefschetz number extends to W HH (U /∂U )

(id,n)∗

P

W HH (B1 /S1 )

p∗ 2

ε

H0H (B1 + )

  H0H S0

id

W (B2 /S2 ) HH

W HH (B1 /S1 )

∼ =

∼ =

 W W HH S

 W W HH S

The horizontal map in the middle is induced by  N  : B1 /S1 → B2 /S2 ,

w →

∗ w−n(w) 2

w−n(w) 2

∈ / B2

else.

This is, up to the identification B1 /S1 ∼ = SW , the one-point compactification SW → SW of the map id −n : W → W . Using the representing spectra, it is clear that the diagram  W W HH S

P

H0H (B1 + )

  H0H S0 ∼ =

σ −1

 0 0 HH S

ε∗

∼ =

A(H)

W commutes. In the initial diagram for the equivariant Lefschetz number, the Thom class in HH (U /∂U ) W W maps to the unit element in HH (S ) under the vertical isomorphism on the left. Then going right is just W precomposition with N  , so the unit is mapped to the class of N  in {S0 , S0 }H ∼ (SW ). It follows that = HH the equivariant Lefschetz number is the stable equivariant homotopy class of the one-point compactification of the map id −n (after identifying A(H) with {S0 , S0 }H ). Under the assumption that id −T0 n is invertible, the Pontryagin–Thom constructions of id −T0 n and id −n are H-homotopic. Indeed, we just have to show that there exists a ball B(W ) in W and a sphere S(W ), both centered at 0, such that w − h(t, w) ∈ / B(W ) for w ∈ S(W ), where

  h(t, w) = w − t · n(w) + (1 − t) · T0 n(w) .

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39

Then h induces an H-homotopy of the one-point compactifications. The derivative of ht = h(t, · ) is given by id −T0 n, so ht is invertible in a small ball around 0. Since ht (0) = 0, this shows that ht (w) > ε > 0 for all t, all w in a small sphere around 0 and some ε > 0. The claim follows. 2 Lemmas 3.6 and 3.7 together prove Theorem 3.5. Using this theorem, we immediately obtain the following corollary. Corollary 3.8. Let G/H be a G-orbit and f : G/H → G/H be a G-map. Then  LG (f ) =

1 · [G/H] ∈ A(G) 0

f  idG/H else

4. Comparison results for equivariant Lefschetz numbers In this short section, we want to compare the equivariant Lefschetz number we constructed in Section 3 with some existing notions. In [17], Lück and Rosenberg define a Lefschetz invariant for cocompact actions of discrete groups. So we can compare the two invariants provided G is finite. Proposition 4.1. Let G be a finite group. Then the equivariant Lefschetz number equals the equivariant Lefschetz class with values in the Burnside ring, as defined in [17, Definition 4.1]. Proof. The fundamental class O of a V -manifold M is characterized by the property that its image under the restriction HVG (M ) → H|V H | (M H ) is a fundamental class for M H for every subgroup of G. Following this restriction map through the defining diagram of the Lefschetz number, we obtain HVG (M ) ⊗ HVG (M )

H0G (∗)

    H|V H | M H ⊗ H|V H | M H

H0 (∗)

∼ =

∼ =

A(G)

Z,

and the map on the right is given by sending a generator [G/K] to |(G/K)H | or equivalently, sending a H H stable G-map f : SW → SW to the degree of f H : S|W | → S|W | . Since the lower row determines the non-equivariant Lefschetz number L(f H ) of f H , the equivariant Lefschetz number is an element of A(G) with the property that its non-equivariant restriction to H-fixed points is L(f H ). Similarly one can check that this holds true for general G-manifolds M and self maps f : M → M such that the equivariant Lefschetz number is defined. It is shown in [21] that this property uniquely determines the element of A(G). The Lefschetz class of Definition (4.1) in [17] is defined via this property, so it is equal to our equivariant Lefschetz number. 2 Since it is interesting in its own right, we note the following Corollary, which has been proven along the lines, even without the assumption that G is finite. Corollary 4.2. Let M be a G-manifold of finite orbit type and f : M → M a G-map with compact fixed point set. Then for every closed subgroup H of G and the fixed point homomorphism ηH : A(G) → Z, there holds     ηH LG (f ) = L f H , the right hand side being the non-equivariant Lefschetz number of f H .

40

P. Wruck / Topology and its Applications 180 (2015) 16–43

For our second result, we recall the definition of the Lewis–May–Steinberger Lefschetz number. A finite G-CW complex X has a dual DX in the category of equivariant spectra. This dual is uniquely characterized, up to equivariant homotopy equivalence, by the existence of two maps η : S → X ∧ DX,

ε : DX ∧ X → S,

satisfying certain conditions (see [16, Chapter III] for the exact formulation). The equivariant Lefschetz number of a map f : X → X is then defined as the stable equivariant homotopy class of the map η

τ

id ∧f

ε

S −→ X ∧ DX −→ DX ∧ X −→ DX ∧ X −→ S, τ being the coordinate twist. To avoid confusion, we will call it the Lewis–May–Steinberger trace of f for the remainder of the section. Proposition 4.3. The equivariant Lefschetz number equals the Lewis–May–Steinberger trace. Proof. We want to use the axioms in [22]. It is easily checked that both the equivariant Lefschetz number and the Lewis–May–Steinberger trace are linear equivariant Lefschetz numbers in the sense of [22]. Therefore we just need to calculate the value of the trace on orbit maps. For smooth compact G-manifolds, the maps η and ε from the definition of the trace are calculated in [16, Theorem III.5.1]. It is immediately clear that the identity map of an orbit has Lefschetz number 1 · [G/H], since this is the classical tom Dieck transfer (see [21]). Furthermore, the map ε is obtained using a tubular neighbourhood of the diagonal embedding. Thus, any map without fixed points has Lefschetz invariant 0. In particular, any orbit map ϕ : G/H → G/H which is not homotopic to the identity has trace equal to 0. According to Corollary 3.8, our invariant has the same value on orbit maps. Using the version of our invariant defined on finite G-CW complexes, as explained after Definition 3.2, we can conclude that it agrees with the Lewis–May–Steinberger invariant. 2 5. The equivariant Fuller index In this section we construct an equivariant Fuller index, using similar homological techniques as for the equivariant Lefschetz number. For the basic theory of the Fuller index, see [3,11,12]. Our approach is an equivariant generalization of [11]. The initial definition on the path to an equivariant Fuller index resembles the definition of the equivariant Lefschetz number for non-compact manifolds. We recall that the equivariant Lefschetz number, in the case where M is a compact orientable V -manifold, equals the image of the Thom class τ of the diagonal embedding of M into M × M under the sequence of maps  M ×M  ψ∗   (id,f )∗ V ∩OM ε V V HG T νΔ −→ HG (M × M )+ −→ HG (M+ ) −→ H0G (M ) −→ A(G), where the first map is induced by the Pontryagin–Thom map. We now use ideas of Franzosa [11] to assign an index to an equivariant flow on a V -manifold, related to an isolated compact set of periodic points. A periodic point of a flow ϕ : M × R → M is a pair (x, T ) ∈ M × R such that ϕ(x, T ) = x. We assume that M is an orientable V -manifold and C ⊆ M × R is a compact subset of periodic points. Moreover, we assume that C is isolated, i.e. there exists an open set Ω ⊆ M with compact closure, and a, b ∈ R, a, b > 0 such that C ⊆ Ω × (a, b) and the only periodic points in Ω × [a, b] are the points in C. Ω can be chosen to be a manifold with boundary, and in particular, the image of ∂Ω under the map

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41

(π1 , ϕ) : M × R → M × M does not meet the diagonal. We find a small invariant tubular neighbourhood U of the diagonal such that ∂Ω does not meet U . M ×M M ×M Then we can define a map Φ : ΣΩ/∂Ω → T νΔ , where νΔ is the normal bundle of the diagonal embedding with total space U . We rescale the suspension variable to run between a and b, and define ⎧ (x, t) ∈ / Ω × (a, b) ⎨∗ M ×M Φ : ΣΩ/∂Ω → T νΔ , (x, t) → ∗ (x, ϕ(x, t)) ∈ /U ⎩ (x, ϕ(x, t)) else. This is well defined by the choice of U . With this map in place, we imitate the definition of the equivariant Lefschetz class. We define the homological index of ϕ with respect to C to be the image of a Thom class of M ×M T νΔ under the sequence of maps  M ×M  Φ∗ ∩OΩ V −1 V V HG T νΔ −→ HG (ΣΩ/∂Ω) ∼ (Ω/∂Ω) −→ H1G (Ω+ ) → H1G (M+ ), = HG where the middle isomorphism is the suspension isomorphism and the last map is induced by the inclusion. There are obvious similarities with the equivariant Lefschetz number, but also significant differences. For one, the index lives in a group which depends on the manifold M , rather than a universal group like A(G) in the case of the Lefschetz number. Furthermore, the index must be zero if H1G(M+ ) is trivial. Thinking non-equivariantly, this would mean that the index is zero for all simply connected manifolds. But certainly, one has interesting flows and periodic orbit structures on such manifolds as well. The same problem arises for the ordinary Fuller index, and there is a standard construction by Fuller, generalized by Franzosa in [11], which tackles this issue. We first note that, as in the case of the Lefschetz number, we can drop orientability assumptions on M . If ϕ is a flow on any G-manifold with finite orbit type and C is an isolated compact set of periodic orbits, we can define an index for ϕ with respect to C as follows. We embed M into a G-representation V and let U be an invariant tubular neighbourhood of the embedding. We can assume that ϕ is induced by a vector field ξ on M and we can extend ξ to a vector field ξ0 on U by defining ξ0 (x) = ξ(r(x)) − (x − r(x)), where r : U → M is the tubular retraction and we identify tangential spaces with V in the canonical way. Then ξ0 is a vector field on U with the same periodic orbits as ξ. In particular, C is still an isolated compact set of periodic points for the flow ϕ0 of ξ0 , and we define the index of ϕ with respect to C to be the index of ϕ0 with respect to C. In the next step, we start with any G-manifold of finite orbit type. For a prime number p, M p is a G-manifold with the diagonal action, and Zp acts freely on the set M p \ Δf , where    Δf = (x1 , . . . , xp ) ∈ M p  ∃1 ≤ i < j ≤ p : xi = xj is the thickened diagonal. We denote the quotient manifold under this action by Mp . A flow ϕ on M induces a flow ϕp on Mp by acting diagonally. For a periodic point (x, T ) ∈ M × R, we write (xp , Tp ) for the equivalence class of         T 2T (p − 1)T T x, ϕ x, , ϕ x, , . . . , ϕ x, , p p p p in Mp × R. Taking p to be a large prime number, such that kT p is not a period of the point x for any k = 1, . . . , p − 1, we see that (xp , Tp ) is a periodic point of ϕp . Moreover it can be shown, compare [11], that if C is compact, p can be chosen in a way such that kT p is not a period for any periodic point (x, T ) ∈ C. We therefore can define, for p large, the set Cp to be the set consisting of all (xp , Tp ) with (x, T ) ∈ C. We remark that there is a one-to-one correspondence of periodic points of ϕ with the periodic points of ϕp . Indeed, if (x, T ) is a periodic point of ϕ, then (xp , Tp ) clearly is a periodic point of ϕp . Conversely, if ([x1 , . . . , xp ], T ) is any periodic point of ϕp , then

P. Wruck / Topology and its Applications 180 (2015) 16–43

42





ϕ(x1 , T ), . . . , ϕ(xp , T ) = (x1 , . . . , xp ) ,

and so there is an index j such that ϕ(xj+i , T ) = xi , i = 1, . . . , p, where we calculate modulo p in the index. It follows that x1 = ϕ(xi·j+1 , i · T ). In particular it follows that, (x1 , p · T ) is a periodic point of ϕ. With all this preparation, we have a homological index defined for the flow ϕp with respect to Cp , for p sufficiently large. It is an element of H1G (Mp + ). We now pass to the group 

HG (M ) =

H1G (Mp + )/

p prime



H1G (Mp + ).

p prime

The various homological indices define an equivariant homological Fuller index of the flow ϕ with respect to C, which we write as I(ϕ) = I(ϕ, C). The restriction maps define homomorphisms 

     H1 MpK + / H1 MpK + .

p prime

p prime

ψK : HG (M ) →

The latter group admits, via covering space theory, a map μK :



     H1 MpK + / H1 MpK + →

p prime

p prime

 p prime

Zp /



Zp .

p prime

We denote the group on the right by Z. Next, we recall from [16], that the Burnside ring of a compact Lie group admits an inclusion A(G) → C(ΦG, Z). Here, ΦG is the set of conjugacy classes of subgroups of G with finite Weyl group. It is topologized as the quotient of a subset of the space of subgroups, which carries the topology induced by the Hausdorff metric. In this topology, ΦG is compact and totally disconnected. In particular, C(ΦG, Z) consists of the locally constant functions. The composition maps μK ◦ ψK constitute an element in C(ΦG, Z), namely the element ΦG → Z,

  (H) → μK ◦ ψK I(ϕ) .

C(ΦG, Z) is isomorphic to C(ΦG, Z) ⊗ Z. The image of the homological Fuller index I(ϕ) under the map μK ◦ ψK is the homological Fuller index of the fixed point flow ϕK . This follows as in the case of the Lefschetz number by applying restriction to the defining diagram of the homological indices. Franzosa has shown that this element is actually a rational number, identified via the embedding Q → Z,

r → {ap }p q

prime ,

r = ap · q

mod p.

It follows that the image of the homological equivariant Fuller index I(ϕ) ∈ HG (M ) in C(ΦG, Z) ⊗ Z actually constitutes an element in C(ΦG, Z) ⊗ Q. By Lemma 2.10 of [16], this ring is isomorphic to the rationalized Burnside ring A(G) ⊗ Q. We define the equivariant Fuller index FG (ϕ) to be the image of the homological equivariant Fuller index in the rationalized Burnside ring via this identification. The following result is now straightforward.

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Theorem 5.1. There exists an equivariant Fuller index FG for a flow ϕ with respect to an isolated set C of periodic points. It is a G-homotopy invariant with the following properties. i) It takes values in the rationalized Burnside ring A(G) ⊗ Q. ii) If C consists of finitely many periodic orbits γ1 , . . . , γn and ϕi is the flow ϕ, restricted to an isolating neighbourhood of the orbit γi , then FG (ϕ) =

n 

FG (ϕi ).

i=1 1 iii) If ϕ has a single periodic orbit of multiplicity m, then FG (ϕ) = LG (P m ) ⊗ m ∈ A(G) ⊗ Q, where P is an equivariant Poincaré map for the orbit, considered with multiplicity one. iv) If ηH (FG (ϕ)) = 0, then ϕ has a periodic orbit of orbit type (K) with (H) ≤ (K).

Proof. i) is immediately clear and ii) and iv) follow from iii), the equivariant Lefschetz fixed point Theorem 3.4 and the fact that every equivariant flow is G-homotopic to a flow with finitely many G-orbits of periodic orbits, see [23]. So it remains to prove iii). The evident non-equivariant analogue has been proven by Franzosa in [11]. We use some basic theory of equivariant Poincaré maps as in e.g. [23]. P H is a Poincaré H m map for the fixed point flow ϕH , so by Franzosa’s result, FG (ϕ) restricts under ηH to F (ϕH ) = L((Pm ) ) . 1 Clearly, LG (P m ) ⊗ m restricts to the same element. Therefore, the two elements are equal in A(G) ⊗ Q. 2 References [1] G.E. Bredon, Introduction to Compact Transformation Groups, Academic Press, 1972. [2] B. Chorny, Equivariant cellular homology and its applications, in: High-Dimensional Manifold Topology, World Scientific, 2003, pp. 125–138. [3] S.-N. Chow, J. Mallet-Paret, Fuller index and Hopf bifurcation, J. Differ. Equ. 29 (1978) 66–85. [4] S.R. Costenoble, S. Waner, Ordinary equivariant homology and cohomology, arXiv:math/0310237, 2007, Version 2. [5] S.R. Costenoble, S. Waner, Equivariant Poincaré duality, Mich. Math. J. 39 (2) (1992) 325–351. [6] Z. Dzedzej, Fixed orbit index for equivariant maps, Nonlinear Anal. 47 (2001) 2835–2840. [7] S.R. Costenoble, J.P. May, S. Waner, Equivariant orientation theory, Homol. Homotopy Appl. 3 (2) (2001) 265–339. [8] M. Crabb, Fixed orbit indices, J. Fixed Point Theory Appl. 11 (2012) 65–92. [9] M. Field, Dynamics and Symmetry, Imperial College Press, 2007. [10] M. Field, Local Structure of Equivariant Dynamics, LNM, vol. 1463, Springer, 1991, pp. 142–166. [11] R. Franzosa, An homological index generalizing Fuller’s index for periodic orbits, J. Diff. Equ. 84 (1990) 1–14. [12] F.B. Fuller, An index of fixed point type for periodic orbits, Am. J. Math. 89 (1967) 133–148. [13] D. Gonçalves, J. Weber, Axioms for the equivariant Lefschetz number and the equivariant Reidemeister trace, J. Fixed Point Theory Appl. 2 (2007) 55–72. [14] J.P.C. Greenlees, P. May, Equivariant stable homotopy theory, in: Handbook of Algebraic Topology, North-Holland, Amsterdam, 1995, pp. 277–323. [15] E. Laitinen, W. Lück, Equivariant Lefschetz classes, Osaka J. Math. 26 (1989) 491–525. [16] L.G. Lewis, J.P. May, M. Steinberger, Equivariant Stable Homotopy Theory, Springer LNM 1213 (1980). [17] W. Lück, J. Rosenberg, The equivariant Lefschetz fixed point theorem for proper cocompact G-manifolds, in: HighDimensional Manifold Topology, World Scientific, 2003, pp. 322–361. [18] J.P. May, et al., Equivariant homotopy and cohomology theory, in: CBMS, vol. 91, 1996. [19] M. Murayama, On G-ANRs and their homotopy type, Osaka J. Math. 20 (1983) 479–512. [20] R. Nussbaum, Generalizing the fixed point index, Math. Ann. 228 (1977) 259–278. [21] T. tom Dieck, Transformation Groups, de Gruyter, 1987. [22] P. Wruck, Axiomatic description of Lefschetz type equivariant homotopy invariants, arXiv:1301.7308, 2013, version 1. [23] P. Wruck, Genericity in equivariant dynamical systems and equivariant Fuller index theory, Dyn. Syst. 29 (3) (2014) 399–423.