We interpret certain equivariant Kasparov groups as equivariant representable K-theory groups. We compute these groups via a classifying space and as K-theory groups of suitable σ-C-algebras. We also relate equivariant vector bundles to these σ-C-algebras and provide sufficient conditions for equivariant vector bundles to generate representable K-theory. Mostly we work in the generality of loca...

We give an equivariant version of the homotopy theory of crossed complexes. The applications generalize work on equivariant Eilenberg-Mac Lane spaces, including the non abelian case of dimension 1, and on local systems. It also generalizes the theory of equivariant 2-types, due to Moerdijk and Svensson. Further, we give results not just on the homotopy classification of maps but also on the hom...

We derive a formula for the η-invariants of equivariant Dirac operators on quotients of compact Lie groups, and for their infinitesimally equivariant extension. As an example, we give some computations for spheres. Quotients M = G/H of compact Lie groups form a very special class of manifolds, but yet they provide many important examples of Riemannian manifolds with non-negative sectional curva...

A refined form of the ‘Folk Theorem’ that a smooth action by a compact Lie group can be (canonically) resolved, by iterated blow up, to have unique isotropy type was established in [1] in the context of manifolds with corners; the canonical construction induces fibrations on the boundary faces of the resolution resulting in an ‘equivariant resolution structure’. Here, equivariant K-theory and t...

Abstract We give a formulation of deformation Dirac operator along orbits group action on possibly noncompact manifold to get an equivariant index and K-homology cycle representing the index. apply this framework Hamiltonian torus manifolds define geometric quantization from viewpoint theory. two applications. The first one is proof [Q,R]=0 type theorem, which can be regarded as Vergne conjectu...

We discuss several natural metrics on spaces of unbounded self– adjoint operators and their relations, among them the Riesz and the graph metric. We show that the topologies of the spaces of Fredholm operators resp. invertible operators depend heavily on the metric. Nevertheless we prove that in all cases the spectral flow is up to a normalization the only integer invariant of paths which is pa...

In this paper we study the Roe index of signature operator manifolds bounded geometry. Our main result is proof uniform homotopy invariance index. other words show that, given an orientation-preserving equivalence $$f:(M,g) \longrightarrow (N,h)$$ between two oriented geometry, $$f_\star (Ind_{Roe}D_M) = Ind_{Roe}(D_N)$$ . Moreover also that same holds if a group $$\Gamma $$ acts on M and N by ...

In this paper we characterize the fiber representations of equivariant complex vector bundles over a circle and classify these bundles. We also treat the triviality of equivariant complex vector bundles over a circle by investigating the extensions of representations. As a corollary of our results, we calculate the reduced equivariant K-group of a circle for any compact Lie group.

We give positive formulas for the restriction of a Schubert Class to a T -fixed point in the equivariant K-theory and equivariant cohomology of the Lagrangian Grassmannian. Our formulas rely on a result of GhorpadeRaghavan, which gives an equivariant Gröbner degeneration of a Schubert variety in the neighborhood of a T -fixed point of the Lagrangian Grassmannian.