The equivariant Lefschetz fixed point theorem for proper cocompact G-manifolds

Wolfgang Lück and Jonathan Rosenberg^{∗}

April 1, 2003

Abstract

Suppose one is given a discrete group G, a cocompact proper Gmanifold M, and a G-self-map f: M → M. Then we introduce the equivariant Lefschetz class of f, which is globally defined in terms of cellular chain complexes, and the local equivariant Lefschetz class of f, which is locally defined in terms of fixed point data. We prove the equivariant Lefschetz fixed point theorem, which says that these two classes agree. As a special case, we prove an equivariant Poincaré-Hopf Theorem, computing the universal equivariant Euler characteristic in terms of the zeros of an equivariant vector field, and also obtain an orbifold Lefschetz fixed point theorem. Finally, we prove a realization theorem for universal equivariant Euler characteristics. Key words: equivariant Lefschetz class, equivariant Lefschetz fixed point theorem, proper cocompact G-manifold, equivariant vector field. Mathematics subject classification 2000: Primary 57R91. Secondary 57S30, 55P91, 58C30, 57R25.

0. Introduction

Let us recall the classical Lefschetz fixed point theorem. Let f: M → M be a smooth self-map of a compact smooth manifold M, such that Fix(f)∩∂M = ∅ and for each x ∈ Fix(f), the determinant of the linear map (id −T_{x}f): T_{x}M → T_{x}M is different from zero. Denote by T_{x}M^{c} the one-point compactification of T_{x}M, which is homeomorphic to a sphere. Let (id −T_{x}f)^{c}: T_{x}M^{c} → T_{x}M^{c} be the homeomorphism induced by the self-homeomorphism (id −T_{x}f): T_{x}M → T_{x}M. Denote by deg((id −T_{x}f)^{c}) its degree, which is 1 or −1, depending on whether det(id −T_{x}f) is positive or negative. Let

L^{ℤ[{1}]}(f) := ∑_{p≥0}(−1)^{p} · tr_{ℚ}(H_{p}(f; ℚ)) = ∑_{p≥0}(−1)^{p} · tr_{ℤ}(C_{p}(f))

∗Partially supported by NSF grants DMS-9625336 and DMS-0103647.

be the classical Lefschetz number of f, where H_{p}(f; ℚ) is the map on the singular homology with rational coefficients and C_{p}(f) is the chain map on the cellular ℤchain complex induced by f for some smooth triangulation of M. The Lefschetz fixed point theorem says that under the conditions above the fixed point set Fix(f) = {x ∈ M | f(x) = x} is finite and

L^{ℤ[{1}]}(f) = ∑_{x∈Fix(f)} deg((id −T_{x}f)^{c}).  (0.1)

For more information about it we refer for instance to [1].

The purpose of this paper is to generalize this to the following equivariant setting. Let G be a (not necessarily finite) discrete group G. A smooth Gmanifold M is a smooth manifold with an action of G by diffeomorphisms. It is called cocompact if the quotient space G\M is compact. It is proper if the map G × M → M × M, (g, m) ↦ (g · m, m) is proper; when the action is cocompact, this happens if and only if all isotropy groups are finite. One can equip M with the structure of a proper finite G-CW -complex by an equivariant smooth triangulation [4]. The main result of this paper is Theorem 0.2 (Equivariant Lefschetz fixed point theorem) Let G be a discrete group. Let M be a cocompact proper G-manifold (possibly with boundary) and let f: M → M be a smooth G-map. Suppose that Fix(f)∩∂M = ∅ and for each x ∈ Fix(f) the determinant of the linear map id −T_{x}f: T_{x}M → T_{x}M is different from zero.

Then G\ Fix(f) is finite, the equivariant Lefschetz class of f (see Definition 3.6)

Λ^{G}(f) ∈ U^{G}(M)

is defined in terms of cellular chain complexes, and the local equivariant Lefschetz class of f (see Definition 4.6)

Λ_{loc}^{G}(f) ∈ U^{G}(M)

is defined. Also Λ^{G}(f) and Λ_{loc}^{G}(f) depend only on the differentials T_{x}f for x ∈ Fix(f), and

Λ^{G}(f) = Λ_{loc}^{G}(f).

If G is trivial, Theorem 0.2 reduces to (0.1). We emphasize that we want to treat arbitrary discrete groups and take the component structure of the various fixed point sets into account.

In Section 1 we will define the orbifold Lefschetz number, which can also be viewed as an L^{2}-Lefschetz number, and prove the orbifold Lefschetz fixed point theorem 2.1 in Section 2. It is both a key ingredient in the proof of and a special case of the equivariant Lefschetz fixed point theorem 0.2.

In Section 3 we introduce the equivariant Lefschetz class Λ^{G}(f), which is globally defined in terms of cellular chain complexes, and in Section 4 we introduce the local equivariant Lefschetz class Λ_{loc}^{G}(f), which is locally defined in terms of the differentials at the fixed points. These two are identified by the equivariant Lefschetz fixed point theorem 0.2, whose proof is completed in Section 5.

A classical result (the Poincaré-Hopf Theorem) says that the Euler characteristic of a compact smooth manifold can be computed by counting (with signs) the zeros of a vector field which is transverse to the zero-section and points outward at the boundary. This is a corollary of the classical Lefschetz fixed point theorem (0.1) via the associated flow. In Section 6 we will extend this result to the equivariant setting for proper cocompact G-manifolds by defining the universal equivariant Euler characteristic, defining the index of an equivariant vector field which is transverse to the zero-section and points outward at the boundary, and proving their equality in Theorem 6.6. As an illustration we explicitly compute the universal equivariant Euler characteristic and the local equivariant index of an equivariant vector field for the standard action of the infinite dihedral group on ℝ in Example 6.9.

To prove Theorem 6.6 was one motivation for this paper, since it is a key ingredient in [10]. There a complete answer is given to the question of what information is carried by the element Eul^{G}(M) ∈ KO_{0}^{G}(M), the class defined by the equivariant Euler operator for a proper cocompact G-manifold M. Rosenberg [11] has already settled this question in the non-equivariant case by perturbing the Euler operator by a vector field and using the classical result that the Euler characteristic can be computed by counting the zeros of a vector field. The equivariant version of this strategy will be applied in [10], which requires having Theorem 6.6 available.

In Section 7 we discuss the problem whether there exists a proper smooth Gmanifold M with prescribed sets π_{0}(M^{H}) for H ⊆ G such that χ^{G}(M) realizes a given element in U^{G}(M). A necessary and sufficient condition for this is given in Theorem 7.6. The sufficiency part of the proof is based on a construction called multiplicative induction or coinduction. Again these results will have applications in [10].

The paper is organized as follows:

1. The orbifold Lefschetz number
2. The orbifold Lefschetz fixed point theorem
3 The equivariant Lefschetz classes
4. The local equivariant Lefschetz class
5. The proof of the equivariant Lefschetz fixed point theorem
6. Euler characteristic and index of a vector field in the equivariant setting
7. Constructing equivariant manifolds with given component structure and

universal equivariant Euler characteristic References

1. The orbifold Lefschetz number

In order to define the various Lefschetz classes and prove the various Lefschetz fixed point theorems for cocompact proper G-manifolds, we need some input about traces.

Let R be a commutative associative ring with unit, for instance R = ℤ or R = ℚ. Let u: P → P be an endomorphism of a finitely generated projective RG-module. Choose a finitely generated projective RG-module Q and an isomorphism v: P ⊕ Q ⟶^{≅} ⨁_{i∈I} RG for some finite index set I. We obtain an RG-endomorphism

v ∘ (u ⊕ 0) ∘ v^{−1} : ⨁_{i∈I} RG → ⨁_{i∈I} RG.

Let A = (a_{i,j})_{i,j∈I} be the matrix associated to this map, i.e.,

v ∘ (u ⊕ 0) ∘ v^{−1}({w_{i} | i ∈ I}) = {∑_{i∈J} w_{i} · a_{i,j} | j ∈ I} .

Define

tr_{RG}: RG → R, ∑_{g∈G} r_{g} · g ↦ r_{1}  (1.1)

where r_{1} is the coefficient of the unit element 1 ∈ G. Define the RG-trace of u by

tr_{RG}(u) := ∑_{i∈I} tr_{RG}(a_{ii}) ∈ R.  (1.2)

We omit the easy and well-known proof that this definition is independent of the various choices such as Q and v and that the following Lemma 1.3 is true. Lemma 1.3 (a) Let u: P → Q and v: Q → P be RG-maps of finitely gener-

ated projective RG-modules. Then

tr_{RG}(v ∘ u) = tr_{RG}(u ∘ v);

(b) Let P_{1} and P_{2} be finitely generated projective RG-modules. Let

( \begin{matrix}u_{1,1} & u_{1,2} \\ u_{2,1} & u_{2,2} \end{matrix} ) : P_{1} ⊕ P_{2} → P_{1} ⊕ P_{2}

be a RG-self-map. Then

tr_{RG} ( \begin{matrix}u_{1,1} & u_{1,2} \\ u_{2,1} & u_{2,2} \end{matrix} ) = tr_{RG}(u_{1,1}) + tr_{RG}(u_{2,2});

(c) Let u_{1}, u_{2} : P → P be RG-endomorphisms of a finitely generated projective RG-module and r_{1}, r_{2} ∈ R. Then

tr_{RG}(r_{1} · u_{1} + r_{2} · u_{2}) = r_{1} · tr_{RG}(u_{1}) + r_{2} · tr_{RG}(u_{2});

(d) Let α: G → K be an inclusion of groups and u: P → P be an endomorphism of a finitely generated projective RG-module. Then induction with α yields an endomorphism α_{∗}u of a finitely generated projective RK-module, and

tr_{RK}(α_{∗}u) = tr_{RG}(u);

(e) Let α: H → G be an inclusion of groups with finite index [G : H] and u: P → P be an endomorphism of a finitely generated projective RGmodule. Then the restriction to RH with α yields an endomorphism α^{∗}u of a finitely generated projective RH-module, and

tr_{RH}(α^{∗}u) = [G : H] · tr_{RG}(u);

(f ) Let H ⊆ G be finite such that |H| is invertible in R. Let u: R[G/H] → R[G/H] be a RG-map which sends 1H to ∑_{gH∈G/H} r_{gH} · gH. Then R[G/H] is a finitely generated projective RG-module and

tr_{RG}(u) = |H|^{−1} · r_{1H};
tr_{RG}(id_{R[G/H]}) = |H|^{−1}.

Let G be a discrete group. A relative G-CW -complex (X, A) is finite if and only if X is obtained from A by attaching finitely many equivariant cells, or, equivalently, G\(X/A) is compact. A relative G-CW -complex (X, A) is proper if and only if the isotropy group G_{x} of each point x ∈ X − A is finite (see for instance [7, Theorem 1.23 on page 18]). Let (f, f_{0}): (X, A) → (X, A) be a cellular G-self-map of a finite proper relative G-CW -complex (X, A). Let R be a commutative ring such that for any x ∈ X − A the order of its isotropy group G_{x} is invertible in R. Then the cellular RG-chain complex C_{∗}(X, A) is finite projective, i.e., each chain module is finitely generated projective and C_{p}(X, A) = 0 for p ≥ d for some integer d.

Definition 1.4 Define the orbifold Lefschetz number of (f, f_{0}) by

L^{RG}(f, f_{0}) := ∑_{p≥0}(−1)^{p} · tr_{RG}(C_{p}(f, f_{0})) ∈ R.  (1.5)

One easily proves using Lemma 1.3

Lemma 1.6 Let (f, f_{0}): (X, A) → (X, A) be a cellular G-self-map of a finite proper relative G-CW -complex such that |G_{x}| is invertible in R for each x ∈ X − A. Then:

(a) The equivariant Lefschetz number L^{RG}(f, f_{0}) depends only on the G-homo-topy class of (f, f_{0});

(b) Let (g, g_{0}): (X, A) → (Y, B) and (h, h_{0}): (Y, B) → (X, A) be cellular Gmaps of finite proper relative G-CW -complexes such that |G_{x}| is invertible in R for each x ∈ X − A and |G_{y}| is invertible in R for each y ∈ Y − B. Then

L^{RG}(g ∘ h, g_{0} ∘ h_{0}) = L^{RG}(h ∘ g, h_{0} ∘ g_{0});

(c) Let α: G → K be an inclusion of groups. Then induction with α yields a cellular K-self-map α_{∗}(f, f_{0}) of a finite proper relative K-CW -complex, and

L^{RK}(α_{∗}(f, f_{0})) = L^{RG}(f, f_{0});

(d) Let α: H → G be an inclusion of groups with finite index [G : H]. Then restriction with α yields a cellular H-self-map α^{∗}(f, f_{0}) of a finite proper relative H-CW -complex, and

L^{RH}(α^{∗}f) = [G : H] · L^{RG}(f).

Remark 1.7 The rational number L^{ℚG}(f, f_{0}) agrees with the L^{2}-Lefschetz number L^{(2)}(f, f_{0}; N (G)) introduced in [9, Section 6.8]. It can be read off from the map induced by (f, f_{0}) on the L^{2}-homology of (X, X_{0}) by the analog of the usual formula, namely by

L^{ℚG}(f, f_{0}) = L^{(2)}(f, f_{0}; N (G)) = ∑_{p≥0}(−1)^{p} · tr_{N (G)} (H_{p}^{(2)}(f, f_{0}; N (G))) ,

where tr_{N (G)} is the standard trace of the group von Neumann algebra N (G). A similar formula exists in terms of H_{n}(X, X_{0}; ℚ) only under the very restrictive assumption, that each ℚG-module H_{p}(X, X_{0}; ℚ) is finitely generated projective. If G is finite, then (X, X_{0}) is a finite relative CW -complex and

L^{ℚG}(f, f_{0}) = \frac{1}{|G|} · L^{ℤ[\{1\}]}(f, f_{0}).

The following description of L^{G}(f) will be useful later. Let I_{p}(X, A) be the set of path components of X_{p} − X_{p−1}. This is the same as the set of open cells of (X, A) regarded as relative CW -complex (after forgetting the group action). The group G acts on I_{p}(X, A). For an open p-cell e let G_{e} be its isotropy group, \overline{e} be its closure and ∂e = e − \overline{e}. Then \overline{e}/∂e is homeomorphic to S^{p} and there is a homeomorphism

h: ⋁_{e′∈I_{p}(X,A)} \overline{e′}/∂e′ ⟶^{≅} X_{p}/X_{p−1}.

For an open cell e ∈ I_{p}(X, A) define the incidence number

inc(f, e) ∈ ℤ  (1.8)

to be the degree of the composition

\overline{e}/∂e ⟶^{i_{e}} ⋁_{e′∈I_{p}(X,A)} \overline{e′}/∂e′ ⟶^{h} X_{p}/X_{p−1}

⟶^{f} X_{p}/X_{p−1} ⟶^{h^{−1}} ⋁_{e′∈I_{p}(X,A)} \overline{e′}/∂e′ ⟶^{pr_{e}} \overline{e}/∂e,

where i_{e} is the obvious inclusion and pr_{e} is the obvious projection. Obviously inc(f, e) = inc(f, ge) for g ∈ G. One easily checks using Lemma 1.3

Lemma 1.9 Let (X, A) be a finite proper relative G-CW -complex. Consider a cellular G-map (f, f_{0}): (X, A) → (X, A). Then

L^{ℚG}(f, f_{0}) = ∑_{p≥0}(−1)^{p} · ∑_{Ge∈G\I_{p}(X,A)} |G_{e}|^{−1} · inc(f, e).

\underline{\underline{Proof}} : By definition,

L^{ℚG}(f, f_{0}) = ∑_{p≥0}(−1)^{p} · tr_{ℚG}(C_{p}(f, f_{0})).

But C_{p}(f, f_{0}) is the ℚG-endomorphism induced by (f, f_{0}) on ℚI_{p}(X, A). This ℚG-module splits as a direct sum of submodules, one for each G-orbit Ge in I_{p}(X, A). And tr_{ℚG} of the G-action on the submodule corresponding to Ge is |G_{e}|^{−1} · inc(f, e).

2. The orbifold Lefschetz fixed point theorem

This section is devoted to the proof of: Theorem 2.1 (The orbifold Lefschetz fixed point theorem) Let M be a cocompact proper G-manifold (possibly with boundary) and let f: M → M be a smooth G-map. Suppose that Fix(M) ∩ ∂M = ∅ and for any x ∈ F ix(f) the determinant of the map (id_{T_{x}M} −T_{x}f) is different from zero. Then G\ Fix(f) is finite, and

L^{ℚG}(f) = ∑_{G\ Fix(f)} |G_{x}|^{−1} · deg ((id_{T_{x}M} −T_{x}f))^{c}) .

Theorem 2.1 above will be a key ingredient in the proof of the equivariant Lefschetz fixed point theorem 0.2. On the other hand Theorem 0.2 implies Theorem 2.1.

Let us first consider as an illustration the easy case, where G is finite. Then

L^{ℚG}(f) = |G|^{−1} · L^{ℚ[{1}]}(f) = |G|^{−1} · L^{ℤ[{1}]}(f)

by Lemma 1.6 (d) and L^{ℤ[{1}]}(f) is the (ordinary) Lefschetz number of the selfmap f: M → M of the compact manifold M. The non-equivariant Lefschetz fixed point theorem says

L^{ℤ{1}}(f) = ∑_{Fix(f)} deg ((id_{T_{x}M} −T_{x}f)^{c}) .

Thus Theorem 2.1 follows for finite G. The proof in the case of an infinite group cannot be reduced to the non-equivariant case in such an easy way since M is not compact anymore. Instead we extend the proof in the non-equivariant case to the equivariant setting. Proof of Theorem 2.1:

Since the argument is a bit complicated technically, even though no individual step is that hard, we start by giving the reader an outline of the proof. The idea is to construct a good G-invariant simplicial structure with respect to which we can compute the equivariant Lefschetz number. This structure will have the property that it will be clear that simplices away from Fix(f) do not contribute anything to the answer. Thus we will be reduced to a local calculation around the fixed points, where smoothness of f and the non-degeneracy condition on (id_{T_{x}M} −T_{x}f) will reduce the calculation to linear algebra.

Fix a G-invariant Riemannian metric on M. Choose ϵ_{1} > 0 such that for all x ∈ M the exponential map is defined on D_{ϵ_{1}}T_{x}M = {v ∈ T_{x}M | ∥v∥ ≤ ϵ_{1}}, where ∥v∥ for v ∈ T_{x}M is the norm coming from the Riemannian metric. Such ϵ_{1} > 0 exists because G\M is compact. The image N_{x,ϵ_{1}} of the exponential map on D_{ϵ_{1}}T_{x}M is a G_{x}-submanifold of M and a compact neighborhood of x. The exponential map induces a G_{x}-diffeomorphism

exp_{x,ϵ_{1}} : D_{ϵ_{1}}T_{x}M ⟶^{≅} N_{x,ϵ_{1}}

with exp_{x,ϵ_{1}}(0) = x whose differential at 0 is the identity under the canonical identification T_{0}D_{ϵ_{1}}T_{x}M = T_{x}M.

Since G\M is compact, we can choose ϵ_{2} > 0 such that f(N_{x,ϵ_{2}}) ⊆ N_{x,ϵ_{1}} and T_{x}f(D_{ϵ_{2}}T_{x}M) ⊆ D_{ϵ_{1}}T_{x}M holds for all x ∈ Fix(f). Notice that exp_{x,ϵ_{1}} restricted to D_{ϵ_{2}}T_{x}M is exp_{x,ϵ_{2}}. We want to change f up to G-homotopy without changing Fix(f) such that exp_{x,ϵ_{1}}^{−1} ∘f ∘exp_{x,ϵ_{2}} and T_{x}f agree on D_{ϵ_{3}}T_{x}M for some positive number ϵ_{3} > 0 and all x ∈ Fix(f). Consider x ∈ Fix(f). Notice that exp_{x,ϵ_{1}}^{−1} ∘f ∘ exp_{x,ϵ_{2}} sends 0 to 0 and has T_{x}f as differential at 0 under the canonical identification T_{0}D_{ϵ_{1}}T_{x}M = T_{x}M. By Taylor’s theorem we can find a constant C_{1} > 0 such that with respect to the norm on T_{x}M induced by the Riemannian metric on M

|| exp_{x,ϵ_{1}}^{−1} ∘f ∘ exp_{x,ϵ_{2}}(v) − T_{x}f(v)|| ≤ C_{1} · ||v||^{2} for v ∈ D_{ϵ_{2}}T_{x}M. (2.2)

Since det(id −T_{x}f) ≠ 0, we can find a constant C_{2} > 0 such that

||T_{x}f(v) − v|| ≥ C_{2} · ||v|| for v ∈ T_{x}M.  (2.3)

Choose a smooth function ϕ: [0, ϵ_{2}] → [0, 1] with the properties that ϕ(t) = 1 for t ≤ min{C_{2}/3C_{1}, ϵ_{2}/3} and ϕ(t) = 0 for t ≥ min{C_{2}/2C_{1}, ϵ_{2}/2}. Define

h: D_{ϵ_{2}}T_{x}M × [0, 1] → D_{ϵ_{1}}T_{x}M

by

h(y, t) := (1 − tϕ(||v||)) · exp_{x,ϵ_{1}}^{−1} ∘f ∘ exp_{x,ϵ_{2}}(v) + tϕ(||v||) · T_{x}f(v).

Obviously h is a G_{x}-homotopy from h_{0} = exp_{x,ϵ_{1}}^{−1} ∘f ∘ exp_{x,ϵ_{2}} to a G_{x}-map h_{1}. The homotopy h is stationary outside D_{min{C_{2}/2C_{1},ϵ_{2}/2}}T_{x}M and h_{1} agrees with T_{x}f on D_{ϵ_{3}}T_{x}M if we put ϵ_{3} = min{C_{2}/3C_{1}, ϵ_{2}/3}. Each map h_{t} has on D_{min{C_{2}/2C_{1},ϵ_{2}/2}}T_{x}M only one fixed point, namely 0. This follows from the following estimate based on (2.2) and (2.3) for v ∈ D_{min{C_{2}/2C_{1},ϵ_{2}/2}}T_{x}M:

||h_{t}(v) − v||

≥ ||T_{x}f(v) − v|| −

||(1 − tϕ(||v||)) · exp_{x,ϵ_{1}}^{−1} ∘f ∘ exp_{x,ϵ_{2}}(v) − (1 − tϕ(||v||)) · T_{x}f(v)||

= ||T_{x}f(v) − v|| − (1 − tϕ(||v||)) · || exp_{x,ϵ_{1}}^{−1} ∘f ∘ exp_{x,ϵ_{2}}(v) − T_{x}f(v)||
≥ C_{2} · ||v|| − (1 − tϕ(||v||)) · C_{1} · ||v||^{2}
≥ (C_{2} − (1 − tϕ(||v||)) · C_{1} · ||v||) · ||v||
≥ C_{2} · ||v||/2.

In particular we see that the only fixed point of f on N_{min{C_{2}/2C_{1},ϵ_{2}/2},x} is x. After possibly decreasing ϵ_{1} we can assume without loss of generality that N_{ϵ_{1},x} ∩ N_{ϵ_{1},y} = ∅ for x, y ∈ Fix(f), x ≠ y. Since M is cocompact, G\ Fix(f) is finite.

Since h_{t} = h_{0} has no fixed points outside D_{min{C_{2}/2C_{1},ϵ_{2}/2}}T_{x}M, each map h_{t} has only one fixed point, namely 0. Since the G_{x}-homotopy h is stationary outside D_{min{C_{2}/2C_{1},ϵ_{2}/2}}T_{x}M, it extends to a G-homotopy from f to a G-map f′ such that Fix(f) = Fix(f′) and

exp_{x,ϵ_{1}}^{−1} ∘f′ ∘ exp_{x,ϵ_{2}}(v) = T_{x}f′(v) = T_{x}f(v)

holds for each x ∈ Fix(f) and each v ∈ D_{ϵ_{3}}T_{x}M. In the sequel we will identify D_{ϵ_{1}}T_{x}M with the compact neighborhood N_{ϵ_{1},x} of x by exp_{x,ϵ_{1}} for x ∈ Fix(f). Since L^{ℚG}(f) depends only on the G-homotopy class of f, we can assume in the sequel that f agrees with T_{x}f: D_{ϵ_{3}}T_{x}M → D_{ϵ_{1}}T_{x}M on D_{ϵ_{3}}T_{x}M for each x ∈ Fix(f).

Next we analyze the G_{x}-linear map T_{x}f: T_{x}M → T_{x}M for x ∈ Fix(f). We can decompose the orthogonal G_{x}-representation T_{x}M as

T_{x}M = ⨁_{i=1}^{n} V_{i}^{m_{i}}

for pairwise non-isomorphic irreducible G_{x}-representations V_{1}, V_{2}, …, V_{n} and positive integers m_{1}, m_{2}, …, m_{n}. The G_{x}-linear automorphism T_{x}f splits as ⊕_{i=1}^{n}f_{i} for G_{x}-linear automorphisms f_{i}: V_{i}^{m_{i}} → V_{i}^{m_{i}}. Let D_{i} = End_{ℝG_{x}}(V_{i}) be the skew-field of G_{x}-linear endomorphisms of V_{i}. It is either the field of real numbers ℝ, the field of complex numbers ℂ or the skew-field of quaternions ℍ. There is a canonical isomorphism of normed vector spaces

End_{ℝG_{i}}(V_{i}^{m_{i}}) ≅ M_{m_{i}}(D_{i}).

Since the open subspace GL_{m_{i}}(D_{i}) ⊆ M_{m_{i}}(D_{i}) is connected for D_{i} = ℂ and D_{i} = ℍ and the sign of the determinant induces a bijection π_{0}(GL_{m_{i}}(ℝ)) → {±1}, we can connect f_{i} ∈ Aut_{G_{x}}(V_{i}) by a (continuous) path to either id: V_{i}^{m_{i}} → V_{i}^{m_{i}} or to − id_{V_{i}} ⊕ id_{V_{i}^{m_{i}−1}} : V_{i}^{m_{i}} → V_{i}^{m_{i}}. This implies that we can find a decomposition

T_{x}M = V_{x} ⊕ W_{x}

of the orthogonal G_{x}-representation T_{x}M into orthogonal G_{x}-subrepresentations and a (continuous) path w_{t}: T_{x}M → T_{x}M of linear G_{x}-maps from T_{x}f to 2 · id_{V_{x}} ⊕ 0_{W_{x}} such that id −w_{t} is an isomorphism for all t ∈ [0, 1]. Since w_{t} is continuous on the compact set [0, 1], there is a constant C_{3} ≥ 1 such that for each v ∈ T_{x}M and each t ∈ [0, 1],

||w_{t}(v)|| ≤ C_{3} · ||v||.

Choose a smooth function ψ: [0, ϵ_{3}/C_{3}] → [0, 1] such that ψ(t) = 1 for t ≤ ϵ_{3}/3C_{3} and ψ(t) = 0 for t ≥ 2ϵ_{3}/3C_{3}. Define a G_{x}-homotopy

u: D_{ϵ_{3}/C_{3}}T_{x}M × [0, 1] → D_{ϵ_{3}}T_{x}M, (y, t) ↦ w_{t·ψ(||v||)}(v).

This is a G_{x}-homotopy from f|_{D_{ϵ_{3}} T_{x}M} = T_{x}f|_{D_{ϵ_{3}} T_{x}M} = w_{0}|_{D_{ϵ_{3}} T_{x}M} to a linear G_{x}-map u_{1}. The map u_{1} and the map 2 id_{V_{x}} ⊕ 0_{W_{x}}: T_{x}M → T_{x}M agree on D_{ϵ_{3}/3C_{3}}T_{x}M. For each t ∈ [0, 1] the map u_{t}: D_{ϵ_{3}/C_{3}}T_{x}M → D_{ϵ_{3}}T_{x}M has only one fixed point, namely 0, since this is true for w_{t} for each t ∈ [0, 1] by construction. The G_{x}-homotopy u is stationary outside D_{2ϵ/3C_{3}}T_{x}M. Hence it can be extended to a G-homotopy U: M × [0, 1] → M which is stationary outside D_{2ϵ/3C_{3}}T_{x}M. Since

L^{ℚG}(f) = L^{ℚG}(U_{1});
deg((id −T_{x}f)^{c}) = deg((id −T_{x}U_{1})^{c}),

we can assume without loss of generality that f looks on D_{ϵ_{3}/C_{3}}T_{x}M like

2 id_{V_{x}} ⊕ 0_{W_{x}}: T_{x}M = V_{x} ⊕ W_{x} → T_{x}M = V_{x} ⊕ W_{x}

for each x ∈ Fix(f). By scaling the metric with a constant, we can arrange that we can take ϵ/3C_{3} = 1/2 and ϵ_{1} = 2, in other words, we can identify D_{2}T_{x}M with a neighborhood of x in M and f is given on D_{1/2}T_{x}M by T_{x}f = 2 · id_{V_{x}} ⊕0_{W_{x}}.

Let d be the metric on M coming from the Riemannian metric. Choose an integer δ > 0 such that the inequality d(y, f(y)) ≥ δ holds for each y ∈ M, which does not lie in D_{1/2}T_{x}M for each x ∈ Fix(f). Consider x ∈ Fix(f). Choose G_{x}-equivariant triangulations on the unit spheres SV_{x} and SW_{x} such that the diameter of each simplex measured with respect to the metric d is smaller than δ/8. Equip [0, 1] with the CW -structure whose 0-skeleton is \{\frac{i}{2n} | i = 0, 1, 2, …, 2n} for some positive integer n which will be specified later. Equip D_{1}V_{x} with the G_{x}-CW -structure which is induced from the product G_{x}- CW -structure on SV_{x} × [0, 1] by the quotient map

SV_{x} × [0, 1] → DV_{x}, (y, t) ↦ t · y.

This is not yet the structure of a simplicial G_{x}-complex since the cells look like cones over simplices or products of simplices. The cones over simplices are again simplices and will not be changed. There is a standard way of subdividing a product of simplices to get a simplicial structure again. We use the resulting simplicial G_{x}-structure on D_{1}V_{x}. It is actually a G_{x}-equivariant triangulation. Define analogously a G_{x}-simplicial structure on D_{1}W.

Notice that D_{1/2}V_{x} ⊆ D_{1}V_{x} inherits a G_{x}-CW -simplicial substructure. We will also use a second G_{x}-simplicial structure on D_{1/2}V_{x}, which will be denoted by D_{1/2}V′_{x}. It is induced by the product G_{x}-CW -structure on SV_{x} ×[0, 1] above together with the the quotient map

SV_{x} × [0, 1] → D_{1/2}V_{x}, (y, t) ↦ t/2 · y.

The G_{x}-simplicial -structure on D_{1/2}V′_{x} is finer than the one on D_{1/2}V_{x} but agrees with the one on D_{1/2}V_{x} on the boundary. The map 2 id: V_{x} → V_{x}

induces an isomorphism of G_{x}-simplicial complexes 2 id: D_{1/2}V′_{x} ⟶^{≅} D_{1/2}V_{x}, but it does not induce a simplicial map 2 id: D_{1/2}V_{x} → D_{1}V_{x}. The latter map is at least cellular with respect to the G_{x}-CW -structures induced from the G_{x}- simplicial structures since the p-skeleton of D_{1/2}V_{x} is contained in the p-skeleton of D_{1/2}V′_{x}.

We equip D_{1}V_{x} × D_{1}W_{x}, D_{1/2}V_{x} × D_{1}W_{x} and D_{1/2}V_{x} × D_{1}W_{x} with the product G_{x}-simplicial structure. Again this requires subdividing products of simplices (except for products of a simplex with a vertex).

Recall that we have identified D_{2}T_{x}M with its image under the exponential map. Choose a complete set of representatives {x_{1}, x_{2}, …, x_{k}} for the G-orbits in Fix(f). By the construction above we get a G-triangulation on the G-submanifold ∐_{i=1}^{k} G ×_{G_{x_{i}}} D_{1}V_{x_{i}} × D_{1}W_{x_{i}} of M such that the diameter of each simplex is smaller than δ/4 if we choose the integer n above small enough. It can be extended to a G-triangulation K of M such that each simplex has a diameter less than δ/4. Let K′ be the refinement of K which agrees with K outside D_{1/2}V_{x_{i}} × D_{1}W and is D_{1/2}V′_{x_{i}} × D_{1}W on the subspace D_{1/2}V_{x_{i}} × D_{1}W. Then f: K′ → K is a G-map which is simplicial on ∐_{i=1}^{k} G ×_{G_{x_{i}}} (D_{1/2}V′_{x_{i}} × D_{1}W_{x_{i}}T_{x}M). The construction in the proof of the (non-equivariant) simplicial approximation theorem yields a subdivision K′′ of K′ such that K′′ and K′ agree on ∐_{i=1}^{k} G ×_{G_{x_{i}}} (D_{1/2}V′_{x_{i}} × D_{1}W_{x_{i}}) and a Ghomotopy h: M × [0, 1] → M from h_{0} = f to a simplicial map h_{1} : K′′ → K such that h is stationary on ∐_{i=1}^{k} G ×_{G_{x_{i}}} (D_{1/2}V′_{x_{i}} × D_{1}W_{x_{i}}) and the track of the homotopy for each point in M lies within a simplex of K. Recall that any simplex of K has diameter at most δ/4 and d(y, f(y)) ≥ δ holds for y ∈ M which does not lie in ∐_{i=1}^{k} G ×_{G_{x_{i}}} (D_{1/2}V′_{x_{i}} × D_{1}W_{x_{i}}). Hence for any simplex e ∈ K′′ outside ∐_{i=1}^{k} G ×_{G_{x_{i}}} (D_{1/2}V′_{x_{i}} × D_{1}W_{x_{i}}) we have h_{1}(e) ∩ e = ∅. The G-map h_{1} : K′′ → K′′ is not simplicial anymore but at least cellular with respect to the G-CW -structure on M coming from K′′. This comes from the fact that each skeleton of K′′ is larger than the one of K′.

Next we compute inc(h_{1}, e) for cells e in M with respect to the G-CW - structure induced by K′′. Obviously inc(h_{1}, e) = 0 if e does not belong to ∐_{i=1}^{k} G×_{G_{x_{i}}} (D_{1/2}V′_{x_{i}} ×D_{1}W_{x_{i}}) since for such cells e we have h_{1}(e)∩e = ∅. If e belongs to D_{1/2}V′_{x_{i}} × D_{1}W_{x_{i}} its image under h_{1} = f = 2 id_{V_{x_{i}}} ⊕ 0_{W_{x_{i}}} does not meet the interior of e unless it is the zero simplex sitting at (0, 0) ∈ V_{x_{i}} ⊕W_{x_{i}} or a simplex of the shape {t · x | t ∈ [0, 1/4n], x ∈ e} × {0} for some simplex in e ∈ SV_{x_{i}}. Hence among the cells in D_{1/2}V′_{x_{i}} ×D_{1}W_{x_{i}} only the zero simplex sitting at (0, 0) ∈ V_{x_{i}} ⊕ W_{x_{i}} and the simplex of the shape {t · x | t ∈ [0, 1/4n], x ∈ e} × {0} for some simplex in e ∈ SV_{x_{i}} can have non-zero incidence numbers inc(f, e) and one easily checks that these incidence numbers are all equal to 1. Hence, using Lemma 1.9 and the equality inc(f, e) = inc(f, ge), we get:

L^{ℚG}(f) = L^{ℚG}(h_{1})
= ∑_{p≥0}(−1)^{p} · ∑_{Ge∈G\I_{p}(K′′)} |G_{e}|^{−1} · inc(f, e)

= ∑_{p≥0}(−1)^{p} · ∑_{i=1}^{k} ∑_{\begin{matrix}G_{x_{i}} e∈ \\ G_{x_{i}} \I_{p}(D_{1/2}V′_{x_{i}} ×D_{1}W) \end{matrix}} |(G_{x_{i}})_{e}|^{−1} · inc(f, e)

= ∑_{p≥0}(−1)^{p} · ∑_{i=1}^{k} |G_{x_{i}}|^{−1} ∑_{\begin{matrix}G_{x_{i}} e∈ \\ G_{x_{i}} \I_{p}(D_{1/2}V′_{x_{i}} ×D_{1}W) \end{matrix}} |G_{x_{i}}/(G_{x_{i}})_{e}| · inc(f, e)

= ∑_{p≥0}(−1)^{p} · ∑_{i=1}^{k} |G_{x_{i}}|^{−1} ∑_{e∈I_{p}(D_{1/2}V′_{x_{i}} ×D_{1}W)} inc(f, e),

and

L^{ℚG}(f) = ∑_{i=1}^{k} |G_{x_{i}}|^{−1} ∑_{p≥0}(−1)^{p} · ∑_{e∈I_{p}(D_{1/2}V′_{x_{i}} ×D_{1}W)} inc(f, e)

= ∑_{i=1}^{k} |G_{x_{i}}|^{−1} (1 + ∑_{p≥1}(−1)^{p} · |I_{p−1}(SV_{x_{i}})|)

= ∑_{i=1}^{k} |G_{x_{i}}|^{−1} (1 − χ(SV_{x_{i}}))

= ∑_{i=1}^{k} |G_{x_{i}}|^{−1}(−1)^{dim(V_{x_{i}})}

= ∑_{i=1}^{k} |G_{x_{i}}|^{−1} \frac{det(id −T_{x_{i}}f)}{| det(id −T_{x_{i}}f)|}

= ∑_{i=1}^{k} |G_{x_{i}}|^{−1} deg((id −T_{x_{i}}f)^{c})
= ∑_{Gx∈G\ Fix(f)} |G_{x}|^{−1} · deg((id −T_{x_{i}}f)^{c}).

This finishes the proof of Theorem 2.1.

3. The equivariant Lefschetz classes

In this section we define the equivariant Lefschetz class appearing in the equivariant Lefschetz fixed point Theorem 0.2. We will use the following notation in the sequel. Notation 3.1 Let G be a discrete group and H ⊆ G be a subgroup. Let NH = {g ∈ G | gHg^{−1} = H} be its normalizer and let WH := NH/H be its Weyl group.

Denote by consub(G) the set of conjugacy classes (H) of subgroups H ⊆ G. Let X be a G-CW -complex. Put

X^{H} := {x ∈ X | H ⊆ G_{x}};
X^{>H} := {x ∈ X | H ⊊ G_{x}},

where G_{x} is the isotropy group of x under the G-action.

Let x: G/H → X be a G-map. Let X^{H}(x) be the component of X^{H} containing x(1H). Put

X^{>H}(x) = X^{H}(x) ∩ X^{>H}.

Let WH_{x} be the isotropy group of X^{H}(x) ∈ π_{0}(X^{H}) under the WH-action.

Next we define the group U^{G}(X), where the equivariant Lefschetz class will take its values.

Let Π_{0}(G, X) be the component category of the G-space X in the sense of tom Dieck [2, I.10.3]. Objects are G-maps x: G/H → X. A morphism σ from x: G/H → X to y: G/K → X is a G-map σ: G/H → G/K such that y ∘ σ and x are G-homotopic. A G-map f: X → Y induces a functor Π_{0}(G, f): Π_{0}(G, X) → Π_{0}(G, Y) by composition with f. Denote by Is Π_{0}(G, X) the set of isomorphism classes [x] of objects x: G/H → X in Π_{0}(G, X). Define

U^{G}(X) := ℤ[Is Π_{0}(G, X)],  (3.2)

where for a set S we denote by ℤ[S] the free abelian group with basis S. Thus we obtain a covariant functor from the category of G-spaces to the category of abelian groups. Obviously U^{G}(f) = U^{G}(g) if f, g: X → Y are G-homotopic.

There is a natural bijection

Is Π_{0}(G, X) ⟶^{≅} ∐_{(H)∈consub(G)} WH\π_{0}(X^{H}),  (3.3)

which sends x: G/H → X to the orbit under the WH-action on π_{0}(X^{H}) of the component X^{H}(x) of X^{H} which contains the point x(1H). It induces a natural isomorphism

U^{G}(X) ⟶^{≅} ⨁_{(H)∈consub(G)} ℤ[WH\π_{0}(X^{H})].  (3.4)

Let α: G → K be a group homomorphism and X be a G-CW -complex. We obtain from α a functor

α_{∗}: Π_{0}(G, X) → Π_{0}(K, α_{∗}X)

which sends an object x: G/H → X to the object α_{∗}(x): K/α(H) = α_{∗}(G/H) → α_{∗}X and similarly for morphisms. Thus we obtain an induction homomorphism of abelian groups

α_{∗}: U^{G}(X) → U^{K}(α_{∗}X).  (3.5)

Next we define the equivariant Lefschetz class. Let X be a finite proper G-CW -complex. Let f: X → X be a cellular G-map such that for each subgroup K ⊆ G the map π_{0}(f^{K}): π_{0}(X^{K}) → π_{0}(X^{K}) is the identity. For any G-map x: G/H → X it induces a map

(f^{H}(x), f^{>H}(x)): (X^{H}(x), X^{>H}(x)) → (X^{H}(x), X^{>H}(x))

of pairs of finite proper WH_{x}-CW -complexes. Then

L^{ℤWH_{x}}(f^{H}(x), f^{>H}(x)) ∈ ℤ

is defined (see (1.5)) since the isotropy group under the WH_{x}-action of any point in X^{H}(x) − X^{>H}(x) is trivial.

Definition 3.6 We define the equivariant Lefschetz class of f

Λ^{G}(f) ∈ U^{G}(X)

by assigning to [x: G/H → X] ∈ Is Π_{0}(G, X) the integer

L^{ℤWH_{x}} (f^{H}(x), f^{>H}(x): (X^{H}(x), X^{>H}(x)) → (X^{H}(x), X^{>H}(x))) ,

if f^{H}: X^{H} → X^{H} maps X^{H}(x) to itself, and zero otherwise.

Since X^{>H}(x) ≠ X^{H}(x) and therefore L^{ℤWH_{x}}(f^{H}(x), f^{>H}(x)) ≠ 0 holds only for finitely many elements [x] in Is Π_{0}(G, X), Definition 3.6 makes sense. Notice for the sequel that f^{H}(X^{H}(x)) ∩ X^{H}(x) ≠ ∅ implies f^{H}(X^{H}(x)) ⊆ X^{H}(x). The elementary proof that Lemma 1.6 implies the following lemma is left to the reader.

Lemma 3.7 Let X be a finite proper G-CW -complex. Let f: X → X be a cellular G-map. Then

(a) The equivariant Lefschetz class Λ^{G}(f) depends only on the cellular Ghomotopy class of f;

(b) If f′: Y → Y is a cellular G-self-map of a finite G-CW -complex and h: X → Y is a cellular G-homotopy equivalence satisfying h ∘ f ≃_{G} f′ ∘ h, then U^{G}(h): U^{G}(X) ⟶^{≅} U^{G}(Y) is bijective and sends Λ^{G}(f) to Λ^{G}(f′);

(c) Let α: G → K be an inclusion of groups. Denote by α_{∗}f the cellular K-self-map obtained by induction with α. Then

Λ^{K}(α_{∗}f) = α_{∗}Λ^{G}(f).

By the equivariant cellular approximation theorem (see for instance [7, Theorem 2.1 on page 32]) any G-map of G-CW -complexes is G-homotopic to a cellular G-map and two cellular G-maps which are G-homotopic are actually cellularly G-homotopic. Hence we can drop the assumption cellular in the sequel because of G-homotopy invariance of the equivariant Lefschetz class (see Lemma 3.7 (a)).

4. The local equivariant Lefschetz class

In this section we introduce the local equivariant Lefschetz class in terms of fixed point data. Before we can define it, we recall the classical notion of the equivariant Lefschetz class with values in the Burnside ring for a finite group.

Let K be a finite group. The abelian group U^{K}({∗}) is canonically isomorphic to the abelian group which underlies the Burnside ring A(K). Recall that the Burnside ring is the Grothendieck ring of finite K-sets with the additive structure coming from disjoint union and the multiplicative structure coming from the Cartesian product.

Let X be a finite K-CW -complex. Define the equivariant Lefschetz class with values in the Burnside ring of f

Λ_{0}^{K}(f) ∈ A(K) = U^{K}({∗})  (4.1)

by

Λ_{0}^{K}(f) := ∑_{(H)∈consub(K)} L^{ℤWH}(f^{H}, f^{>H}) · [K/H].

(Here and elsewhere the subscript _{0} shall indicate that the corresponding invariant takes values in the Burnside ring and the component structure of the various fixed point sets is not taken into account.) Denote by

ch_{0}^{K}: A(K) → ∏_{(H)∈consub(K)} ℤ  (4.2)

the character map which sends the class of a finite set S to the collection {|S^{H}| | (H) ∈ consub(K)} given by the orders of the various H-fixed point sets. The character map is a ring homomorphism, and it is injective (see Lemma 5.3). The equivariant Lefschetz class Λ_{0}^{K}(f) is characterized by the property (see for instance [5, Theorem 2.19 on page 504]), [6, Lemma 3.3 on page 138])

ch_{0}^{K}(Λ_{0}^{K}(f)) = {L^{ℤ[{1}]}(f^{H}) | (H) ∈ consub(K)}.  (4.3)

If pr: X → {∗} is the projection, then U^{K}(pr): U^{K}(X) → U^{K}({∗}) = A(K) sends Λ^{K}(f) (see Definition 3.6) to Λ_{0}^{K}(f) defined in (4.1).

Let V be a (finite-dimensional) K-representation and let f: V^{c} → V^{c} be a K-self-map of the one-point-compactification V^{c}. Define its equivariant degree

Deg_{0}^{K}(f) ∈ A(K) = U^{K}({∗})  (4.4)

by

Deg_{0}^{K}(f) := (Λ_{0}^{K}(f) − 1) · (Λ_{0}^{K}(id_{V^{c}}) − 1).

Since the character map (4.2) is an injective ring homomorphism, we conclude from (4.3) above that Deg_{0}^{K}(f) is uniquely characterized by the equality

ch_{0}^{K}(Deg_{0}^{K}(f)) = {deg(f^{H}) | (H) ∈ consub(K)},  (4.5)

where deg(f^{H}) is the degree of the self-map f^{H}: (V^{c})^{H} → (V^{c})^{H} of the connected closed orientable manifold (V^{c})^{H}, if dim((V^{c})^{H}) ≥ 1, and deg(f^{H}) is defined to be 1, if dim((V^{c})^{H}) = 0. The equivariant degree of (4.4) induces an isomorphism from the K-equivariant stable cohomotopy of a point to the Burnside ring A(K) [3, Theorem 7.6.7 on page 190], [13].

Let M be a cocompact proper G-manifold (possibly with boundary). Let f: M → M be a smooth G-map. Denote by Fix(f) = {x ∈ X | f(x) = x} the set of fixed points of f. Suppose that for any x ∈ Fix(f) the determinant of the linear map id −T_{x}f: T_{x}M → T_{x}M is different from zero. (One can always find a representative in the G-homotopy class of f which satisfies this assumption.) Then G\ Fix(f) is finite. Consider an element x ∈ Fix(f). Let α_{x}: G_{x} → G be the inclusion. We obtain from (α_{x})_{∗} (see (3.5)) and U^{G}(x) for x ∈ Fix(f) interpreted as a G-map x: G/G_{x} → X a homomorphism

U^{G_{x}}({∗}) ⟶^{(α_{x})_{∗}} U^{G}(G/G_{x}) ⟶^{U^{G}(x)} U^{G}(X).

Thus we can assign to x ∈ Fix(f) the element U^{G}(x)∘(α_{x})_{∗}(Deg_{0}^{G_{x}}((id −T_{x}f)^{c})), where Deg_{0}^{G_{x}}((id −T_{x}f)^{c}) is the equivariant degree (see (4.4)) of the map induced on the one-point-compactifications by the isomorphism (id −T_{x}f): T_{x}M → T_{x}M. One easily checks that this element depends only on the G-orbit of x ∈ Fix(X). Definition 4.6 We can define the local equivariant Lefschetz class by

Λ_{loc}^{G}(f)

:= ∑_{Gx∈G\ Fix(f)} U^{G}(x) ∘ (α_{x})_{∗} (Deg_{0}^{G_{x}} ((id −T_{x}f)^{c}))) ∈ U^{G}(M).

Now have defined all the ingredients appearing in the Equivariant Lefschetz fixed point theorem 0.2. Before we give its proof, we discuss the following example

Example 4.7 Let G be a discrete group and M be a cocompact proper Gmanifold (possibly with boundary). Suppose that the isotropy group G_{x} of each point x ∈ M has odd order. This holds automatically if G itself is a finite group of odd order. Let f: M → M be a smooth G-map. Suppose that Fix(f) ∩ ∂M = ∅ and for each x ∈ Fix(f) the determinant of the linear map id −T_{x}f: T_{x}M → T_{x}M is different from zero. If H is a finite group of odd order, then the multiplicative group of units A(H)^{∗} of the Burnside ring is known to be {±1} [3, Proposition 1.5.1]. The element Deg_{0}^{G_{x}}((id −T_{x}f)^{c}) ∈ A(G_{x}) =

U^{G_{x}}({∗}) satisfies (Deg_{0}^{G_{x}}((id −T_{x}f)^{c}))^{2} = 1 since this holds for its image un-

der the injective ring homomorphism ch_{0}^{G_{x}}: A(G_{x}) → ∏_{(H)∈consub(G_{x})} ℤ, whose coefficient at (H) ∈ consub(G_{x}) is deg((id −T_{x}f)^{c})^{H}) ∈ {±1} (see (4.5)). Hence Deg_{0}^{G_{x}}((id −T_{x}f)^{c}) belongs to A(G_{x})^{∗} = {±1}. This implies that

Deg_{0}^{G_{x}}((id −T_{x}f)^{c}) = \frac{det(id −T_{x}f: T_{x}M → T_{x}M)}{| det(id −T_{x}f: T_{x}M → T_{x}M)|} · [G_{x}/G_{x}].

Hence the definition of the local equivariant Lefschetz class reduces to

Λ_{loc}^{G}(f) := ∑_{Gx∈G\ Fix(f)} \frac{det(id −T_{x}f: T_{x}M → T_{x}M)}{| det(id −T_{x}f: T_{x}M → T_{x}M)|} · [x: G/G_{x} → M].

where x: G/G_{x} → M sends g · G_{x} to gx.

Remark 4.8 Equivariant Lefschetz classes for compact Lie groups were studied in [5]. In the non-equivariant setting, universal Lefschetz classes with values in certain K-groups were defined and analyzed in [8]. It seems to be possible to combine the K-theoretic invariants there with the equivariant versions presented here to obtain a universal equivariant Lefschetz class.

5. The proof of the equivariant Lefschetz fixed point theorem

This section is devoted to the proof of the equivariant Lefschetz fixed point Theorem 0.2.

First we define the character map for a proper G-CW -complex X:

ch^{G}(X): U^{G}(X) → ⨁_{Is Π_{0}(G,X)} ℚ.  (5.1)

We have to define for an isomorphism class [x] of objects x: G/H → X in Π_{0}(G, X) the component ch^{G}(X)([x])_{[y]} of ch^{G}(X)([x]) which belongs to an isomorphism class [y] of objects y: G/K → X in Π_{0}(G, X), and check that χ^{G}(X)([x])_{[y]} is different from zero for at most finitely many [y]. Denote by mor(y, x) the set of morphisms from y to x in Π_{0}(G, X). We have the left operation

Aut(y) × mor(y, x) → mor(y, x), (σ, τ) ↦ τ ∘ σ^{−1}.

There is an isomorphism of groups

WK_{y} ⟶^{≅} Aut(y)

which sends gK ∈ WK_{y} to the automorphism of y given by the G-map

R_{g^{−1}} : G/K → G/K, g′K ↦ g′g^{−1}K.

Thus mor(y, x) becomes a left WK_{y}-set.

The WK_{y}-set mor(y, x) can be rewritten as

mor(y, x) = {g ∈ G/H^{K} | g · x(1H) ∈ X^{K}(y)},

where the left operation of WK_{y} on {g ∈ G/H^{K} | g · x(1H) ∈ Y^{K}(y)} comes from the canonical left action of G on G/H. Since H is finite and hence contains only finitely many subgroups, the set WK\(G/H^{K}) is finite for each K ⊆ G and is non-empty for only finitely many conjugacy classes (K) of subgroups K ⊆ G. This shows that mor(y, x) ≠ ∅ for at most finitely many isomorphism classes [y] of objects y ∈ Π_{0}(G, X) and that the WK_{y}-set mor(y, x) decomposes into finitely many WK_{y} orbits with finite isotropy groups for each object y ∈ Π_{0}(G, X). We define

ch^{G}(X)([x])_{[y]} := ∑_{\begin{matrix}WK_{y}·σ∈ \\ WK_{y}\ mor(y,x) \end{matrix}} |(WK_{y})_{σ}|^{−1},  (5.2)

where (WK_{y})_{σ} is the isotropy group of σ ∈ mor(y, x) under the WK_{y}-action. Notice that ch^{G}(X)([x])_{[y]} is the same as dim_{ℚWK_{y}}(ℚ(mor(y, x))), if one defines dim_{ℚWK_{y}}(P) of a finitely generated ℚWK_{y}-module P by tr_{ℚWK_{y}}(id_{P}) (see Lemma 1.3 (f)). This agrees with the more general notion of the von Neumann dimension of the finitely generated Hilbert N (WK_{y})-module l^{2}(mor(y, x)).

The character map ch^{G}(X) of (5.1) should not be confused with the isomorphism appearing in (3.4). If G is finite and X = {∗}, then character map ch^{G}(X) of (5.1) and the character map ch_{0}^{G} of (4.2) are related under the identifications U^{G}({∗}) = A(G) and Is Π_{0}(G, {∗}) = consub(G) by

(ch_{0}^{G})_{(H)} = |WH| · ch^{G}({∗})_{(H)}

for (H) ∈ consub(G). Lemma 5.3 The map ch^{G} of (5.1) is injective.

\underline{\underline{Proof}} : Consider u ∈ U^{G}(X) with ch^{G}(X)(u) = 0. We can write u as a finite sum

u = ∑_{i=1}^{n} m_{i} · [x_{i}: G/H_{i} → X]

for some integer n ≥ 1 and integers m_{i} such that [x_{i}] = [x_{j}] implies i = j and such that H_{i} is subconjugate to H_{j} only if i ≥ j or (H_{i}) = (H_{j}). We have to show that u = 0. It suffices to prove

ch^{G}(X)([x_{i}])_{[x_{1}]} = \{ \begin{matrix}0 & if i > 1 \\ 1 & if i = 1 \end{matrix}

because then m_{1} = 0 follows and one can show inductively that m_{i} = 0 for i = 1, 2, …, n. Suppose that ch^{G}(X)([x_{i}])_{[x_{1}]} ≠ 0. Then mor(x_{1}, x_{i}) is nonempty. This implies that im(x_{i}) ∩ X^{H_{1}} is non-empty and hence that H_{1} is subconjugate to H_{i}. From the way we have enumerated the H_{i}’s we conclude (H_{i}) = (H_{1}). Since im(x_{i})∩X^{H_{1}}(x_{1}) is non-empty, we get [x_{i}] = [x_{1}] and hence i = 1. Since by definition ch^{G}(X)([x_{1}])_{[x_{1}]} = 1, the claim follows.

Lemma 5.4 Let f: X → X be a G-self-map of a finite proper G-CW -complex X. Let [y] be an isomorphism class of objects y: G/K → X in Π_{0}(G, X). Then

ch^{G}(X)(Λ^{G}(f))_{[y]} = L^{ℚWK_{y}} (f|_{X^{K}(y)} : X^{K}(y) → X^{K}(y)) ,

if f^{K}(X^{K}(y)) ⊆ X^{K}(y) and

ch^{G}(X)(Λ^{G}(f))_{[y]} = 0

otherwise.

\underline{\underline{Proof}} : We first consider the case f^{K}(X^{K}(y)) ⊆ X^{K}(y). Let X_{p} be the p-th skeleton of X. Then we can write X_{p} as a G-pushout

∐_{i=1}^{n_{p}} G/H_{i} × S^{p−1} \underrightarrow{∐_{i=1}^{np} q_{p,i}} X_{p−1}
↓  ↓
∐_{i=1}^{n_{p}} G/H_{i} × D^{p} \overrightarrow{∐_{i=1}^{np} Q_{p,i}} X_{p}

for an integer n_{p} ≥ 0 and finite subgroups H_{i} ⊆ G. For each i ∈ I_{p} let x_{p,i}: G/H_{i} → X be the G-map obtained by restricting the characteristic map Q_{p,i} to G/H_{i} × {0}. For i = 1, 2, …, p define

inc(f, p, i) = inc(f, e_{i})  (5.5)

where e_{i} is the open cell Q_{p,i}(gH_{i} × (D^{p} − S^{p−1})) for any choice of gH_{i} ∈ G/H_{i} and inc(f, e_{i}) is the incidence number defined in (1.8). Since f is G-equivariant, the choice of gH_{i} ∈ G/H_{i} does not matter.

Now the G-CW -structure on X induces a WK_{y}-CW -structure on X^{K}(y) whose p-skeleton X^{K}(y)_{p} is X^{K}(y) ∩ X_{p}. Let x: G/H → X be an object in Π_{0}(G, X). Recall that mor(y, x) is the set of morphisms from y to x in Π_{0}(G, X) and carries a canonical WK_{y}-operation. One easily checks that there is a WK_{y}- pushout diagram

∐_{i=1}^{n_{p}} mor(y, x_{p,i}) × S^{p−1} \underrightarrow{∐_{i=1}^{np} q_{p,i}^{K}(y)} X^{K}(y)_{p−1}
↓  ↓
∐_{i=1}^{n_{p}} mor(y, x_{p,i}) × D^{p} \overrightarrow{∐_{i=1}^{np} Q_{p,i}^{K}(y)} X^{K}(y)_{p} ,

where the maps q_{p,i}^{K}(y) and Q_{p,i}^{K}(y) are induced by the maps q_{p,i} and Q_{p,i} in the obvious way. We conclude from Lemma 1.9

L^{ℚ[WK_{y}]}(f^{K}(y))

= ∑_{p≥0}(−1)^{p} · ∑_{i=1}^{n_{p}} ∑_{\begin{matrix}WK_{y}·σ∈ \\ WK_{y}\ mor(y,x_{p,i}) \end{matrix}} |(WK_{y})_{σ}|^{−1} · inc(f, p, i), (5.6)

where the incidence number inc(f, p, i) has been defined in (5.5).

Analogously we get for any object x: G/H → X a WH_{x}-pushout diagram

∐\begin{matrix}i=1,2...,n_{p} \\ [x_{p,i}]=[x] \end{matrix} Aut(x, x) × S^{p−1} \underrightarrow{∐_{i=1}^{np} q_{p,i}^{H}(x)} X^{H}(x)_{p−1} ∪ X^{>H}(x)
↓  ↓
∐\begin{matrix}i=1,2...,n_{p} \\ [x_{p,i}]=[x] \end{matrix} Aut(x, x) × D^{p} \overrightarrow{∐_{i=1}^{np} Q_{p,i}^{H}(x)} X^{H}(x)_{p} ∪ X^{>H}(x)

describing how the p-skeleton of the relative WH_{x}-CW -complex (X^{H}(x), X^{>H}(x)) is obtained from its (p − 1)-skeleton. We conclude from Lemma 1.9

L^{ℤ[WH_{x}]} (f^{H}(x), f^{>H}(x)): (X^{H}(x), X^{>H}(x)) → (X^{H}(x), X^{>H}(x)))

= ∑_{p≥0}(−1)^{p} · ∑_{\begin{matrix}i=1,2,...,n_{p} \\ [x_{p,i}]=[x] \end{matrix}} inc(f, p, i), (5.7)

where inc(f, p, i) is the number introduced in (5.5). We get from the Definition 3.6 of Λ^{G}(f) and (5.7)

ch^{G}(X)(Λ^{G}(f))_{[y]} = ∑_{p≥0}(−1)^{p} · ∑_{i=1}^{n_{p}} inc(f, p, i) · ch^{G}(X)([x_{p,i}])_{[y]}. (5.8)

Now Lemma 5.4 follows from the definition (5.2) of ch^{G}(X)([x_{p,i}])_{[y]} and equations (5.6) and (5.8) in the case f^{K}(X^{K}(y)) ⊆ X^{K}(y).

Now suppose that f^{K}(X^{K}(y)) ∩ X^{K}(y) = ∅. For any object x: G/H → X with mor(y, x) ≠ ∅ we conclude f^{H}(X^{H}(x)) ∩ X^{H}(x) = ∅ and hence Λ^{G}(f) assigns to [x] zero by definition. We conclude ch^{G}(X)(Λ^{G}(f))_{[y]} = 0 from the definition of ch^{G}. This finishes the proof of Lemma 5.4.

Lemma 5.9 Let M be a cocompact smooth proper G-manifold. Let f: M → M be a smooth G-map. Suppose that for any x ∈ F ix(f) the determinant det(id_{T_{x}f} −T_{x}f) is different from zero. Let y: G/K → M be an object in Π_{0}(G, M). Denote by (WK_{y})_{x} the isotropy group for x ∈ Fix(f|_{M^{K}(y)}) under the WK_{y}-action on M^{K}(y).

Then G\ Fix(f) is finite and we get

ch^{G}(M)_{[y]}(Λ_{loc}^{G}(f))

= ∑_{\begin{matrix}WK_{y}·x∈ \\ WK_{y}\ Fix(f|_{M^{K} (y)}) \end{matrix}} |(WK_{y})_{x}|^{−1} · deg ((id_{T_{x}M^{K}(y)} −T_{x}(f|_{M^{K}(y)}))^{c}) .

\underline{\underline{Proof}} : We have already shown in Theorem 2.1 that G\ Fix(f) is finite.

Consider x ∈ M. Let G_{x} be its isotropy group under the G-action and denote by x: G/G_{x} → M the G-map g ↦ gx. Let α_{x}: G_{x} → G be the inclusion. Given a morphism σ: y → x, let (WK_{y})_{σ} be its isotropy group under the WK_{y}-action and let (K_{σ}) be the element in consub(G_{x}) given by g^{−1}Kg for any element g ∈ G for which σ: G/K → G/G_{x} sends g′K to g′gG_{x}. Notice that (K_{σ}) depends only on WK_{y} · σ ∈ WK_{y}\ mor(y, x). We first show for the composition

U^{G_{x}}({∗}) ⟶^{(α_{x})_{∗}} U^{G}(G/G_{x}) ⟶^{U^{G}(x)} U^{G}(M) ⟶^{ch^{G}(M)_{[y]}} ℚ

that for each u ∈ U^{G_{x}}({∗})

ch^{G}(M)_{[y]} ∘ U^{G}(x) ∘ (α_{x})_{∗}(u)

= ∑_{\begin{matrix}WK_{y}·σ∈ \\ WK_{y}\ mor(y,x) \end{matrix}} |(WK_{y})_{σ}|^{−1} · ch_{0}^{G_{x}}(u)_{(K_{σ})}. (5.10)

It suffices to check this for a basis element u = [G_{x}/L] ∈ U^{G}({∗}) = A(G_{x}) for some subgroup L ⊆ G_{x}. Let pr: G/L → G/G_{x} be the projection. We get from the definitions

ch^{G}(M)_{[y]} ∘ U^{G}(x) ∘ (α_{x})_{∗}([G/L])
= ch^{G}(M)_{[y]}([x ∘ pr: G/L → M])
= ∑_{\begin{matrix}WK_{y}·τ∈ \\ WK_{y}\ mor(y,x∘pr) \end{matrix}} |(WK_{y})_{τ} |^{−1}.  (5.11)

We get a WK_{y}-map

q: mor(y, x ∘ pr) → mor(y, x), τ ↦ pr ∘τ.

We can write

mor(y, x ∘ pr) = ∐_{\begin{matrix}WK_{y}·σ∈ \\ WK_{y}\ mor(y,x) \end{matrix}} WK_{y} ×_{(WK_{y})_{σ}} q^{−1}(σ).

The (WK_{y})_{σ}-set q^{−1}(σ) is a finite disjoint union of orbits

q^{−1}(σ) = ∐_{i∈I(σ)}(WK_{y})_{σ}/A_{i}.

This implies

mor(y, x ∘ pr) = ∐_{\begin{matrix}WK_{y}·σ∈ \\ WK_{y}\ mor(y,x) \end{matrix}} ∐_{i∈I(σ)} WK_{y}/A_{i}

and hence

∑_{\begin{matrix}WK_{y}·τ∈ \\ WK_{y}\ mor(y,x∘pr) \end{matrix}} |(WK_{y})_{τ} |^{−1} = ∑_{\begin{matrix}WK_{y}·σ∈ \\ WK_{y}\ mor(y,x) \end{matrix}} ∑_{i∈I(σ)} |A_{i}|^{−1}.

Putting this into (5.11) yields

ch^{G}(M)_{[y]} ∘ U^{G}(x) ∘ (α_{x})_{∗}([G/L])
= ∑_{\begin{matrix}WK_{y}·σ∈ \\ WK_{y}\ mor(y,x) \end{matrix}} ∑_{i∈I(σ)} |A_{i}|^{−1}.  (5.12)

Obviously

|q^{−1}(σ)| = |(WK_{y})_{σ}| · ∑_{i∈I(σ)} |A_{i}|^{−1}.  (5.13)

If σ: G/K → G/G_{x} sends g′K to g′gG_{x} for appropriate g ∈ G with g^{−1}Kg ⊆ G_{x}, then there is a bijection

G_{x}/L^{g^{−1}Kg} ⟶^{≅} q^{−1}(σ)

which maps g′L to the morphism R_{gg′}: G/K → G/L, g′′K → g′′gg′L. This implies

|q^{−1}(σ)| = |G_{x}/L^{g^{−1}Kg}| = ch_{0}^{G_{x}}([G_{x}/L])_{(K_{σ})}.  (5.14)

We conclude from the equations (5.12), (5.13) and (5.14) above that (5.10) follows for u = [G_{x}/L] and hence for all u ∈ U^{G_{x}}({∗}) = A(G_{x}).

Now we are ready to prove Lemma 5.9. We get from the Definition 4.6 of Λ_{loc}^{G}(f) and from (5.10)

ch^{G}(M)_{[y]}(Λ_{loc}^{G}(v)) = ∑_{Gx∈G\ Fix(f)} ∑_{\begin{matrix}WK_{y}·σ∈ \\ WK_{y}\ mor(y,x) \end{matrix}}

|(WK_{y})_{σ}|^{−1} · ch_{0}^{G_{x}} (Deg_{0}^{G_{x}} ((id −T_{x}f)^{c})))_{(K_{σ})} . (5.15)

We conclude from (4.5)

ch_{0}^{G_{x}} (Deg_{0}^{G_{x}} ((id −T_{x}f)^{c})))_{(K_{σ})} = deg ((id_{T_{x}M^{Kσ}} −T_{x}f^{K_{σ}})^{c}) .(5.16)

If σ: G/K → G/G_{x} is of the form g′K ↦ g′gG_{x} for appropriate g ∈ G with g^{−1}Kg ⊆ G_{x}, then

deg ((id_{T_{x}M^{Kσ}} −T_{x}f^{K_{σ}})^{c}) = deg ((id_{T_{gx}M^{K}(y)} −T_{gx}(f|_{M^{K}(y)}))^{c}) (5.17)

since f is G-equivariant. There is a bijection of WK_{y}-sets

∐_{Gx∈G\ Fix(f)} mor(y, x) ⟶^{≅} Fix(f|_{M^{K}(y)})

which sends σ ∈ mor(y, x) to σ(1K) · x. Now Lemma 5.9 follows from (5.15), (5.16) and (5.17).

Because of Lemma 5.3, Lemma 5.4 and Lemma 5.9 the equivariant Lefschetz fixed point Theorem 0.2 follows from the orbifold Lefschetz fixed point Theorem 2.1.

6. Euler characteristic and index of a vector field in the equivariant setting

Definition 6.1 Let X be a finite G-CW -complex X. We define the universal equivariant Euler characteristic of X

χ^{G}(X) ∈ U^{G}(X)

by assigning to [x: G/H → X] ∈ Is Π_{0}(G, X) the (ordinary) Euler characteristic of the pair of finite CW -complexes (WH_{x}\X^{H}(x), WH_{x}\X^{>H}(x)). If X is proper, we define its orbifold Euler characteristic

χ^{ℚG}(X) := ∑_{p≥0} (−1)^{p} ∑_{G·e∈G\I_{p}(X)} |G_{e}|^{−1} ∈ ℚ,

where I_{p}(X) is the set of the open cells of the CW-complex X (after forgetting the group action) and G_{e} is the isotropy group of e under the G-action on I_{p}(X).

One easily checks that χ^{G}(X) agrees with the equivariant Lefschetz class Λ^{G}(id_{X}) (see Definition 3.6). It can also be expressed by counting equivariant cells. If x: G/H → X is an object in Π_{0}(G, X) and ♯_{p}([x]) is the number of equivariant p-dimensional cells of orbit type G/H which meet the component X^{H}(x), then

χ^{G}(X)([x]) = ∑_{p≥0}(−1)^{p} · ♯_{p}([x]).  (6.2)

The universal equivariant Euler characteristic is the universal additive invariant for finite G-CW -complexes in the sense of [7, Theorem 6.7].

The orbifold Euler characteristic χ^{ℚG}(X) can be identified with L^{ℚG}(id_{X}) (see Definition 1.4) or with the more general notion of the L^{2}-Euler characteristic χ^{(2)}(X; N (G)). In analogy with L^{(2)}(f, f_{0}; N (G)) (see Remark 1.7), one can compute χ^{(2)}(X; N (G)) in terms of L^{2}-homology

χ^{(2)}(X; N (G)) = ∑_{p≥0}(−1)^{p} · dim_{N (G)} (H_{p}^{(2)}(X; N (G)) ,

where dim_{N (G)} denotes the von Neumann dimension (see for instance [9, Section 6.6]).

We conclude from Lemma 5.4:

Lemma 6.3 Let X be a finite proper G-CW -complex X. Let [y] be an isomorphism class of objects y: G/K → X in π_{0}(G, X). Then

ch^{G}(X)_{[y]}(χ^{G}(X)) = χ^{ℚWK_{y}}(X^{K}(y)).

Next we want to express the universal equivariant Euler characteristic of a cocompact proper G-manifold M, possibly with boundary, in terms of the zeros of an equivariant vector field.

Consider an equivariant vector field Ξ on M, i.e., a G-equivariant section of the tangent bundle T M of M. Suppose that Ξ is transverse to the zero-section i: M → T M, i.e., if Ξ(x) = 0, then T_{Ξ(x)}T M is the sum of the subspaces given by the images of T_{x}Ξ: T_{x}M → T_{Ξ(x)}(T M) and T_{x}i: T_{x}M → T_{Ξ(x)}(T M). Any equivariant vector field on M can be changed by an arbitrary small perturbation into one which is transverse to the zero-section. We want to assign to such a Ξ its equivariant index as follows.

Since Ξ is transverse to the zero-section, the set Zero(Ξ) of points x ∈ M with Ξ(x) = 0 is discrete. Hence G\ Zero(Ξ) is finite, since G acts properly on M and G\M is compact by assumption. Fix x ∈ Zero(Ξ). The zero-section i: M → T M and the inclusion j_{x}: T_{x}M → T M induce an isomorphism of G_{x}-representations

T_{x}i ⊕ T_{x}j_{x}: T_{x}M ⊕ T_{x}M ⟶^{≅} T_{i(x)}(T M)

if we identify T_{i(x)}(T_{x}M) = T_{x}M in the obvious way. If pr_{k} denotes the projection onto the k-th factor for k = 1, 2 we obtain a linear G_{x}-equivariant isomorphism

d_{x}Ξ: T_{x}M ⟶^{T_{x}Ξ} T_{Ξ(x)}(T M) ⟶^{(T_{x}i⊕T_{x}j_{x})^{−1}} T_{x}M ⊕ T_{x}M ⟶^{pr_{2}} T_{x}M.(6.4)

Notice that we obtain the identity if we replace pr_{2} by pr_{1} in the expression (6.4) above. The G_{x}-map d_{x}Ξ induces a G_{x}-map (d_{x}Ξ)^{c}: T_{x}M^{c} → T_{x}M^{c} on the onepoint compactification. Define analogously to the local equivariant Lefschetz class (see Definition 4.6) the equivariant index of Ξ

i^{G}(Ξ) ∈ U^{G}(M)  (6.5)

by

i^{G}(Ξ) := ∑_{Gx∈G\ Zero(Ξ)} U^{G}(x) ∘ (α_{x}) (Deg_{0}^{G_{x}}((d_{x}Ξ)^{c})) .

We say that Ξ points outward at the boundary if for each x ∈ ∂M the tangent vector Ξ(x) ∈ T_{x}M does not lie in the subspace T_{x}∂M and is contained in the half space T_{x}M^{+} of tangent vectors u ∈ T_{x}M for which there is a path w: [−1, 0] → M with w(0) = x and w′(0) = u. If ∂M = ∅, this condition is always satisfied.

Theorem 6.6 (Equivariant Euler characteristic and vector fields) Let M be a cocompact proper smooth G-manifold. Let Ξ be a G-equivariant vector field which is transverse to the zero-section and points outward at the boundary. Then we get in U^{G}(M)

χ^{G}(M) = i^{G}(Ξ).

\underline{\underline{Proof}} : Let Φ: M × (−∞, 0] → M be the flow associated to the vector field Ξ. It is defined on M × (−∞, 0] since Ξ is equivariant and points outward and M is cocompact. Moreover, each map Φ_{−ϵ}: M → M is a G-diffeomorphism and G-homotopic to id_{M} for ϵ > 0. This implies

Λ^{G}(Φ_{−ϵ}) = Λ^{G}(id_{M}) = χ^{G}(M).

Because of the equivariant Lefschetz fixed point Theorem 0.2 it remains to prove for some ϵ > 0

i^{G}(Ξ) = Λ_{loc}^{G}(Φ_{−ϵ}).

If we choose ϵ > 0 small enough, the diffeomorphism ϕ_{−ϵ}: M → M will have as set of fixed points Fix(ϕ_{−ϵ}) precisely Zero(Ξ). It suffices to prove for x ∈ Zero(Ξ) = Fix(Φ_{−ϵ})

Deg_{0}^{G_{x}}((d_{x}Ξ)^{c}) = Deg_{0}^{G_{x}}((id −T_{x}Φ_{−ϵ})^{c}).

Recall that the character map ch_{0}^{G_{x}}: A(G_{x}) → ∏_{(H)∈consub(G_{x})} ℤ of (4.2) is injective. We conclude from (4.5) that it suffices to prove for any subgroup H ⊆ G_{x} the equality of degrees of self-maps of the closed orientable manifold ((T_{x}M)^{c})^{H} = ((T_{x}M)^{H})^{c}

deg (((d_{x}Ξ)^{c})^{H}) = deg (((id −T_{x}Φ_{−ϵ})^{c})^{H}) .  (6.7)

It suffices to treat the case H = {1} and dim(M) ≥ 1 — the other cases are completely analogous or follow directly from the definitions. Since (6.7) is of local nature, we may assume M = ℝ^{n} and x = 0. In the sequel we use the standard identification T ℝ^{n} = ℝ^{n} × ℝ^{n}. Then the vector field Ξ becomes a smooth map Ξ: ℝ^{n} → ℝ^{n} with Ξ(0) = 0 and d_{0}Ξ becomes the differential T_{0}Ξ. Let Φ be the flow associated to Ξ. Choose ϵ ≥ 0 and an open neighborhood U ⊆ ℝ^{n} of 0 such that Φ is defined on U × [−ϵ, 0]. By Taylor’s theorem we can find a smooth map η: U × [−ϵ, 0] → ℝ^{n} such that for t ∈ [−ϵ, 0] and u ∈ U,

ϕ_{t}(u) = u + t · Ξ(u) + t^{2} · η_{t}(u) .

This implies

T_{0}ϕ_{t} = id +t · (T_{0}Ξ + t · T_{0}η_{t}) .

Since [−ϵ, 0] is compact, we can find a constant C independent of t such that the operator norm of T_{0}η_{t} satisfies ∥T_{0}η_{t}∥ < C for t ∈ [−ϵ, 0]. The differential T_{0}Ξ is an isomorphism by assumption. Hence T_{0}Ξ + t · T_{0}η_{t} is invertible for t ∈ [−D, 0] if we put D := min{ϵ, C^{−1} · ||(T_{0}Ξ)^{−1}||^{−1}}. Hence we get for t ∈ [−D, 0] that id −T_{0}Φ_{t} is invertible and

\frac{det(id −T_{0}Φ_{t})}{| det(id −T_{0}Φ_{t})|} = \frac{det(T_{0}Ξ + t · T_{0}η_{t})}{| det(T_{0}Ξ + t · T_{0}η_{t})|} = \frac{det(T_{0}Ξ)}{| det(T_{0}Ξ)|}.

Hence (6.7) follows if we take ϵ > 0 small enough. This finishes the proof that Theorem 6.6 follows from Theorem 0.2.

Remark 6.8 Let M be a proper cocompact G-manifold without boundary. If M possesses a nowhere-vanishing equivariant vector field, then χ^{G}(M) = 0 by the Theorem 6.6. The converse is true if M satisfies the weak gap hypothesis that dim(M^{>G_{x}}(x)) ≤ dim(M^{G_{x}}(x)) − 2 holds for each x ∈ G_{x}. The proof is done by induction over the orbit bundles and the induction step is reduced to the non-equivariant case. The weak gap condition ensures that M^{G_{x}}(x)−M^{>G_{x}}(x) is connected for x ∈ M. It is satisfied if all isotropy groups of M have odd order. For finite groups G more information about this question can be found in [12, Remark 2.5 (iii) on page 32].

Example 6.9 Let D be the infinite dihedral group D = ℤ ⋊ ℤ/2 = ℤ/2 ∗ ℤ/2. We use the presentation D = ⟨s, t | s^{2} = 1, s^{−1}ts = t^{−1}⟩ in the sequel. The subgroups H_{0} = ⟨t⟩ and H_{1} = ⟨ts⟩ have order two and {{1}, H_{0}, H_{1}} is a complete system of representatives for the conjugacy classes of finite subgroups of D. The infinite dihedral group D acts on ℝ by s · r = −r and t · r = r + 1 for r ∈ ℝ. The interval [0, 1/2] is a fundamental domain for the D-action. There is a D-CW -structure on ℝ with {n | n ∈ ℤ} ∐{n + 1/2 | n ∈ ℤ} as zero-skeleton. Let x_{i}: D/H_{i} → ℝ be the D-map sending 1H_{i} to 0 for i = 0 and to 1/2 for i = 1. Let y: D → ℝ be the D-map sending 1 to 0. Then ℝ has two equivariant 0-cells, which have x_{0} and x_{1} as their characteristic maps, and one equivariant 1-cell of orbit type D/{1}.

We get Is Π_{0}(D; ℝ) = {[x_{0}], [x_{1}], [y]}. Recall that U^{D}(ℝ) is the free ℤ-module with basis Is Π_{0}(D; ℝ). Hence we write

U^{D}(ℝ) = ℤ⟨[x_{0}]⟩ ⊕ ℤ⟨[x_{1}]⟩ ⊕ ℤ⟨[y]⟩.

We conclude from (6.2)

χ^{D}(ℝ) = [x_{0}] + [x_{1}] − [y].

This is consistent with the original Definition 6.1, since

χ(W_{D}H_{i}\(ℝ^{H_{i}}(x_{i}), ℝ^{>H_{i}}(x_{i}))) = χ({pt.}) = 1;
χ(W_{D}{1}\(ℝ^{{1}}(y), ℝ^{>{1}}(y)) = χ([0, 1/2], {0, 1/2}) = −1.

The character map (5.1) is given by

χ^{D}(ℝ) = \{ \begin{matrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 1/2 & 1/2 & 1 \end{matrix} \} : U^{D}(ℝ) = ℤ⟨[x_{0}]⟩ ⊕ ℤ⟨[x_{1}]⟩ ⊕ ℤ⟨[y]⟩

→ ℤ⟨[x_{0}]⟩ ⊕ ℤ⟨[x_{1}]⟩ ⊕ ℤ⟨[y]⟩

since the D-set mor(y, x_{i}) is D/H_{i} and the sets mor(x_{i}, y) and mor(x_{i}, x_{j}) for i ≠ j are empty. The character map sends χ^{D}(ℝ) to the various orbifold Euler characteristics which therefore must be

χ^{ℚW_{D}H_{i}}(ℝ^{H_{i}}(x_{i})) = 1;
χ^{ℚD}(ℝ) = 0.

One easily checks that this is consistent with the Definition 6.1 of the orbifold Euler characteristics.

Let Ξ be a vector field on ℝ. Under the standard identification T ℝ = ℝ × ℝ this is the same as a function Ξ: ℝ → ℝ. The vector field Ξ is transverse to the zero-section if and only if the function Ξ satisfies Ξ(z) = 0 ⇒ Ξ′(z) ≠ 0. The vector field Ξ is D-equivariant if and only if Ξ(−z) = −Ξ(z) and Ξ(z) = Ξ(z+1) holds for all z ∈ ℝ. Let Ξ be a D-equivariant vector field transverse to the zerosection. For example, we can take Ξ(z) = sin(2πz).

We conclude Ξ(0) = 0 from Ξ(−z) = −Ξ(z) and Ξ(1/2) = 0 from Ξ(1−z) = −Ξ(z). Let z_{0} = 0 < z_{1} < z_{2} < … < z_{r} = 1/2 be the points z ∈ [0, 1/2] for which Ξ(z) = 0. (The example of sin(2πz) shows the minimum value of r is 1.) For i ∈ \{0, 1, …, r\} put δ_{i} = \frac{Ξ′(z_{i})}{|Ξ′(z_{i})|}. Since Ξ is different from zero on (z_{i}, z_{i+1}), we have δ_{i+1} = −δ_{i} for i ∈ {0, 1, …, r − 1}. Let d_{z_{i}}Ξ: T_{z_{i}}ℝ → T_{z_{i}}ℝ be the map associated to Ξ at z_{i} (see (6.4)); this is simply multiplication by Ξ′(z_{i}) under the standard identification T_{z_{i}}ℝ = ℝ. The degree of the map (d_{z_{i}}Ξ)^{c} induced on ℝ^{c} = S^{1} is δ_{i}. For i = 0, r the isotropy group D_{z_{i}} = ℤ/2 acts on ℝ by − id. Hence the degree deg ((d_{z_{i}}Ξ)^{D_{z_{i}}})^{c}) is by definition 1 since dim(ℝ^{D_{z_{i}}}) = 0. We conclude from (4.5) for i = 0, r

Deg^{D_{z_{i}}} (d_{z_{i}}Ξ) = [H/H] + \frac{−1 + δ_{i}}{2} · [H/\{1\}] ∈ A(D_{z_{i}})

and for i ∈ {1, 2, …, r − 1}

Deg^{D_{z_{i}}} (d_{z_{i}}Ξ) = δ_{i} · [{1}] ∈ A({1}).

Hence we get for the equivariant index

i^{D}(Ξ) = [x_{0}] + \frac{−1 + δ_{0}}{2} · [y] + [x_{1}] + \frac{−1 + δ_{r}}{2} · [y] + ∑_{i=1}^{r−1} δ_{i} · [y]

= [x_{0}] + [x_{1}] + (−1 + \frac{δ_{0}}{2} + \frac{δ_{r}}{2} + ∑_{i=1}^{r−1} δ_{i}) · [y]

= [x_{0}] + [x_{1}] − [y] + δ_{0} · (\frac{1}{2} + \frac{(−1)^{r}}{2} + ∑_{i=1}^{r−1}(−1)^{i}) · [y_{0}]

= [x_{0}] + [x_{1}] − [y] + δ_{0} · 0 · [y_{0}]
= [x_{0}] + [x_{1}] − [y].

This is consistent with Theorem 6.6.

7. Constructing equivariant manifolds with given component structure and universal equivariant Euler characteristic

In this section we discuss the problem of whether there exists a proper smooth G-manifold M with prescribed sets π_{0}(M^{H}) for H ⊆ G, and whether χ^{G}(M) can realize a given element in U^{G}.

Let Or(G; Fin) be the orbit category for the family of finite subgroups, i.e., objects are homogeneous spaces G/H for finite subgroups H ⊆ G and morphisms are G-maps. A contravariant Or(G; Fin)-set S is a contravariant functor from Or(G; Fin) to the category of sets. Given a G-space X, define a contravariant Or(G; Fin)-set \underline{π_{0}(X)} by sending G/H to π_{0}(X^{H}) = π_{0}(map_{G}(G/H, X)). Next we investigate under which conditions a contravariant Or(G; Fin)-set S arises from a finite proper G-CW -complex. Lemma 7.1 For a contravariant Or(G; Fin)-set S: Or(G; Fin) → Sets the following assertions are equivalent:

(a) There are only finitely many elements (H) ∈ consub(H) with S(G/H) ≠ ∅. For any finite subgroup H ⊆ G the set WH\S(G/H) is finite and the isotropy group WH_{s} of each element s ∈ S(G/H) is finitely generated;

(b) There is a proper finite G-CW -complex X such that there exists a natural equivalence T: \underline{π_{0}(X)} ⟶^{≅} S.

\underline{\underline{Proof}} : (b) ⇒ (a) Since X is finite, there are finitely many elements (K_{1}), (K_{2}), …, (K_{m}) in {(K) ∈ consub(G) | |K| < ∞} such that for each equivariant cell G/H × D^{n} there is i ∈ {1, 2, …, m} with (H) = (K_{i}). Hence any subgroup H ⊆ G with X^{H} ≠ ∅ is conjugate to a subgroup of one of the K_{i}’s. Since a finite group has only finitely many subgroups, the set {(H) ∈ consub(G) | X^{H} ≠ ∅} is finite.

Since X is finite and proper and WH\(G/K^{H}) is finite for each finite group H ⊆ G and subgroup K ⊆ G, the WH-CW -complex X^{H} is finite proper for each finite subgroup H ⊆ G. Hence the quotient space WH\X^{H} is a finite CW -complex and has only finitely many components. Since WH\π_{0}(X^{H}) ≅ π_{0}(WH\X^{H}), the set WH\π_{0}(X^{H}) is finite.

Consider a finite subgroup H ⊆ G and a component C ∈ π_{0}(X^{H}). Let WH_{C} be the isotropy group C. Then C is a connected proper finite WH_{C}-CW - complex. The long exact homotopy sequence of the fibration

C → EWH_{C} ×_{WH_{C}} C → BWH_{C}

shows that WH_{C} is a quotient of π_{1}(EWH_{C} ×_{WH_{C}} C). Since for any finite subgroup K ⊆ WH_{C} the WH_{C}-space EWH_{C}×_{WH_{C}} WH_{C}/K is homotopy equivalent to BK and hence to a CW -complex of finite type (a CW -complex for which each skeleton is finite), C is built out of finitely many cells G/K × D^{i}, K ⊆ G finite, and EWH_{C} ×_{WH_{C}} WH_{C}/K has the homotopy type of CW -complex of finite type. This implies that π_{1}(EWH_{C} ×_{WH_{C}} C) is finitely generated. Hence WH_{C} is finitely generated.

(a) ⇒ (b) Choose an ordering (H_{1}), (H_{2}), …, (H_{r}) on the set {(H) ∈ consub(G) | S(G/H) ≠ ∅} such that H_{i} is subconjugate to H_{j} only if i ≥ j holds. Define

X_{0} := ∐_{i=1}^{s} ∐_{WH_{i}\S(G/H_{i})} G/H_{i}.

Define a transformation ϕ_{0} : \underline{π_{0}(X_{0})} → S as follows. Fix an object G/K ∈ Or(G; Fin). Then \underline{π_{0}(X_{0})} evaluated at this object G/K is given by

∐_{i=1}^{s} ∐_{WH_{i}\S(G/H)} map_{G}(G/K, G/H_{i})

since π_{0}(G/H_{i}^{K}) = map_{G}(G/K, G/H_{i}). Now require that ϕ_{0}(G/K) sends a G-map σ ∈ map_{G}(G/K, G/H_{i}) belonging to the summand for WH_{i} · s ∈ WH_{i}\S(G/H_{i}) to S(σ)(s). One easily checks that ϕ_{0}(G/H): π_{0}(X^{H}) → S(G/H) is surjective for all H ⊆ G.

In the next step we attach equivariant one-cells to X_{0} to get a G-CW - complex X together with a transformation ϕ: \underline{π_{0}(X)} → S, such that the composition of ϕ with the transformation \underline{π_{0}(X_{0})} → \underline{π_{0}(X)} induced by the inclusion is ϕ_{0}, and ϕ(G/H) is bijective for all H ⊆ G. We do this by constructing by induction a sequence of proper cocompact G-CW -complexes X_{0} ⊆ X_{1} ⊆ … ⊆ X_{r}, together with transformations ϕ_{i}: \underline{π_{0}(X_{i})} → S such that the composition of ϕ_{i} with the transformation \underline{π_{0}(X_{i−1})} → \underline{π_{0}(X_{i})} induced by the inclusion is ϕ_{i−1}, ϕ_{i}(G/H_{j}) is bijective for all j ≤ i, and X_{i} is obtained from X_{i−1} by attaching finitely many equivariant cells of the type G/H_{i} × D^{1}. Then we can take X = X_{r} and ϕ = ϕ_{r}.

The induction beginning is X_{0} together with ϕ_{0}. The induction step from i − 1 to i is done as follows. Consider WH_{i} · s ∈ WH_{i}\S(G/H_{i}). Let (WH_{i})_{s} be the isotropy group of s under the WH_{i}-action on S(G/H_{i}). The preimage of s under ϕ_{i−1}(G/H_{i}): π_{0}(X_{i−1}^{H_{i}}) → S(G/H_{i}) consists of finitely many (WH_{i})_{s}- orbits since WH_{i}\π_{0}(X_{i−1}^{H_{i}}) is finite by the implication (b) ⇒ (a) which we have already proved. Let u(s)_{1}, u(s)_{2}, …, u(s)_{v} be a system of generators of (WH_{i})_{s} which contains the unit element 1 ∈ (WH_{i})_{s}. Fix elements C(s)_{1}, C(s)_{2}, …, C(s)_{n(s)} in ϕ_{i−1}^{−1}(G/H_{i})(s) such that

(WH_{i})_{s}\ϕ_{i−1}(G/H_{i})^{−1}(s) = {(WH_{i})_{s} · C(s)_{i} | i = 1, 2 …, n(s)}

and (WH_{i})_{s}·C(s)_{j} = (WH_{i})_{s}·C(s)_{k} implies j = k. Now attach for each WH_{i}·s ∈ WH_{i}\S(G/H_{i}), each generator u(s)_{i}, each C(s)_{j} an equivariant cell G/H_{i} × D^{1} to X_{i−1} such that {1H_{i}} × D^{1} connects C_{1}(s) and u(s)_{i} · C_{j}(s). The resulting G-CW -complex is the desired G-CW -complex X_{i}, one easily constructs the desired transformation ϕ_{i} out of ϕ_{i−1}.

Notice that for a finite group G the statement (a) in Lemma 7.1 is equivalent to the statement that S(G/H) is finite for all subgroups H ⊆ G. If we take as Or(G; Fin)-set the functor S which sends G/{1} to {∗} and G/H for H ≠ {1} to ∅, then Lemma 7.1 boils down to the statement that there exists a connected finite free G-CW -complex X if and only if G is finitely generated.

Now given a contravariant Or(G; Fin)-set S, we define U^{G}(S) to be the free abelian group on ∐_{(H)∈consub(G)} W H\S(G/H). If S satisfies the equivalent conditions of Lemma 7.1, then clearly this is naturally isomorphic to U^{G}(X), with X as in part (b) of Lemma 7.1.

Next we prove that is X is a finite proper G-CW -complex, then any element u ∈ U^{G}(X) can be realized from χ^{G} of a manifold. More precisely, there is a G-map f: M → X with M a G-manifold, such that U^{G}(f) is an isomorphism sending χ^{G}(M) to u. We are grateful to Tammo tom Dieck for pointing out to us the use of the multiplicative induction in the proof of the next result.

Lemma 7.2 Let X be a finite proper G-CW -complex and u ∈ U^{G}(X). Then there is a proper cocompact G-manifold M without boundary together with a G-map f: M → X with the following properties:

For any x ∈ M the G_{x}-representation T_{x}M is a multiple of the regular G_{x}- representation ℝ[G_{x}] for G_{x} the isotropy group of x ∈ X. The dimensions of the components C ∈ π_{0}(M) are all equal. The components of M^{H} are orientable manifolds for each H ⊆ G. The induced map π_{0}(f^{H}): π_{0}(M^{H}) → π_{0}(X^{H}) is bijective for each finite subgroup H ⊆ G. The induced map

U^{G}(f): U^{G}(M) ⟶^{≅} U^{G}(X)

is bijective and sends χ^{G}(M) to u.

\underline{\underline{Proof}} : In the first step we want to reduce the claim to the case, where X is a finite proper 1-dimensional G-CW -complex such that for any x: G/H → X there is a zero cell G/H which meets X^{H}.

If the G-map f: X → Y of proper G-CW -complexes induces bijections π_{0}(f^{H}): π_{0}(X^{H}) ⟶^{≅} π_{0}(Y^{H}) for all finite subgroups H ⊆ G, then the induced map U^{G}(f): U^{G}(X) → U^{G}(Y) is bijective by (3.3). In particular the inclusion of the 1-skeleton i_{1} : X_{1} → X induces a bijection U^{G}(i_{1}).

Since X_{1} is finite and proper, WH\π_{0}(X^{H}) is finite for all finite subgroups H ⊆ G and the set {(H) ∈ consub(G) | X^{H} ≠ ∅} is finite (see Lemma 7.1). We conclude from (3.3) that Is π_{0}(G, X_{1}) is finite. Fix for any [x: G/H → X_{1}] a representative x: G/H → X whose image lies in X_{0}. This is possible by the equivariant cellular approximation theorem. Define a finite proper G-CW - complex Y by the G-pushout diagram

∐\begin{matrix}[x: G/H→X]∈ \\ Is π_{0}(G,X) \end{matrix} G/H × \{0\} \underrightarrow{∐\begin{matrix}[x: G/H→X]∈ \\ Is π_{0}(G,X) \end{matrix} x} X_{1}
i_{2}↓  ↓^{i_{3}}
∐\begin{matrix}[x: G/H→X]∈ \\ Is π_{0}(G,X) \end{matrix} G/H × [0, 1]  ⟶  Y

where i_{2} is the inclusion. Since i_{2} is a G-homotopy equivalence, i_{3} is a Ghomotopy equivalence. Let i_{3}^{−1} : Y → X_{1} be a G-homotopy inverse. Then U^{G}(i_{1} ∘ i_{3}^{−1}): U^{G}(Y) → U^{G}(X) is a bijection and Y is a finite proper G-CW - complex such that for each G-map y: G/H → Y there is a zero cell G/H which meets Y^{H}(y), namely G/H × {1} ⊆ G/H × [0, 1] for the corresponding [x] ∈ Is Π_{0}(G, X) in the pushout diagram above. Hence we can assume without loss of generality X = Y in the sequel.

Fix a number n such for any H ⊆ G with X^{H} ≠ ∅ the order |H| divides n. Let G/H_{1}, G/H_{2}, …, G/H_{r} be the equivariant zero-cells of X. Let N_{i} be a 4n/|H_{i}|-dimensional closed oriented manifold. Let ∏_{H_{i}} N_{i} be the closed H_{i}- manifold with the H_{i}-action coming from permuting the factors. This is called the multiplicative induction or coinduction of N_{i}. One easily checks that for K ⊆ H_{i} the K-fixed point sets of ∏_{H_{i}} N_{i} is diffeomorphic to ∏_{k=1}^{|H_{i}/K|} N_{i} and hence a closed connected orientable manifold which is non-empty. Moreover, the (H_{i})_{x}-representation T_{x} (∏_{H_{i}} N_{i}) is a multiple of the regular real (H_{i})_{x}- representation for all x ∈ ∏_{H_{i}} N_{i}. The manifolds N_{i} will be specified later. Given an orthogonal H-representation V of a finite group H, we denote by DV and SV the unit disk and the unit sphere and by int(DV) = DV − SV the interior of DV. Define the 4n-dimensional proper cocompact G-manifold M_{0} to be

M_{0} = ∐_{i=1}^{r} G ×_{H_{i}} (∏_{H_{i}} N_{i}) .

Let f_{0} : M_{0} → X be the map which is on G×_{H_{i}} (∏_{H_{i}} N_{i}) given by the canonical projection onto the cell G/H_{i} = G ×_{H_{i}} {∗}.

Fix a G-pushout describing, how X = X_{1} is obtained from X_{0} by attaching equivariant one-cells

∐_{j=1}^{s} G/K_{j} × S^{0} ⟶^{q} X_{0}
↓  ↓
∐_{j=1}^{s} G/K_{j} × D^{1} ⟶_{Q} X

Consider a G-map σ: G/K → G/H for finite subgroups H, K ⊆ G. Suppose that |H| and |K| divide n. Choose g ∈ G such that g^{−1}Kg ⊆ H and σ sends g′K to g′gH. Let c_{g}: K → H be the injective group homomorphism g′ ↦ g^{−1}g′g. The K-representations ℝ[K]^{4n/|K|} and c_{g}^{∗}ℝ[H]^{4n/|H|}, which is obtained from the H-representation ℝ[H]^{4n/|H|} by restriction with c_{g}, are isomorphic. Choose an isometric c_{g}-equivariant linear isomorphism

ϕ: ℝ[K]^{4n/|K|} → ℝ[H]^{4n/|H|}.

Choose a small number ϵ > 0 and an element w ∈ ℝ[H]^{4n/|H|} such that ||w|| = 1, the H-isotropy group of w is g^{−1}Kg and the distance of two distinct points in the H-orbit through w is larger than 3ϵ. The following map is a G-embedding

ψ: G ×_{K} Dℝ[K]^{4n/|K|} → G ×_{H} ℝ[H]^{4n/|H|}, (g′, v) ↦ (g′g, ϵ · ϕ(v) + w)

such that the following diagram with the canonical projections as vertical arrows commutes:

G ×_{K} Dℝ[K]^{4n/|K|} ⟶^{ψ} G ×_{H} ℝ[H]^{4n/|H|}
pr↓  ↓^{pr}
G/K  ⟶_{σ}  G/H

Using the construction above with appropriate choices of w and ϵ we can find a G-embedding

Ψ: ∐_{j=1}^{s} G ×_{K_{j}} (Dℝ[K_{j}]^{4n/|K_{j}|} × S^{0}) → M_{0}

such that the following diagram commutes:

∐_{j=1}^{s} G ×_{K_{j}} (Dℝ[K_{j}]^{4n/|K|} × S^{0}) ⟶^{Ψ} M_{0}
pr↓  ↓^{f_{0}}
∐_{j=1}^{s} G/K_{j} × S^{0}  ⟶_{q} X_{0}

where pr is the obvious projection. Let M′_{0} be the proper cocompact G-manifold with boundary

M′_{0} = M_{0} − Ψ (∐_{j=1}^{s} G ×_{K_{j}} (int (Dℝ[K_{j}]^{4n/|K_{j}|}) × S^{0})) .

Define a smooth G-manifold M by the following pushout

∐_{j=1}^{s} G ×_{K_{j}} (Sℝ[K_{j}]^{4n/|K_{j}|} × S^{0}) ⟶^{Ψ|} M′_{0}
↓  ↓
∐_{j=1}^{s} G ×_{K_{j}} (Sℝ[K_{j}]^{4n/|K_{j}|} × D^{1}) ⟶_{\overline{Ψ|}} M

where Ψ| denotes the restriction of Ψ. The map f_{0} : M_{0} → X_{0} defines a map f′_{0} : M′_{0} → X_{0} by restriction. Since (Sℝ[K_{j}]^{4n/|K|})^{L} for each L ⊆ K_{j} and (∏_{H_{i}} N_{i})^{L} for each L ⊆ H_{i} are non-empty and connected, the maps f′_{0} and the projections pr: ∐_{j=1}^{s} G ×_{K_{j}} (Sℝ[K_{j}]^{4n/|K|} × S^{0}) → ∐_{j=1}^{s} G/K_{j} × S^{0} and pr: ∐_{j=1}^{s} G ×_{K_{j}} (Sℝ[K_{j}]^{4n/|K|} × D^{1}) → ∐_{j=1}^{s} G/K_{j} × D^{1} induce on the L-fixed point set 1-connected maps for each L ⊆ G. Hence the G-map f: M → X, which is induced by these three maps and the pushout property, induces a 1-connected map on the L-fixed point sets for each L ⊆ G. In particular π_{0}(f^{L}): π_{0}(M^{L}) → π_{0}(X^{L}) is bijective for each L ⊆ G and the map U^{G}(f): U^{G}(M) → U^{G}(X) is bijective by (3.4). Obviously the G_{x}- representation T_{x}M for any point x ∈ M is a multiple of the regular G_{x}- representation ℝ[G_{x}].

The following diagram commutes

U^{G}(M_{0}) ⟵_{≅}^{U^{G}(i_{0})} U^{G}(M′_{0}) ⟶^{U^{G}(i_{4})} U^{G}(M)
U^{G}(f_{0})↓≅  ^{U^{G}(f′_{0})}↓≅  ^{U^{G}(f)}↓≅
U^{G}(X_{0}) ⟶_{≅}^{id} U^{G}(X_{0}) ⟶^{U^{G}(i_{5})} U^{G}(X)

where i_{0}, i_{4} and i_{5} denote the inclusions.

Next we compute U^{G}(f)(χ^{G}(M)). We get by the sum formula for the universal equivariant Euler characteristic [7, Theorem 5.4 on page 100]

U^{G}(f)(χ^{G}(M))

= U^{G}(i_{5} ∘ f_{0}) (χ^{G}(M_{0}))

− U^{G}(i_{5} ∘ f_{0} ∘ Ψ) (χ^{G} (∐_{j=1}^{s} G ×_{K_{j}} ((Dℝ[K_{j}]^{4n/|K_{j}|}) × S^{0})))

+ U^{G}(i_{5} ∘ f_{0} ∘ Ψ ∘ i_{6}) (χ^{G} (∐_{j=1}^{s} G ×_{K_{j}} ((Sℝ[K_{j}]^{4n/|K_{j}|}) × S^{0})))

− U^{G}(i_{5} ∘ f′_{0} ∘ Ψ|) (χ^{G} (∐_{j=1}^{s} G ×_{K_{j}} (Sℝ[K_{j}]^{4n/|K_{j}|} × S^{0})))

+ U^{G}(f ∘ \overline{Ψ|}) (χ^{G} (∐_{j=1}^{s} G ×_{K_{j}} (Sℝ[K_{j}]^{4n/|K_{j}|} × D^{1}))) ,

where i_{6} is the inclusion. Since (Sℝ[K_{j}]^{4n/|K_{j}|})^{L} is an odd-dimensional sphere and hence has vanishing (non-equivariant) Euler characteristic for each L ⊆ K_{j}, the element χ^{K_{j}} (Sℝ[K_{j}]^{4n/|K_{j}|}) in U^{K_{j}}({∗}) = A(K_{j}) is sent to zero under the injective map χ_{0}^{K_{j}}: A(K_{j}) → ∏_{(L)∈consub(K_{j})} ℤ. This implies

χ^{K_{j}} (Sℝ[K_{j}]^{4n/|K_{j}|}) = 0

and hence

χ^{G} (∐_{j=1}^{s} G ×_{K_{j}} (Sℝ[K_{j}]^{4n/|K_{j}|} × S^{0})) = 0.

The space Dℝ[K_{j}]^{4n/|K_{j}|} is K_{j}-homotopy equivalent to {∗}. Hence

U^{G}(i_{5} ∘ f_{0} ∘ Ψ) (χ^{G} (∐_{j=1}^{s} G ×_{K_{j}} (Dℝ[K_{j}]^{4n/|K_{j}|} × S^{0})))

= U^{G}(i_{5} ∘ f_{0} ∘ Ψ ∘ i_{7}) (χ^{G} (∐_{j=1}^{s} G ×_{K_{j}} S^{0}))

= 2 · ∑_{j=1}^{s} U^{G}(x_{j})(χ^{G}(G/K_{j})),

where i_{7} is the inclusion and x_{j}^{1} : G/K_{j} → X is the restriction of the characteristic map of the one cell G/K_{j} × D^{1} to G/K_{j} × {1/2}. If x_{i}^{0} : G/H_{i} → X is the characteristic map of the 0-cell G/H_{i} and α_{i}: H_{i} → G the inclusion, we conclude

U^{G}(f)(χ^{G}(M)) = ∑_{i=1}^{r} U^{G}(x_{i}^{0}) ∘ (α_{i})_{∗} (χ^{H_{i}} (∏_{H_{i}} N_{i}))

− 2 · ∑_{j=1}^{s} U^{G}(x_{j})(χ^{G}(G/K_{j})). (7.3)

The element χ^{H_{i}} (∏_{H_{i}} N_{i}) ∈ A(H_{i}) is sent under the injective character map

ch_{0}^{H_{i}}: A(H_{i}) → ∏_{(K) consub(H_{i})} ℤ, [S] ↦ |S^{K}|

to (χ(N_{i})^{|H/K|} | (K) ∈ consub(H_{i})}. This implies

χ^{H_{i}} (∏_{H_{i}} N_{i}) = χ(N_{i}) · [H_{i}/H_{i}]

+ ∑_{\begin{matrix}(K)∈consub(H_{i}), \\ K≠H_{i} \end{matrix}} λ_{(K)}(χ(N_{i})) · [H_{i}/K] (7.4)

for appropriate functions λ_{(K)} : ℤ → ℤ. We can order the equivariant zero-cells G/H_{i} of X_{0} such that H_{i} is subconjugate to H_{j} only if i ≥ j holds. We conclude from (7.4) that for an appropriate map μ, the composition

⨁_{i=1}^{r} ℤ ⟶^{μ} ⨁_{i=1}^{r} ℤ = U^{G}(X_{0}) ⟶^{U^{G}(i_{8})} U^{G}(X_{1})

sends {χ(N_{i}) | i = 1, 2 …, r} to ∑_{i=1}^{r} U^{G}(x_{i}^{0}) ∘ (α_{i})_{∗} (χ^{H_{i}} (∏_{H_{i}} N_{i})), where i_{8} : X_{0} → X is the inclusion and μ is given by

μ(a_{1}, a_{2}, …, a_{r}) = (∑_{j=1}^{r} μ_{1,j}(a_{j}), ∑_{j=1}^{r} μ_{2,j}(a_{j}), …, ∑_{j=1}^{r} μ_{r,j}(a_{j}), )

for (not necessarily linear) maps μ_{i,j}: ℤ → ℤ for which μ_{i,j} = 0 for i > j and μ_{i,i} = id. The map U^{G}(i_{8}) is surjective and the map μ bijective. Since for any integer k and any positive integer l there is a closed connected 4l-dimensional manifold N with χ(N) = k, we can find appropriate N_{i} with the right Euler characteristics χ(N_{i}) such that for a given element u ∈ U^{G}(X)

∑_{i=1}^{r} U^{G}(x_{i}^{0}) ∘ (α_{i})_{∗} (χ^{H_{i}} (∏_{H_{i}} N_{i})) = u + 2 · ∑_{j=1}^{s} U^{G}(x_{j})(χ^{G}(G/K_{j})). (7.5)

We conclude from (7.3) and (7.5) that

U^{G}(f)(χ^{G}(M)) = u.

This finishes the proof of Lemma 7.2.

Putting Lemmas 7.1 and 7.2 together gives the following result on the realization problem:

Theorem 7.6 Let S be a contravariant Or(G; Fin)-set, and suppose that there are only finitely many elements (H) ∈ consub(H) with S(G/H) ≠ ∅. Also assume that for any finite subgroup H ⊆ G, the set WH\S(G/H) is finite, and that the isotropy group WH_{s} of each element s ∈ S(G/H) is finitely generated. Let u ∈ U^{G}(S). Then there is a proper cocompact G-manifold M (without boundary) and there is a natural equivalence T: \underline{π_{0}(M)} ⟶^{≅} S sending χ^{G}(M) to u.

References

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[2] tom Dieck, T.: “Transformation groups”, Studies in Math. 8, de Gruyter (1987).

[3] tom Dieck, T.: “Transformation groups and representation theory”, Lecture notes in mathematics 766, Springer (1979).

[4] Illman, S.: “Existence and uniqueness of equivariant triangulations of smooth proper G-manifolds with some applications to equivariant Whitehead torsion”, J. Reine Angew. Math. 524, 129–183 (2000).

[5] Laitinen, E. and Lück, W.: “Equivariant Lefschetz classes”, Osaka Journal of Mathematics 26, 491–525 (1989).

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[11] Rosenberg, J.: “The K-homology class of the Euler characteristic operator is trivial”, Proc. Amer. Math. Soc. 127, 3467–3474, (1999).

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Addresses:

Wolfgang Lück Jonathan Rosenberg Institut für Mathematik und Informatik Department of Mathematics Westfälische Wilhelms-Universtität University of Maryland Einsteinstr. 62 College Park, MD 20742-4015 48149 Münster U.S.A Germany

lueck@math.uni-muenster.de                        jmr@math.umd.edu
wwwmath.uni-muenster.de/math/u/lueck         www.math.umd.edu/~jmr