Witten deformation and the equivariant index

Let M be a compact Riemannian manifold endowed with an isometric action of a compact, connected Lie group. The method of the Witten deformation is used to compute the virtual representation-valued equivariant index of a transversally elliptic, first order differential operator on M. The multiplicities of irreducible representations in the index are expressed in terms of local quantities associated to the isolated singular points of an equivariant bundle map that is locally Clifford multiplication by a Killing vector field near these points.      

The Chern character of a transversally elliptic symbol and the equivariant index

Let G be a compact Lie group acting on a compact manifold M. In this article, we associate to a G-transversally elliptic symbol on M a G-invariant generalized function on G, constructed in terms of equivariant closed differential forms on the cotangent bundle T ^* M. Oblatum 24-VII-1995     

Equivariant \eta-invariants on homogeneous spaces

Let D be a homogeneous Dirac operator on the quotient M = G/H of two compact connected Lie groups. We construct a deformation ofD and calculate its equivariant -invariant explicitly on the dense subset of G that acts freely onM. On , and differ only by a virtual character of . Moreover, if is a symmetric pair or if D is the untwisted Dirac operator, then on . We also sketch some applications of . Received August 7, 1998     

Actions of totally disconnected groups and equivariant singular homology

We study equivariant singular homology in the case of actions of totally disconnected locally compact groups on topological spaces. Theorem A says that if G is a totally disconnected locally compact group and X is a G-space, then any short exact sequence of covariant coefficient systems for G induces a long exact sequence of corresponding equivariant singular homology groups of the G-space X. In particular we consider the case where G is a totally disconnected compact group, i.e., a profinite group, and G acts freely on X. Of special interest is the case where G is a p-adic group, p a prime. The conjecture that no p-adic group, p a prime, can act effectively on a connected topological manifold, is namely known to be equivalent to the famous Hilbert-Smith conjecture. The Hilbert-Smith conjecture is the statement that, if a locally compact group G acts effectively on a connected topological manifold M, then G is a Lie group.     

Geometric lattice actions, entropy and fundamental groups

Let be a lattice in a noncompact simple Lie Group G, where . Suppose acts analytically and ergodically on a compact manifold M preserving a unimodular rigid geometric structure (e.g. a connection and a volume). We show that either the action is isometric or there exists a "large image" linear representation of . Under an additional assumption on the dynamics of the action, we associate to a virtual arithmetic quotient of full entropy. Received: December 14, 2000     

The periodicity in stable equivariant surgery

The classical surgery theory (see [5] and [23]) computes the structure set S_m (M, rel ?) of manifolds homotopy equivalent to M relative to the boundary. Siebenmann showed that in topological category, the structure set is 4-periodic: S_m(M, rel ?) ? S_m+4(M × D⁴, rel ?) up to a copy of ?; see [12]. Cappell and Weinberger gave a geometric interpretation of this periodicity in [8]. By using Weinberger's stratified surgery theory (see [24]), we extend this to an equivariant periodicity result for topological manifolds with homotopically stratified actions by compact Lie groups, with D⁴ replaced by the unit ball of certain group representations. In particular, if G is an odd order group acting on a topological manifold M, then the equivariant stable structure sets satisfy S (M, rel ?) ? S(M × D(?⁴ ? ?G), rel ?) up to copies of ?. © 1993 John Wiley & Sons, Inc.     

Equivariant Lefschetz number of differential operators

Let G be a compact Lie group acting on a compact complex manifold M by holomorphic transformations. We prove a trace density formula for the G-Lefschetz number of a holomorphic differential operator on M. We generalize the recent results of Engeli and the first author to orbifolds.     

Isometric Splitting for Actions of Simple Lie Groups on Pseudo-Riemannian Manifolds

We study the action of a connected noncompact simple Lie group G on a connected finite volume pseudo-Riemannian manifold M. Our main result provides a geometric splitting of the metric on such M that involves natural metrics on G. For suitable G such splitting is a warped product. For any G the splitting turns out to be a direct product in the ergodic case.     

Equivariant cohomology and localization for Lie algebroids

Let M be a manifold carrying the action of a Lie group G, and let A be a Lie algebroid on M equipped with a compatible infinitesimal G-action. Using these data, we construct an equivariant cohomology of A and prove a related localization formula for the case of compact G. By way of application, we prove an analog of the Bott formula.     

The degree of maps of free G-manifolds and the average

Suppose that G is a compact Lie group, M and N are orientable, free G-manifolds and f : M → N is an equivariant map. We show that the degree of f satisfies a formula involving data given by the classifying maps of the orbit spaces M/G and N/G. In particular, if the generator of the top dimensional cohomology of M/G with integer coefficients is in the image of the cohomology map induced by the classifying map for M, then the degree is one. The condition that the map be equivariant can be relaxed: it is enough to require that it be “nearly equivariant”, up to a positive constant. We will also discuss the G-average construction and show that the requirement that the map be equivariant can be replaced by a somewhat weaker condition involving the average of the map. These results are applied to maps into real, complex and quaternionic Stiefel manifolds. In particular, we show that a nearly equivariant map of a complex or quaternionic Stiefel manifold into itself has degree one. Dedicated to the memory of Jean Leray     

Equivariant geometric K-homology for compact Lie group actions

Let G be a compact Lie-group, X a compact G-CW-complex. We define equivariant geometric K-homology groups $K^{G}_{*}(X)$ , using an obvious equivariant version of the (M,E,f)-picture of Baum-Douglas for K-homology. We define explicit natural transformations to and from equivariant K-homology defined via KK-theory (the “official” equivariant K-homology groups) and show that these are isomorphisms.     

The Levi problem on strongly pseudoconvex G-bundles

Let G be a unimodular Lie group, X a compact manifold with boundary, and M the total space of a principal bundle G → M → X so that M is also a strongly pseudoconvex complex manifold. In this study, we show that if G acts by holomorphic transformations satisfying a local property, then the space of square-integrable holomorphic functions on M is infinite-dimensional.     

Géométries Lorentziennes de dimension 3 : classification et complétude

We classify three-dimensional Lorentz homogeneous spaces G/I having a compact manifold locally modeled on them. We prove a completeness result: any compact locally homogeneous Lorentz threefold M is isometric to a quotient of a Lorentz homogeneous space G/I by a discrete subgroup Γ of G acting properly and freely on G/I. Moreover, if I is noncompact, G/I is isometric to a Lie group L endowed with a left invariant Lorentz metric, where L is isomorphic to one of the following Lie groups:

Version kählérienne d'une conjecture de Robert J. Zimmer

Let M be a compact Kähler manifold. Let G be a connected simple real Lie group. Let Γ be a lattice in G. We prove the following: if the R-rank of G is strictly larger than the complex dimension of M any morphism from Γ to the group of holomorphic diffeomorphisms of M has finite image. This is a particular case in a conjecture of Robert J. Zimmer     

Ergodic Actions of Semisimple Lie Groups on Compact Principal Bundles

Let G = SL(n, ?) (or, more generally, let G be a connected, noncompact, simple Lie group). For any compact Lie group K, it is easy to find a compact manifold M, such that there is a volume-preserving, connection-preserving, ergodic action of G on some smooth, principal K-bundle P over M. Can M can be chosen independent of K? We show that if M = H/Λ is a homogeneous space, and the action of G on M is by translations, then P must also be a homogeneous space H′Λ′. Consequently, there is a strong restriction on the groups K that can arise over this particular M.     

Generalized Dirichlet to Neumann operator on invariant differential forms and equivariant cohomology

In recent work, Belishev and Sharafutdinov show that the generalized Dirichlet to Neumann (DN) operator Λ on a compact Riemannian manifold M with boundary ∂M determines de Rham cohomology groups of M. In this paper, we suppose G is a torus acting by isometries on M. Given X in the Lie algebra of G and the corresponding vector field XM on M, Witten defines an inhomogeneous coboundary operator dXM=d+ιXM on invariant forms on M. The main purpose is to adapt Belishev-Sharafutdinov?s boundary data to invariant forms in terms of the operator dXM in order to investigate to what extent the equivariant topology of a manifold is determined by the corresponding variant of the DN map. We define an operator ΛXM on invariant forms on the boundary which we call the XM-DN map and using this we recover the XM-cohomology groups from the generalized boundary data (∂M,ΛXM). This shows that for a Zariski-open subset of the Lie algebra, ΛXM determines the free part of the relative and absolute equivariant cohomology groups of M. In addition, we partially determine the ring structure of XM-cohomology groups from ΛXM. These results explain to what extent the equivariant topology of the manifold in question is determined by ΛXM.     

Nonnegatively curved fixed point homogeneous 5-manifolds

Let G be a compact Lie group acting effectively by isometries on a compact Riemannian manifold M with nonempty fixed point set Fix(M, G). We say that the action is fixed point homogeneous if G acts transitively on a normal sphere to some component of Fix(M, G), equivalently, if Fix(M, G) has codimension one in the orbit space of the action. We classify up to diffeomorphism closed, simply connected 5-manifolds with nonnegative sectional curvature and an effective fixed point homogeneous isometric action of a compact Lie group.     

Multiplicities of the integrable discrete series: The case of a nonuniform lattice in an R-rank one semisimple group

Let G be a noncompact connected simple Lie group of split-rank 1. Assume that G possesses a compact Cartan subgroup so that the discrete series for G is not empty. Let Γ be a nonuniform lattice in G. In this paper, we give an explicit formula for the multiplicity with which an integrable discrete series representation of G occurs in the space of cusp forms in L²(G/Γ).     

An effective algorithm for the cohomology ring of symplectic reductions

Let G be a compact torus acting on a compact symplectic manifold M in a Hamiltonian fashion, and T a subtorus of G. We prove that the kernel of is generated by a small number of classes satisfying very explicit restriction properties. Our main tool is the equivariant Kirwan map, a natural map from the G-equivariant cohomology of M to the G/T-equivariant cohomology of the symplectic reduction of M by T . We show this map is surjective. This is an equivariant version of the well-known result that the (nonequivariant) Kirwan map is surjective. We also compute the kernel of the equivariant Kirwan map, generalizing the result due to Tolman and Weitsman [TW] in the case T = G and allowing us to apply their methods inductively. This result is new even in the case that dim T = 1. We close with a worked example: the cohomology ring of the product of two , quotiented by the diagonal 2-torus action. Submitted: September 2001, Revised: December 2001, Revised: February 2002.