Equivariant Euler characteristics and K- Euler classes for proper cocompact G-

Wolfgang Luck¨ Jonathan Rosenberg

Institut fur¨ Mathematik und Informatik Westf¨alische Wilhelms-Universtit¨at Einsteinstr. 62 48149 Munster,¨ Germany and Department of University of Maryland College Park, MD 20742, USA Email: [email protected] and [email protected] URL: wwwmath.uni-muenster.de/u/lueck, www.math.umd.edu/~jmr

Abstract

Let G be a countable discrete and let M be a smooth proper cocompact G- without boundary. The Euler operator defines via Kasparov theory an element, called the equivariant , in the equivariant KO-homology of M. The universal equivariant of M, which lives in a group U G(M), counts the equivariant cells of M , taking the component structure of the various fixed point sets into account. We construct a natural homomorphism from U G(M) to the equivariant KO-homology of M. The main result of this paper says that this map sends the universal equivariant Euler characteristic to the equivariant Euler class. In particular this shows that there are no “higher” equivariant Euler characteristics. We show that, rationally, the equivariant Euler class carries the same information as the collection of the Euler characteristics of the components of the L-fixed point sets M L, where L runs through the finite cyclic subgroups of G. However, we give an example of an action of the symmetric group S3 on the 3- for which the equivariant Euler class has order 2, so there is also some torsion information.

AMS Classification numbers Primary: 19K33

1 Secondary: 19K35, 19K56, 19L47, 58J22, 57R91, 57S30, 55P91

Keywords: equivariant K -homology, de Rham operator, signature operator, Kasparov theory, equivariant Euler characteristic, fixed sets, cyclic subgroups, Burnside ring, Euler operator, equivariant Euler class, universal equivariant Euler characteristic

2 0 Background and statements of results

Given a countable discrete group G and a cocompact proper smooth G-manifold M without boundary and with G- Riemannian metric, the Euler char- acteristic operator defines via Kasparov theory an element, the equivariant Eu- ler class, in the equivariant real K -homology group of M EulG(M) KOG(M). (0.1) ∈ 0 The Euler characteristic operator is the minimal closure, or equivalently, the maximal closure, of the densely defined operator

2 2 (d + d∗): Ω∗(M) L Ω∗(M) L Ω∗(M), ⊆ → with the Z/2-grading coming from the degree of a differential p-form. The equivariant signature operator is the same underlying operator, but with a dif- ferent grading coming from the Hodge star operator. The signature operator also defines an element SignG(M) KG(M), ∈ 0 which carries a lot of geometric information about the action of G on M . (Rationally, when G = 1 , Sign(M) is the Poincar´e dual of the total - { } L class, the Atiyah-Singer L-class, which differs from the Hirzebruch L-class only by certain well-understood powers of 2, but in addition, it also carries quite interesting integral information [11], [22], [27]. A partial analysis of the class SignG(M) for G finite may be found in [26] and [24].) We want to study how much information EulG(M) carries. This has already been done by the second author [23] in the non-equivariant case. Namely, given a closed Riemannian manifold M , not necessarily connected, let

e: Z = KO0( ) KO0(M) {∗} → πM0(M) πM0(M) be the map induced by the various inclusions M . This map is split {∗} → injective; a splitting is given by the various projections C for C π (M), → {∗} ∈ 0 and sends χ(C) C π (M) to Eul(M). Hence Eul(M) carries precisely the { | ∈ 0 } same information as the Euler characteristics of the various components of M, and there are no “higher” Euler classes. Thus the situation is totally different from what happens with the signature operator. We will see that in the equivariant case there are again no “higher” Euler characteristics and that EulG(M) is determined by the universal equivariant

3 Euler characteristic (see Definition 2.5) G G χ (M) U (M) = Z. ∈ H (H) consub(G) WH π0(M ) ∈ M \M Here and elsewhere consub(G) is the of conjugacy classes of subgroups of G and NH = g G g 1Hg = H is the normalizer of the subgroup H G { ∈ | − } ⊆ and WH := NH/H is its Weyl group. The component of χG(M) associated to (H) consub(G) and WH C WH π (M H ) is the (ordinary) Euler ∈ · ∈ \ 0 characteristic χ(WH (C, C M >H )), where WH is the isotropy group of C \ ∩ C C π (M H ) under the WH -action. There is a natural homomorphism ∈ 0 eG(M): U G(M) KOG(M). (0.2) → 0 It sends the basis element associated to (H) consub(G) and WH C H ⊆ · ∈ WH π0(M ) to the image of the class of the trivial H -representation R under \ the composition

(α) KOG(x) R (H) = KOH ( ) ∗ KOG(G/H) 0 KOG(M), R 0 {∗} −−→ 0 −−−−−→ 0 where (α) is the isomorphism coming from induction via the inclusion α: H ∗ G and x: G/H M is any G-map with x(1H) C . The main result of → → ∈ this paper is

Theorem 0.3 (Equivariant Euler class and Euler characteristic) Let G be a countable discrete group and let M be a cocompact proper smooth G-manifold without boundary. Then eG(M)(χG(M)) = EulG(M).

The proof of Theorem 0.3 involves two independent steps. Let Ξ be an equiv- ariant vector field on M which is transverse to the zero-section. Let Zero(Ξ) be the set of points x M with Ξ(x) = 0. Then G Zero(Ξ) is finite. The ∈ \ zero-section i: M T M and the inclusion j : T M T M induce an iso- → x x → morphism of Gx -representations T i T j : T M T M ∼= T (T M) x ⊕ 0 x x ⊕ x −→ i(x) if we identify T0(TxM) = TxM in the obvious way. If pri denotes the projection onto the i-th factor for i = 1, 2 we obtain a linear Gx -equivariant isomorphism −1 T Ξ (Txi Txjx) pr d Ξ: T M x T (T M) ⊕ T M T M 2 T M. (0.4) x x −−→ i(x) −−−−−−−−−→ x ⊕ x −−→ x Notice that we obtain the identity if we replace pr2 by pr1 in the expression (0.4) above. One can even achieve that Ξ is canonically transverse to the zero- section, i.e., it is transverse to the zero-section and dxΞ induces the identity

4 Gx on TxM/(TxM) for Gx the isotropy group of x under the G-action. This is proved in [29, Theorem 1A on page 133] in the case of a finite group and the argument directly carries over to the proper cocompact case. Define the index of Ξ at a zero x by

Gx Gx Gx det (dxΞ) : (TxM) (TxM) s(Ξ, x) = → 1 . det ((d Ξ)Gx : (T M)Gx (T M)Gx ) ∈ {§ } | ¡ x x → x ¢| Gx For x M let αx : Gx G be the inclusion, (αx) : RR(Gx) = KO0 ( ) ∈ → ∗ {∗} → KOG(G/G ) be the map induced by induction via α and let x: G/G M 0 x x x → be the G-map sending g to g x. By perturbing the equivariant Euler operator · using the vector field Ξ we will show :

Theorem 0.5 (Equivariant Euler class and vector fields) Let G be a count- able discrete group and let M be a cocompact proper smooth G-manifold without boundary. Let Ξ be an equivariant vector field which is canonically transverse to the zero-section. Then

G G Eul (M) = s(Ξ, x) KO0 (x) (αx) ([R]), · ◦ ∗ Gx G Zero(Ξ) ∈ X\

Gx where [R] R (Gx) = K ( ) is the class of the trivial Gx -representation ∈ R 0 {∗} R, we consider x as a G-map G/Gx M and αx : Gx G is the inclusion. → →

In the second step one has to prove

G G G e (M)(χ (M)) = s(Ξ, x) KO0 (x) (αx) ([R]). (0.6) · ◦ ∗ Gx G Zero(Ξ) ∈ X\ This is a direct conclusion of the equivariant Poincar´e-Hopf theorem proved in [20, Theorem 6.6] (in turn a consequence of the equivariant Lefschetz fixed point theorem proved in [20, Theorem 0.2]), which says

χG(M) = iG(Ξ). (0.7) where iG(Ξ) is the equivariant index of the vector field Ξ defined in [20, (6.5)]. Since we get directly from the definitions

G G G e (M)(i (Ξ)) = s(Ξ, x) KO0 (x) (αx) ([R]), (0.8) · ◦ ∗ Gx G Zero(Ξ) ∈ X\ equation (0.6) follows from (0.7) and (0.8). Hence Theorem 0.3 is true if we can prove Theorem 0.5, which will be done in Section 1.

5 We will factorize eG(M) as

eG(M) eG(M) eG(M): U G(M) 1 HOr(G)(M; R ) 2 HOr(G)(M; R ) −−−−→ 0 Q −−−−→ 0 R eG(M) 3 KOG(M), −−−−→ 0 Or(G) where H0 (M; RF ) is the Bredon homology of M with coefficients in the coefficient system which sends G/H to the representation ring RF (H) for the G G field F = Q, R. We will show that e2 (M) and e3 (M) are rationally injective G (see Theorem 3.6). We will analyze the map e1 (M), which is not rationally injective in general, in Theorem 3.21. The rational information carried by EulG(M) can be expressed in terms of orbifold Euler characteristics of the various components of the L-fixed point H sets for all finite cyclic subgroups L G. For a component C π0(M ) ⊆ ∈ H denote by WHC its isotropy group under the WH -action on π0(M ). For H G finite WH acts properly and cocompactly on C and its orbifold Euler ⊆ C characteristic (see Definition 2.5), which agrees with the more general notion of L2 -Euler characteristic, WH χQ C (C) Q, ∈ is defined. Notice that for finite WHC the orbifold Euler characteristic is given in terms of the ordinary Euler characteristic by χ(C) χQWHC (C) = . WH | C | There is a character map (see (2.6)) G G ch (M): U (M) Q → H (H) consub(G) WH π0(M ) ∈ M \M which sends χG(M) to the various L2 -Euler characteristics χQWHC (C) for L G (H) consub(G) and WH C WH π0(M ). Recall that rationally Eul (M) ∈ · ∈ G\ G carries the same information as e1 (M)(χ (M)) since the rationally injec- G G G G G tive map e3 (M) e2 (M) sends e1 (M)(χ (M)) to Eul (M). Rationally G G ◦ e1 (M)(χ (M)) is the same as the collection of all these orbifold Euler charac- teristics χQWHC (C) if one restricts to finite cyclic subgroups H . Namely, we will prove (see Theorem 3.21):

Theorem 0.9 There is a bijective natural map

G ∼= Or(G) γQ : Q Q Z H0 (X; RQ) L −→ ⊗ (L) consub(G) WL π0(M ) L finite∈ Mcyclic \M

6 which maps χQWLC )(C) (L) consub(G), L finite cyclic, WL C WL π (M L) { | ∈ · ∈ \ 0 } to 1 eG(M)(χG(M)). ⊗Z 1 However, we will show that EulG(M) does carry some torsion information. Namely, we will prove:

Theorem 0.10 There exists an action of the symmetric group S3 of order 3! on the 3-sphere S3 such that EulS3 (S3) KOS3 (S3) has order 2. ∈ 0 The relationship between EulG(M) and the various notions of equivariant Euler characteristic is clarified in sections 2 and 4.3. The paper is organized as follows:

1. Perturbing the equivariant Euler operator by a vector field 2. Review of notions of equivariant Euler characteristic 3. The transformation eG(M) 4. Examples 4.1. Finite groups and connected non-empty fixed point sets 4.2. The equivariant Euler class carries torsion information G G 4.3. Independence of Eul (M) and χs (M) 4.4. The image of the equivariant Euler class under assembly References

This paper subsumes and replaces the preprint [25], which gave a much weaker version of Theorem 0.3. This research was supported by Sonderforschungsbereich 478 (“Geometrische Strukturen in der Mathematik”) of the University of Munster.¨ Jonathan Rosen- berg was partially supported by NSF grants DMS-9625336 and DMS-0103647.

1 Perturbing the equivariant Euler operator by a vector field

Let M n be a complete Riemannian manifold without boundary, equipped with an isometric action of a discrete group G. Recall that the de Rham operator D = d + d∗ , acting on differential forms on M (of all possible degrees) is a formally self-adjoint elliptic operator, and that on the Hilbert space of L2

7 forms, it is essentially self-adjoint [8]. With a certain grading on the form bundle (coming from the Hodge -operator), D becomes the signature operator; with ∗ the more obvious grading of forms by parity of the degree, D becomes the Euler characteristic operator or simply the Euler operator. When M is compact and G is finite, the kernel of D, the space of harmonic forms, is naturally identified with the real or complex1 of M by the Hodge Theorem, and in this way one observes that the (equivariant) index of D (with respect to the parity grading) in the real representation ring of G is simply the (equivariant homological) Euler characteristic of M , whereas the index with respect to the other grading is the G-signature [2]. Now by Kasparov theory (good general references are [4] and [9]; for the de- tailed original papers, see [12] and [13]), an elliptic operator such as D gives rise to an equivariant K -homology class. In the case of a compact manifold, the equivariant index of the operator is recovered by looking at the image of this class under the map collapsing M to a point. However, the K -homology class usually carries far more information than the index alone; for example, it determines the G-index of the operator with coefficients in any G-, and even determines the families index in KG∗ (Y ) of a family of twists of the operator, as determined by a G-vector bundle on M Y . (Y here is × an auxiliary parameter space.) When M is non-compact, things are similar, except that usually there is no index, and the class lives in an appropriate G Kasparov group KG−∗(C0(M)), which is locally finite K -homology, i.e., the relative group KG(M, ), where M is the one-point compactification of ∗ {∞} M .2 We will be restricting attention to the case where the action of G is proper and cocompact, in which case KG−∗(C0(M)) may be viewed as a kind of orbifold K -homology for the compact orbifold G M (see [4, Theorem 20.2.7].) \ We will work throughout with real scalars and real K -theory, and use a variant of the strategy found in [23] to prove Theorem 0.5.

Proof of Theorem 0.5 Recall that since Ξ is transverse to the zero-section, its zero set Zero(Ξ) is discrete, and since M is assumed G-cocompact, Zero(Ξ) consists of only finitely many G-orbits. Write Zero(Ξ) = Zero(Ξ)+ Zero(Ξ) , q − 1depending on what scalars one is using 2 Here C0(M) denotes continuous real- or complex-valued functions on M vanishing at infinity, depending on whether one is using real or complex scalars. This algebra is contravariant in M , so a contravariant functor of C0(M) is covariant in M . Excision in −∗ −∗ Kasparov theory identifies KG (C0(M)) with KG (C(M), C(pt)), which is identified with relative KG -homology. When M does not have finite G- type, K G - homology here means Steenrod KG -homology, as explained in [10].

8 according to the signs of the indices s(Ξ, x) of the zeros x Zero(Ξ). We ∈ fix a G-invariant Riemannian metric on M and use it to identify the form bundle of M with the Clifford algebra bundle Cliff(T M) of the , with its standard grading in which vector fields are sections of Cliff(T M)− , and D with the Dirac operator on Cliff(T M).3 (This is legitimate by [15, II, + 2 Theorem 5.12].) Let = − be the Z/2-graded Hilbert space of L H H ⊕ H sections of Cliff(T M). Let A be the operator on defined by right Clifford H multiplication by Ξ on Cliff(T M)+ (the even part of Cliff(T M)) and by right Clifford multiplication by Ξ on Cliff(T M) (the odd part). We use right − − Clifford multiplication since it commutes with the symbol of D. Observe that A is self-adjoint, with square equal to multiplication by the non-negative function Ξ(x) 2 . Furthermore, A is odd with respect to the grading and commutes with | | multiplication by scalar-valued functions. For λ 0, let D = D + λA. As in [23], each D defines an unbounded G- ≥ λ λ equivariant Kasparov module in the same Kasparov class as D. In the “bounded picture” of Kasparov theory, the corresponding operator is

1 1 2 2 1 1 1 2 − B = D 1 + D − 2 = D + D . (1.1) λ λ λ λ λ λ2 λ2 λ µ ¶ The axioms satisfied by¡this op¢erator that insure that it defines a Kasparov KG -homology class (in the “bounded picture”) are the following: (B1) It is self-adjoint, of norm 1, and commutes with the action of G. ≤ (B2) It is odd with respect to the grading of Cliff(T M). (B3) For f C (M), fB B f and fB2 f , where denotes equality ∈ 0 λ ∼ λ λ ∼ ∼ modulo compact operators. We should point out that (B1) is somewhat stronger than it needs to be when G is infinite. In that case, we can replace invariance of Bλ under G by “G- continuity,” the requirement (see [13] and [4, 20.2.1]) that § (B1 ) f(g B B ) 0 for f C (M), g G. 0 · λ − λ ∼ ∈ 0 ∈ In order to simplify the calculations that are coming next, we may assume with- out loss of generality that we’ve chosen the G-invariant Riemannian metric on M so that for each z Zero(Ξ), in some small open G -invariant neighborhood ∈ z Uz of z, M is Gz -equivariantly isometric to a , say of radius 1, about the n origin in R with an orthogonal Gz -action, with z correspond- ing to the origin. This can be arranged since the exponential map induces a 3Since sign conventions differ, we emphasize that for us, unit tangent vectors on M have square 1 in the Clifford algebra. −

9 G -diffeomorphism of a small G -invariant neighborhood of 0 T M onto a z z ∈ z Gz -invariant neighborhood of z such that 0 is mapped to z and its differential at 0 is the identity on TzM under the standard identification T0(TzM) = TzM . Thus the usual coordinates x1, x2, . . . , xn in Euclidean space give local coor- dinates in M for x < 1, and ∂ , ∂ , . . . , ∂ define a local orthonormal | | ∂x1 ∂x2 ∂xn frame in T M near z. We can arrange that (Rn)Gz contains the points with n Gz x2 = . . . = xn = 0 if (R ) is different from 0 . In these exponential local { } coordinates, the point x = x = = x = 0 corresponds to z. We may 1 2 · · · n assume we have chosen the vector field Ξ so that in these local coordinates, Ξ is given by the radial vector field ∂ ∂ ∂ x1 + x2 + + xn (1.2) ∂x1 ∂x2 · · · ∂xn if z Zero(Ξ)+ , or by the vector field ∈ ∂ ∂ ∂ x1 + x2 + + xn (1.3) − ∂x1 ∂x2 · · · ∂xn if z Zero(Ξ) . Thus Ξ(x) = 1 on ∂U for each z, and we can assume ∈ − | | z (rescaling Ξ if necessary) that Ξ 1 on the of U . | | ≥ z Zero(Ξ) z Recall that D = D + λA. ∈ λ S Lemma 1.4 Fix a small number ε > 0, and let Pλ denote the spectral pro- 2 jection of Dλ corresponding to [0, ε]. Then for λ sufficiently large, range Pλ is G-isomorphic to L2(Zero(Ξ)) (a Hilbert space with Zero(Ξ) as orthonormal basis, with the obvious unitary action of G coming from the action of G on Zero(Ξ)), and there is a constant C > 0 such that (1 P )D2 Cλ. (In − λ λ ≥ other words, (ε, Cλ) (spec D2) = .) Furthermore, the functions in range P ∩ λ ∅ λ become increasingly concentrated near Zero(Ξ) as λ . → ∞ Proof First observe that in Euclidean space Rn , if Ξ is defined by (1.2) or 2 (1.3) and A and Dλ are defined from Ξ as on M , then Sλ = Dλ is basically a Schr¨odinger operator for a harmonic oscillator, so one can compute its spectral 2 decomposition explicitly. (For example, if n = 1, then S = d + λ2x2 λ, λ − dx2 § the sign depending on whether z Zero(Ξ)+ or z Zero(Ξ) and whether one ∈ ∈ − considers the action on + or .) When z Zero(Ξ)+ , the kernel of S in H H− ∈ λ L2 sections of Cliff(T Rn) is spanned by the Gaussian function λ x 2/2 (x , x , . . . , x ) e− | | , 1 2 n 7→ 2 and if z Zero(Ξ)− , the L kernel is spanned by a similar section of Cliff(T M)−, 2 ∈ e λ x /2 ∂ . Also, in both cases, S has discrete spectrum lying on an arith- − | | ∂x1 λ metic progression, with one-dimensional kernel (in L2 ) and first non-zero eigen- value given by 2nλ.

10 Now let’s go back to the operator on M . Just as in [23, Lemma 2], we have the estimate Kλ D2 (D2 + λ2A2) Kλ, (1.5) − ≤ λ − ≤ where K > 0 is some constant (depending on the size of the covariant deriva- tives of Ξ).4 But D2 0, and also, from (1.5), λ ≥ 1 1 K D2 A2 + D2 , (1.6) λ2 λ ≥ λ2 − λ which implies that 1 K D2 multiplication by Ξ(x) 2 . (1.7) λ2 λ ≥ | | − λ So if ξλ is a unit vector in range Pλ , we have ε 1 K D2ξ , ξ Ξ(x) 2 ξ (x) 2 dvol ξ 2. (1.8) λ2 ≥ λ2 λ λ λ ≥ | | | λ | − λ λ ¿ À ZM ° ° Now ξ = 1, and if we fix η > 0, we only make the integral° ° smaller by k λk replacing Ξ(x) 2 by η on the set E = x : Ξ(x) 2 η and by 0 elsewhere. | | η { | | ≥ } So ε K + η ξ (x) 2 dvol λ2 ≥ − λ | λ | ZEη or K ε ξ χ 2 + . (1.9) λ Eη ≤ ηλ ηλ2 This being true for any η, °we hav°e verified that as λ , ξ becomes in- ° ° → ∞ λ creasingly concentrated near the zeros of Ξ, in the sense that the L2 norm of its restriction to the complement of any neighborhood of Zero(Ξ) goes to 0.

It remains to compute range Pλ (as a unitary representation space of G) and to 2 2 prove that Dλ has the desired spectral gap. Define a C cut-off function ϕ(t), 1 0 t < , so that 0 ϕ(t) 1, ϕ(t) = 1 for 0 t 2 , ϕ(t) = 0 for t 1, ≤ ∞ ≤ ≤ 1 ≤ ≤ ≥ and ϕ is decreasing on the 2 , 1 . In other words, ϕ is supposed to have a graph like this: £ ¤ 1 0.8 0.6 0.4 0.2

0.5 1 1.5 2

4We are using the cocompactness of the G-action to obtain a uniform estimate.

11 We can arrange that ϕ (t) 3 and that ϕ (t) 20. For each element z | 0 | ≤ | 00 | ≤ of Zero(Ξ), recall that we have a Gz -invariant neighborhood Uz that can be n identified with the unit ball in R equipped with an orthogonal Gz -action. So λr2/2 the function ψz,λ(x) = ϕ(r)e− , where r = x is the radial coordinate in n 2 | | R , makes sense as a function in C (M), with in Uz . For simplicity suppose z Zero(Ξ)+ ; the other case is exactly analogous except that we need ∈ 2 a 1-form instead of a function. Then Dλ , acting on radial functions, becomes ∂2 1 ∂ ∆ + λ2 x 2 nλ = (n 1) + λ2r2 nλ. − | | − −∂r2 − − r ∂r − n λr2/2 As we mentioned before, this operator on R annihilates x e− , so we 7→ have 2 2 D ψ D ψz,λ, ψz,λ k λ z,λk = λ ψ 2 ψ , ψ k z,λk ­ h z,λ z,λi ® 1 2 λr2 n 2 ϕ(r) rϕ00(r) + (1 n + 2r λ)ϕ0(r) e− r − dr = 0 − − 1 2 λr2 n 1 R ¡ 0 ϕ(r) e− r − dr ¢ 1 λr2 n 2 1/2(20r + 6λ)e−R r − dr . (1.10) ≤ 1/2 λr2 n 1 R 0 e− r − dr k The expression (1.10) goRes to 0 faster than λ− for any k 1, since the ≥ n/2 numerator dies rapidly and the denominator behaves like a constant times λ− for large λ, so Pλψz,λ is non-zero and very close to ψz,λ . Rescaling constructs a unit vector in range Pλ concentrated near z, regardless of the value of ε, provided λ is sufficiently large (depending on ε). And the action of g G ∈ sends this unit vector to the corresponding unit vector concentrated near g z. 2 · In particular, range Pλ contains a Hilbert space G-isomorphic to L (Zero(Ξ)). To complete the proof of the Lemma, it will suffice to show that if ξ is a unit vector in the domain of D which is orthogonal to each ψ , then D ξ 2 Cλ z,λ k λ k ≥ for some constant C > 0, provided λ is sufficiently large Let E = z Zero(Ξ) Vz , 1 ∈ where Vz corresponds to the ball about the origin of radius 2 when we identify n S Uz with the ball about the origin in R of radius 1. Let χE be the characteristic function of E . Then 1 = ξ 2 = χ ξ 2 + (1 χ )ξ 2. k k k E k k − E k Hence we must be in one of the following two cases: (a) (1 χ )ξ 2 1 . k − E k ≥ 2 (b) χ ξ 2 1 . k E k ≥ 2

12 1 In case (1), we can argue just as in the inequalities (1.8) and (1.9) with η = 4 , since E is precisely the set where Ξ(x) 2 < 1 . So we obtain | | 4 1 1 K 1 1 K D ξ 2 = D2ξ, ξ + (1 χ )ξ 2 , λ2 k λ k λ2 λ ≥ − λ 4 k − E k ≥ 8 − λ ¿ À 2 2 which gives Dλξ const λ once λ is sufficiently large. So now consider k k º · 2 1 case (2). Then for some z, we must have χG Vz ξ 2 G Zero(Ξ) . But by k · k ≥ | \ | assumption, ξ ψg z,λ (for this same z and all g G). Assume for simplicity ⊥ · ∈ that ξ + and z Zero(Ξ)+ . If ξ + and z Zero(Ξ) , there is no ∈ H ∈ ∈ H ∈ − essential difference, and if ξ , the calculations are similar, but we need ∈ H− 1-forms in place of functions. Anyway, if we let ξg denote ξ Ug z transported | · to Rn , we have

λ x 2/2 0 = ϕ( x )ξg(x)e− | | dx ( g), n | | ∀ ZR 2 2 1 1 ϕ( x ) ξg(x) dx . ≥ x 1 | | ≥ 2 G Zero(Ξ) g Z 2 X | |≤ ¯ ¯ | \ | ¯ ¯ n Now we use the fact that the Schr¨odinger operator Sλ on R has one-dimen- 2 sional kernel in L2 spanned by x e λ x /2 (if z Zero(Ξ)+ ), and spectrum 7→ − | | ∈ bounded below by 2nλ on the orthogonal complement of this kernel. (If z ∈ Zero(Ξ) , the entire spectrum of S on + is bounded below by 2nλ.) So − λ H compute as follows: D ϕ( x )ξ 2 = D2 ϕ( x )ξ , ϕ( x )ξ λ | | g λ | | g | | g ° ¡ ¢° 2­nλ¡ϕ( x )ξg,¢ ϕ( x )ξg ®. (1.11) ° ° ≥ h | | | | i Let ω be the function on M which is 0 on the complement of g Ug z and given · by ϕ( x ) on Ug z (when we use the local coordinate system there centered at | | · S g z). Then: · D ξ 2 = D ωξ 2 + D 1 ω ξ 2 k λ k λ λ − 2 ° + ¡2 D¢λ° 1 ° ω ¡¡ξ , ωξ ¢. ¢° (1.12) ° ° −° ° Since Dλ is local and ω is supported­ on¡¡the U¢g z¢, g ®G, the first term on the · ∈ right is simply 2nλ D ωξ 2 = D ϕ( x )ξ 2 (1.13) λ λ | | g ≥ 2 G Zero(Ξ) g | \ | ° ¡ ¢° X ° ¡ ¢° ° ° ° ° by (1.11). In the inner product term in (1.12), since ωξ is a sum of pieces with disjoint supports Ug z , we can split this as a sum over terms we can transfer to ·

13 Rn , getting 2 S 1 ϕ( x ) ξ , ϕ( x )ξ = 2 D2 1 ϕ( x ) ξ , ϕ( x )ξ λ − | | g | | g − | | g | | g g g X ­ ¡¡ ¢ ¢ ® X ­ ¡¡ ¢ ¢ ® 2 2 2 + 2 (λ x + T λ) ϕ( x ) 1 ϕ( x ) ξg(x) dx, 1 x 1 | | | | − | | | | g Z 2 X ≤| |≤ ¡ ¢ where K T K . Since λ2 x 2 +T λ > 0 on 1 x 1 for large enough λ, − ≤ ≤ | | 2 ≤ | | ≤ the integral here is nonnegative, and the only possible negative contributions to D ξ 2 are the terms 2 D2 1 ϕ( x ) ξ , ϕ( x )ξ , which do not grow k λ k − | | g | | g with λ. So from (1.12), (1.13), and (1.11), D ξ 2 const λ for large enough ­ ¡¡ k¢ λ¢ k ≥ ® · λ, which completes the proof.

Proof of Theorem 0.5, continued We begin by defining a continuous field E of Z/2-graded Hilbert spaces over the closed interval [0, + ]. Over the open ∞ interval [0, + ), the field is just the trivial one, with fiber = , the L2 ∞ Eλ H sections of Cliff(T M). But the fiber over + will be the direct sum of E∞ ∞ V , where V = L2(Zero(Ξ)) is a Hilbert space with orthonormal basis v , H ⊕ z z Zero(Ξ). We put a Z/2-grading on V by letting V + = L2(Zero(Ξ)+), ∈ 2 V − = L (Zero(Ξ)−). To define the continuous field structure, it is enough by [7, Proposition 10.2.3] to define a suitable total set of continuous sections near the exceptional point λ = . We will declare ordinary continuous functions ∞ [0, ] to be continuous, but will also allow additional continuous fields ∞ → H that become increasingly concentrated near the points of Zero(Ξ). Namely, suppose z Zero(Ξ). By Lemma 1.4, for λ large, D has an element ψ ∈ λ z,λ in its “approximate kernel” increasingly supported close to z, and we have a formula for it. So we declare (ξ(λ))λ< to define a continuous field converging to cv at λ = if for any neighborho∞od U of z, z ∞ ξ(λ)(m) 2 dvol(m) 0 as λ , | | → → ∞ ZMrU and if (assuming z Zero(Ξ)+ ) ξ(λ) + and ∈ ∈ H n λ 4 ξ(λ) c ψz,λ 0 as λ . (1.14) ° − π ° → → ∞ ° µ ¶ ° ° ° 2 n The constant reflects° the fact that the°L2 -norm of e λ x /2 is π 4 . If z ° ° − | | λ ∈ Zero(Ξ) , we use the same definition, but require ξ(λ) . − ∈ H− ¡ ¢ This concludes the definition of the continuous field of Hilbert spaces , which E we can think of as a Hilbert C -module over C(I), I the interval [0, + ]. We ∗ ∞

14 will use this to define a Kasparov (C0(M), C(I))-bimodule, or in other words, a homotopy of Kasparov (C0(M), R)-modules. The action of C0(M) on is E the obvious one: C (M) acts on the usual way, and it acts on V (the other 0 H summand of ) by evaluation of functions at the points of Zero(Ξ): E∞ f v = f(z)v , z Zero(Ξ), f C (M). · z z ∈ ∈ 0 We define a field T of operators on as follows. For λ < , T ( ) = E ∞ λ ∈ L Eλ ( ) is simply B as defined in (1.1), where recall that D = D + λA. For L H λ λ λ = , = V and T is 0 on V and is given on by the operator B = ∞ E∞ H⊕ ∞ Ξ(x)H ∞ A/ A which is right Clifford multiplication by Ξ(x) (an L∞ , but possibly | | | | + Ξ(x) 2 discontinuous, vector field) on and by −Ξ(x) on − . Note that T is 0 on H H ∞ V and the identity on . While 1 is not compact| | on V if Zero(Ξ) is infinite, H V this is not a problem since for f C (M), the action of f on V has finite rank ∈ c (since f annihilates v for z supp f ). z 6∈ Now we check the axioms for ( , T ) to define a homotopy of Kasparov modules E from [D] to the class of

C0(M), , T = C0(M), , B C0(M), V, 0 . E∞ ∞ H ∞ ⊕ But C0(M¡), , B is a de¢ gener¡ ate Kasparov¢module,¡ since B¢ commutes H ∞ ∞ with multiplication by functions and has square 1. So the class of C0(M), , ¡ ¢ E∞ T is just the class of C0(M), V, 0 , which (essentially by definition) is the ∞ ¡ image under the inclusion Zero(Ξ) , M of the sum (over G Zero(Ξ)) of +1 ¢ ¡ G →¢ \ + times the canonical class KO0 (z) (αz) ([R]) for G z Zero(Ξ) and of 1 ◦ ∗ · ⊆ − times this class if G z Zero(Ξ) . This will establish Theorem 0.5, assuming · ⊆ − we can verify that we have a homotopy of Kasparov modules. The first thing to check is that the action of C (M) on is continuous, i.e., 0 E given by a -homomorphism C (M) ( ). The only issue is continuity at ∗ 0 → L E λ = . In other words, since the action on is constant, we just need to know ∞ H that if ξ is a continuous field converging as λ to a vector v in V , then → ∞ for f C (M), f ξ(λ) f v. But it’s enough to consider the special kinds ∈ 0 · → · of continuous fields discussed above, since they generate the structure, and if ξ(λ) cv , then ξ(λ) becomes increasingly concentrated at z (in the sense of → z L2 norm), and hence f ξ(λ) cf(z)v , as required. · → z Next, we need to check that T ( ). Again, the only issue is (strong operator) ∈ L E continuity at λ = . Because of the way continuous fields are defined at ∞ λ = , there are basically two cases to check. First, if ξ , we need to check ∞ ∈ H that Bλξ B ξ as λ . Since the Bλ ’s all have norm 1, we also only → ∞ → ∞ ≤ need to check this on a dense set of ξ’s. First, fix ε > 0 small and suppose ξ

15 is smooth and supported on the open set where Ξ(x) 2 > ε. Then for λ large, | | Lemma 1.4 implies that there is a constant C > 0 (depending on ε but not on λ) such that D2ξ, ξ > Cλ ξ 2 . In fact, if P is the spectral projection of D2 h λ i k k λ λ for the interval [0, Cλ], Lemma 1.4 implies that P ξ ε ξ for λ sufficiently k λ k ≤ k k large. (This is because the condition on the support of ξ forces ξ to be almost 2 + orthogonal to the spectral subspace where D ε.) Now let E and E− λ ≤ λ λ be the spectral projections for D corresponding to the intervals (0, ) and λ ∞ ( , 0), and let F + and F be the spectral projections for A corresponding −∞ − to the same intervals. Since the vector field Ξ vanishes only on a discrete set, + the operator A has no kernel, and hence F + F − = 1. Now we appeal to two results in Chapter VIII of [14]: Corollary 1.6 in 1, and Theorem 1.15 in 2. 1 § § The former shows that the operators A + λ D, all defined on dom D, “converge strongly in the generalized sense” to A. Since the positive and negative spectral 1 subspaces for A+ λ D are the same as for Dλ (since the operators only differ by + + a homothety), [14, Chapter VIII, 2, Theorem 1.15] then shows that Eλ F + § → and E− F in the strong operator . Note that the fact that A has λ → no kernel is needed in these results. Now since P ξ ε ξ for λ sufficiently large, we also have k λ k ≤ k k + + B ξ = F ξ F −ξ, and Bλξ (E ξ E−ξ) 2ε ∞ − k − λ − λ k ≤ for λ sufficiently large. Hence + + Bλξ B ξ 2ε + (E ξ F ξ) (E−ξ F −ξ) 2ε. k − ∞ k ≤ k λ − − λ − k → Now let ε 0. Since, with ε tending to zero, ξ’s satisfying our support → condition are dense, we have the required strong convergence. There is one other case to check, that where ξ(λ) cv in the sense of the → z continuous field structure of . In this case, we need to show that B ξ(λ) 0. E λ → This case is much easier: ξ(λ) cv means → z n λ 4 ξ(λ) c ψz,λ 0 by (1.14), ° − π ° → ° µ ¶ ° ° ° while B 1 and° ° k λk ≤ ° ° n λ 4 Dλ ψz,λ 0 by (1.10), ° Ã π !° → ° µ ¶ ° ° ° so B ξ(λ) 0 in norm.° ° λ → ° ° Thus T ( ). Obviously, T satisfies (B1) and (B2) of page 9, so we need to ∈ L E check the analogues of (B3), which are that f(1 T 2) and [T, f] lie in ( ) − K E

16 for f C (M). First consider 1 T 2 . 1 T 2 is locally compact (i.e., compact ∈ 0 − − λ after multiplying by f C (M)) for each λ, since ∈ c 2 2 2 1 2 1 1 T = 1 B = (1 + D )− = 1 + (D + λA) − − λ − λ λ is locally compact for λ < , and 1 T 2 is¡just projection¢ onto V , where ∞ − ∞ functions f of compact support act by finite-rank operators. So we just need to check that 1 T 2 is a norm-continuous field of operators on . Continuity − λ E for λ < is routine, and implicit in [3, Remarques 2.5]. To check continuity ∞ at λ = , we use Lemma 1.4, which shows that (1 + D2) 1 = P + O 1 , and ∞ λ − λ λ also that Pλ is increasingly concentrated near Zero(Ξ). So near λ = , we 2 1 ¡ ∞¢ can write the field of operators (1 + Dλ)− as a sum of rank-one projections onto vector fields converging to the various vz ’s (in the sense of our continuous field structure) and another locally compact operator converging in norm to 0. This leaves just one more thing to check, that for f C (M), [f, T ] lies in ∈ 0 λ ( ). We already know that [f, B ] ( ) for fixed λ and is norm-continuous K E λ ∈ K H in λ for λ < , so since T commutes with multiplication operators, it suffices ∞ ∞ to show that [f, B ] converges to 0 in norm as λ 0. We follow the method λ → of proof in [23, p. 3473], pointing out the changes needed because of the zeros of the vector field Ξ. We can take f C (M) with critical points at all of the points of the set ∈ c∞ Zero(Ξ), since such functions are dense in C0(M). Then estimate as follows: 2 1/2 [f, Bλ] = f, Dλ(1 + Dλ)−

h 2 1/2i 2 1/2 = [f, Dλ](1 + Dλ)− + Dλ f, (1 + Dλ)− . (1.15) h i We have [f, Dλ] = [f, D], which is a 0’th order operator determined by the derivatives of f , of compact support since f has compact support, and we’ve seen that (1 + D2) 1/2 converges as λ (in the norm of our continuous λ − → ∞ field) to projection onto the space V = L2(Zero(Ξ)). Since the derivatives of 2 1/2 f vanish on Zero(Ξ), the product [f, Dλ](1 + Dλ)− , which is the first term in (1.15), goes to 0 in norm. As for the second term, we have (following [4, p. 199])

2 1/2 1 ∞ 1 2 1 Dλ f, (1 + Dλ)− = µ− 2 Dλ f, (1 + Dλ + µ)− dµ, (1.16) π 0 h i Z and £ ¤ 2 1 2 1 2 2 1 Dλ f, (1 + Dλ + µ)− = Dλ(1 + Dλ + µ)− 1 + Dλ + µ, f (1 + Dλ + µ)− . Now£use the fact that ¤ £ ¤ 2 2 1 + Dλ + µ, f = Dλ, f = Dλ[Dλ, f] + [Dλ, f]Dλ = Dλ[D, f] + [D, f]Dλ. £ ¤ £ ¤ 17 We obtain that

2 1 Dλ f, (1 + Dλ + µ)− 2 £ Dλ ¤ 1 Dλ Dλ = 2 [D, f] 2 + 2 [D, f] 2 . (1.17) 1 + Dλ + µ 1 + Dλ + µ 1 + Dλ + µ 1 + Dλ + µ

Again a slight modification of the argument in [23, p. 3473] is needed, since Dλ has an “approximate kernel” concentrated near the points of Zero(Ξ). So we estimate the norm of the right side of (1.17) as follows:

2 2 1 Dλ 1 D f, (1 + D + µ)− [D, f] (1.18) λ λ ≤ 1 + D2 + µ 1 + D2 + µ ° λ λ ° ° £ ¤° ° D D ° ° ° + ° λ [D, f] λ °. (1.19) °1 + D2 + µ 1 + D2 + µ ° ° λ λ ° ° ° ° ° The first term, (1.18), is bounded° by the second, (1.19), plus °an additional commutator term:

D 1 λ [D, [D, f]] . (1.20) 1 + D2 + µ 1 + D2 + µ ° λ λ ° ° ° ° ° Now the contribution° of the term (1.19) is estimated °by observing that the function x , < x < 1 + x2 + µ −∞ ∞

1 has maximum value 2√1+µ at x = √1 + µ, is increasing for 0 < x < √1 + µ, and is decreasing to 0 for x > √1 + µ. Fix ε > 0 small. Since, by Lemma 1.4, D has spectrum contained in [0, √ε] [√Cλ, ), we find that | λ| ∪ ∞

1 2√1+µ ,  √Cλ √1 + µ, or µ Cλ 1, Dλ  ≤ ≥ −  (1.21) 1 + D2 + µ ≤  ° λ °  √ε √Cλ ° ° max 1+ε+µ , 1+µ+Cλ , ° ° ° °  √³ Cλ √1 + µ,´or 0 µ Cλ 1.  ≥ ≤ ≤ −    18 Thus the contribution of the term (1.19) to the integral in (1.16) is bounded by 2 [D, f] ∞ D 1 k k λ dµ π 1 + D2 + µ √µ Z0 ° λ ° ° ° [D, f] ° ∞ 1 °1 k k ° ° dµ (1.22) ≤ π Cλ 1 √µ 4(1 + µ) µZ − Cλ 1 1 Cλ ε + − max , dµ √µ (1 + µ + Cλ)2 (1 + ε + µ)2 Z0 µ ¶ ¶ Cλ [D, f] 1 ∞ 3 1 Cλ ∞ ε 2 k k µ− dµ + 2 dµ + 2 dµ ≤ π 4 Cλ 1 0 √µ (Cλ) 0 √µ(1 + µ) µ Z − Z Z ¶ [D, f] 1 2 πε [D, f] = k k + + k kε. (1.23) π 2√Cλ 1 √Cλ 2 → 2 µ − ¶ We can make this as small as we like by taking ε small enough. Similarly, the contribution of term (1.20) to the integral in (1.16) is bounded by

[D, [D, f]] ∞ D 1 1 k k λ dµ π 1 + D2 + µ 1 + µ √µ Z0 ° λ ° ° ° [D, [D, f]] ° ∞ 1 ° 1 1 k k ° ° dµ ≤ π Cλ 1 2√1 + µ 1 + µ √µ Z − Cλ 1 ³ − √Cλ 1 1 ∞ ε + dµ + 2 dµ 0 (1 + µ + Cλ) 1 + µ √µ 0 √µ(1 + µ) Z Z ´ [D, [D, f]] ∞ 1 ∞ 1 1 k k 2 dµ + dµ ≤ π Cλ 1 2µ 0 √Cλ √µ(1 + µ) µZ − Z ε + ∞ dµ √µ(1 + µ)2 Z0 ¶ [D, [D, f]] 1 π πε [D, [D, f]] k k + + k kε, (1.24) ≤ π 2(Cλ 1) √Cλ 2 → 2 µ − ¶ which again can be taken as small as we like. This completes the proof.

2 Review of notions of equivariant Euler character- istic

Next we briefly review the universal equivariant Euler characteristic, as well as some other notions of equivariant Euler characteristic, so we can see exactly how they are related to the KOG -Euler class EulG(M). We will use the following notation in the sequel.

19 Notation 2.1 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 H X := x X H Gx ; >H { ∈ | ⊆ } X := x X H ( Gx , { ∈ | } where Gx is the isotropy group of x under the G-action. Let x: G/H X be a G-map. Let XH (x) be the component of XH contain- → ing x(1H). Put X>H (x) = XH (x) X>H . ∩ Let WH be the isotropy group of XH (x) π (XH ) under the WH -action. x ∈ 0 Next we define the group U G(X), in which the universal equivariant Euler characteristic takes its values. Let Π0(G, X) be the component of the G-space X in the sense of tom Dieck [6, 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 (G, f): Π (G, X) Π (G, Y ) by composition → 0 0 → 0 with f . Denote by Is Π0(G, X) the set of isomorphism classes [x] of objects x: G/H X in Π (G, X). Define → 0 G U (X) := Z[Is Π0(G, X)], (2.2) 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 (G, X) ∼= WH π (XH ), (2.3) 0 −→ \ 0 (H) consub(G) ∈ a which sends x: G/H X to the orbit under the WH -action on π (XH ) of → 0 the component XH (x) of XH which contains the point x(1H). It induces a natural isomorphism

G = U (X) ∼ Z. (2.4) −→ H (H) consub(G) WH π0(X ) ∈ M \M

20 Definition 2.5 Let X be a finite G-CW -complex X . We define the universal equivariant Euler characteristic of X χG(X) U G(X) ∈ by assigning to [x: G/H X] Is Π (G, X) the (ordinary) Euler character- → ∈ 0 istic of the pair of finite CW -complexes (WH XH (x), WH X>H (x)). x\ x\ If the action of G on X is proper (so that the isotropy group of any open cell in X is finite), we define the orbifold Euler characteristic of X by:

QG 1 χ (X) := Ge − Q, | | ∈ p 0 G e G Ip(X) X≥ · ∈X\ where Ip(X) is the set of open cells of X (after forgetting the group action).

The orbifold Euler characteristic χQG(X) can be identified with the more gen- eral notion of the L2 -Euler characteristic χ(2)(X; (G)), where (G) is the N N group von Neumann algebra of G. One can compute χ(2)(X; (G)) in terms N of L2 -homology

(2) p (2) χ (X; (G)) = ( 1) dim (G) Hp (X; (G) , N − · N N p 0 X≥ ³ ´ where dim (G) denotes the von Neumann dimension (see for instance [17, Sec- tion 6.6]). N Next we define for a proper G-CW -complex X the character map G G ch (X): U (X) Q = Q. (2.6) → H Is Π0(G,X) (H) consub(G) WH π0(X ) M ∈ M \M We have to define for an isomorphism class [x] of objects x: G/H X in G G → Π0(G, X) the component ch (X)([x])[y] of ch (X)([x]) which belongs to an isomorphism class [y] of objects y : G/K X in Π0(G, X), and check that G → χ (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 Π0(G, X). We have the left operation 1 aut(y, y) mor(y, x) mor(y, x), (σ, τ) τ σ− . × → 7→ ◦ There is an isomorphism of groups WK ∼= aut(y, y) y −→ which sends gK WK to the automorphism of y given by the G-map ∈ y 1 R −1 : G/K G/K, g0K g0g− K. g → 7→

21 Thus mor(y, x) becomes a left WKy -set.

The WKy -set mor(y, x) can be rewritten as mor(y, x) = g G/HK g x(1H) XK (y) , { ∈ | · ∈ } where the left operation of WK on g G/HK g x(1H) Y K (y) comes y { ∈ | · ∈ } 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 sub- ⊆ groups K G. This shows that mor(y, x) = for at most finitely many iso- ⊆ 6 ∅ morphism classes [y] of objects y Π (G, X) and that the WK -set mor(y, x) ∈ 0 y decomposes into finitely many WKy orbits with finite isotropy groups for each object y Π (G, X). We define ∈ 0 G 1 ch (X)([x]) := (WK ) − , (2.7) [y] | y σ| WKy σ · ∈ WKy Xmor(y,x) \ where (WK ) is the isotropy group of σ mor(y, x) under the WK -action. y σ ∈ y Lemma 2.8 Let X be a finite proper G-CW -complex. Then the map chG(X) of (2.6) is injective and satisfies

G G QWKy K ch (X)(χ (X))[y] = χ (X (y)). The induced map

G G ∼= idQ Z ch (X): Q Z U (X) Q ⊗ ⊗ −→ H (H) consub(G) WH π0(X ) ∈ M \M is bijective.

G G G QWKy K Proof Injectivity of χ (X) and ch (X)(χ (X))[y] = χ (X (y)). is proved in [20, Lemma 5.3]. The bijectivity of id chG(X) follows since its Q ⊗Z source and its target are Q-vector spaces of the same finite Q-dimension. Now let us briefly summarize the various notions of equivariant Euler charac- teristic and the relations among them. Since some of these are only defined when M is compact and G is finite, we temporarily make these assumptions for the rest of this section only.

Definition 2.9 If G is a finite group, the Burnside ring A(G) of G is the of the (additive) of finite G-sets, where the addition comes from disjoint . This becomes a ring under the obvious multiplication

22 coming from the Cartesian product of G sets. There is a natural map of rings j : A(G) R (G) = KOG(pt) that comes from sending a finite G-set X to 1 → R 0 the orthogonal representation of G on the finite-dimensional real Hilbert space 2 LR(X). This map can fail to be injective or fail to be surjective, even rationally.

The rank of A(G) is the number of conjugacy classes of subgroups of G, while the rank of RR(G) is the number of R-conjugacy classes in G, where x and y 1 are called R-conjugate if they are conjugate or if x− and y are conjugate [28, 13.2]. Thus rank A (Z/2)3 = 16 > rank R (Z/2)3 = 8; on the other hand, § R rank A( /5) = 2 < rank R ( /5) = 3. Z ¡ R ¢Z ¡ ¢

Definition 2.10 Now let G be a finite group, M a compact G-manifold (without boundary). We define three more equivariant Euler characteristics for G:

(a) the analytic equivariant Euler characteristic χG(M) KOG , the equiv- a ∈ 0 ariant index of the Euler operator on M . Since the index of an operator is computed by pushing its K -homology class forward to K -homology of G G a point, χa (M) = c (Eul (M)), where c: M pt and c is the induced G ∗ → ∗ map on KO0 . (b) the stable homotopy-theoretic equivariant Euler characteristic χG(M) s ∈ A(G). This is discussed, say, in [6], Chapter IV, 2. § (c) a certain unstable homotopy-theoretic equivariant Euler characteristic, G G which we will denote here χu (M) to distinguish it from χs (M). This in- variant is defined in [30], and shown to be the obstruction to existence of an everywhere non-vanishing G-invariant vector field on M . The invari- G G ant χu (M) lives in a group Au (M) (Waner and Wu call it AM (G), but G G the notation Au (M) is more consistent with our notation for U (M)) G defined as follows: Au (M) is the free abelian group on finite G-sets embedded in M , modulo isotopy (if st is a 1-parameter family of finite G-sets embedded in M , all isomorphic to one another as G-sets, then s0 s1 ) and the relation [s t] = [s] + [t]. Waner and Wu define a map ∼ G q d: Au (M) A(G) (defined by forgetting that a G-set s is embedded in → G G M , and just viewing it abstractly) which maps χu (M) to χs (M). Both G G χu (M) and χs (M) may be computed from the virtual finite G-set given by the singularities of a G-invariant canonically transverse vector field, where the signs are given by the the indices at the singularities.

23 Proposition 2.11 Let G be a finite group, and let M be a compact G- manifold (without boundary). The following diagram commutes:

eG(M) G / G / G Au (M) U (M) KO0 (M) (2.12) OO OOO OOO c∗ c∗ d OOO'   G j1 / G A(G) = U (pt) RR(G) = KO0 (pt). The map AG(M) U G(M) in the upper left is an isomorphism if (M, G) u → satisfies the weak gap hypothesis, that is, if whenever H ( K are subgroups of G, each component of GK has at least 2 in the component of GH that contains it [30]. Furthermore, under the maps of this diagram, χG(M) χG(M), eG(M): χG(M) EulG(M), u 7→ 7→ G G G G c : χ (M) χs (M), j1 : χs (M) χa (M), ∗ 7→ 7→ G G G G d: χu (M) χs (M), c : Eul (M) χa (M). 7→ ∗ 7→

Proof This is just a matter of assembling known information. The facts about the map AG(M) U G(M) are in [30, 2] and in [20]. That eG(M) sends u → § χG(M) to EulG(M) is Theorem 0.3. Commutativity of the square follows immediately from the definition of eG(M), since c eG(M) sends the basis ∗ ◦ element associated to (H) consub(G) and WH C WH π (M H ) to the ⊆ · ∈ \ 0 class of the orthogonal representation of G on L2(G/H). But under c , this ∗ same basis element maps to the G-set G/H in A(G), which also maps to the 2 orthogonal representation of G on L (G/H) under j1 .

3 The transformation eG(X)

Next we factorize the transformation eG(M) defined in (0.2) as

eG(M) eG(M) eG(M): U G(M) 1 HOr(G)(M; R ) 2 HOr(G)(M; R ) −−−−→ 0 Q −−−−→ 0 R eG(M) 3 KOG(M), −−−−→ 0 G G where e2 (X) and e3 (X) are rationally injective. Rationally we will identify Or(G) G G H0 (M; RQ) and the element e1 (M)(χ (M)) in terms of the orbifold Euler characteristics χWLC (C), where (L) runs through the conjugacy classes of finite cyclic subgroups L of G and WL C runs through the orbits in WL π (XL). · \ 0

24 L Here WLC is the isotropy group of C π0(X ) under the WL = NL/L- G G ∈ action. Notice that e1 (M)(χ (M)) carries rationally the same information as EulG(M) KOG(M). ∈ 0 Or(G) Here and elsewhere H0 (M; RF ) is the Bredon homology of M with coeffi- cients the covariant functor

RF : Or(G; in) Z-Mod. F → The orbit category Or(G; in) has as objects homogeneous spaces G/H with F finite H and as morphisms G-maps (Since M is proper, it suffices to consider coefficient systems over Or(G; in) instead over the full orbit category Or(G).) F The functor RF into the category Z-Mod of Z-modules sends G/H to the representation ring RF (H) of the group H over the field F = Q, R or C. It sends a morphism G/H G/K given by g H g gK for some g G with → 0 7→ 0 ∈ g 1Hg K to the induction homomorphism R (H) R (K) associated − ⊆ F → F with the group homomorphism H K, h g 1hg. This is independent of the → 7→ − choice of g since an inner automorphism of K induces the identity on RR(K). Given a covariant functor V : Or(G) Z-Mod, the Bredon homology of a → G-CW -complex X with coefficients in V is defined as follows. Consider the cellular (contravariant) ZOr(G)- C (X−): Or(G) Z-Chain ∗ → which assigns to G/H the cellular chain complex of the CW -complex X H = mapG(G/H, X). One can form the tensor product over the orbit category (see for instance [16, 9.12 on page 166]) C (X−) ZOr(G; in) V which is a Z-chain ∗ ⊗ F Or(G) complex and whose homology groups are defined to be Hp (X; V ).

The zero-th Bredon homology can be made more explicit. Let

Q: Π (G; X) Or(G; in) (3.1) 0 → F be the forgetful functor sending an object x: G/H X to G/H . Any covari- → ant functor V : Or(G; in) Z-Mod induces a functor Q∗V : Π0(G; X) F → → Z-Mod by composition with Q. The colimit (= direct limit) of the functor Q∗RF is naturally isomorphic to the Bredon homology

G = Or(G) β (X): lim Q∗R (H) ∼ H (X; R ). (3.2) F Π (G,X) F 0 F −→ 0 −→ G The isomorphism βF (X) above is induced by the various maps

Or(G) H (x;RF ) R (H) = HOr(G)(G/H; R ) 0 HOr(G)(X; R ), F 0 F −−−−−−−−−→ 0 F

25 where x runs through all G-maps x: G/H X . We define natural maps → eG(X): U G(X) HOr(G)(X; R ); (3.3) 1 → 0 Q eG(X): HOr(G)(X; R ) HOr(G)(X; R ); (3.4) 2 0 Q → 0 R eG(X): HOr(G)(X; R ) KOG(X) (3.5) 3 0 R → 0 as follows. The map βG(X) 1 eG(X) sends the basis element [x: G/H X] Q − ◦ 1 → to the image of the trivial representation [Q] R (H) under the canonical ∈ Q map associated to x

R (H) lim Q∗R (H). Q Π (G,X) Q → −→ 0 The map eG(X) is induced by the change of fields homomorphisms R (H) 2 Q → R (H) for H G finite. The map eG(X) βG(X) is the colimit over the R ⊆ 3 ◦ system of maps

(α ) KOG(x) R (H) = KOH ( ) H ∗ KOG(G/H) 0 KOG(X) R 0 {∗} −−−−→ 0 −−−−−→ 0 for the various G-maps x: G/H X , where α : H G is the inclusion. → H → Theorem 3.6 Let X be a proper G-CW -complex. Then (a) The map eG(X) defined in (0.2) factorizes as

eG(X) eG(X) eG(X): U G(X) 1 HOr(G)(X; R ) 2 HOr(G)(X; R ) −−−−→ 0 Q −−−−→ 0 R eG(X) 3 KOG(X); −−−−→ 0 (b) The map G Or(G) Or(G) Q e (X): Q H (X; R ) Q H (X; R ) ⊗Z 2 ⊗Z 0 Q → ⊗Z 0 R is injective; (c) For each n Z there is an isomorphism, natural in X , ∈ G Or(G) G = G chern (X): Q H (X; KO ) ∼ Q KO (X), n ⊗Z p q −→ ⊗Z n p,q Z, p 0M,p∈+q=n ≥ G where KOq is the covariant functor from Or(G; in) to Z-Mod sending G F G/H to KOq (G/H). The map G Or(G) G id e (X): Q H (M; R ) Q KO (M) Q ⊗Z 3 ⊗Z 0 R → ⊗Z 0 G is the restriction of chernn (X) to the summand for p = q = 0 and is hence injective;

26 (d) Suppose dim(X) 4 and one of the following conditions is satisfied: ≤ (a) either dim(XH ) 2 for H G, H = 1 , or ≤ ⊆ 6 { } (b) no subgroup H of G has irreducible representations of complex or quaternionic type. Then eG(X): HOr(G)(M; R ) KOG(M) 3 0 R → 0 is injective.

Proof (a) follows directly from the definitions. (b) will be proved later. (c) An equivariant Chern character chernG for equivariant homology theories such as equivariant K-homology is constructed∗ in [18, Theorem 0.1] (see also G Or(G) [19, Theorem 0.7]). The restriction of chern0 (M) to H0 (X; RR) is just G G e3 (X) under the identification RR = KO0 . (d) Consider the equivariant Atiyah-Hirzebruch spectral sequence which con- G 2 verges to KOp+q(X) (see for instance [5, Theorem 4.7 (1)]). Its E -term is 2 Or(G) G G Ep,q = Hp (X; KOq ). The abelian group KOq (G/H) is isomorphic to the real topological K -theory KOq(RH) of the real C∗ -algebra RH . The real C∗ -algebra RH splits as a product of matrix algebras over R, C and H, with as many summands of a given type as there are irreducible real representa- tions of H of that type (see [28, 13.2]). By Morita invariance of topological § G real K -theory, we conclude that KOq (G/H) is a direct sum of copies of the non-equivariant K-homologies KOq( ) = KOq(R), KUq( ) = KOq(C), and ∗ G∗ KSpq( ) = KOq(H). In particular, we conclude that KO (G/H) = 0 for q ∗ q ≡ G 2 Or(G) G 1 (mod 8). As a consequence, KO = 0 and E = Hp (X; KO ) = − 1 p, 1 1 0. If no subgroup H of G has irreducible− representations− of complex or quater-− G nionic type, then similarly KOq (G/H) = 0 for all subgroups H of G and 2 2 q = 2, 3, and Ep, 2 = 0, Ep, 3 = 0, as well. − − − − Let X>1 X be the x X G = 1 . There is a short exact sequence ⊆ { ∈ | x 6 } HOr(G)(X>1; KOG) HOr(G)(X; KOG) HOr(G)(X, X>1; KOG). p q → p q → p q Since the isotropy group of any point in X X >1 is trivial, we get an isomor- − phism Or(G) >1 G >1 G Hp (X, X ; KOq ) = Hp(C (X, X ) ZG KOq (G/1)) ∗ ⊗

27 G Since KO (G/1)) = KOq(R) vanishes for q 2, 3 , we get for p Z and q ∈ {− − } ∈ q 2, 3 ∈ {− − } Or(G) >1 G Hp (X, X ; KOq ) = 0. So if dim(XH ) 2 for H G, H = 1 , we have dim(X>1) 2. This implies ≤ ⊆ 6 { } ≤ for p 3 and q Z ≥ ∈ Or(G) >1 G Hp (X ; KOq ) = 0. We conclude that for p 3 and q 2, 3 ≥ ∈ {− − } 2 Or(G) G Ep,q = Hp (X; KOq ) = 0, just as in the previous case. r Now there are no non-trivial differentials out of E0,0 , and since dim X 4, r r r ≤ dr,1 r : Er,1 r E0,0 must be zero for r > 4. But we have just seen that 2 − − 2 → 2 r E2, 1 = 0, E3, 2 = 0, and E4, 3 = 0. Hence each differential dp,q which has r − − − E0,0 as source or target is trivial. Hence the homomorphism restricted to 2 G E0,0 is injective. But this map is e3 (X).

Remark 3.7 We conclude from Theorem 0.3 and Theorem 3.6 that EulG(M) carries rationally the same information as the image of the equivariant Eu- ler characteristic χG(X) under the map eG(X): U G(X) HOr(G)(M; R ). 1 → 0 R Moreover, in contrast to the class of the signature operator, the class EulG(M) G ∈ KO0 (M) of the Euler operator does not carry “higher” information because its preimage under the equivariant Chern character is concentrated in the sum- mand corresponding to p = q = 0.

Next we recall the definition of the Hattori-Stallings rank of a finitely generated projective RG-module P , for some commutative ring R and a group G. Let R[con(G)] be the R-module with the set of conjugacy classes con(G) of elements in G as basis. Define the universal RG-trace tru : RG R[con(G)], r g r (g). RG → g · 7→ g · g G g G X∈ X∈ 2 Choose a matrix A = (ai,j) Mn(RG) such that A = A and the image of the n n ∈ map rA : RG RG sending x to xA is RG-isomorphic to P . Define the Hattori-Stallings→rank HS (P ) := n tru (a ) R[con(G)]. (3.8) RG i=1 RG i,i ∈ Let α: H1 H2 be a group homomorphism.P It induces a map → con(H ) con(H ), (h) (α(h)) 1 → 2 7→

28 and thus an R- α : R[con(H1)] R[con(H2)]. If α P is the R[H2]- ∗ → ∗ module obtained by induction from the finitely generated projective R[H1]- module P , then

HSRH (α P ) = α (HSRH (P )). (3.9) 2 ∗ ∗ 1 Next we compute F lim Q R (H) for F = , , . Two elements Z Π (G,X) ∗ F Q R C ⊗ −→ 0 g1 and g2 of a group G are³ called Q-conjugate´if and only if the cyclic subgroup g generated by g and the cyclic subgroup g generated by g are conjugate h 1i 1 h 2i 2 in G. Two elements g1 and g2 of a group G are called R-conjugate if and only 1 if g1 and g2 or g1− and g2 are conjugate in G. Two elements g1 and g2 of a group G are called C-conjugate if and only if they are conjugate in G (in the usual sense). We denote by (g)F the set of elements of G which are F -conjugate to g. Denote by conF (G) the set of F -conjugacy classes (g)F of elements of finite order g G. Let class (G) be the F -vector space generated by the set ∈ F conF (G). This is the same as the F -vector space of functions conF (G) R → whose support is finite. Let pr : con(H) con (H) be the canonical epimorphism for a finite group F → F H . It extends to an F -linear epimorphism F [pr ]: F [con(H)] class (H). F → F Define for a finite-dimensional H -representation V over F for a finite group H HS (V ) := F [pr ](HS (V )) class (H). (3.10) F,H F F H ∈ F Let α: H H be a homomorphism of finite groups. It induces a map 1 → 2 con (H ) con (H ), (h) (α(h)) and thus an F -linear map F 1 → F 2 F 7→ F α : classF (H1) classF (H2). ∗ → If V is a finite-dimensional H1 -representation over F , we conclude from (3.9):

HSF,H (α V ) = α (HSF,H (V )). (3.11) 2 ∗ ∗ 1 Lemma 3.12 Let H be a finite group and F = Q, R or C. Then the Hattori- Stallings rank defines an isomorphism HS : F R (H) ∼= class (H) F,H ⊗Z F −→ F which is natural with respect to induction with respect to group homomorphism α: H H of finite groups. 1 → 2 Proof One easily checks for a finite-dimensional H -representation over F of a finite group H (h) HS (V ) = | | tr (l ). F H H · F h (h) con(H) ∈X | |

29 This explains the relation between the Hattori-Stallings rank and the character of a representation — they contain equivalent information. (We prefer the Hattori-Stallings rank because it behaves better under induction.) We conclude from [28, page 96] that HS (V ) as a function con(H) F is constant on the F H → F -conjugacy classes of elements in H and that HSF,H is bijective. Naturality follows from (3.11). Let class be the covariant functor Or(G; in) F -Mod which sends an ob- F F → ject G/H to class (H). The isomorphisms HS : F R (H) ∼= class (H) F F,H ⊗Z F −→ F yield a natural equivalence of covariant functors from π0(G, X) to F -Mod. Thus we obtain an isomorphism G HS (X): F lim Q∗R F Z Π (G,X) F ⊗ −→ 0 = = ∼ lim Q∗F R ∼ lim Q∗class . (3.13) Π (G,X) Z F Π (G,X) F −→ −→ 0 ⊗ −→ −→ 0 Let f : S S be a map of sets. It extends to an F -linear map F [f]: F [S ] 0 → 1 0 → F [S1]. Suppose that the preimage of any element in S1 is finite. Then we obtain an F -linear map

f ∗ : F [S1] F [S0], s1 s0. (3.14) → 7→ −1 s0 f (s1) ∈X If we view elements in F [S ] as functions S F , then f is given by composing i i → ∗ with f . One easily checks that F [f] f is bijective and that for a second map ◦ ∗ g : S S , for which the preimages of any element in S is finite, we have 1 → 2 2 f g = (g f) . ∗ ◦ ∗ ◦ ∗ Now we can finish the proof of Theorem 3.6 by explaining how assertion (b) is proved.

Proof Let H be a finite group. Let p : con (H) con (H) be the pro- H R → Q jection. If V is a finite-dimensional H -representation over Q, then R V is ⊗Q a finite-dimensional H -representation over R and HSRH (R V ) is the im- ⊗Q age of HSQH (V ) under the obvious map Q[con(H)] R[con(H)]. Recall that → HS (V ) is the image of HSF H (V ) under F [pr ] for pr : con(H) con (H) F,H F F → F the canonical projection; HSF H (V ) is constant on the F -conjugacy classes of elements in H . This implies that the following diagram commutes id HS R ⊗Z Q,H R Z RQ(H) R Q classQ(H) ⊗ −−−−−∼=−−−→ ⊗ ind(H) ind(H)

 HS ,H  R (H) R class(H), R Z y R yR ⊗ −−−∼=−→

30 where the left vertical arrow comes from inducing a Q-representation to a R- representation and the right vertical arrow is ∗ idR Q(pH ) R class (H) = R Q[con (H)] ⊗ R Q[con (H)] = class (H). ⊗Q Q ⊗Q Q −−−−−−−−→ ⊗Q R R Let q(H): class (H) R class (H) R → ⊗Q Q be the map

R[pH ]: R[con (H)] R[con (H)] = R Q[con (H)] R → Q ⊗Q Q The q(H) ind(H) is bijective. ◦ We get natural transformations of functors from Or(G; in) to R-Mod: F ind: R R R R , ⊗Z Q → ⊗Z R ind: R class class , ⊗Q Q → R q : class R class . R → ⊗Q Q Also q ind is a natural equivalence. This implies that ind induces a split ◦ injection on the colimits

lim Q∗ ind: lim Q∗ class lim Q∗class . Π (G;X) Π (G;X) R Q Q Π (G;X) R −→ 0 −→ 0 ⊗ → −→ 0 Since the following diagram commutes and has isomorphisms as vertical arrows G Or(G) R Ze2 (X) Or(G) R H (X; R ) ⊗ R H (X; R ) ⊗Z 0 Q −−−−−−−→ ⊗Z 0 R G R β (X) = R βG(X) = ⊗Z Q ∼ ⊗Z R ∼ x lim Q∗ ind x  Π (G;X)  lim Q R −→ 0 lim Q R Π (G;X) ∗R Z Q Π (G;X) ∗R Z R −→ 0 ⊗ −−−−−−−−−−−→ −→ 0 ⊗ G G R HS (X) = R HS (X) = ⊗Q Q ∼ ⊗Z R ∼  lim Q∗ ind   Π (G;X)  lim Qy class −→ 0 lim yQ class , Π (G;X) ∗R Q Q Π (G;X) ∗ R −→ 0 ⊗ −−−−−−−−−−−→ −→ 0 G the top horizontal arrow is split injective. Hence e2 (X) is rationally split injective. This finishes the proof of Theorem 3.6 (b).

Notation 3.15 Consider g G of finite order. Denote by g the finite cyclic ∈ h i subgroup generated by g. Let y : G/ g X be a G-map. Let F be one of h i → the fields Q, R and C. Define 1 CQ(g) = g0 G, (g0)− gg0 g ; { ∈ 1 ∈ h i}} 1 CR(g) = g0 G, (g0)− gg0 g, g− ; { ∈ 1 ∈ { }} C (g) = g0 G, (g0)− gg0 = g . C { ∈ }

31 Since C (g) is a subgroup of the normalizer N g of g in G and contains g , F h i h i h i we can define a subgroup Z (g) W g by F ⊆ h i Z (g) := C (g)/ g . F F h i Let Z (g) be the intersection of W g (see Notation 2.1) with Z (g), or, F y h iy F equivalently, the subgroup of Z (g) represented by elements g C (g) for F ∈ F which g y(1 g ) and y(1 g ) lie in the same component of X g . · h i h i h i

It is useful to interpret the group CF (g) as follows. Let G act on the set G := g g G, g < by conjugation. Then the projection G con (G) f { | ∈ | | ∞} f → C induces a bijection G G ∼= con (G) and the isotropy group of g G is \ f −→ C ∈ f C (g). Let G / 1 be the quotient of G under the 1 -action given by C f {§ } f {§ } g g 1 . The conjugation action of G on G induces an action on G / 1 . 7→ − f f {§ } The projection G con (G) induces a bijection G (G/ 1 ) ∼= con (G) → R \ {§ } −→ R and the isotropy group of g 1 G / 1 is C (g). The set con (G) · {§ } ∈ f {§ } R Q is the same as the set (L) consub(G) L finite cyclic . For g G the { ∈ | } ∈ f group C (g) agrees with N g and is the isotropy group of g in L G Q h i h i { ⊆ | L finite cyclic under the conjugation action of G. The quotient of L G } { ⊆ | L finite cyclic under the conjugation action of G is by definition con (G) = } Q (L) consub(G) L finite cyclic . { ∈ | } Consider (g) con (G). For the sequel we fix a representative g (g) . F ∈ F ∈ F Consider an object of Π (G, X) of the special form y : G/ g X . Let 0 h i → x: G/H X be any object of Π (G, X). Recall that W g and thus the → 0 h iy subgroup ZF (g)y act on mor(y, x). Define α (y, x): Z (g) mor(y, x) con (H) F F y\ → F by sending sending Z (g) σ for a morphism σ : y x, which given by a F y · → G-map σ : G/ g G/H , to (σ(1 g ) 1gσ(1 g )) . We obtain a map of sets h i → h i − h i F

αF (x) = a(y(C), x):

(g)F con(G)R ZF (g) C ∈a a· ∈hgi ZF (g) π0(X ) \ Z (g) mor(y(C), x) ∼= con (H), (3.16) F y(C)\ −→ F (g)F con(G)F ZF (g) C ∈a a· ∈hgi ZF (g) π0(X ) \ g g where we fix for each ZF (g) C ZF (g) π0(Xh i) a representative C π0(Xh i) · ∈ \ g ∈ and y(C) is a fixed morphism y(C): G/ g X such that X h i(y) = C in g h i → π0(Xh i). The map αF (x) is bijective by the following argument.

32 Consider (h)F conF (H). Let g G be the representative of the class (g)F ∈ ∈ 1 for which (g)F = (h)F holds in conF (G). Choose g0 G with g0− gg0 H 1 ∈ ∈ and (g− gg ) = (h) in con (H). We get a G-map R : G/ g G/H 0 0 F F F g0 h i → by mapping g g to g g H . Let y = y(C): G/ g X be the object chosen 0h i 0 0 h i → above for the fixed representative C of Z (g) X g (x R ) Z (g) π (X g ). F · h i ◦ g0 ∈ F F \ 0 h i Choose g Z (g) such that the G-map R : G/ g G/ g sending g g 1 ∈ F g1 h i → h i 0h i to g g g defines a morphism R : y x R in Π (G, X). Then σ := 0 1h i g1 → ◦ g0 0 R R : y x is a morphism such that g0 ◦ g1 → 1 (σ(1 g )− gσ(1 g ) = (h) h i h i F F holds in conF (H). This shows that α(x) is surjective.

Consider for i = 0, 1 elements (g ) con (G), Z (g ) C Z (g ) π (X gi ) i ∈ F F i · i ∈ F i \ 0 h i and Z (g ) σ Z (g ) mor(y , x) for y = y(C ) such that F i yi · i ∈ F i yi \ i i i 1 1 (σ (1 g )− g σ (1 g )) = (σ (1 g )− g σ (1 g )) 0 h 0i 0 0 h 0i F 1 h 1i 1 1 h 1i F holds in conF (H). So we get two elements in the source of αF (x) which are mapped to the same element under αF (x). We have to show that these elements in the source agree. Choose gi0 G such that σi is given by sending g00 gi 1 ∈ 1 h i to g00gi0H . Then ((g0)− g0g0)F and ((g10 )− g1g10 )F agree in conF (H). This implies (g0)F = (g1)F in con(G)F and hence g0 = g1 . In the sequel we write 1 1 g = g0 = g1 . Since ((g0)− gg0)F and ((g10 )− gg10 )F agree in conF (H), there exists h H with ∈ 1 (g10 )− gg10 if F = Q; 1 1 ∈ h 1 i 1 1 h− (g0)− gg0h (g10 )− gg10 , (g10 )− g− g10 if F = R;  ∈ { 1 }  = (g10 )− gg10 if F = C.

We can assume withoutloss of generality that h = 1, otherwise replace g0 by g h. Put g := g (g ) 1 . Then g is an element in Z (g). Let σ : G/ g 0 2 0 10 − 2 F 2 h i → G/ g be the G-map which sends g g to g g g . We get the equality of h i 00h i 00 2h i G-maps σ = σ σ . Since σ is a morphism y x for i = 0, 1, we conclude 0 1 ◦ 2 i i → g x(1H) X g (y ) for i = 0, 1. This implies that g X g (y ) = X g (y ) in i0 · ∈ h i i 2 · h i 1 h i 0 π (X g ). This shows Z (g) X g (y ) = Z (g) X g (y ) in Z (g) π (X g ) 0 h i F · h i 0 F · h i 1 F \ 0 h i and hence y0 = y1 . Write in the sequel y = y0 = y1 . The G-map σ2 defines a morphism σ : y y. We obtain an equality σ = σ σ of morphisms y x. 2 → 0 1 ◦ 2 → We conclude Z (g) σ = Z (g) σ in Z (g) π (X g ). This finishes the F y · 0 F y · 1 F y\ 0 h i proof that α(x) is bijective.

Let con be the covariant functor from Or(G; in) to the category of finite F F sets which sends an object G/H to conF (H). The map αF (x) is natural in

33 x: G/H X , in other words, we get a natural equivalence of functors from → Π0(G, X) to the category of finite sets. We obtain a bijection of sets

αF : lim ZF g y mor(y, x) x: G/H X Π0(G,X) −→ → ∈ h i \ (g)F conF (G) ZF g C ∈a ah i ∈hgi ZF g π0(X ) h i\ = ∼ lim Q∗con . Π (G,X) F −→ −→ 0 One easily checks that lim ZF g y mor(y, x) consists of one x: G/H X Π0(G,X) h i \ element, namely, the one−→represen→ted∈by Z g id Z g mor(y, y). Thus F h iy · y ∈ F h iy\ we obtain a bijection G g = α (X): Z g π (Xh i) ∼ lim Q∗con , F F 0 Π (G,X) F h i\ −→ −→ 0 (g)F conF (G) ∈a which sends an element Z g C in Z g π (X g ) to the class in the colimit F h i · F h i\ 0 h i represented by Z g id in Z g mor(y, y) for any object y : G/ g X F h iy · y F h iy\ h i → for which Z (g) X g (y) = Z (g) C holds in Z (g) π (X g ). It yields an F · h i F · F \ 0 h i isomorphism of F -vector spaces denoted in the same way

G ∼= αF (X): F lim Q∗classF . (3.17) −→ Π0(G,X) hgi −→ (g)F conF (G) ZF (g) π0(X ) ∈M M\ Let us consider in particular the case F = Q. Recall that consub(G) is the set of conjugacy classes (H) of subgroups of G. Then conQ(G) is the same as the set (L) consub(G) L finite cyclic and Z (g) agrees with W g . Thus { ∈ | } Q h i (3.17) becomes an isomorphism of Q-vector spaces G = α (X): Q ∼ lim Q∗class . (3.18) Q Π0(G,X) Q L −→ −→ (L) consub(G) WL π0(X ) L finite∈ Mcyclic \M Denote by

pr: Q Q H → L (H) consub(G) WH π0(X ) (L) consub(G) WL π0(X ) ∈ M \M L finite∈ Mcyclic \M the obvious projection. Let G D (X): Q Q (3.19) L → L (L) consub(G) WL π0(X ) (L) consub(G) WL π0(X ) L finite∈ Mcyclic \M L finite∈ Mcyclic \M Gen(L) be the automorphism (L) consub(G) | L | id, where Gen(L) is the set of L finite∈ cyclic | | · generators of L. L

34 G G G 1 G G Recall the isomorphisms ch (X), βQ (X), HSQ(X)− , αQ(X) and D (X) from (2.6), (3.2), (3.13) (3.18) and (3.19). We define

G ∼= Or(G) γQ (X): Q Q Z H0 (X; RQ) (3.20) L −→ ⊗ (L) consub(G) WL π0(X ) L finite∈ Mcyclic \M to be the composition γG(X) := βG(X) HSG(X) 1 αG(X) DG(X). Q Q ◦ Q − ◦ Q ◦ Theorem 3.21 The following diagram commutes

G idQ Ze1 (X) Or(G) Q U G(X) ⊗ Q H (X; R ) ⊗Z −−−−−−−−→ ⊗Z 0 Q id chG(X) = γG(X) = Q ⊗Z ∼ Q ∼  pr x H L (H) consub(G)  WH π0(X ) Q (L) consub(G) WL π0(X ) Q ∈ y \ −−−−→ L finite∈ cyclic  \ L L L L and has isomorphisms as vertical arrows. G G The element idQ Ze1 (X)(χ (X)) agrees with the image under the isomor- phism G of the elemen⊗ t γQ χQWLC (C) (L) consub(G), L finite cyclic, WL C WL π (XL) { | ∈ · ∈ \ 0 } given by the various orbifold Euler characteristics of the WLC -CW -complexes C , where WL is the isotropy group of C π (XL) under the WL-action. C ∈ 0 Proof It suffices to prove the commutativity of the diagram above, then the rest follows from Lemma 2.8. Recall that U G(X) is the free abelian group generated by the set of isomorphism classes [x] of objects x: G/H X . Hence it suffices to prove for any G-map → x: G/H X → αG(X) DG(X) pr chG(X) ([x]) Q ◦ ◦ ◦ G G 1 G = HS (X) β (X)− e (X)([x]). (3.22) ¡ ¢ ◦ ◦ 1 Given a finite cyclic subgroup L G and a component C π (XL) the element ⊆ ∈ 0 DG(X) pr chG(X) ([x]) has as entry in the summand belonging to (L) and ◦ ◦ WL C WL π (XC ) the number ¡ · ∈ \ 0 ¢ Gen(L) | | , L (W C )σ WL σ y(C) y(C)· ∈ | | · | | WL Xmor(y(C),x) y(C)\

35 L where y(C): G/L X is some object in Π0(G, X) with X (y) = C in L → π0(X ).

Recall the bijection αF (x) from (3.16). In the case F = Q it becomes the map

αQ(x): WLy(C) mor(y(C), x) L \ (L) consub(G) WL π0(X ) L finite∈ Xcyclic \a ∼= con (H) = (K) consub(H) L cyclic −→ Q { ∈ | } which sends WL σ WL π (XL) to (σ(1L) 1Lσ(1L)). Let · ∈ \ 0 − u class (H) [x] ∈ Q Gen(L) be the element which assigns to (K) con (H) the number | | if Q L (W C )σ ∈ | |·| y(C) | σ mor(y(C), x) represents the preimage of (K) under the bijection α (x). ∈ Q We conclude that αG(X) DG(X) pr chG(X) ([x]) is given by the image Q ◦ ◦ ◦ under the structure³map associated to the object x´: G/H X → class (H) lim Q∗class Q Π (G,X) Q → −→ 0 of the element u class (H) above. [x] ∈ Q Consider (K) con (H). Let σ mor(y(C), x) represent the preimage of (K) ∈ Q ∈ under the bijection α (x). Choose g such that σ : G/ g G/H is given by Q 0 h i → g g g g H . Let N K be the normalizer in H and W K := N K/K be 00h i 7→ 00 0 H H H the Weyl group of K H . Define a bijection ⊆ = 1 1 1 f : (WL ) ∼ W ((g0)− Lg0), g00L (g0)− g00g0 (g0)− Lg0. y(C) σ −→ H 7→ · The map is well-defined because of 1 (WL ) = g00L WL (g0)− g00g0 H y(C) σ { ∈ y(C) | ∈ } and the following calculation

1 1 1 1 1 1 1 1 (g0)− g00g0 − (g0)− Lg0(g0)− g00g0 = (g0)− (g00)− g0(g0)− Lg0(g0)− g00g0 1 1 1 ¡ ¢ = (g0)− (g00)− Lg00g0 = (g0)− Lg0. One easily checks that f is injective. Consider h (g ) 1Lg in W ((g ) 1Lg ). · 0 − 0 H 0 − 0 Define g = g h(g ) 1 . We have g WL. Since h x(1H) = x(1H), we get 0 0 0 − 0 ∈ · g L WL . Hence g L is a preimage of h (g ) 1Lg under f . Hence f is 0 ∈ y 0 · 0 − 0 bijective. This shows 1 (WL ) = W ((g0)− Lg0) . | y(C) σ| | H |

36 We conclude that u[x] is the element Gen(K) con (H) = (K) consub(H) K finite cyclic Q, (K) | | . Q { ∈ | } → 7→ K W K | | · | H | 1 Since right multiplication with H − h H h induces an idempotent QH - | | · ∈ linear map QH QH whose image is Q with the trivial H -action, the element → P HS ,H ([Q]) class (H) is given by Q ∈ Q 1 con (H) Q, (K) h H h (K) . Q → 7→ H · |{ ∈ | h i ∈ }| | | From

h H h (K) = Gen(K0) |{ ∈ | h i ∈ }| ¯ ¯ ¯K0 H,K0 (K) ¯ ¯ ⊆ a ∈ ¯ ¯ ¯ = ¯ K0 H, K0 (K) ¯ Gen(K) ¯{ ⊆ ∈ } ¯· | | H = ¯ | | Gen(K) ¯ ¯N K · | | ¯ | H | H Gen(K) = | | · | | K W K | | · | H | we conclude

u[x] = HS ,H ([Q]) class (H). Q ∈ Q Now (3.22) and hence Theorem 3.21 follow.

4 Examples

In this section we discuss some examples. Recall that we have described the non-equivariant case in the introduction.

4.1 Finite groups and connected non-empty fixed point sets

Next we consider the case where G is a finite group, M is a closed compact G-manifold, and M H is connected and non-empty for all subgroups H G. ⊆ Let i: M be the G-map given by the inclusion of a point into M G . {∗} → Since we have assumed that M G is connected, i is unique up to G-homotopy.

37 Let A(G) be the Burnside ring of formal differences of finite G-sets (Definition 2.9). We have the following commutative diagram:

U G(i) f U G(M) U G( ) 1 A(G) ←−∼=−−− {∗} ←−∼=−−− G G e (M) e ( ) j1 1 1 {∗} Or  H (G)(i;R )   Or(G) 0 Q Or(G)  f2  H0 y(M; RQ) H0 (y ; RQ) RQy(G) ←−−−−∼=−−−−− {∗} ←−∼=−−− G G e (M) e ( ) j2 2 2 {∗} Or  H (G)(i;R )   Or(G) 0 R Or(G)  f3  H0 y(M; RR) H0 (y ; RR) RRy(G) ←−−−∼=−−−−− {∗} ←−∼=−−− G G e (M) e ( ) = j3 = 3 3 {∗} ∼ ∼  G   G KO0 (i) G id G KO0y(M) KO0y( ) KO0y( ), ←−−−−− {∗} ←−∼=−−− {∗} where j1 sends the class of a G-set S in the Burnside ring A(G) to the class of the rational G-representation Q[S] and j2 is the change-of-coefficients ho- momorphism. The homomorphism KOG(i): KOG( ) KOG(M) is split 0 0 {∗} → 0 injective, with a splitting given by the map KOG(pr) induced by pr: M . 0 → {∗} The map eG( ) is bijective since the category Π (G, ) has G as 3 {∗} 0 {∗} → {∗} terminal object. This implies that eG(M): HOr(G)(M; R ) KOG(M) 3 0 R → 0 is split injective. We have already explained in the Section 3 that the map j : R (G) R (G) is rationally injective. Since R (G) is a torsion-free 2 Q → R Q finitely generated abelian group, j2 is injective. Hence eG(M): HOr(G)(M; R ) HOr(G)(M; R ) 2 0 Q → 0 R G G is injective. The upshot of this discussion is, that e1 (χ (M)) carries (inte- grally) the same information as EulG(M) because it is sent to EulG(M) by the injective map eG(M) eG(M). 3 ◦ 2 G G G Analyzing the difference between e1 (χ (M)) and χ (M) is equivalent to an- G alyzing the map j1 : A(G) RepQ(G), which sends χ (M) to the element p → G given by p 0( 1) [Hp(M; Q)]. Recall that χ (M) A(G) is given by ≥ − · ∈ χG(M) =P χ(WH M H , WH M >H ) G/H] = ( 1)p ] (G/H), \ \ · − · p (H) consub(G) p 0 ∈ X X≥ where χ(WH M H , WH M >H )) is the non-equivariant Euler characteristic and \ \ ] (G/H) is the number of equivariant cells of the type G/H Dp appearing in p ×

38 some G-CW -complex structure on M . The following diagram commutes (see Theorem 3.21)

j A(G) 1 R (G) −−−−→ Q G ch HSQ,G   (H) consub(G) (H) consub( H) Q pr  Q ∈ y −−−−→ H cyclicy Q Q where pr is the obvious projection, chG(S) has as entry for (H) consub(G) 1 H ⊆ the number WH S and HSQ,G(V ) has as entry at (H) consub(G) for | | · | | ∈ trQ(lh) cyclic H G the number , where tr(lh) Q is the trace of the endo- ⊆ WH ∈ morphism of the rational vector| | space V given by multiplication with h for some generator h H . The vertical arrows chG and HS are rationally bi- ∈ Q,G jective and χG(M)(χG(M)) has as component belonging to (H) consub(G) H ∈ QWH H χ(M ) the number χ (M ) = WH (see Lemma 2.8 and Lemma 3.12). This G | | implies that ch and HSQ,G are injective because their sources are torsion-free finitely generated abelian groups. Moreover, χG(M) A(G) carries integrally ∈ the same information as all the collection of Euler characteristics χ(M H ) G p { | (H) consub(H) , whereas j1(χ (M)) = ( 1) [Hp(M; Q)] carries ∈ } p 0 − · integrally the same information as the collection≥of the Euler characteristics P χ(M H ) (H) consub(H), H cyclic . In particular j is injective if and only { | ∈ } 1 if G is cyclic. But any element u in A(G) occurs as χG(M) for a closed smooth G-manifold M for which M H is connected and non-empty for all H G (see ⊆ [20, Section 7]). Hence, given a finite group G, the elements χG(M) and G j1(χ (M)) carry the same information for all such G-manifolds M if and only if G is cyclic. From this discussion we conclude that EulG(M) does not carry torsion infor- mation in the case where G is finite and M H is connected and non-empty for all H G, since R (G) is a torsion-free finitely generated abelian group. This ⊆ R is different from the case where one allows non-connected fixed point sets, as the following example shows.

4.2 The equivariant Euler class carries torsion information

Let S3 be the symmetric group on 3 letters. It has the presentation 2 3 1 S = s, t s = 1, t = 1, sts = t− . 3 h | i

39 Let R be the trivial 1-dimensional real representation of S3 . Denote by R− the one-dimensional real representation on which t acts trivially and s acts by id. Denote by V the 2-dimensional irreducible real representation of S3 ; we − 2 can take R as the underlying real vector space of V , with s (r1, r2) = (r2, r1) · and t (r1, r2) = ( r2, r1 r2). Then R, R− and V are the irreducible real · − − representations of S3 , and RS3 is as an RS3 -module isomorphic to R R− ⊕ ⊕ V V . Let L be the cyclic group of order two generated by s and let L ⊕ 2 3 be the cyclic group of order three generated by t. Any finite subgroup of S3 is conjugate to precisely one of the subgroups L1 = 1 , L2 , L3 or S3 . One L2 L3 L2 { } L3 easily checks that V ∼= R, V ∼= 0, (R−) = 0 and (R−) ∼= R as real S3 L2 L3 2 vector spaces. Put W = R R− V . Then W ∼= R, W ∼= W ∼= R and 4 ⊕ ⊕ W ∼= R as real vector spaces. Let M be the closed 3-dimensional S3 -manifold SW . Then 3 M ∼= S ; L2 1 M ∼= S ; L3 1 M ∼= S ; S3 0 M ∼= S . Since χ(M S3 ) = 0, χG(M) U G(M) cannot vanish. But since the fixed sets 6 ∈ for all cyclic subgroups have vanishing Euler characteristic, Theorem 0.9 implies G G that Eul (M) is a torsion element in KO0 (M). We want to show that it has order precisely two.

Let xi : S3/Li X for i = 1, 2, 3 be a G-map. Let x : S3/S3 X and → − → x : S /S X be the two different G-maps for which M S3 is the union of + 3 3 → the images of x and x+ . Then x1 , x2 , x3 , x and x+ form a complete set of − − representatives for the isomorphism classes of objects in Π0(S3, M). Notice for i = 1, 2, 3 that mor(xi, x ) and mor(xi, x+) consist of precisely one element. − Therefore we get an exact sequence

i2 i3  i i  − 2 − 3 R (Z/2) R (Z/3)   R (G) R (G) R ⊕ R −−−−−−−−−−−→ R ⊕ R s−+s+ lim Q∗R 0, (4.1) Π (S ,M) R −−−−→ −→ 0 3 → where i2 : R (Z/2) R (S3) and i3 : R (Z/3) R (S3) are the induction R → R R → R homomorphisms associated to any injective group homomorphism from Z/2 and Z/3 into S3 and s and s+ are the structure maps of the colimit belonging − to the objects x and x+ . Define a map − δ : R (G) Z/2, λ [R] + λ − [R−] + λV [V ] λ + λ − + λV . R → R · R · · 7→ R R

40 If R denotes the trivial and R− denotes the non-trivial one-dimensional real Z/2-representation, then

i2 : R (Z/2) R (S3), R → R µ [R] + µ − [R−] µ [R] + µ − [R−] + (µ + µ − ) [V ]. R · R · 7→ R · R · R R · If R denotes the trivial one-dimensional and W the 2-dimensional irreducible real Z/3-representation, then i3 : R (Z/3) R (S3) µ [R] + µW [W ] µ [R] + µ [R−] + 2µW [V ]. R → R R · · 7→ R · R · · This implies that the following sequence is exact

i2+i3 δ R (Z/2) R (Z/3) R (G) Z/2 0. (4.2) R ⊕ R −−−→ R →− → We conclude from the exact sequences (4.1) and (4.2) above that the epimor- phism s + s : R (G) R (G) lim Q∗R + R R Π (S ,M) R − ⊕ → −→ 0 3 factorizes through the map 1 1 u := : R (G) R (G) R (G) Z/2 0 δ R ⊕ R → R ⊕ µ ¶ to an isomorphism = v : R (G) /2 ∼ lim Q∗R . (4.3) R Z Π (S ,M) R ⊕ −→ −→ 0 3 Define a map S f : U 3 (M) R (G) Z/2 → R ⊕ by

f([x1]) := = ([R[S3]], 0); f([x2]) := = ([R[S3/L2]], 0); f([x3]) := = ([R[S3/L3]], 0); f([x ]) := = ([R], 0); − f([x+]) := = ([R], 1).

The reader may wonder why f does not look symmetric in x and x+ . This − comes from the choice of u which affects the isomorphism v. The composition

eG(M) eG(M) U G(M) 1 HOr(G)(M; R ) 2 HOr(G)(M; R ) −−−−→ 0 Q −−−−→ 0 R agrees with the composition

βG(M) G f v R Or(G) U (M) R (G) /2 lim Q∗R H (M; R ). R Z Π (S ,M) R 0 R −→ ⊕ −→ −→ 0 3 −−−−→

41 We get G G χ (M) = [x+] + [x ] 2 [x2] [x3] + [x1] U (M) (4.4) − − · − ∈ since the image of the element on the right side and the image of χG(M) under the injective character map chG(M) (see (2.6) and Lemma 2.8) agree by the following calculation

S3 S3 L2 G G χ(M (x )) χ(M (x+)) χ(M ) ch (χ (M)) = − [x ] + [x+] + [x2] (WS ) · − (WS ) · WL · | 3 x− | | 3 x+ | | 2| χ(M L3 ) χ(M) + [x3] + [x1] WL3 · WL1 · = [x ] + [x+] − G = ch (M)([x+] + [x ] 2 [x2] [x3] + [x1]). − − · − Now one easily checks G f(χ (M)) = (0, 1) R (S3) Z/2. (4.5) ∈ R ⊕ βG(M) Since R (G) /2 v lim Q R R HOr(G)(M; R ) is a composi- R Z Π (S ,M) ∗ R 0 R ⊕ −→ −→ 0 3 −−−−→ tion of isomorphisms, we conclude that eG(M) eG(M)(χG(M)) is an element 2 ◦ 1 of order two in HOr(G)(M; R ). Since dim(M) 4 and dim(M >1) 2, we 0 R ≤ ≤ conclude from Theorem 3.6 (d) that eG(M): HOr(G)(M; R ) KOG(M) is 3 0 R → 0 injective. We conclude from Theorem 0.3 and Theorem 3.6 (a) that EulG(M) G ∈ KO0 (M) is an element of order two as promised in Theorem 0.10.

4.3 The equivariant Euler class is independent of the stable equivariant Euler characteristic

In this subsection we will give examples to show that EulG(M) is independent of the stable equivariant Euler characteristic with values in the Burnside ring A(G), in the sense that it is possible for either one of these invariants to vanish while the other does not vanish. For the first example, take G = Z/p cyclic of prime order, so that G has only two subgroups (the trivial subgroup and G itself) and A(G) has rank 2. We will see that it is possible for EulG(M) to be non-zero, even rationally, while G χs (M) = 0 in A(G) (see Definition 2.10). To see this, we will construct a closed 4-dimensional G-manifold M with χ(M) = 0 and such that M G has two components of dimension 2, one of which is S2 and the other of which is a N 2 of 2, so that χ(N) = 2. Then − χ(M G) = χ(S2 N 2) = χ(S2) + χ(N 2) = 2 2 = 0, q −

42 1 G while also χ(M { }) = χ(M) = 0, so that χs (M) = 0 in A(G) and hence also G χa (M) = 0. For the construction, simply choose any bordism W 3 between S2 and N 2 , and let 4 2 2 3 1 2 2 M = S D S2 S1 W S N 2 S1 N D . × ∪ × × ∪ × × ¡ ¢ ¡ ¢ ¡ ¢ We give this the G-action which is trivial on the S2 , W 3 , and N 2 factors, and which is rotation by 2π/p on the D2 and S1 factors. Then the action of G is free except for M G , which consists of S2 0 and of N 2 0 . Furthermore, × { } × { } we have χ(M) = χ S2 D2 χ S2 S1 + χ W 3 S1 × − × × χ N 2 S1 + χ N 2 D2 ¡ − ¢ × ¡ ¢ × ¡ ¢ = 2 0 + 0 0 2 = 2 2 = 0. − ¡ − − ¢ −¡ ¢ G G Thus χ(M) = χ(M ) = 0 and χs (M) = 0 in A(G). On the other hand, EulG(M) is non-zero, even rationally, since from it (by Theorem 0.9) we can recover the two (non-zero) Euler characteristics of the two components of M G .

For the second example, take G = S3 and retain the notation of Subsection 4.2. By [20, Theorem 7.6], there is a closed G-manifold M with M H connected for each subgroup H of G, with χ(M H ) = 0 for H cyclic, and with χ(M G) = 0. 6 (Note that G is the only noncyclic subgroup of G.) In fact, we can write down such an example explicitly; simply let Q = W 0 R R, the S3 -representation ⊕ ⊕ R R R R− V , and let M = SW 0 be the unit ball in W 0 . Then each ⊕ ⊕ ⊕ ⊕ fixed set in M is a sphere of dimension bigger by 2 than in the example of 5 L2 3 L3 3 G 2 4.2, so M ∼= S , M ∼= S , M ∼= S , and M ∼= S . Since the fixed sets are all connected and each fixed set of a cyclic subgroup has vanishing Euler characteristic, it follows by Subsection 4.1 that EulG(M) = 0. On the other hand, since χ(M G) = 2, χG(M) = 0 in A(G). s 6

4.4 The image of the equivariant Euler class under the assembly maps

Now let us consider an infinite (discrete) group G. Let EG be a model for the for proper G-actions, i.e., a G-CW -complex EG such that EGH is contractible (and in particular non-empty) for finite H G and ⊆ EGH is empty for infinite H G. It has the universal property that for any ⊆ proper G-CW -complex X there is up to G-homotopy precisely one G-map X EG. This implies that all models for EG are G-homotopy equivalent. If →

43 G is a word-, its Rips complex is a model for EG [21]. If G is a discrete subgroup of the L with finitely many path components, then L/K for a maximal compact subgroup K with the (left) G-action is a model for EG (see [1, Corollary 4.14]). If G is finite, is a model for EG. {∗} Consider a proper smooth G-manifold M . Let f : M EG be a G-map. We → obtain a commutative diagram

U G(f) U G(M) U G(EG) id U G(EG) −−−−→ −−−−→ G G e1 (M) e1 (EG) j1

Or  H (G)(f;R )   Or(G) 0 Q Or(G)  asmb1  H y(M; R ) H (yEG; R ) K0(yQG) 0 Q −−−−−−−−−→ 0 Q −−−−→ G G e2 (M) e2 (EG) j2

Or  H (G)(f;R )   Or(G) 0 R Or(G)  asmb2  H y(M; R ) H (yEG; R ) K0(yRG) 0 R −−−−−−−−−→ 0 R −−−−→ G G e3 (M) e3 (EG) j3  G   G KO0 (f) G asmb3 KO(M) KO (EG) KO0(C∗(G; R)) 0y −−−−−→ 0y −−−−→ yr Here asmbi for i = 1, 2, 3 are the assembly maps appearing in the Farrell-Jones Isomorphism Conjecture and the Baum-Connes Conjecture. The maps asmbi for i = 1, 2 are the obvious maps lim R K (F G) for F = , Or(G; in) F 0 Q R −→ F → under the identifications R (H) = K (F H) for finite H G and βG(M) of F 0 ⊆ F (3.2). The Baum-Connes assembly map is given by the index with values in the reduced real group C∗ -algebra Cr∗(G; R). The Farrell-Jones Isomorphism Conjecture and the Baum-Connes Conjecture predict that asmbi for i = 1, 2, 3 are bijective. The abelian group U G(EG) is the free abelian group with (H) { ∈ consub(G) H < as basis and the map j sends the basis element (H) to | | | ∞} 1 the class of the finitely generated projective QG-module Q[G/H]. The maps G G j2 and j3 are change-of-rings homomorphisms. The maps e2 (M) and e3 (M) are rationally injective (see Theorem 3.6 (b) and (c)). If M H is connected and non-empty for all finite subgroups H G, then the horizontal arrows U G(f), Or(G) Or(G) ⊆ H0 (f; RQ) and H0 (f; RR) are bijective. In contrast to the case where G is finite, the groups K0(QG), K0(RG) and KO0(Cr∗(G; R)) may not be torsion free. The problem whether asmbi is bijec- tive for i = 1, 2 or 3 is a difficult and in general unsolved problem. Moreover, EG is a complicated G-CW -complex for infinite G, whereas for a finite group G we can take as a model for EG. {∗}

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