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

Wolfgang Lück Jonathan Rosenberg

Institut für Mathematik und Informatik Westfälische Wilhelms-Universtität Einsteinstr. 62 48149 Münster, Germany and Department of Mathematics University of Maryland College Park, MD 20742, USA

        Email: lueck@math.uni-muenster.de and jmr@math.umd.edu
      URL: wwwmath.uni-muenster.de/u/lueck, www.math.umd.edu/~jmr

Abstract

Let G be a countable discrete group and let M be a smooth proper cocompact G-manifold without boundary. The Euler operator defines via Kasparov theory an element, called the equivariant Euler class, in the equivariant KO-homology of M. The universal equivariant Euler characteristic 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 orbifold 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 S_{3} on the 3-sphere for which the equivariant Euler class has order 2, so there is also some torsion information.

AMS Classification numbers Primary: 19K33 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

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-invariant Riemannian metric, the Euler characteristic operator defines via Kasparov theory an element, the equivariant Euler class, in the equivariant real K -homology group of M

Eul^{G}(M) ∈ KO_{0}^{G}(M).  (0.1)

The Euler characteristic operator is the minimal closure, or equivalently, the maximal closure, of the densely defined operator

(d + d^{∗}): Ω^{∗}(M) ⊆ L^{2}Ω^{∗}(M) → L^{2}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 different grading coming from the Hodge star operator. The signature operator also defines an element

Sign^{G}(M) ∈ K_{0}^{G}(M),

which carries a lot of geometric information about the action of G on M. (Rationally, when G = {1}, Sign(M) is the Poincaré dual of the total Lclass, 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 Sign^{G}(M) for G finite may be found in [26] and [24].)

We want to study how much information Eul^{G}(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: ⨁_{π_{0}(M)} ℤ = ⨁_{π_{0}(M)} KO_{0}({∗}) → KO_{0}(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_{0}(M), and sends {χ(C) | C ∈ π_{0}(M)} to Eul(M). Hence Eul(M) carries precisely the 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 Eul^{G}(M) is determined by the universal equivariant Euler characteristic (see Definition 2.5)

χ^{G}(M) ∈ U^{G}(M) = ⨁_{(H)∈consub(G)} ⨁_{WH\π_{0}(M^{H})} ℤ.

Here and elsewhere consub(G) is the set of conjugacy classes of subgroups of G and NH = {g ∈ G | g^{−1}Hg = H} is the normalizer of the subgroup H ⊆ G and WH := NH/H is its Weyl group. The component of χ^{G}(M) associated to (H) ∈ consub(G) and WH · C ∈ WH\π_{0}(M^{H}) is the (ordinary) Euler characteristic χ(WH_{C}\(C, C ∩ M^{>H})), where WH_{C} is the isotropy group of C ∈ π_{0}(M^{H}) under the WH -action. There is a natural homomorphism

e^{G}(M): U^{G}(M) → KO_{0}^{G}(M).  (0.2)

It sends the basis element associated to (H) ⊆ consub(G) and WH · C ∈ WH\π_{0}(M^{H}) to the image of the class of the trivial H -representation ℝ under the composition

R_{ℝ}(H) = KO_{0}^{H}({∗}) ⟶^{(α)_{∗}} KO_{0}^{G}(G/H) ⟶^{KO_{0}^{G}(x)} KO_{0}^{G}(M),

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

e^{G}(M)(χ^{G}(M)) = Eul^{G}(M).

The proof of Theorem 0.3 involves two independent steps. Let Ξ be an equivariant 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_{x}: T_{x}M → T M induce an isomorphism of G_{x}-representations

T_{x}i ⊕ T_{0}j_{x}: T_{x}M ⊕ T_{x}M ⟶^{≅} T_{i(x)}(T M)

if we identify T_{0}(T_{x}M) = T_{x}M in the obvious way. If pr_{i} denotes the projection onto the i-th factor for i = 1, 2 we obtain a linear G_{x}-equivariant isomorphism

d_{x}Ξ: T_{x}M ⟶^{T_{x}Ξ} T_{i(x)}(T M) ⟶^{(T_{x}i⊕T_{x}j_{x})^{−1}} T_{x}M ⊕ T_{x}M ⟶^{pr_{2}} T_{x}M. (0.4)

Notice that we obtain the identity if we replace pr_{2} by pr_{1} in the expression (0.4) above. One can even achieve that Ξ is canonically transverse to the zerosection, i.e., it is transverse to the zero-section and d_{x}Ξ induces the identity on T_{x}M/(T_{x}M)^{G_{x}} for G_{x} 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

s(Ξ, x) = det ((d_{x}Ξ)^{G_{x}}: (T_{x}M)^{G_{x}} → (T_{x}M)^{G_{x}}) ; |det ((d_{x}Ξ)^{G_{x}}: (T_{x}M)^{G_{x}} → (T_{x}M)^{G_{x}})| ∈ {±1}.

For x ∈ M let α_{x}: G_{x} → G be the inclusion, (α_{x})_{∗}: R_{ℝ}(G_{x}) = KO_{0}^{G_{x}}({∗}) → KO_{0}^{G}(G/G_{x}) be the map induced by induction via α_{x} and let x: G/G_{x} → M 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 countable 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

Eul^{G}(M) = ∑_{Gx∈G\ Zero(Ξ)} s(Ξ, x) · KO_{0}^{G}(x) ∘ (α_{x})_{∗}([ℝ]),

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

In the second step one has to prove

e^{G}(M)(χ^{G}(M)) = ∑_{Gx∈G\ Zero(Ξ)} s(Ξ, x) · KO_{0}^{G}(x) ∘ (α_{x})_{∗}([ℝ]). (0.6)

This is a direct conclusion of the equivariant Poincaré-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

χ^{G}(M) = i^{G}(Ξ).  (0.7)

where i^{G}(Ξ) is the equivariant index of the vector field Ξ defined in [20, (6.5)]. Since we get directly from the definitions

e^{G}(M)(i^{G}(Ξ)) = ∑_{Gx∈G\ Zero(Ξ)} s(Ξ, x) · KO_{0}^{G}(x) ∘ (α_{x})_{∗}([ℝ]), (0.8)

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.

We will factorize e^{G}(M) as

e^{G}(M): U^{G}(M) ⟶^{e_{1}^{G}(M)} H_{0}^{Or(G)}(M; \underline{R_{ℚ}}) ⟶^{e_{2}^{G}(M)} H_{0}^{Or(G)}(M; \underline{R_{ℝ}})

⟶^{e_{3}^{G}(M)} KO_{0}^{G}(M),

where H_{0}^{Or(G)}(M; \underline{R_{F}}) is the Bredon homology of M with coefficients in the coefficient system which sends G/H to the representation ring R_{F} (H) for the field F = ℚ, ℝ. We will show that e_{2}^{G}(M) and e_{3}^{G}(M) are rationally injective (see Theorem 3.6). We will analyze the map e_{1}^{G}(M), which is not rationally injective in general, in Theorem 3.21.

The rational information carried by Eul^{G}(M) can be expressed in terms of orbifold Euler characteristics of the various components of the L-fixed point sets for all finite cyclic subgroups L ⊆ G. For a component C ∈ π_{0}(M^{H}) denote by WH_{C} its isotropy group under the WH -action on π_{0}(M^{H}). For H ⊆ G finite WH_{C} acts properly and cocompactly on C and its orbifold Euler characteristic (see Definition 2.5), which agrees with the more general notion of L^{2}-Euler characteristic,

χ^{ℚWH_{C}}(C) ∈ ℚ,

is defined. Notice that for finite WH_{C} the orbifold Euler characteristic is given in terms of the ordinary Euler characteristic by

χ^{ℚWH_{C}}(C) = \frac{χ(C)}{|WH_{C}|}.

There is a character map (see (2.6))

ch^{G}(M): U^{G}(M) → ⨁_{(H)∈consub(G)} ⨁_{WH\π_{0}(M^{H})} ℚ

which sends χ^{G}(M) to the various L^{2}-Euler characteristics χ^{ℚWH_{C}}(C) for (H) ∈ consub(G) and WH ·C ∈ WH\π_{0}(M^{L}). Recall that rationally Eul^{G}(M) carries the same information as e_{1}^{G}(M)(χ^{G}(M)) since the rationally injective map e_{3}^{G}(M) ∘ e_{2}^{G}(M) sends e_{1}^{G}(M)(χ^{G}(M)) to Eul^{G}(M). Rationally e_{1}^{G}(M)(χ^{G}(M)) is the same as the collection of all these orbifold Euler characteristics χ^{ℚWH_{C}}(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

γ_{ℚ}^{G}: ⨁_{(L)∈consub(G) L finite cyclic} ⨁_{WL\π_{0}(M^{L})} ℚ ⟶^{≅} ℚ ⊗_{ℤ} H_{0}^{Or(G)}(X; \underline{R_{ℚ}})

which maps

{χ^{ℚWL_{C})}(C) | (L) ∈ consub(G), L finite cyclic, WL · C ∈ WL\π_{0}(M^{L})}

to 1 ⊗_{ℤ} e_{1}^{G}(M)(χ^{G}(M)).

However, we will show that Eul^{G}(M) does carry some torsion information. Namely, we will prove:

Theorem 0.10 There exists an action of the symmetric group S_{3} of order 3! on the 3-sphere S^{3} such that Eul^{S_{3}}(S^{3}) ∈ KO_{0}^{S_{3}}(S^{3}) has order 2.

The relationship between Eul^{G}(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 e^{G}(M)
4. Examples

4.1. Finite groups and connected non-empty fixed point sets 4.2. The equivariant Euler class carries torsion information 4.3. Independence of Eul^{G}(M) and χ_{s}^{G}(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 Münster. Jonathan Rosenberg 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 L^{2} 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 complex^{1} cohomology 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 detailed 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-vector bundle, and even determines the families index in K_{G}^{∗}(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 Kasparov group K_{G}^{−∗}(C_{0}(M)), which is locally finite K^{G}-homology, i.e., the relative group K_{∗}^{G}(\overline{M}, \{∞\}), where \overline{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 K_{G}^{−∗}(C_{0}(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)^{−}, 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 tangent bundle, 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, Theorem 5.12].) Let H = H^{+} ⊕ H^{−} be the ℤ/2-graded Hilbert space of L^{2} sections of Cliff(T M). Let A be the operator on H defined by right Clifford 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.

1depending on what scalars one is using

2Here C_{0}(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 C_{0}(M) is covariant in M. Excision in Kasparov theory identifies K_{G}^{−∗}(C_{0}(M)) with K_{G}^{−∗}(C(\overline{M}), C(pt)), which is identified with relative K^{G} -homology. When \overline{M} does not have finite G-homotopy type, K^{G} - homology here means Steenrod K^{G} -homology, as explained in [10].

For λ ≥ 0, let D_{λ} = D + λA. As in [23], each D_{λ} defines an unbounded Gequivariant Kasparov module in the same Kasparov class as D. In the “bounded picture” of Kasparov theory, the corresponding operator is

B_{λ} = D_{λ}(1 + D_{λ}^{2})^{− \frac{1}{2}} = \frac{1}{λ}D_{λ} (\frac{1}{λ^{2}} + \frac{1}{λ^{2}} D_{λ}^{2})^{− \frac{1}{2}} .  (1.1)

The axioms satisfied by this operator that insure that it defines a Kasparov K^{G}-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_{0}(M), fB_{λ} ∼ B_{λ}f and fB_{λ}^{2} ∼ f, where ∼ denotes equality

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 B_{λ} under G by “Gcontinuity,” the requirement (see [13] and [4, §20.2.1]) that

(B1′) f(g · B_{λ} − B_{λ}) ∼ 0 for f ∈ C_{0}(M), g ∈ G.

In order to simplify the calculations that are coming next, we may assume without 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_{z} -invariant neighborhood U_{z} of z, M is G_{z} -equivariantly isometric to a ball, say of radius 1, about the origin in Euclidean space ℝ^{n} with an orthogonal G_{z} -action, with z corresponding to the origin. This can be arranged since the exponential map induces a G_{z} -diffeomorphism of a small G_{z} -invariant neighborhood of 0 ∈ T_{z}M onto a G_{z} -invariant neighborhood of z such that 0 is mapped to z and its differential at 0 is the identity on T_{z}M under the standard identification T_{0}(T_{z}M) = T_{z}M. Thus the usual coordinates x_{1}, x_{2}, …, x_{n} in Euclidean space give local coordinates in M for |x| < 1, and \frac{∂}{∂x_{1}} , \frac{∂}{∂x_{2}} , …, \frac{∂}{∂x_{n}} define a local orthonormal frame in T M near z. We can arrange that (ℝ^{n})^{G_{z}} contains the points with x_{2} = … = x_{n} = 0 if (ℝ^{n})^{G_{z}} is different from {0}. In these exponential local coordinates, the point x_{1} = x_{2} = ⋯ = x_{n} = 0 corresponds to z. We may assume we have chosen the vector field Ξ so that in these local coordinates, Ξ is given by the radial vector field

3Since sign conventions differ, we emphasize that for us, unit tangent vectors on M have square −1 in the Clifford algebra.

x_{1} \frac{∂}{∂x_{1}} + x_{2} \frac{∂}{∂x_{2}} + ⋯ + x_{n} \frac{∂}{∂x_{n}}  (1.2)

if z ∈ Zero(Ξ)^{+}, or by the vector field

−x_{1} \frac{∂}{∂x_{1}} + x_{2} \frac{∂}{∂x_{2}} + ⋯ + x_{n} \frac{∂}{∂x_{n}}  (1.3)

if z ∈ Zero(Ξ)^{−}. Thus |Ξ(x)| = 1 on ∂U_{z} for each z, and we can assume (rescaling Ξ if necessary) that |Ξ| ≥ 1 on the complement of ⋃_{z∈Zero(Ξ)} U_{z}. Recall that D_{λ} = D + λA.

Lemma 1.4 Fix a small number ε > 0, and let P_{λ} denote the spectral projection of D_{λ}^{2} corresponding to [0, ε]. Then for λ sufficiently large, range P_{λ} is G-isomorphic to L^{2}(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_{λ})D_{λ}^{2} ≥ Cλ. (In other words, (ε, Cλ) ∩ (spec D_{λ}^{2}) = ∅.) Furthermore, the functions in range P_{λ} become increasingly concentrated near Zero(Ξ) as λ → ∞.

Proof First observe that in Euclidean space ℝ^{n}, if Ξ is defined by (1.2) or (1.3) and A and D_{λ} are defined from Ξ as on M, then S_{λ} = D_{λ}^{2} is basically a Schrödinger operator for a harmonic oscillator, so one can compute its spectral decomposition explicitly. (For example, if n = 1, then S_{λ} = −\frac{d^{2}}{dx^{2}} + λ^{2}x^{2} ± λ, the sign depending on whether z ∈ Zero(Ξ)^{+} or z ∈ Zero(Ξ)^{−} and whether one considers the action on H^{+} or H^{−}.) When z ∈ Zero(Ξ)^{+}, the kernel of S_{λ} in L^{2} sections of Cliff(T ℝ^{n}) is spanned by the Gaussian function

(x_{1}, x_{2}, …, x_{n}) ↦ e^{−λ|x|^{2}/2},

and if z ∈ Zero(Ξ)^{−}, the L^{2} kernel is spanned by a similar section of Cliff(T M)^{−}, e^{−λ|x|^{2}/2} \frac{∂}{∂x_{1}} . Also, in both cases, S_{λ} has discrete spectrum lying on an arithmetic progression, with one-dimensional kernel (in L^{2}) and first non-zero eigenvalue given by 2nλ.

Now let’s go back to the operator on M. Just as in [23, Lemma 2], we have the estimate

−Kλ ≤ D_{λ}^{2} − (D^{2} + λ^{2}A^{2}) ≤ Kλ,  (1.5)

where K > 0 is some constant (depending on the size of the covariant derivatives of Ξ).^{4} But D_{λ}^{2} ≥ 0, and also, from (1.5),

\frac{1}{λ^{2}} D_{λ}^{2} ≥ A^{2} + \frac{1}{λ^{2}} D^{2} − \frac{K}{λ} ,  (1.6)

which implies that

\frac{1}{λ^{2}} D_{λ}^{2} ≥ multiplication by |Ξ(x)|^{2} − \frac{K}{λ} .  (1.7)

So if ξ_{λ} is a unit vector in range P_{λ}, we have

\frac{ε}{λ^{2}} ≥ ⟨\frac{1}{λ^{2}} D_{λ}^{2}ξ_{λ}, ξ_{λ}⟩ ≥ ∫_{M} |Ξ(x)|^{2} |ξ_{λ}(x)|^{2} dvol − \frac{K}{λ} ∥ξ_{λ}∥^{2}.  (1.8)

Now ∥ξ_{λ}∥ = 1, and if we fix η > 0, we only make the integral smaller by replacing |Ξ(x)|^{2} by η on the set E_{η} = {x : |Ξ(x)|^{2} ≥ η} and by 0 elsewhere. So

\frac{ε}{λ^{2}} ≥ −\frac{K}{λ} + η ∫_{E_{η}} |ξ_{λ}(x)|^{2} dvol

or

∥ξ_{λ}χ_{E_{η}} ∥^{2} ≤ \frac{K}{ηλ} + \frac{ε}{ηλ^{2}} .  (1.9)

This being true for any η, we have verified that as λ → ∞, ξ_{λ} becomes increasingly concentrated near the zeros of Ξ, in the sense that the L^{2} norm of its restriction to the complement of any neighborhood of Zero(Ξ) goes to 0.

It remains to compute range P_{λ} (as a unitary representation space of G) and to prove that D_{λ}^{2} has the desired spectral gap. Define a C^{2} cut-off function φ(t), 0 ≤ t < ∞, so that 0 ≤ φ(t) ≤ 1, φ(t) = 1 for 0 ≤ t ≤ \frac{1}{2} , φ(t) = 0 for t ≥ 1, and φ is decreasing on the interval [\frac{1}{2}, 1]. In other words, φ is supposed to have a graph like this:

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

We can arrange that |φ′(t)| ≤ 3 and that |φ′′(t)| ≤ 20. For each element z of Zero(Ξ), recall that we have a G_{z} -invariant neighborhood U_{z} that can be identified with the unit ball in ℝ^{n} equipped with an orthogonal G_{z} -action. So the function ψ_{z,λ}(x) = φ(r)e^{−λr^{2}/2}, where r = |x| is the radial coordinate in ℝ^{n}, makes sense as a function in C^{2}(M), with support in U_{z}. For simplicity suppose z ∈ Zero(Ξ)^{+}; the other case is exactly analogous except that we need a 1-form instead of a function. Then D_{λ}^{2}, acting on radial functions, becomes

−Δ + λ^{2}|x|^{2} − nλ = −\frac{∂^{2}}{∂r^{2}} − (n − 1)\frac{1}{r} \frac{∂}{∂r} + λ^{2}r^{2} − nλ.

As we mentioned before, this operator on ℝ^{n} annihilates x ↦ e^{−λr^{2}/2}, so we have

\frac{∥D_{λ}ψ_{z,λ}∥^{2}}{∥ψ_{z,λ}∥^{2}} = \frac{⟨D_{λ}^{2}ψ_{z,λ}, ψ_{z,λ}⟩}{⟨ψ_{z,λ}, ψ_{z,λ}⟩}

= \frac{∫_{0}^{1} φ(r) (−rφ′′(r) + (1 − n + 2r^{2}λ)φ′(r)) e^{−λr^{2}}r^{n−2} dr}{∫_{0}^{1} φ(r)^{2}e^{−λr^{2}}r^{n−1} dr}

≤ \frac{∫_{1/2}^{1}(20r + 6λ)e^{−λr^{2}}r^{n−2} dr}{∫_{0}^{1/2} e^{−λr^{2}}r^{n−1} dr} .  (1.10)

The expression (1.10) goes to 0 faster than λ^{−k} for any k ≥ 1, since the numerator dies rapidly and the denominator behaves like a constant times λ^{−n/2} for large λ, so P_{λ}ψ_{z,λ} is non-zero and very close to ψ_{z,λ}. Rescaling constructs a unit vector in range P_{λ} 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. In particular, range P_{λ} contains a Hilbert space G-isomorphic to L^{2}(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_{z,λ}, then ∥D_{λ}ξ∥^{2} ≥ Cλ for some constant C > 0, provided λ is sufficiently large Let E = ⋃_{z∈Zero(Ξ)} V_{z}, where V_{z} corresponds to the ball about the origin of radius \frac{1}{2} when we identify U_{z} with the ball about the origin in ℝ^{n} of radius 1. Let χ_{E} be the characteristic function of E. Then

1 = ∥ξ∥^{2} = ∥χ_{E}ξ∥^{2} + ∥(1 − χ_{E})ξ∥^{2}.

Hence we must be in one of the following two cases:

(a) ∥(1 − χ_{E})ξ∥^{2} ≥ \frac{1}{2} .

(b) ∥χ_{E}ξ∥^{2} ≥ \frac{1}{2} .

In case (1), we can argue just as in the inequalities (1.8) and (1.9) with η = \frac{1}{4} , since E is precisely the set where |Ξ(x)|^{2} < \frac{1}{4} . So we obtain

\frac{1}{λ^{2}} ∥D_{λ}ξ∥^{2} = ⟨\frac{1}{λ^{2}} D_{λ}^{2}ξ, ξ⟩ ≥ −\frac{K}{λ} + \frac{1}{4} ∥(1 − χ_{E})ξ∥^{2} ≥ \frac{1}{8} − \frac{K}{λ} ,

which gives ∥D_{λ}ξ∥^{2} ⪰ const · λ^{2} once λ is sufficiently large. So now consider case (2). Then for some z, we must have ∥χ_{G·V_{z}}ξ∥^{2} ≥ \frac{1}{2|G\Zero(Ξ)|}. But by assumption, ξ ⊥ ψ_{g·z,λ} (for this same z and all g ∈ G). Assume for simplicity that ξ ∈ H^{+} and z ∈ Zero(Ξ)^{+}. If ξ ∈ H^{+} and z ∈ Zero(Ξ)^{−}, there is no essential difference, and if ξ ∈ H^{−}, the calculations are similar, but we need 1-forms in place of functions. Anyway, if we let ξ_{g} denote ξ|_{U_{g}·z} transported to ℝ^{n}, we have

\begin{matrix}0 = & ∫_{ℝ^{n}} φ(|x|)ξ_{g}(x)e^{−λ|x|^{2}/2} dx (∀g), \\ 1 ≥ & ∑_{g} ∫_{|x|≤ \frac{1}{2}} φ(|x|)^{2}|ξ_{g}(x)|^{2} dx ≥ \frac{1}{2|G\Zero(Ξ)|}. \end{matrix}

Now we use the fact that the Schrödinger operator S_{λ} on ℝ^{n} has one-dimen-sional kernel in L^{2} spanned by x ↦ e^{−λ|x|^{2}/2} (if z ∈ Zero(Ξ)^{+}), and spectrum bounded below by 2nλ on the orthogonal complement of this kernel. (If z ∈ Zero(Ξ)^{−}, the entire spectrum of S_{λ} on H^{+} is bounded below by 2nλ.) So compute as follows:

∥D_{λ}(φ(|x|)ξ_{g})∥^{2} = ⟨D_{λ}^{2}(φ(|x|)ξ_{g}), φ(|x|)ξ_{g}⟩
≥ 2nλ ⟨φ(|x|)ξ_{g}, φ(|x|)ξ_{g}⟩ .  (1.11)

Let ω be the function on M which is 0 on the complement of ⋃_{g} U_{g·z} and given by φ(|x|) on U_{g·z} (when we use the local coordinate system there centered at g · z). Then:

∥D_{λ}ξ∥^{2} = ∥D_{λ}(ωξ)∥^{2} + ∥D_{λ}((1 − ω)ξ)∥^{2}
+ 2 ⟨D_{λ}^{2}((1 − ω)ξ), ωξ⟩ .  (1.12)

Since D_{λ} is local and ω is supported on the U_{g·z}, g ∈ G, the first term on the right is simply

∥D_{λ}(ωξ)∥^{2} = ∑_{g} ∥D_{λ}(φ(|x|)ξ_{g})∥^{2} ≥ \frac{2nλ}{2|G\Zero(Ξ)|}  (1.13)

by (1.11). In the inner product term in (1.12), since ωξ is a sum of pieces with disjoint supports U_{g·z}, we can split this as a sum over terms we can transfer to ℝ^{n}, getting

\begin{aligned}2 & ∑_{g} ⟨S_{λ}((1 − φ(|x|))ξ_{g}), φ(|x|)ξ_{g}⟩ = 2 ∑_{g} ⟨D^{2}((1 − φ(|x|))ξ_{g}), φ(|x|)ξ_{g}⟩ \\ & + 2 ∑_{g} ∫_{\frac{1}{2} ≤|x|≤1}(λ^{2}|x|^{2} + T λ) φ(|x|)(1 − φ(|x|))|ξ_{g}(x)|^{2} dx, \end{aligned}

where −K ≤ T ≤ K. Since λ^{2}|x|^{2} +T λ > 0 on \frac{1}{2} ≤ |x| ≤ 1 for large enough λ, the integral here is nonnegative, and the only possible negative contributions to ∥D_{λ}ξ∥^{2} are the terms 2 ⟨D^{2}((1 − φ(|x|))ξ_{g}), φ(|x|)ξ_{g}⟩, which do not grow with λ. So from (1.12), (1.13), and (1.11), ∥D_{λ}ξ∥^{2} ≥ const · λ for large enough 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 E_{λ} = H, the L^{2} sections of Cliff(T M). But the fiber E_{∞} over +∞ will be the direct sum of H ⊕ V , where V = L^{2}(Zero(Ξ)) is a Hilbert space with orthonormal basis v_{z}, z ∈ Zero(Ξ). We put a ℤ/2-grading on V by letting V^{+} = L^{2}(Zero(Ξ)^{+}), V^{−} = L^{2}(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, ∞] → H to be continuous, but will also allow additional continuous fields that become increasingly concentrated near the points of Zero(Ξ). Namely, suppose z ∈ Zero(Ξ). By Lemma 1.4, for λ large, D_{λ} has an element ψ_{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_{z} at λ = ∞ if for any neighborhood U of z,

∫_{M∖U} |ξ(λ)(m)|^{2} dvol(m) → 0 as λ → ∞,

and if (assuming z ∈ Zero(Ξ)^{+}) ξ(λ) ∈ H^{+} and

∥ξ(λ) − c (\frac{λ}{π})^{\frac{n}{4}} ψ_{z,λ}∥ → 0  as λ → ∞.  (1.14)

The constant reflects the fact that the L^{2}-norm of e^{−λ|x|^{2}/2} is (\frac{π}{λ})^{\frac{n}{4}} . If z ∈ Zero(Ξ)^{−}, we use the same definition, but require ξ(λ) ∈ H^{−}.

This concludes the definition of the continuous field of Hilbert spaces E, which we can think of as a Hilbert C^{∗}-module over C(I), I the interval [0, +∞]. We will use this to define a Kasparov (C_{0}(M), C(I))-bimodule, or in other words, a homotopy of Kasparov (C_{0}(M), ℝ)-modules. The action of C_{0}(M) on E is the obvious one: C_{0}(M) acts on H the usual way, and it acts on V (the other summand of E_{∞}) by evaluation of functions at the points of Zero(Ξ):

f · v_{z} = f(z)v_{z}, z ∈ Zero(Ξ), f ∈ C_{0}(M).

We define a field T of operators on E as follows. For λ < ∞, T_{λ} ∈ L(E_{λ}) = L(H) is simply B_{λ} as defined in (1.1), where recall that D_{λ} = D + λA. For λ = ∞, E_{∞} = H⊕V and T_{∞} is 0 on V and is given on H by the operator B_{∞} = A/|A| which is right Clifford multiplication by \frac{Ξ(x)}{|Ξ(x)|} (an L^{∞}, but possibly

discontinuous, vector field) on H^{+} and by \frac{−Ξ(x)}{|Ξ(x)|} on H^{−}. Note that T_{∞}^{2} is 0 on V and the identity on H. While 1_{V} is not compact on V if Zero(Ξ) is infinite, this is not a problem since for f ∈ C_{c}(M), the action of f on V has finite rank (since f annihilates v_{z} for z ∉ supp f).

Now we check the axioms for (E, T) to define a homotopy of Kasparov modules from [D] to the class of

(C_{0}(M), E_{∞}, T_{∞}) = (C_{0}(M), H, B_{∞}) ⊕ (C_{0}(M), V, 0).

But (C_{0}(M), H, B_{∞}) is a degenerate Kasparov module, since B_{∞} commutes with multiplication by functions and has square 1. So the class of (C_{0}(M), E_{∞}, T_{∞}) is just the class of (C_{0}(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 times the canonical class KO_{0}^{G}(z) ∘ (α_{z})_{∗}([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_{0}(M) on E is continuous, i.e., given by a ∗-homomorphism C_{0}(M) → L(E). The only issue is continuity at λ = ∞. In other words, since the action on H is constant, we just need to know that if ξ is a continuous field converging as λ → ∞ to a vector v in V , then for f ∈ C_{0}(M), f · ξ(λ) → f · v. But it’s enough to consider the special kinds of continuous fields discussed above, since they generate the structure, and if ξ(λ) → cv_{z}, then ξ(λ) becomes increasingly concentrated at z (in the sense of L^{2} norm), and hence f · ξ(λ) → cf(z)v_{z}, as required.

Next, we need to check that T ∈ L(E). Again, the only issue is (strong operator) continuity at λ = ∞. Because of the way continuous fields are defined at λ = ∞, there are basically two cases to check. First, if ξ ∈ H, we need to check that B_{λ}ξ → B_{∞}ξ as λ → ∞. Since the B_{λ}’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 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 ⟨D_{λ}^{2}ξ, ξ⟩ > Cλ∥ξ∥^{2}. In fact, if P_{λ} is the spectral projection of D_{λ}^{2} for the interval [0, Cλ], Lemma 1.4 implies that ∥P_{λ}ξ∥ ≤ ε∥ξ∥ for λ sufficiently large. (This is because the condition on the support of ξ forces ξ to be almost orthogonal to the spectral subspace where D_{λ}^{2} ≤ ε.) Now let E_{λ}^{+} and E_{λ}^{−} be the spectral projections for D_{λ} corresponding to the intervals (0, ∞) and (−∞, 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. The former shows that the operators A + \frac{1}{λ}D, all defined on dom D, “converge strongly in the generalized sense” to A. Since the positive and negative spectral subspaces for A+ \frac{1}{λ}D are the same as for D_{λ} (since the operators only differ by a homothety), [14, Chapter VIII, §2, Theorem 1.15] then shows that E_{λ}^{+} → F^{+} and E_{λ}^{−} → F^{+} in the strong operator topology. Note that the fact that A has no kernel is needed in these results.

Now since ∥P_{λ}ξ∥ ≤ ε∥ξ∥ for λ sufficiently large, we also have

B_{∞}ξ = F^{+}ξ − F^{−}ξ, and ∥B_{λ}ξ − (E_{λ}^{+}ξ − E_{λ}^{−}ξ)∥ ≤ 2ε

for λ sufficiently large. Hence

∥B_{λ}ξ − B_{∞}ξ∥ ≤ 2ε + ∥(E_{λ}^{+}ξ − F^{+}ξ) − (E_{λ}^{−}ξ − F^{−}ξ)∥ → 2ε.

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_{z} in the sense of the continuous field structure of E. In this case, we need to show that B_{λ}ξ(λ) → 0. This case is much easier: ξ(λ) → cv_{z} means

∥ξ(λ) − c (\frac{λ}{π})^{\frac{n}{4}} ψ_{z,λ}∥ → 0 by (1.14),

while ∥B_{λ}∥ ≤ 1 and

∥D_{λ} ((\frac{λ}{π})^{\frac{n}{4}} ψ_{z,λ})∥ → 0 by (1.10),

so B_{λ}ξ(λ) → 0 in norm.

Thus T ∈ L(E). Obviously, T satisfies (B1) and (B2) of page 9, so we need to check the analogues of (B3), which are that f(1 − T^{2}) and [T, f] lie in K(E) for f ∈ C_{0}(M). First consider 1 − T^{2}. 1 − T_{λ}^{2} is locally compact (i.e., compact after multiplying by f ∈ C_{c}(M)) for each λ, since

1 − T_{λ}^{2} = 1 − B_{λ}^{2} = (1 + D_{λ}^{2})^{−1} = (1 + (D + λA)^{2})^{−1}

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_{λ}^{2} is a norm-continuous field of operators on E. Continuity 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 + D_{λ}^{2})^{−1} = P_{λ} + O(\frac{1}{λ}), and also that P_{λ} is increasingly concentrated near Zero(Ξ). So near λ = ∞, we can write the field of operators (1 + D_{λ}^{2})^{−1} as a sum of rank-one projections onto vector fields converging to the various v_{z} ’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_{0}(M), [f, T_{λ}] lies in K(E). We already know that [f, B_{λ}] ∈ K(H) for fixed λ and is norm-continuous in λ for λ < ∞, so since T_{∞} commutes with multiplication operators, it suffices to show that [f, B_{λ}] converges to 0 in norm as λ → 0. We follow the method 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_{c}^{∞}(M) with critical points at all of the points of the set Zero(Ξ), since such functions are dense in C_{0}(M). Then estimate as follows:

[f, B_{λ}] = [f, D_{λ}(1 + D_{λ}^{2})^{−1/2}]
= [f, D_{λ}](1 + D_{λ}^{2})^{−1/2} + D_{λ} [f, (1 + D_{λ}^{2})^{−1/2}] .  (1.15)

We have [f, D_{λ}] = [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 + D_{λ}^{2})^{−1/2} converges as λ → ∞ (in the norm of our continuous field) to projection onto the space V = L^{2}(Zero(Ξ)). Since the derivatives of f vanish on Zero(Ξ), the product [f, D_{λ}](1 + D_{λ}^{2})^{−1/2}, which is the first term in (1.15), goes to 0 in norm. As for the second term, we have (following [4, p. 199])

D_{λ} [f, (1 + D_{λ}^{2})^{−1/2}] = \frac{1}{π} ∫_{0}^{∞} μ^{− \frac{1}{2}} D_{λ} [f, (1 + D_{λ}^{2} + μ)^{−1}] dμ,  (1.16)

and

D_{λ} [f, (1 + D_{λ}^{2} + μ)^{−1}] = D_{λ}(1 + D_{λ}^{2} + μ)^{−1} [1 + D_{λ}^{2} + μ, f] (1 + D_{λ}^{2} + μ)^{−1}.

Now use the fact that

[1 + D_{λ}^{2} + μ, f] = [D_{λ}^{2}, f] = D_{λ}[D_{λ}, f] + [D_{λ}, f]D_{λ} = D_{λ}[D, f] + [D, f]D_{λ}.

We obtain that

D_{λ} [f, (1 + D_{λ}^{2} + μ)^{−1}]

= \frac{D_{λ}^{2}}{1 + D_{λ}^{2} + μ}[D, f]\frac{1}{1 + D_{λ}^{2} + μ} + \frac{D_{λ}}{1 + D_{λ}^{2} + μ}[D, f]\frac{D_{λ}}{1 + D_{λ}^{2} + μ}. (1.17)

Again a slight modification of the argument in [23, p. 3473] is needed, since D_{λ} 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:

∥D_{λ} [f, (1 + D_{λ}^{2} + μ)^{−1}]∥ ≤ ∥\frac{D_{λ}^{2}}{1 + D_{λ}^{2} + μ}[D, f]\frac{1}{1 + D_{λ}^{2} + μ}∥  (1.18)
+ ∥\frac{D_{λ}}{1 + D_{λ}^{2} + μ}[D, f]\frac{D_{λ}}{1 + D_{λ}^{2} + μ}∥ .  (1.19)

The first term, (1.18), is bounded by the second, (1.19), plus an additional commutator term:

∥\frac{D_{λ}}{1 + D_{λ}^{2} + μ}[D, [D, f]]\frac{1}{1 + D_{λ}^{2} + μ}∥ .  (1.20)

Now the contribution of the term (1.19) is estimated by observing that the function

\frac{x}{1 + x^{2} + μ}, −∞ < x < ∞

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

∥\frac{D_{λ}}{1 + D_{λ}^{2} + μ}∥ ≤ \begin{cases}\frac{1}{2\sqrt{1+μ}}, \\ \sqrt{Cλ} ≤ \sqrt{1 + μ}, or μ ≥ Cλ − 1, \\ max (\frac{\sqrt{ε}}{1+ε+μ}, ^{\frac{\sqrt{Cλ}}{1+μ+Cλ}}) , \\ \sqrt{Cλ} ≥ \sqrt{1 + μ}, or 0 ≤ μ ≤ Cλ − 1. \end{cases}  (1.21)

Thus the contribution of the term (1.19) to the integral in (1.16) is bounded by

\frac{∥[D, f]∥}{π} ∫_{0}^{∞} ∥\frac{D_{λ}}{1 + D_{λ}^{2} + μ}∥^{2} \frac{1}{\sqrt{μ}} dμ

≤ \frac{∥[D, f]∥}{π} (∫_{Cλ−1}^{∞} \frac{1}{\sqrt{μ}} \frac{1}{4(1 + μ)} dμ  (1.22)

+ ∫_{0}^{Cλ−1} \frac{1}{\sqrt{μ}} max (\frac{Cλ}{(1 + μ + Cλ)^{2}} , \frac{ε}{(1 + ε + μ)^{2}} ) dμ)

≤ \frac{∥[D, f]∥}{π} (\frac{1}{4} ∫_{Cλ−1}^{∞} μ^{− \frac{3}{2}} dμ + ∫_{0}^{Cλ} \frac{1}{\sqrt{μ}} \frac{Cλ}{(Cλ)^{2}} dμ + ∫_{0}^{∞} \frac{ε}{\sqrt{μ}(1 + μ)^{2}} dμ)

= \frac{∥[D, f]∥}{π} (\frac{1}{2\sqrt{Cλ − 1}} + \frac{2}{\sqrt{Cλ}} + \frac{πε}{2}) → \frac{∥[D, f]∥}{2}ε.  (1.23)

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

\frac{∥[D, [D, f]]∥}{π} ∫_{0}^{∞} ∥\frac{D_{λ}}{1 + D_{λ}^{2} + μ}∥ \frac{1}{1 + μ} \frac{1}{\sqrt{μ}} dμ

≤ \frac{∥[D, [D, f]]∥}{π}(∫_{Cλ−1}^{∞} \frac{1}{2\sqrt{1 + μ}} \frac{1}{1 + μ} \frac{1}{\sqrt{μ}} dμ

+ ∫_{0}^{Cλ−1} \frac{\sqrt{Cλ}}{(1 + μ + Cλ)} \frac{1}{1 + μ} \frac{1}{\sqrt{μ}} dμ + ∫_{0}^{∞} \frac{ε}{\sqrt{μ}(1 + μ)^{2}} dμ)

≤ \frac{∥[D, [D, f]]∥}{π} (∫_{Cλ−1}^{∞} \frac{1}{2μ^{2}} dμ + ∫_{0}^{∞} \frac{1}{\sqrt{Cλ}} \frac{1}{\sqrt{μ}(1 + μ)} dμ

+ ∫_{0}^{∞} \frac{ε}{\sqrt{μ}(1 + μ)^{2}} dμ)

≤ \frac{∥[D, [D, f]]∥}{π} (\frac{1}{2(Cλ − 1)} + \frac{π}{\sqrt{Cλ}} + \frac{πε}{2}) → \frac{∥[D, [D, f]]∥}{2}ε,  (1.24)

which again can be taken as small as we like. This completes the proof.

2 Review of notions of equivariant Euler characteristic

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 KO^{G}-Euler class Eul^{G}(M). We will use the following notation in the sequel.

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

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), in which the universal equivariant Euler characteristic takes its values. Let Π_{0}(G, X) be the component category 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_{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)],  (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_{0}(G, X) ⟶^{≅} ∐_{(H)∈consub(G)} WH\π_{0}(X^{H}),  (2.3)

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)} ⨁_{WH\π_{0}(X^{H})} ℤ.  (2.4)

Definition 2.5 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 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:

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

where I_{p}(X) is the set of open cells of X (after forgetting the group action).

The orbifold Euler characteristic χ^{ℚG}(X) can be identified with the more general notion of the L^{2}-Euler characteristic χ^{(2)}(X; N (G)), where N (G) is the group von Neumann algebra of G. 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 [17, Section 6.6]).

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

ch^{G}(X): U^{G}(X) → ⨁_{Is Π_{0}(G,X)} ℚ = ⨁_{(H)∈consub(G)} ⨁_{WH\π_{0}(X^{H})} ℚ. (2.6)

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

There is an isomorphism of groups

WK_{y} ⟶^{≅} aut(y, 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},  (2.7)

where (WK_{y})_{σ} is the isotropy group of σ ∈ mor(y, x) under the WK_{y} -action.

Lemma 2.8 Let X be a finite proper G-CW -complex. Then the map ch^{G}(X) of (2.6) is injective and satisfies

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

The induced map

id_{ℚ} ⊗_{ℤ} ch^{G}(X): ℚ ⊗_{ℤ} U^{G}(X) ⟶^{≅} ⨁_{(H)∈consub(G)} ⨁_{WH\π_{0}(X^{H})} ℚ

is bijective.

Proof Injectivity of χ^{G}(X) and ch^{G}(X)(χ^{G}(X))_{[y]} = χ^{ℚWK_{y}}(X^{K}(y)). is proved in [20, Lemma 5.3]. The bijectivity of id_{ℚ} ⊗_{ℤ} ch^{G}(X) follows since its 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 characteristic 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 Grothendieck group of the (additive) monoid of finite G-sets, where the addition comes from disjoint union. This becomes a ring under the obvious multiplication coming from the Cartesian product of G sets. There is a natural map of rings j_{1} : A(G) → R_{ℝ}(G) = KO_{0}^{G}(pt) that comes from sending a finite G-set X to the orthogonal representation of G on the finite-dimensional real Hilbert space L_{ℝ}^{2}(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 R_{ℝ}(G) is the number of ℝ-conjugacy classes in G, where x and y are called ℝ-conjugate if they are conjugate or if x^{−1} and y are conjugate [28, §13.2]. Thus rank A((ℤ/2)^{3}) = 16 > rank R_{ℝ}((ℤ/2)^{3}) = 8; on the other hand, rank A(ℤ/5) = 2 < rank R_{ℝ}(ℤ/5) = 3.

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 χ_{a}^{G}(M) ∈ KO_{0}^{G}, the equivariant 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 a point, χ_{a}^{G}(M) = c_{∗}(Eul^{G}(M)), where c: M → pt and c_{∗} is the induced map on KO_{0}^{G}.

(b) the stable homotopy-theoretic equivariant Euler characteristic χ_{s}^{G}(M) ∈ A(G). This is discussed, say, in [6], Chapter IV, §2.

(c) a certain unstable homotopy-theoretic equivariant Euler characteristic, which we will denote here χ_{u}^{G}(M) to distinguish it from χ_{s}^{G}(M). This invariant is defined in [30], and shown to be the obstruction to existence of an everywhere non-vanishing G-invariant vector field on M. The invariant χ_{u}^{G}(M) lives in a group A_{u}^{G}(M) (Waner and Wu call it A_{M}(G), but the notation A_{u}^{G}(M) is more consistent with our notation for U^{G}(M)) defined as follows: A_{u}^{G}(M) is the free abelian group on finite G-sets embedded in M, modulo isotopy (if s_{t} is a 1-parameter family of finite G-sets embedded in M, all isomorphic to one another as G-sets, then s_{0} ∼ s_{1}) and the relation [s ⨿ t] = [s] + [t]. Waner and Wu define a map d: A_{u}^{G}(M) → A(G) (defined by forgetting that a G-set s is embedded in M, and just viewing it abstractly) which maps χ_{u}^{G}(M) to χ_{s}^{G}(M). Both χ_{u}^{G}(M) and χ_{s}^{G}(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.

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

\xymatrix{A_{u}^{G}(M) \ar[r] \ar@{-->}[dr]_{d} & U^{G}(M) \ar[r]^{e^{G}(M)} \ar[d]^{c_{∗}} & KO_{0}^{G}(M) \ar[d]^{c_{∗}} \\  & A(G) = U^{G}(pt) \ar[r]^{j_{1}} & R_{ℝ}(G) = KO_{0}^{G}(pt).}  (2.12)

The map A_{u}^{G}(M) → U^{G}(M) in the upper left is an isomorphism if (M, G) satisfies the weak gap hypothesis, that is, if whenever H ⊊ K are subgroups of G, each component of G^{K} has codimension at least 2 in the component of G^{H} that contains it [30]. Furthermore, under the maps of this diagram,

\begin{matrix}χ_{u}^{G}(M) & ↦ χ^{G}(M), \\ c_{∗}: χ^{G}(M) & ↦ χ_{s}^{G}(M), \\ d: χ_{u}^{G}(M) & ↦ χ_{s}^{G}(M), \end{matrix}  \begin{matrix}e^{G}(M): χ^{G}(M) & ↦ Eul^{G}(M), \\ j_{1} : χ_{s}^{G}(M) & ↦ χ_{a}^{G}(M), \\ c_{∗}: Eul^{G}(M) & ↦ χ_{a}^{G}(M). \end{matrix}

Proof This is just a matter of assembling known information. The facts about the map A_{u}^{G}(M) → U^{G}(M) are in [30, §2] and in [20]. That e^{G}(M) sends χ^{G}(M) to Eul^{G}(M) is Theorem 0.3. Commutativity of the square follows immediately from the definition of e^{G}(M), since c_{∗} ∘ e^{G}(M) sends the basis element associated to (H) ⊆ consub(G) and WH · C ∈ WH\π_{0}(M^{H}) to the class of the orthogonal representation of G on L^{2}(G/H). But under c_{∗}, this same basis element maps to the G-set G/H in A(G), which also maps to the orthogonal representation of G on L^{2}(G/H) under j_{1}.

3 The transformation e^{G}(X)

Next we factorize the transformation e^{G}(M) defined in (0.2) as

e^{G}(M): U^{G}(M) ⟶^{e_{1}^{G}(M)} H_{0}^{Or(G)}(M; \underline{R_{ℚ}}) ⟶^{e_{2}^{G}(M)} H_{0}^{Or(G)}(M; \underline{R_{ℝ}})

⟶^{e_{3}^{G}(M)} KO_{0}^{G}(M),

where e_{2}^{G}(X) and e_{3}^{G}(X) are rationally injective. Rationally we will identify H_{0}^{Or(G)}(M; \underline{R_{ℚ}}) and the element e_{1}^{G}(M)(χ^{G}(M)) in terms of the orbifold Euler characteristics χ^{WL_{C}}(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_{0}(X^{L}). Here WL_{C} is the isotropy group of C ∈ π_{0}(X^{L}) under the WL = NL/Laction. Notice that e_{1}^{G}(M)(χ^{G}(M)) carries rationally the same information as Eul^{G}(M) ∈ KO_{0}^{G}(M).

Here and elsewhere H_{0}^{Or(G)}(M; \underline{R_{F}}) is the Bredon homology of M with coefficients the covariant functor

\underline{R_{F}} : Or(G; Fin) → ℤ-Mod .

The orbit category Or(G; Fin) has as objects homogeneous spaces G/H with finite H and as morphisms G-maps (Since M is proper, it suffices to consider coefficient systems over Or(G; Fin) instead over the full orbit category Or(G).) The functor \underline{R_{F}} into the category ℤ-Mod of ℤ-modules sends G/H to the representation ring R_{F} (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 g^{−1}Hg ⊆ K to the induction homomorphism R_{F} (H) → R_{F} (K) associated with the group homomorphism H → K, h ↦ g^{−1}hg. This is independent of the choice of g since an inner automorphism of K induces the identity on R_{ℝ}(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)-chain complex C_{∗}(X^{−}): Or(G) → ℤ-Chain which assigns to G/H the cellular chain complex of the CW -complex X^{H} = map_{G}(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;Fin )} V which is a ℤ-chain complex and whose homology groups are defined to be H_{p}^{Or(G)}(X; V).

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

Q: Π_{0}(G; X) → Or(G; Fin)  (3.1)

be the forgetful functor sending an object x: G/H → X to G/H. Any covariant functor V : Or(G; Fin) → ℤ-Mod induces a functor Q^{∗}V : Π_{0}(G; X) → ℤ-Mod by composition with Q. The colimit (= direct limit) of the functor Q^{∗}\underline{R_{F}} is naturally isomorphic to the Bredon homology

β_{F}^{G}(X): lim_{⟶}_{Π_{0}(G,X)}Q^{∗}R_{F} (H) ⟶^{≅} H_{0}^{Or(G)}(X; \underline{R_{F}}).  (3.2)

The isomorphism β_{F}^{G}(X) above is induced by the various maps

R_{F} (H) = H_{0}^{Or(G)}(G/H; \underline{R_{F}}) ⟶^{H_{0}^{Or (G)}(x;\underline{R_{F}})} H_{0}^{Or(G)}(X; \underline{R_{F}}),

where x runs through all G-maps x: G/H → X. We define natural maps

e_{1}^{G}(X): U^{G}(X) → H_{0}^{Or(G)}(X; \underline{R_{ℚ}});  (3.3)
e_{2}^{G}(X): H_{0}^{Or(G)}(X; \underline{R_{ℚ}}) → H_{0}^{Or(G)}(X; \underline{R_{ℝ}});  (3.4)
e_{3}^{G}(X): H_{0}^{Or(G)}(X; \underline{R_{ℝ}}) → KO_{0}^{G}(X)  (3.5)

as follows. The map β_{ℚ}^{G}(X)^{−1} ∘ e_{1}^{G}(X) sends the basis element [x: G/H → X] to the image of the trivial representation [ℚ] ∈ R_{ℚ}(H) under the canonical map associated to x

R_{ℚ}(H) → lim_{⟶}_{Π_{0}(G,X)}Q^{∗}R_{ℚ}(H).

The map e_{2}^{G}(X) is induced by the change of fields homomorphisms R_{ℚ}(H) → R_{ℝ}(H) for H ⊆ G finite. The map e_{3}^{G}(X) ∘ β^{G}(X) is the colimit over the system of maps

R_{ℝ}(H) = KO_{0}^{H}({∗}) ⟶^{(α_{H})_{∗}} KO_{0}^{G}(G/H) ⟶^{KO_{0}^{G}(x)} KO_{0}^{G}(X)

for the various G-maps x: G/H → X, where α_{H}: H → G is the inclusion.

Theorem 3.6 Let X be a proper G-CW -complex. Then

(a) The map e^{G}(X) defined in (0.2) factorizes as

e^{G}(X): U^{G}(X) ⟶^{e_{1}^{G}(X)} H_{0}^{Or(G)}(X; \underline{R_{ℚ}}) ⟶^{e_{2}^{G}(X)} H_{0}^{Or(G)}(X; \underline{R_{ℝ}})

⟶^{e_{3}^{G}(X)} KO_{0}^{G}(X);

(b) The map

ℚ ⊗_{ℤ} e_{2}^{G}(X): ℚ ⊗_{ℤ} H_{0}^{Or(G)}(X; \underline{R_{ℚ}}) → ℚ ⊗_{ℤ} H_{0}^{Or(G)}(X; \underline{R_{ℝ}})

is injective;

(c) For each n ∈ ℤ there is an isomorphism, natural in X,

chern_{n}^{G}(X): ⨁_{\begin{matrix}p,q∈ℤ, \\ p≥0,p+q=n \end{matrix}} ℚ ⊗_{ℤ} H_{p}^{Or(G)}(X; \underline{KO}_{q}^{G}) ⟶^{≅} ℚ ⊗_{ℤ} KO_{n}^{G}(X),

where \underline{KO}_{q}^{G} is the covariant functor from Or(G; Fin) to ℤ-Mod sending G/H to KO_{q}^{G}(G/H). The map

id_{ℚ} ⊗_{ℤ}e_{3}^{G}(X): ℚ ⊗_{ℤ} H_{0}^{Or(G)}(M; \underline{R_{ℝ}}) → ℚ ⊗_{ℤ} KO_{0}^{G}(M)

is the restriction of chern_{n}^{G}(X) to the summand for p = q = 0 and is hence injective;

(d) Suppose dim(X) ≤ 4 and one of the following conditions is satisfied:

(a) either dim(X^{H}) ≤ 2 for H ⊆ G, H ≠ {1}, or

(b) no subgroup H of G has irreducible representations of complex or quaternionic type.

Then

e_{3}^{G}(X): H_{0}^{Or(G)}(M; \underline{R_{ℝ}}) → KO_{0}^{G}(M)

is injective.

Proof (a) follows directly from the definitions.

(b) will be proved later.

(c) An equivariant Chern character chern_{∗}^{G} for equivariant homology theories such as equivariant K-homology is constructed in [18, Theorem 0.1] (see also [19, Theorem 0.7]). The restriction of chern_{0}^{G}(M) to H_{0}^{Or(G)}(X; \underline{R_{ℝ}}) is just e_{3}^{G}(X) under the identification \underline{R_{ℝ}} = \underline{KO}_{0}^{G}.

(d) Consider the equivariant Atiyah-Hirzebruch spectral sequence which converges to KO_{p+q}^{G}(X) (see for instance [5, Theorem 4.7 (1)]). Its E^{2}-term is E_{p,q}^{2} = H_{p}^{Or(G)}(X; \underline{KO_{q}^{G}}). The abelian group KO_{q}^{G}(G/H) is isomorphic to the real topological K -theory KO_{q}(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 representations of H of that type (see [28, §13.2]). By Morita invariance of topological real K -theory, we conclude that KO_{q}^{G}(G/H) is a direct sum of copies of the non-equivariant K-homologies KO_{q}(∗) = KO_{q}(ℝ), KU_{q}(∗) = KO_{q}(ℂ), and KSp_{q}(∗) = KO_{q}(ℍ). In particular, we conclude that KO_{q}^{G}(G/H) = 0 for q ≡ −1 (mod 8). As a consequence, \underline{KO_{−1}^{G}} = 0 and E_{p,−1}^{2} = H_{p}^{Or(G)}(X; \underline{KO_{−1}^{G}}) = 0. If no subgroup H of G has irreducible representations of complex or quaternionic type, then similarly KO_{q}^{G}(G/H) = 0 for all subgroups H of G and q = −2, −3, and E_{p,−2}^{2} = 0, E_{p,−3}^{2} = 0, as well.

Let X^{>1} ⊆ X be the subset {x ∈ X | G_{x} ≠ 1}. There is a short exact sequence

H_{p}^{Or(G)}(X^{>1}; \underline{KO_{q}^{G}}) → H_{p}^{Or(G)}(X; \underline{KO_{q}^{G}}) → H_{p}^{Or(G)}(X, X^{>1}; \underline{KO_{q}^{G}}).

Since the isotropy group of any point in X − X^{>1} is trivial, we get an isomorphism

H_{p}^{Or(G)}(X, X^{>1}; \underline{KO_{q}^{G}}) = H_{p}(C_{∗}(X, X^{>1}) ⊗_{ℤG} KO_{q}^{G}(G/1))

Since KO_{q}^{G}(G/1)) = KO_{q}(ℝ) vanishes for q ∈ {−2, −3}, we get for p ∈ ℤ and q ∈ {−2, −3}

H_{p}^{Or(G)}(X, X^{>1}; \underline{KO_{q}^{G}}) = 0.

So if dim(X^{H}) ≤ 2 for H ⊆ G, H ≠ {1}, we have dim(X^{>1}) ≤ 2. This implies for p ≥ 3 and q ∈ ℤ

H_{p}^{Or(G)}(X^{>1}; \underline{KO_{q}^{G}}) = 0.

We conclude that for p ≥ 3 and q ∈ {−2, −3}

E_{p,q}^{2} = H_{p}^{Or(G)}(X; \underline{KO_{q}^{G}}) = 0,

just as in the previous case.

Now there are no non-trivial differentials out of E_{0,0}^{r}, and since dim X ≤ 4, d_{r,1−r}^{r}: E_{r,1−r}^{r} → E_{0,0}^{r} must be zero for r > 4. But we have just seen that E_{2,−1}^{2} = 0, E_{3,−2}^{2} = 0, and E_{4,−3}^{2} = 0. Hence each differential d_{p,q}^{r} which has E_{0,0}^{r} as source or target is trivial. Hence the edge homomorphism restricted to E_{0,0}^{2} is injective. But this map is e_{3}^{G}(X).

Remark 3.7 We conclude from Theorem 0.3 and Theorem 3.6 that Eul^{G}(M) carries rationally the same information as the image of the equivariant Euler characteristic χ^{G}(X) under the map e_{1}^{G}(X): U^{G}(X) → H_{0}^{Or(G)}(M; \underline{R_{ℝ}}). Moreover, in contrast to the class of the signature operator, the class Eul^{G}(M) ∈ KO_{0}^{G}(M) of the Euler operator does not carry “higher” information because its preimage under the equivariant Chern character is concentrated in the summand 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

tr_{RG}^{u}: RG → R[con(G)], ∑_{g∈G} r_{g} · g ↦ ∑_{g∈G} r_{g} · (g).

Choose a matrix A = (a_{i,j}) ∈ M_{n}(RG) such that A^{2} = A and the image of the map r_{A}: RG^{n} → RG^{n} sending x to xA is RG-isomorphic to P. Define the Hattori-Stallings rank

HS_{RG}(P) := ∑_{i=1}^{n} tr_{RG}^{u}(a_{i,i}) ∈ R[con(G)].  (3.8)

Let α: H_{1} → H_{2} be a group homomorphism. It induces a map

con(H_{1}) → con(H_{2}), (h) ↦ (α(h))

and thus an R-linear map α_{∗}: R[con(H_{1})] → R[con(H_{2})]. If α_{∗}P is the R[H_{2}]- module obtained by induction from the finitely generated projective R[H_{1}]- module P, then

HS_{RH_{2}}(α_{∗}P) = α_{∗}(HS_{RH_{1}}(P)).  (3.9)

Next we compute F ⊗_{ℤ}(lim_{⟶}_{Π_{0}(G,X)}Q^{∗}R_{F} (H)) for F = ℚ, ℝ, ℂ. Two elements g_{1} and g_{2} of a group G are called ℚ-conjugate if and only if the cyclic subgroup ⟨g_{1}⟩ generated by g_{1} and the cyclic subgroup ⟨g_{2}⟩ generated by g_{2} are conjugate in G. Two elements g_{1} and g_{2} of a group G are called ℝ-conjugate if and only if g_{1} and g_{2} or g_{1}^{−1} and g_{2} are conjugate in G. Two elements g_{1} and g_{2} 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 con_{F} (G) the set of F -conjugacy classes (g)_{F} of elements of finite order g ∈ G. Let class_{F} (G) be the F -vector space generated by the set con_{F} (G). This is the same as the F -vector space of functions con_{F} (G) → ℝ whose support is finite.

Let pr_{F} : con(H) → con_{F} (H) be the canonical epimorphism for a finite group H. It extends to an F -linear epimorphism F [pr_{F}]: F [con(H)] → class_{F} (H). Define for a finite-dimensional H -representation V over F for a finite group H

HS_{F,H}(V) := F [pr_{F}](HS_{F H}(V)) ∈ class_{F} (H).  (3.10)

Let α: H_{1} → H_{2} be a homomorphism of finite groups. It induces a map con_{F} (H_{1}) → con_{F} (H_{2}), (h)_{F} ↦ (α(h))_{F} and thus an F -linear map

α_{∗}: class_{F} (H_{1}) → class_{F} (H_{2}).

If V is a finite-dimensional H_{1}-representation over F, we conclude from (3.9):

HS_{F,H_{2}}(α_{∗}V) = α_{∗}(HS_{F,H_{1}}(V)).  (3.11)

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,H}: F ⊗_{ℤ} R_{F} (H) ⟶^{≅} class_{F} (H)

which is natural with respect to induction with respect to group homomorphism α: H_{1} → H_{2} of finite groups.

Proof One easily checks for a finite-dimensional H -representation over F of a finite group H

HS_{F H}(V) = ∑_{(h)∈con(H)} \frac{|(h)|}{|H|} · tr_{F} (l_{h}).

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_{F H}(V) as a function con(H) → F is constant on the F -conjugacy classes of elements in H and that HS_{F,H} is bijective. Naturality follows from (3.11).

Let \underline{class_{F}} be the covariant functor Or(G; Fin) → F -Mod which sends an object G/H to class_{F} (H). The isomorphisms HS_{F,H}: F ⊗_{ℤ}R_{F} (H) ⟶^{≅} class_{F} (H) yield a natural equivalence of covariant functors from π_{0}(G, X) to F -Mod . Thus we obtain an isomorphism

HS_{F}^{G}(X): F ⊗_{ℤ} lim_{⟶}_{Π_{0}(G,X)}Q^{∗}\underline{R_{F}}

⟶^{≅} lim_{⟶}_{Π_{0}(G,X)}Q^{∗}F ⊗_{ℤ} \underline{R_{F}} ⟶^{≅} lim_{⟶}_{Π_{0}(G,X)}Q^{∗}\underline{class_{F}} . (3.13)

Let f: S_{0} → S_{1} be a map of sets. It extends to an F -linear map F [f]: F [S_{0}] → F [S_{1}]. Suppose that the preimage of any element in S_{1} is finite. Then we obtain an F -linear map

f^{∗}: F [S_{1}] → F [S_{0}], s_{1} ↦ ∑_{s_{0}∈f^{−1}(s_{1})} s_{0}.  (3.14)

If we view elements in F [S_{i}] as functions S_{i} → F, then f^{∗} is given by composing with f. One easily checks that F [f] ∘ f^{∗} is bijective and that for a second map g: S_{1} → S_{2}, for which the preimages of any element in S_{2} is finite, we have 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_{H}: con_{ℝ}(H) → con_{ℚ}(H) be the projection. If V is a finite-dimensional H -representation over ℚ, then ℝ ⊗_{ℚ} V is a finite-dimensional H -representation over ℝ and HS^{ℝH}(ℝ ⊗_{ℚ} V) is the image of HS^{ℚH}(V) under the obvious map ℚ[con(H)] → ℝ[con(H)]. Recall that HS_{F,H}(V) is the image of HS^{F H}(V) under F [pr_{F}] for pr_{F} : con(H) → con_{F} (H) the canonical projection; HS^{F H}(V) is constant on the F -conjugacy classes of elements in H. This implies that the following diagram commutes

ℝ ⊗_{ℤ} R_{ℚ}(H) ⟶_{≅}^{id_{ℝ} ⊗_{ℤ} HS_{ℚ,H}} ℝ ⊗_{ℚ} class_{ℚ}(H)
ind(H)↓  ^{ind(H)}↓
ℝ ⊗_{ℤ} R_{ℝ}(H) ⟶_{≅}^{HS_{ℝ,H}}  class_{ℝ}(H),

where the left vertical arrow comes from inducing a ℚ-representation to a ℝrepresentation and the right vertical arrow is

ℝ⊗_{ℚ}class_{ℚ}(H) = ℝ⊗_{ℚ}ℚ[con_{ℚ}(H)] ⟶^{id_{ℝ} ⊗_{ℚ}(p_{H})^{∗}} ℝ⊗_{ℚ}ℚ[con_{ℝ}(H)] = class_{ℝ}(H).

Let

q(H): class_{ℝ}(H) → ℝ ⊗_{ℚ} class_{ℚ}(H)

be the map

ℝ[p_{H}]: ℝ[con_{ℝ}(H)] → ℝ[con_{ℚ}(H)] = ℝ ⊗_{ℚ} ℚ[con_{ℚ}(H)]

The q(H) ∘ ind(H) is bijective.

We get natural transformations of functors from Or(G; Fin) to ℝ-Mod :

ind: \underline{ℝ ⊗_{ℤ} R_{ℚ}} → \underline{ℝ ⊗_{ℤ} R_{ℝ}},
ind: \underline{ℝ ⊗_{ℚ} class_{ℚ}} → \underline{class_{ℝ}},
q: \underline{class_{ℝ}} → \underline{ℝ ⊗_{ℚ} class_{ℚ}}.

Also q ∘ ind is a natural equivalence. This implies that ind induces a split injection on the colimits

lim_{⟶}_{Π_{0}(G;X)}Q^{∗} ind: lim_{⟶}_{Π_{0}(G;X)}Q^{∗}\underline{ℝ ⊗_{ℚ} class_{ℚ}} → lim_{⟶}_{Π_{0}(G;X)}Q^{∗}\underline{class_{ℝ}}.

Since the following diagram commutes and has isomorphisms as vertical arrows

ℝ ⊗_{ℤ} H_{0}^{Or(G)}(X; \underline{R_{ℚ}})  ⟶^{ℝ⊗_{ℤ}e_{2}^{G}(X)}  ℝ ⊗_{ℤ} H_{0}^{Or(G)}(X; \underline{R_{ℝ}})
ℝ⊗_{ℤ}β_{ℚ}^{G}(X)↑≅  ^{ℝ⊗_{ℤ}β_{ℝ}^{G}(X)}↑≅
lim_{⟶}_{Π_{0}(G;X)}Q^{∗}\underline{ℝ ⊗_{ℤ} R_{ℚ}} ⟶^{lim_{⟶}_{Π_{0}(G;X)}Q^{∗} ind} lim_{⟶}_{Π_{0}(G;X)}Q^{∗}\underline{ℝ ⊗_{ℤ} R_{ℝ}}
ℝ⊗_{ℚ}HS_{ℚ}^{G}(X)↓≅  ^{ℝ⊗_{ℤ}HS_{ℝ}^{G}(X)}↓≅
lim_{⟶}_{Π_{0}(G;X)}Q^{∗}\underline{ℝ ⊗_{ℚ} class_{ℚ}} ⟶^{lim_{⟶}_{Π_{0}(G;X)}Q^{∗} ind} lim_{⟶}_{Π_{0}(G;X)}Q^{∗}\underline{class_{ℝ}} ,

the top horizontal arrow is split injective. Hence e_{2}^{G}(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 subgroup generated by g. Let y: G/⟨g⟩ → X be a G-map. Let F be one of the fields ℚ, ℝ and ℂ. Define

C_{ℚ}(g) = {g′ ∈ G, (g′)^{−1}gg′ ∈ ⟨g⟩}};
C_{ℝ}(g) = {g′ ∈ G, (g′)^{−1}gg′ ∈ {g, g^{−1}}};
C_{ℂ}(g) = {g′ ∈ G, (g′)^{−1}gg′ = g}.

Since C_{F} (g) is a subgroup of the normalizer N⟨g⟩ of ⟨g⟩ in G and contains ⟨g⟩, we can define a subgroup Z_{F} (g) ⊆ W⟨g⟩ by

Z_{F} (g) := C_{F} (g)/⟨g⟩.

Let Z_{F} (g)_{y} be the intersection of W⟨g⟩_{y} (see Notation 2.1) with Z_{F} (g), or, equivalently, the subgroup of Z_{F} (g) represented by elements g ∈ C_{F} (g) for which g · y(1⟨g⟩) and y(1⟨g⟩) lie in the same component of X^{⟨g⟩}.

It is useful to interpret the group C_{F} (g) as follows. Let G act on the set G_{f} := {g | g ∈ G, |g| < ∞} by conjugation. Then the projection G_{f} → con_{ℂ}(G) induces a bijection G\G_{f} ⟶^{≅} con_{ℂ}(G) and the isotropy group of g ∈ G_{f} is C_{ℂ}(g). Let G_{f}/{±1} be the quotient of G_{f} under the {±1}-action given by g ↦ g^{−1}. The conjugation action of G on G_{f} induces an action on G_{f}/{±1}. The projection G → con_{ℝ}(G) induces a bijection G\(G/{±1}) ⟶^{≅} con_{ℝ}(G) and the isotropy group of g · {±1} ∈ G_{f}/{±1} is C_{ℝ}(g). The set con_{ℚ}(G) is the same as the set {(L) ∈ consub(G) | L finite cyclic}. For g ∈ G_{f} the group C_{ℚ}(g) agrees with N⟨g⟩ and is the isotropy group of ⟨g⟩ in {L ⊆ G | 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_{ℚ}(G) = {(L) ∈ consub(G) | L finite cyclic}.

Consider (g)_{F} ∈ con_{F} (G). For the sequel we fix a representative g ∈ (g)_{F}. Consider an object of Π_{0}(G, X) of the special form y: G/⟨g⟩ → X. Let x: G/H → X be any object of Π_{0}(G, X). Recall that W⟨g⟩_{y} and thus the subgroup Z_{F} (g)_{y} act on mor(y, x). Define

α_{F} (y, x): Z_{F} (g)_{y}\ mor(y, x) → con_{F} (H)

by sending sending Z_{F} (g)_{y} · σ for a morphism σ: y → x, which given by a G-map σ: G/⟨g⟩ → G/H, to (σ(1⟨g⟩)^{−1}gσ(1⟨g⟩))_{F}. We obtain a map of sets

α_{F} (x) = ∐_{(g)_{F} ∈con(G)_{ℝ}} ∐_{\begin{matrix}Z_{F} (g)·C∈ \\ Z_{F} (g)\π_{0}(X^{⟨g⟩}) \end{matrix}} a(y(C), x):

∐_{(g)_{F} ∈con(G)_{F}} ∐_{\begin{matrix}Z_{F} (g)·C∈ \\ Z_{F} (g)\π_{0}(X^{⟨g⟩}) \end{matrix}} Z_{F} (g)_{y(C)}\ mor(y(C), x) ⟶^{≅} con_{F} (H), (3.16)

where we fix for each Z_{F} (g)·C ∈ Z_{F} (g)\π_{0}(X^{⟨g⟩}) a representative C ∈ π_{0}(X^{⟨g⟩}) and y(C) is a fixed morphism y(C): G/⟨g⟩ → X such that X^{⟨g⟩}(y) = C in π_{0}(X^{⟨g⟩}). The map α_{F} (x) is bijective by the following argument.

Consider (h)_{F} ∈ con_{F} (H). Let g ∈ G be the representative of the class (g)_{F} for which (g)_{F} = (h)_{F} holds in con_{F} (G). Choose g_{0} ∈ G with g_{0}^{−1}gg_{0} ∈ H and (g_{0}^{−1}gg_{0})_{F} = (h)_{F} in con_{F} (H). We get a G-map R_{g_{0}} : G/⟨g⟩ → G/H by mapping g′⟨g⟩ to g′g_{0}H. Let y = y(C): G/⟨g⟩ → X be the object chosen above for the fixed representative C of Z_{F} (g)·X^{⟨g⟩}(x∘R_{g_{0}}) ∈ Z_{F} (g)_{F} \π_{0}(X^{⟨g⟩}). Choose g_{1} ∈ Z_{F} (g) such that the G-map R_{g_{1}} : G/⟨g⟩ → G/⟨g⟩ sending g′⟨g⟩ to g′g_{1}⟨g⟩ defines a morphism R_{g_{1}} : y → x ∘ R_{g_{0}} in Π_{0}(G, X). Then σ := R_{g_{0}} ∘ R_{g_{1}} : y → x is a morphism such that

(σ(1⟨g⟩)^{−1}gσ(1⟨g⟩)_{F} = (h)_{F}

holds in con_{F} (H). This shows that α(x) is surjective.

Consider for i = 0, 1 elements (g_{i}) ∈ con_{F} (G), Z_{F} (g_{i}) · C_{i} ∈ Z_{F} (g_{i})\π_{0}(X^{⟨g_{i}⟩}) and Z_{F} (g_{i})_{y_{i}} · σ_{i} ∈ Z_{F} (g_{i})_{y_{i}}\ mor(y_{i}, x) for y_{i} = y(C_{i}) such that

(σ_{0}(1⟨g_{0}⟩)^{−1}g_{0}σ_{0}(1⟨g_{0}⟩))_{F} = (σ_{1}(1⟨g_{1}⟩)^{−1}g_{1}σ_{1}(1⟨g_{1}⟩))_{F}

holds in con_{F} (H). So we get two elements in the source of α_{F} (x) which are mapped to the same element under α_{F} (x). We have to show that these elements in the source agree. Choose g′_{i} ∈ G such that σ_{i} is given by sending g′′⟨g_{i}⟩ to g′′g′_{i}H. Then ((g′_{0})^{−1}g_{0}g′_{0})_{F} and ((g′_{1})^{−1}g_{1}g′_{1})_{F} agree in con_{F} (H). This implies (g_{0})_{F} = (g_{1})_{F} in con(G)_{F} and hence g_{0} = g_{1}. In the sequel we write g = g_{0} = g_{1}. Since ((g′_{0})^{−1}gg′_{0})_{F} and ((g′_{1})^{−1}gg′_{1})_{F} agree in con_{F} (H), there exists h ∈ H with

h^{−1}(g′_{0})^{−1}gg′_{0}h \{ \begin{matrix}∈ & ⟨(g′_{1})^{−1}gg′_{1}⟩ & if F = ℚ; \\ ∈ & \{(g′_{1})^{−1}gg′_{1}, (g′_{1})^{−1}g^{−1}g′_{1}\} & if F = ℝ; \\ = & (g′_{1})^{−1}gg′_{1} & if F = ℂ. \end{matrix}

We can assume without loss of generality that h = 1, otherwise replace g′_{0} by g′_{0}h. Put g_{2} := g′_{0}(g′_{1})^{−1}. Then g_{2} is an element in Z_{F} (g). Let σ_{2} : G/⟨g⟩ → G/⟨g⟩ be the G-map which sends g′′⟨g⟩ to g′′g_{2}⟨g⟩. We get the equality of G-maps σ_{0} = σ_{1} ∘ σ_{2}. Since σ_{i} is a morphism y_{i} → x for i = 0, 1, we conclude g′_{i} · x(1H) ∈ X^{⟨g⟩}(y_{i}) for i = 0, 1. This implies that g_{2} · X^{⟨g⟩}(y_{1}) = X^{⟨g⟩}(y_{0}) in π_{0}(X^{⟨g⟩}). This shows Z_{F} (g) · X^{⟨g⟩}(y_{0}) = Z_{F} (g) · X^{⟨g⟩}(y_{1}) in Z_{F} (g)\π_{0}(X^{⟨g⟩}) and hence y_{0} = y_{1}. Write in the sequel y = y_{0} = y_{1}. The G-map σ_{2} defines a morphism σ_{2} : y → y. We obtain an equality σ_{0} = σ_{1} ∘σ_{2} of morphisms y → x. We conclude Z_{F} (g)_{y} · σ_{0} = Z_{F} (g)_{y} · σ_{1} in Z_{F} (g)_{y}\π_{0}(X^{⟨g⟩}). This finishes the proof that α(x) is bijective.

Let \underline{con_{F}} be the covariant functor from Or(G; Fin) to the category of finite sets which sends an object G/H to con_{F} (H). The map α_{F} (x) is natural in x: G/H → X, in other words, we get a natural equivalence of functors from Π_{0}(G, X) to the category of finite sets. We obtain a bijection of sets

α_{F} : lim_{⟶}_{x: G/H→X∈Π_{0}(G,X)} ∐_{(g)_{F} ∈con_{F} (G)} ∐_{\begin{matrix}Z_{F} ⟨g⟩C∈ \\ Z_{F} ⟨g⟩\π_{0}(X^{⟨g⟩}) \end{matrix}} Z_{F} ⟨g⟩_{y}\ mor(y, x)

⟶^{≅} lim_{⟶}_{Π_{0}(G,X)}Q^{∗}\underline{con_{F}} .

One easily checks that lim_{⟶}_{x: G/H→X∈Π_{0}(G,X)} Z_{F} ⟨g⟩_{y}\ mor(y, x) consists of one element, namely, the one represented by Z_{F} ⟨g⟩_{y} ·id_{y} ∈ Z_{F} ⟨g⟩_{y}\ mor(y, y). Thus we obtain a bijection

α_{F}^{G}(X): ∐_{(g)_{F} ∈con_{F} (G)} Z_{F} ⟨g⟩\π_{0}(X^{⟨g⟩}) ⟶^{≅} lim_{⟶}_{Π_{0}(G,X)}Q^{∗}\underline{con_{F}} ,

which sends an element Z_{F} ⟨g⟩ · C in Z_{F} ⟨g⟩\π_{0}(X^{⟨g⟩}) to the class in the colimit represented by Z_{F} ⟨g⟩_{y} · id_{y} in Z_{F} ⟨g⟩_{y}\ mor(y, y) for any object y: G/⟨g⟩ → X for which Z_{F} (g) · X^{⟨g⟩}(y) = Z_{F} (g) · C holds in Z_{F} (g)\π_{0}(X^{⟨g⟩}). It yields an isomorphism of F -vector spaces denoted in the same way

α_{F}^{G}(X): ⨁_{(g)_{F} ∈con_{F} (G)} ⨁_{Z_{F} (g)\π_{0}(X^{⟨g⟩})} F ⟶^{≅} lim_{⟶}_{Π_{0}(G,X)}Q^{∗}\underline{class_{F}} . (3.17)

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 con_{ℚ}(G) is the same as the set {(L) ∈ consub(G) | L finite cyclic} and Z_{ℚ}(g) agrees with W⟨g⟩. Thus (3.17) becomes an isomorphism of ℚ-vector spaces

α_{ℚ}^{G}(X): ⨁_{(L)∈consub(G) L finite cyclic} ⨁_{WL\π_{0}(X^{L})} ℚ ⟶^{≅} lim_{⟶}_{Π_{0}(G,X)}Q^{∗}\underline{class_{ℚ}}. (3.18)

Denote by

pr: ⨁_{(H)∈consub(G)} ⨁_{WH\π_{0}(X^{H})} ℚ → ⨁_{(L)∈consub(G) L finite cyclic} ⨁_{WL\π_{0}(X^{L})} ℚ

the obvious projection. Let

D^{G}(X): ⨁_{(L)∈consub(G) L finite cyclic} ⨁_{WL\π_{0}(X^{L})} ℚ → ⨁_{(L)∈consub(G) L finite cyclic} ⨁_{WL\π_{0}(X^{L})} ℚ (3.19)

be the automorphism ⨁\begin{matrix}(L)∈consub(G) \\ L finite cyclic \end{matrix} \frac{| Gen(L)|}{|L|} · id, where Gen(L) is the set of

generators of L.

Recall the isomorphisms ch^{G}(X), β_{ℚ}^{G}(X), HS_{ℚ}^{G}(X)^{−1}, α_{ℚ}^{G}(X) and D^{G}(X) from (2.6), (3.2), (3.13) (3.18) and (3.19). We define

γ_{ℚ}^{G}(X): ⨁_{(L)∈consub(G) L finite cyclic} ⨁_{WL\π_{0}(X^{L})} ℚ ⟶^{≅} ℚ ⊗_{ℤ} H_{0}^{Or(G)}(X; \underline{R_{ℚ}}) (3.20)

to be the composition γ_{ℚ}^{G}(X) := β_{ℚ}^{G}(X) ∘ HS_{ℚ}^{G}(X)^{−1} ∘ α_{ℚ}^{G}(X) ∘ D^{G}(X).

Theorem 3.21 The following diagram commutes

ℚ ⊗_{ℤ} U^{G}(X)  ⟶^{id_{ℚ} ⊗_{ℤ}e_{1}^{G}(X)}  ℚ ⊗_{ℤ} H_{0}^{Or(G)}(X; \underline{R_{ℚ}})
id_{ℚ} ⊗_{ℤ} ch^{G}(X)↓≅  ^{γ_{ℚ}^{G}(X)}↑≅
⨁_{(H)∈consub(G)} ⨁_{WH\π_{0}(X^{H})} ℚ ⟶^{pr} ⨁\begin{matrix}(L)∈consub(G) \\ L finite cyclic \end{matrix} ⨁_{WL\π_{0}(X^{L})} ℚ

and has isomorphisms as vertical arrows.

The element id_{ℚ} ⊗_{ℤ}e_{1}^{G}(X)(χ^{G}(X)) agrees with the image under the isomorphism γ_{ℚ}^{G} of the element

{χ^{ℚWL_{C}}(C) | (L) ∈ consub(G), L finite cyclic, WL · C ∈ WL\π_{0}(X^{L})}

given by the various orbifold Euler characteristics of the WL_{C} -CW -complexes C, where WL_{C} is the isotropy group of C ∈ π_{0}(X^{L}) under the WL-action.

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

(α_{ℚ}^{G}(X) ∘ D^{G}(X) ∘ pr ∘ ch^{G}(X)) ([x])

= HS^{G}(X) ∘ β^{G}(X)^{−1} ∘ e_{1}^{G}(X)([x]). (3.22)

Given a finite cyclic subgroup L ⊆ G and a component C ∈ π_{0}(X^{L}) the element (D^{G}(X) ∘ pr ∘ ch^{G}(X)) ([x]) has as entry in the summand belonging to (L) and WL · C ∈ WL\π_{0}(X^{C}) the number

∑_{\begin{matrix}WL_{y(C)}·σ∈ \\ WL_{y(C)}\ mor(y(C),x) \end{matrix}} \frac{| Gen(L)|}{|L| · |(W C_{y(C)})_{σ}|},

where y(C): G/L → X is some object in Π_{0}(G, X) with X^{L}(y) = C in π_{0}(X^{L}).

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

α_{ℚ}(x): ∑_{(L)∈consub(G) L finite cyclic} ∐_{WL\π_{0}(X^{L})} WL_{y(C)}\ mor(y(C), x)

⟶^{≅} con_{ℚ}(H) = {(K) ∈ consub(H) | L cyclic}

which sends WL · σ ∈ WL\π_{0}(X^{L}) to (σ(1L)^{−1}Lσ(1L)). Let

u_{[x]} ∈ class_{ℚ}(H)

be the element which assigns to (K) ∈ con_{ℚ}(H) the number \frac{| Gen(L)|}{|L|·|(W C_{y(C)})_{σ}|} if σ ∈ mor(y(C), x) represents the preimage of (K) under the bijection α_{ℚ}(x). We conclude that (α_{ℚ}^{G}(X) ∘ D^{G}(X) ∘ pr ∘ ch^{G}(X)) ([x]) is given by the image under the structure map associated to the object x: G/H → X

class_{ℚ}(H) → lim_{⟶}_{Π_{0}(G,X)}Q^{∗}\underline{class_{ℚ}}

of the element u_{[x]} ∈ class_{ℚ}(H) above.

Consider (K) ∈ con_{ℚ}(H). Let σ ∈ mor(y(C), x) represent the preimage of (K) under the bijection α_{ℚ}(x). Choose g′ such that σ: G/⟨g⟩ → G/H is given by g′′⟨g⟩ ↦ g′′g′H. Let N_{H}K be the normalizer in H and W_{H}K := N_{H}K/K be the Weyl group of K ⊆ H. Define a bijection

f: (WL_{y(C)})_{σ} ⟶^{≅} W_{H}((g′)^{−1}Lg′), g′′L ↦ (g′)^{−1}g′′g′ · (g′)^{−1}Lg′.

The map is well-defined because of

(WL_{y(C)})_{σ} = {g′′L ∈ WL_{y(C)} | (g′)^{−1}g′′g′ ∈ H}

and the following calculation

((g′)^{−1}g′′g′)^{−1} (g′)^{−1}Lg′(g′)^{−1}g′′g′ = (g′)^{−1}(g′′)^{−1}g′(g′)^{−1}Lg′(g′)^{−1}g′′g′

= (g′)^{−1}(g′′)^{−1}Lg′′g′ = (g′)^{−1}Lg′.

One easily checks that f is injective. Consider h · (g′)^{−1}Lg′ in W_{H}((g′)^{−1}Lg′). Define g_{0} = g′h(g′)^{−1}. We have g_{0} ∈ WL. Since h · x(1H) = x(1H), we get g_{0}L ∈ WL_{y}. Hence g_{0}L is a preimage of h · (g′)^{−1}Lg′ under f. Hence f is bijective. This shows

|(WL_{y(C)})_{σ}| = |W_{H}((g′)^{−1}Lg′)|.

We conclude that u_{[x]} is the element

con_{ℚ}(H) = \{(K) ∈ consub(H) | K finite cyclic\} → ℚ, (K) ↦ \frac{| Gen(K)|}{|K| · |W_{H}K|}.

Since right multiplication with |H|^{−1} · ∑_{h∈H} h induces an idempotent ℚH - linear map ℚH → ℚH whose image is ℚ with the trivial H -action, the element HS_{ℚ,H}([ℚ]) ∈ class_{ℚ}(H) is given by

con_{ℚ}(H) → ℚ, (K) ↦ \frac{1}{|H|} · |\{h ∈ H | ⟨h⟩ ∈ (K)\}|.

From

|{h ∈ H | ⟨h⟩ ∈ (K)}| = | ∐_{K′⊆H,K′∈(K)} Gen(K′)|
= |{K′ ⊆ H, K′ ∈ (K)}| · | Gen(K)|
= \frac{|H|}{|N_{H}K|} · | Gen(K)|
= \frac{|H| · | Gen(K)|}{|K| · |W_{H}K|}

we conclude

u_{[x]} = HS_{ℚ,H}([ℚ]) ∈ class_{ℚ}(H).

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.

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}(M)  ⟵_{≅}^{U^{G}(i)}  U^{G}({∗})  ⟵_{≅}^{f_{1}} A(G)
e_{1}^{G}(M)↓  ^{e_{1}^{G}({∗})}↓  ^{j_{1}}↓
H_{0}^{Or(G)}(M; \underline{R_{ℚ}}) ⟵_{≅}^{H_{0}^{Or (G)}(i;\underline{R_{ℚ}})} H_{0}^{Or(G)}(\{∗\}; \underline{R_{ℚ}}) ⟵_{≅}^{f_{2}} R_{ℚ}(G)
e_{2}^{G}(M)↓  ^{e_{2}^{G}({∗})}↓  ^{j_{2}}↓
H_{0}^{Or(G)}(M; \underline{R_{ℝ}}) ⟵_{≅}^{H_{0}^{Or (G)}(i;\underline{R_{ℝ}})} H_{0}^{Or(G)}(\{∗\}; \underline{R_{ℝ}}) ⟵_{≅}^{f_{3}} R_{ℝ}(G)
e_{3}^{G}(M)↓  ^{e_{3}^{G}({∗})}↓≅  ^{j_{3}}↓≅
KO_{0}^{G}(M)  ⟵^{KO_{0}^{G}(i)}  KO_{0}^{G}({∗}) ⟵_{≅}^{id} KO_{0}^{G}({∗}),

where j_{1} 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 j_{2} is the change-of-coefficients homomorphism. The homomorphism KO_{0}^{G}(i): KO_{0}^{G}({∗}) → KO_{0}^{G}(M) is split injective, with a splitting given by the map KO_{0}^{G}(pr) induced by pr: M → {∗}. The map e_{3}^{G}({∗}) is bijective since the category Π_{0}(G, {∗}) has G → {∗} as terminal object. This implies that

e_{3}^{G}(M): H_{0}^{Or(G)}(M; \underline{R_{ℝ}}) → KO_{0}^{G}(M)

is split injective. We have already explained in the Section 3 that the map j_{2} : R_{ℚ}(G) → R_{ℝ}(G) is rationally injective. Since R_{ℚ}(G) is a torsion-free finitely generated abelian group, j_{2} is injective. Hence

e_{2}^{G}(M): H_{0}^{Or(G)}(M; \underline{R_{ℚ}}) → H_{0}^{Or(G)}(M; \underline{R_{ℝ}})

is injective. The upshot of this discussion is, that e_{1}^{G}(χ^{G}(M)) carries (integrally) the same information as Eul^{G}(M) because it is sent to Eul^{G}(M) by the injective map e_{3}^{G}(M) ∘ e_{2}^{G}(M).

Analyzing the difference between e_{1}^{G}(χ^{G}(M)) and χ^{G}(M) is equivalent to analyzing the map j_{1} : A(G) → Rep_{ℚ}(G), which sends χ^{G}(M) to the element given by ∑_{p≥0}(−1)^{p} · [H_{p}(M; ℚ)]. Recall that χ^{G}(M) ∈ A(G) is given by

χ^{G}(M) = ∑_{(H)∈consub(G)} χ(WH\M^{H}, WH\M^{>H})·G/H] = ∑_{p≥0}(−1)^{p}·♯_{p}(G/H),

where χ(WH\M^{H}, WH\M^{>H})) is the non-equivariant Euler characteristic and ♯_{p}(G/H) is the number of equivariant cells of the type G/H × D^{p} appearing in some G-CW -complex structure on M. The following diagram commutes (see Theorem 3.21)

A(G)  ⟶^{j_{1}}  R_{ℚ}(G)
ch^{G}↓  ↓^{HS}_{ℚ,G}
∏_{(H)∈consub(H)} ℚ ⟶_{pr} ∏\begin{matrix}(H) consub(G) \\ H cyclic \end{matrix} ℚ

where pr is the obvious projection, ch^{G}(S) has as entry for (H) ⊆ consub(G) the number \frac{1}{|WH|} · |S^{H}| and HS_{ℚ,G}(V) has as entry at (H) ∈ consub(G) for

cyclic H ⊆ G the number \frac{tr_{ℚ}(l_{h})}{|WH|}, where tr(l_{h}) ∈ ℚ is the trace of the endomorphism of the rational vector space V given by multiplication with h for some generator h ∈ H. The vertical arrows ch^{G} and HS_{ℚ,G} are rationally bijective and χ^{G}(M)(χ^{G}(M)) has as component belonging to (H) ∈ consub(G) the number χ^{ℚWH}(M^{H}) = \frac{χ(M^{H})}{|WH|} (see Lemma 2.8 and Lemma 3.12). This implies that ch^{G} and HS_{ℚ,G} are injective because their sources are torsion-free finitely generated abelian groups. Moreover, χ^{G}(M) ∈ A(G) carries integrally the same information as all the collection of Euler characteristics {χ(M^{H}) | (H) ∈ consub(H)}, whereas j_{1}(χ^{G}(M)) = ∑_{p≥0}(−1)^{p} · [H_{p}(M; ℚ)] carries integrally the same information as the collection of the Euler characteristics {χ(M^{H}) | (H) ∈ consub(H), H cyclic}. In particular j_{1} is injective if and only if G is cyclic. But any element u in A(G) occurs as χ^{G}(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 χ^{G}(M) and j_{1}(χ^{G}(M)) carry the same information for all such G-manifolds M if and only if G is cyclic.

From this discussion we conclude that Eul^{G}(M) does not carry torsion information in the case where G is finite and M^{H} is connected and non-empty for all H ⊆ G, since R_{ℝ}(G) is a torsion-free finitely generated abelian group. This 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 S_{3} be the symmetric group on 3 letters. It has the presentation

S_{3} = ⟨s, t | s^{2} = 1, t^{3} = 1, sts = t^{−1}⟩.

Let ℝ be the trivial 1-dimensional real representation of S_{3}. 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 S_{3}; we can take ℝ^{2} as the underlying real vector space of V , with s · (r_{1}, r_{2}) = (r_{2}, r_{1}) and t · (r_{1}, r_{2}) = (−r_{2}, r_{1} − r_{2}). Then ℝ, ℝ^{−} and V are the irreducible real representations of S_{3}, and ℝS_{3} is as an ℝS_{3}-module isomorphic to ℝ ⊕ ℝ^{−} ⊕ V ⊕ V . Let L_{2} be the cyclic group of order two generated by s and let L_{3} be the cyclic group of order three generated by t. Any finite subgroup of S_{3} is conjugate to precisely one of the subgroups L_{1} = {1}, L_{2}, L_{3} or S_{3}. One easily checks that V^{L_{2}} ≅ ℝ, V^{L_{3}} ≅ 0, (ℝ^{−})^{L_{2}} = 0 and (ℝ^{−})^{L_{3}} ≅ ℝ as real vector spaces. Put W = ℝ ⊕ ℝ^{−} ⊕ V . Then W^{S_{3}} ≅ ℝ, W^{L_{2}} ≅ W^{L_{3}} ≅ ℝ^{2} and W ≅ ℝ^{4} as real vector spaces. Let M be the closed 3-dimensional S_{3}-manifold SW. Then

M ≅ S^{3};
M^{L_{2}} ≅ S^{1};
M^{L_{3}} ≅ S^{1};
M^{S_{3}} ≅ S^{0}.

Since χ(M^{S_{3}}) ≠ 0, χ^{G}(M) ∈ U^{G}(M) cannot vanish. But since the fixed sets for all cyclic subgroups have vanishing Euler characteristic, Theorem 0.9 implies that Eul^{G}(M) is a torsion element in KO_{0}^{G}(M). We want to show that it has order precisely two.

Let x_{i}: S_{3}/L_{i} → X for i = 1, 2, 3 be a G-map. Let x_{−}: S_{3}/S_{3} → X and x_{+} : S_{3}/S_{3} → X be the two different G-maps for which M^{S_{3}} is the union of the images of x_{−} and x_{+}. Then x_{1}, x_{2}, x_{3}, x_{−} and x_{+} form a complete set of representatives for the isomorphism classes of objects in Π_{0}(S_{3}, M). Notice for i = 1, 2, 3 that mor(x_{i}, x_{−}) and mor(x_{i}, x_{+}) consist of precisely one element. Therefore we get an exact sequence

R_{ℝ}(ℤ/2) ⊕ R_{ℝ}(ℤ/3) \underrightarrow{( \begin{matrix}i_{2} & i_{3} \\ −i_{2} & −i_{3} \end{matrix} )} R_{ℝ}(G) ⊕ R_{ℝ}(G)

⟶^{s_{−}+s_{+}} lim_{⟶}_{Π_{0}(S_{3},M)}Q^{∗}\underline{R_{ℝ}} → 0, (4.1)

where i_{2} : R_{ℝ}(ℤ/2) → R_{ℝ}(S_{3}) and i_{3} : R_{ℝ}(ℤ/3) → R_{ℝ}(S_{3}) are the induction homomorphisms associated to any injective group homomorphism from ℤ/2 and ℤ/3 into S_{3} and s_{−} and s_{+} are the structure maps of the colimit belonging to the objects x_{−} and x_{+}. Define a map

δ: R_{ℝ}(G) → ℤ/2, λ_{ℝ} · [ℝ] + λ_{ℝ^{−}} · [ℝ^{−}] + λ_{V} · [V] ↦ \overline{λ_{ℝ}} + \overline{λ_{ℝ^{−}}} + \overline{λ_{V}} .

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

\begin{matrix}i_{2} : R_{ℝ}(ℤ/2) → R_{ℝ}(S_{3}), \\ μ_{ℝ} · [ℝ] + μ_{ℝ^{−}} · [ℝ^{−}] ↦ μ_{ℝ} · [ℝ] + μ_{ℝ^{−}} · [ℝ^{−}] + (μ_{ℝ} + μ_{ℝ^{−}}) · [V]. \end{matrix}

If ℝ denotes the trivial one-dimensional and W the 2-dimensional irreducible real ℤ/3-representation, then

i_{3} : R_{ℝ}(ℤ/3) → R_{ℝ}(S_{3}) μ_{ℝ} · [ℝ] + μ_{W} · [W] ↦ μ_{ℝ} · [ℝ] + μ_{ℝ} · [ℝ^{−}] + 2μ_{W} · [V].

This implies that the following sequence is exact

R_{ℝ}(ℤ/2) ⊕ R_{ℝ}(ℤ/3) ⟶^{i_{2}+i_{3}} R_{ℝ}(G) ⟶^{δ} ℤ/2 → 0.  (4.2)

We conclude from the exact sequences (4.1) and (4.2) above that the epimorphism

s_{−} + s_{+} : R_{ℝ}(G) ⊕ R_{ℝ}(G) → lim_{⟶}_{Π_{0}(S_{3},M)}Q^{∗}\underline{R_{ℝ}}

factorizes through the map

u := ( \begin{matrix}1 & 1 \\ 0 & δ \end{matrix} ) : R_{ℝ}(G) ⊕ R_{ℝ}(G) → R_{ℝ}(G) ⊕ ℤ/2

to an isomorphism

v: R_{ℝ}(G) ⊕ ℤ/2 ⟶^{≅} lim_{⟶}_{Π_{0}(S_{3},M)}Q^{∗}\underline{R_{ℝ}}.  (4.3)

Define a map

f: U^{S_{3}}(M) → R_{ℝ}(G) ⊕ ℤ/2

by

f([x_{1}]) := = ([ℝ[S_{3}]], 0);
f([x_{2}]) := = ([ℝ[S_{3}/L_{2}]], 0);
f([x_{3}]) := = ([ℝ[S_{3}/L_{3}]], 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

U^{G}(M) ⟶^{e_{1}^{G}(M)} H_{0}^{Or(G)}(M; \underline{R_{ℚ}}) ⟶^{e_{2}^{G}(M)} H_{0}^{Or(G)}(M; \underline{R_{ℝ}})

agrees with the composition

U^{G}(M) ⟶^{f} R_{ℝ}(G) ⊕ ℤ/2 ⟶^{v} lim_{⟶}_{Π_{0}(S_{3},M)}Q^{∗}\underline{R_{ℝ}} ⟶^{β_{ℝ}^{G}(M)} H_{0}^{Or(G)}(M; \underline{R_{ℝ}}).

We get

χ^{G}(M) = [x_{+}] + [x_{−}] − 2 · [x_{2}] − [x_{3}] + [x_{1}] ∈ U^{G}(M)  (4.4)

since the image of the element on the right side and the image of χ^{G}(M) under the injective character map ch^{G}(M) (see (2.6) and Lemma 2.8) agree by the following calculation

ch^{G}(χ^{G}(M)) = \frac{χ(M^{S_{3}}(x_{−}))}{|(WS_{3})_{x_{−}}|} · [x_{−}] + \frac{χ(M^{S_{3}}(x_{+}))}{|(WS_{3})_{x_{+}}|} · [x_{+}] + \frac{χ(M^{L_{2}})}{|WL_{2}|} · [x_{2}]

+\frac{χ(M^{L_{3}})}{WL_{3}} · [x_{3}] + \frac{χ(M)}{WL_{1}} · [x_{1}]

= [x_{−}] + [x_{+}]
= ch^{G}(M)([x_{+}] + [x_{−}] − 2 · [x_{2}] − [x_{3}] + [x_{1}]).

Now one easily checks

f(χ^{G}(M)) = (0, 1) ∈ R_{ℝ}(S_{3}) ⊕ ℤ/2.  (4.5)

Since R_{ℝ}(G) ⊕ ℤ/2 ⟶^{v} lim_{⟶}_{Π_{0}(S_{3},M)}Q^{∗}\underline{R_{ℝ}} ⟶^{β_{ℝ}^{G}(M)} H_{0}^{Or(G)}(M; \underline{R_{ℝ}}) is a composi-

tion of isomorphisms, we conclude that e_{2}^{G}(M) ∘ e_{1}^{G}(M)(χ^{G}(M)) is an element of order two in H_{0}^{Or(G)}(M; \underline{R_{ℝ}}). Since dim(M) ≤ 4 and dim(M^{>1}) ≤ 2, we conclude from Theorem 3.6 (d) that e_{3}^{G}(M): H_{0}^{Or(G)}(M; \underline{R_{ℝ}}) → KO_{0}^{G}(M) is injective. We conclude from Theorem 0.3 and Theorem 3.6 (a) that Eul^{G}(M) ∈ KO_{0}^{G}(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 Eul^{G}(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 Eul^{G}(M) to be non-zero, even rationally, while χ_{s}^{G}(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 S^{2} and the other of which is a surface N^{2} of genus 2, so that χ(N) = −2. Then

χ(M^{G}) = χ(S^{2} ⨿ N^{2}) = χ(S^{2}) + χ(N^{2}) = 2 − 2 = 0,

while also χ(M^{{1}}) = χ(M) = 0, so that χ_{s}^{G}(M) = 0 in A(G) and hence also χ_{a}^{G}(M) = 0.

For the construction, simply choose any bordism W^{3} between S^{2} and N^{2}, and let

M^{4} = (S^{2} × D^{2}) ∪_{S^{2}×S^{1}} (W^{3} × S^{1}) ∪_{N^{2}×S^{1}} (N^{2} × D^{2}) .

We give this the G-action which is trivial on the S^{2}, W^{3}, and N^{2} factors, and which is rotation by 2π/p on the D^{2} and S^{1} factors. Then the action of G is free except for M^{G}, which consists of S^{2} × {0} and of N^{2} × {0}. Furthermore, we have

χ(M) = χ (S^{2} × D^{2}) − χ (S^{2} × S^{1}) + χ (W^{3} × S^{1})
−χ (N^{2} × S^{1}) + χ (N^{2} × D^{2})
= 2 − 0 + 0 − 0 − 2 = 2 − 2 = 0.

Thus χ(M) = χ(M^{G}) = 0 and χ_{s}^{G}(M) = 0 in A(G). On the other hand, Eul^{G}(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 = S_{3} 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. (Note that G is the only noncyclic subgroup of G.) In fact, we can write down such an example explicitly; simply let Q = W′ ⊕ ℝ ⊕ ℝ, the S_{3}-representation ℝ ⊕ ℝ ⊕ ℝ ⊕ ℝ^{−} ⊕ V , and let M = SW′ be the unit ball in W′. Then each fixed set in M is a sphere of dimension bigger by 2 than in the example of 4.2, so M ≅ S^{5}, M^{L_{2}} ≅ S^{3}, M^{L_{3}} ≅ S^{3}, and M^{G} ≅ S^{2}. 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 Eul^{G}(M) = 0. On the other hand, since χ(M^{G}) = 2, χ_{s}^{G}(M) ≠ 0 in A(G).

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

Now let us consider an infinite (discrete) group G. Let \underline{E}G be a model for the classifying space for proper G-actions, i.e., a G-CW -complex \underline{E}G such that \underline{E}G^{H} is contractible (and in particular non-empty) for finite H ⊆ G and \underline{E}G^{H} 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 → \underline{E}G. This implies that all models for \underline{E}G are G-homotopy equivalent. If G is a word-hyperbolic group, its Rips complex is a model for \underline{E}G [21]. If G is a discrete subgroup of the Lie group L with finitely many path components, then L/K for a maximal compact subgroup K with the (left) G-action is a model for \underline{E}G (see [1, Corollary 4.14]). If G is finite, \{∗\} is a model for \underline{E}G.

Consider a proper smooth G-manifold M. Let f: M → \underline{E}G be a G-map. We obtain a commutative diagram

U^{G}(M)  ⟶^{U^{G}(f)}  U^{G}(\underline{E}G)  ⟶^{id}  U^{G}(\underline{E}G)
e_{1}^{G}(M)↓  ^{e_{1}^{G}(\underline{E}G)}↓  ^{j_{1}}↓
H_{0}^{Or(G)}(M; \underline{R_{ℚ}}) ⟶^{H_{0}^{Or (G)}(f;\underline{R_{ℚ}})} H_{0}^{Or(G)}(\underline{E}G; \underline{R_{ℚ}}) ⟶^{asmb_{1}}  K_{0}(ℚG)
e_{2}^{G}(M)↓  ^{e_{2}^{G}(\underline{E}G)}↓  ^{j_{2}}↓
H_{0}^{Or(G)}(M; \underline{R_{ℝ}}) ⟶^{H_{0}^{Or (G)}(f;\underline{R_{ℝ}})} H_{0}^{Or(G)}(\underline{E}G; \underline{R_{ℝ}}) ⟶^{asmb_{2}}  K_{0}(ℝG)
e_{3}^{G}(M)↓  ^{e_{3}^{G}(\underline{E}G)}↓  ^{j_{3}}↓
KO_{0}^{G}(M)  ⟶^{KO_{0}^{G}(f)}  KO_{0}^{G}(\underline{E}G) ⟶^{asmb_{3}} KO_{0}(C_{r}^{∗}(G; ℝ))

Here asmb_{i} for i = 1, 2, 3 are the assembly maps appearing in the Farrell-Jones Isomorphism Conjecture and the Baum-Connes Conjecture. The maps asmb_{i} for i = 1, 2 are the obvious maps lim_{⟶}_{Or(G;Fin )}\underline{R_{F}} → K_{0}(F G) for F = ℚ, ℝ

under the identifications R_{F} (H) = K_{0}(F H) for finite H ⊆ G and β_{F}^{G}(M) of (3.2). The Baum-Connes assembly map is given by the index with values in the reduced real group C^{∗}-algebra C_{r}^{∗}(G; ℝ). The Farrell-Jones Isomorphism Conjecture and the Baum-Connes Conjecture predict that asmb_{i} for i = 1, 2, 3 are bijective. The abelian group U^{G}(\underline{E}G) is the free abelian group with \{(H) ∈ consub(G) | |H| < ∞} as basis and the map j_{1} sends the basis element (H) to the class of the finitely generated projective ℚG-module ℚ[G/H]. The maps j_{2} and j_{3} are change-of-rings homomorphisms. The maps e_{2}^{G}(M) and e_{3}^{G}(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), H_{0}^{Or(G)}(f; \underline{R_{ℚ}}) and H_{0}^{Or(G)}(f; \underline{R_{ℝ}}) are bijective.

In contrast to the case where G is finite, the groups K_{0}(ℚG), K_{0}(ℝG) and KO_{0}(C_{r}^{∗}(G; ℝ)) may not be torsion free. The problem whether asmb_{i} is bijective for i = 1, 2 or 3 is a difficult and in general unsolved problem. Moreover, \underline{E}G is a complicated G-CW -complex for infinite G, whereas for a finite group G we can take \{∗\} as a model for \underline{E}G.

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