The Furstenberg lemma characterizes amenability

We characterize amenability in terms of the existence of equivariant assignments of measures for cocycles into the homeomorphism group of a single compact metric space.

     

Equivariant Brauer and Picard groups and a Chase–Harrison–Rosenberg exact sequence

The classical Chase–Harrison–Rosenberg exact sequence relates the Picard and Brauer groups of a Galois extension S of a commutative ring R to the group cohomology of the Galois group. We associate to each action of a locally compact group G on a locally compact space X two groups which we call the equivariant Picard group and the equivariant Brauer group. We then prove an analogue of the Chase–Harrison–Rosenberg exact sequence in the which the roles of the Picard and Brauer groups are played by their equivariant analogues.     

Equivariant Lefschetz maps for simplicial complexes and smooth manifolds

Let X be a locally compact space with a continuous proper action of a locally compact group G. Assuming that X satisfies a certain kind of duality in equivariant bivariant Kasparov theory, we can enrich the classical construction of Lefschetz numbers for self-maps to an equivariant K-homology class. We compute the Lefschetz invariants for self-maps of finite-dimensional simplicial complexes and smooth manifolds. The resulting invariants are independent of the extra structure used to compute them. Since smooth manifolds can be triangulated, we get two formulas for the same Lefschetz invariant in this case. The resulting identity is closely related to the equivariant Lefschetz Fixed Point Theorem of Lück and Rosenberg.     

Invariants de Maslov équivariants

We construct two cohomological invariants associated to pairs of Lagrangian sub-bundles of a symplectic bundle on a compact manifold upon which a compact Lie group is acting. The first invariant, which we call the classical equivariant Maslov H-invariant, provides an obstruction to Lagrangian transversality and lives in the Borel cohomology. The second invariant, which we call the equivariant Maslov U-invariant, generalises the author's results in K-Theory 13 (1998), 347–361 to the equivariant context and provides a necessary and sufficient condition for equivariant Lagrangian transversality, up to homotopic stability, and lives in the U-theory (intermediate between the real complex K-theories). As an application, we show that two Lagrangian sub-bundles of a symplectic bundle on a homogeneous space are always stably transverse.     

K-theory of equivariant quantization

Using an equivariant version of Connes? Thom isomorphism, we prove that equivariant K-theory is invariant under strict deformation quantization for a compact Lie group action.     

The Amenability of Measure Algebras

In this paper we shall prove that the measure algebra M(G) ofa locally compact group G is amenable as a Banach algebra ifand only if G is discrete and amenable as a group. Our contributionis to resolve a conjecture by proving that M(G) is not amenablein the case where the group G is not discrete. Indeed, we shallprove a much stronger result: the measure algebra of a non-discrete,locally compact group has a non-zero, continuous point derivationat a certain character on the algebra.     

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.      

Actions of Totally Disconnected Groups and Equivariant Singular Cohomology

In this work, we study the special properties of the equivariant singular cohomology of a G-space X, where G is a totally disconnected, locally compact group. We prove that any short exact sequence of coefficient systems for G, over a ring R, gives a long exact sequence of the associated equivariant singular cohomology modules. We establish the relationship between the ordinary singular cohomology modules and the equivariant singular cohomology modules with the natural contravariant coefficient system. Moreover, under some conditions, we give an isomorphism of the equivariant singular cohomology modules of the G-space X onto the ordinary singular cohomology modules of the orbit space X/G.     

Equivariant–Bivariant Chern Character for Profinite Groups

For the action of a locally compact and totally disconnected group G on a pair of locally compact spaces X and Y we construct, by sheaf theoretic means, a new equivariant and bivariant cohomology theory. If we take for the first space Y an universal proper G-action then we obtain for the second space its delocalized equivariant homology. This is in exact formal analogy to the definition of equivariant K-homology by Baum, Connes, Higson starting from the bivariant equivariant Kasparov KK-theory. Under certain basic finiteness conditions on the first space Y we conjecture the existence of a Chern character from the equivariant Kasparov KK-theory of Y and X into our cohomology theory made two-periodic which becomes an isomorphism upon tensoring the KK-theory with the complex numbers. This conjecture is proved for profinite groups G. An essential role in our construction is played by a bivariant version of Segal localization which we establish for KK-theory.     

On the Structure of Minimal Left Ideals in the Largest Compactification of a Locally Compact Group

This paper is centred around a single question: can a minimalleft ideal L in G^LUC, the largest semi-group compactificationof a locally compact group G, be itself algebraically a group?Our answer is no (unless G is compact). In deriving this conclusion,we obtain for nearly all groups the stronger result that nomaximal subgroup in L can be closed. A feature of our work isthat completely different techniques are required for the connectedand totally disconnected cases. For the former, we can relyon the extensive structure theory of connected, non-compact,locally compact groups to derive the solution from the commutativecase, using some reduction lemmas. The latter directly involvestopological dynamics; we construct a compact space and an actionof G on it which has pathological properties. We obtain otherresults as tools towards our main goal or as consequences ofour methods. Thus we find an extension to earlier work on therelationship between minimal left ideals in G^LUC and H^LUC whenH is a closed subgroup of G with G/H compact. We show that thedistal compactification of G is finite if and only if the almostperiodic compactification of G is finite. Finally, we use ourmethods to show that there is no finite subset of G^LUC invariantunder the right action of G when G is an almost connected groupor an IN-group.     

Equivariant strong shape

In this paper we propose a construction of the equivariant strong shape for compact metrizable G-spaces using an equivariant version of so-called cotelescopes and the concept of a fibrant G-space.     

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.     

The first homology of the group of equivariant diffeomorphisms and its applications

Let V be a representation space of a finite group G. We determinethe group structure of the first homology of the equivariantdiffeomorphism group of V. Then we can apply it to the calculationof the first homology of the corresponding automorphism groupsof smooth orbifolds, compact Hausdorff foliations, codimensionone or two compact foliations and the locally free S¹-actionson 3-manifolds. Received November 5, 2007.     

Cohomology Actions and Centralisers in Unitary Reflection Groups

In an earlier work, the second author proved a general formulafor the equivariant Poincaré polynomial of a linear transformationg which normalises a unitary reflection group G, acting on thecohomology of the corresponding hyperplane complement. Thisformula involves a certain function (called a Z-function below)on the centraliser CG(g), which was proved to exist only incertain cases, for example, when g is a reflection, or is G-regular,or when the centraliser is cyclic. In this work we prove theexistence of Z-functions in full generality. Applications includereduction and product formulae for the equivariant Poincarépolynomials. The method is to study the poset L(C_G(g)) of subspaceswhich are fixed points of elements of C_G(g). We show that thisposet has Euler characteristic 1, which is the key propertyrequired for the definition of a Z-function. The fact aboutthe Euler characteristic in turn follows from the ‘join-atom’property of L(C_G(g)), which asserts that if [X₁,..., X_k} isany set of elements of L(C_G(g)) which are maximal (set theoretically)then their setwise intersection lies in L(C_G(g)). 2000 Mathematical Subject Classification:primary 14R20, 55R80; secondary 20C33, 20G40.     

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.     

Continuity of Actions of Groups and Semigroups on Banach Spaces

It is shown that if a locally compact group acts isometricallyon a Banach space X leaving a closed subspace M invariant, andif the induced actions on M and X/M are strongly continuous,then the action on X is strongly continuous. Since this maybe of interest for one-parameter semigroups, similar resultsare proved for actions of suitable topological semigroups. Othergeneralizations are given for (suitable) non-isometric actions,non-locally compact groups, and non-Banach spaces; corollariesconcerning 1-cocycles and uniformly continuous actions are given.     

Fibrant extensions of free G-spaces

We show that any equivariant fibrant extension of a compact free G-space is also free. This result allows us to prove that the orbit space of any equivariant fibrant compact space E is also fibrant, provided that E has only one orbit type.     

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     

Reconstructing the orbit type stratification of a torus action from its equivariant cohomology

We investigate what information on the orbit type stratification of a torus action on a compact space is contained in its rational equivariant cohomology algebra. Regarding the (labelled) poset structure of the stratification, we show that equivariant cohomology encodes the subposet of ramified elements. For equivariantly formal actions, we also examine what cohomological information of the stratification is encoded. In the smooth setting, we show that under certain conditions—which in particular hold for a compact orientable manifold with discrete fixed point set—the equivariant cohomologies of the strata are encoded in the equivariant cohomology of the manifold.