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Register a new userBordisms of manifolds with proper action of a discrete group: signatures and descriptions of $G$-bundles
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In this work the equivariant signature of a manifold with proper action of a discrete group is defined as an invariant of equivariant bordisms. It is shown that the computation of this signature can be reduced to its computation on fixed points sets equipped with their tubular neighborhoods. It is given a description of the equivariant vector bundles with action of a discrete group $G$ for the case when the action over the base is proper quasi-free, i.e. the stationary subgroup of any point is finite. The description is given in terms of some classifying space.
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Let $G$ be a Lie group and $GtoAut(G)$ be the canonical group homomorphism induced by the adjoint action of a group on itself. We give an explicit description of a 1-1 correspondence between Morita equivalence classes of, on the one hand, principal 2-group $[GtoAut(G)]$-bundles over Lie groupoids and, on the other hand, $G$-extensions of Lie groupoids (i.e. between principal $[GtoAut(G)]$-bundles over differentiable stacks and $G$-gerbes over differentiable stacks). This approach also allows us to identify $G$-bound gerbes and $[Z(G)to 1]$-group bundles over differentiable stacks, where $Z(G)$ is the center of $G$. We also introduce universal characteristic classes for 2-group bundles. For groupoid central $G$-extensions, we introduce Dixmier--Douady classes that can be computed from connection-type data generalizing the ones for bundle gerbes. We prove that these classes coincide with universal characteristic classes. As a corollary, we obtain further that Dixmier--Douady classes are integral.
We give an algebraic proof for the result of Eilenberg and Mac Lane that the second cohomology group of a simplicial group G can be computed as a quotient of a fibre product involving the first two homotopy groups and the first Postnikov invariant of G. Our main tool is the theory of crossed module extensions of groups.
We construct a regularized index of a generalized Dirac operator on a complete Riemannian manifold endowed with a proper action of a unimodular Lie group. We show that the index is preserved by a certain class of non-compact cobordisms and prove a gluing formula for the regularized index. The results of this paper generalize our previous construction of index for compact group action and the recent paper of Mathai and Hochs who studied the case of a Hamiltonian action on a symplectic manifold. As an application of the cobordism invariance of the index we give an affirmative answer to a question of Mathai and Hochs about the independence of the Mathai-Hochs quantization of the metric, connection and other choices.
In this paper, by use of techniques associated to Cobordism theory and Morse theory, we give a proof of Space-Form-Conjecture, i.e. a free action of a finite group on 3-manifold is equivalent to a linear action.
Let $G=SU(2)$ and let $Omega G$ denote the space of based loops in SU(2). We explicitly compute the $R(G)$-module structure of the topological equivariant $K$-theory $K_G^*(Omega G)$ and in particular show that it is a direct product of copies of $K^*_G(pt) cong R(G)$. (We intend to describe in detail the $R(G)$-algebra (i.e. product) structure of $K^*_G(Omega G)$ in a forthcoming companion paper.) Our proof uses the geometric methods for analyzing loop spaces introduced by Pressley and Segal (and further developed by Mitchell). However, Pressley and Segal do not explicitly compute equivariant $K$-theory and we also need further analysis of the spaces involved since we work in the equivariant setting. With this in mind, we have taken this opportunity to expand on the original exposition of Pressley-Segal in the hope that in doing so, both our results and theirs would be made accessible to a wider audience.
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