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We define and study an equivariant version of Farber's topological complexity for spaces with a given compact group action. This is a special case of the equivariant sectional category of an equivariant map, also defined in this paper. The…

This is the third paper of a series relating the equivariant twisted $K$-theory of a compact Lie group $G$ to the ``Verlinde space'' of isomorphism classes of projective lowest-weight representations of the loop groups. Here, we treat…

It is known that the Picard group of a complex manifold can be expressed as a Deligne cohomology group. One may wonder if the same holds for the Picard group of a smooth algebraic variety and Deligne-Beilinson cohomology but this is not…

For any compact and connected Lie group $G$ and any free abelian or free nilpotent group $\Gamma$ , we determine the cohomology of the path component of the trivial representation of the representation space (character variety)…

Let G be a locally compact group, let X be a universal proper G-space, and let Z be a G-equivariant compactification of X that is H-equivariantly contractible for each compact subgroup H of G. Let W be the resulting boundary. Assuming the…

We introduce and compare two approaches to equivariant homotopy theory in a topological or ordinary Quillen model category. For the topological model category of spaces, we generalize Piacenza's result that the categories of topological…

In this paper we define twisted equivariant K-theory for actions of Lie groupoids. For a Bredon-compatible Lie groupoid, this defines a periodic cohomology theory on the category of finite CW-complexes with equivariant stable projective…

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 sigma-C*-algebras. We also relate equivariant vector bundles…

An action of a complex reductive group $\mathrm G$ on a smooth projective variety $X$ is regular when all regular unipotent elements in $\mathrm G$ act with finitely many fixed points. Then the complex $\mathrm G$-equivariant cohomology…

The primary purpose of this paper concerns the relation of (compact) generalized manifolds to finite Poincar\'{e} duality complexes (PD complexes). The problem is that an arbitrary generalized manifold $X$ is always an ENR space, but it is…

We consider twisted equivariant K--theory for actions of a compact Lie group $G$ on a space $X$ where all the isotropy subgroups are connected and of maximal rank. We show that the associated rational spectral sequence \`a la Segal has a…

We show that for any cohomogeneity one continuous action of a compact connected Lie group $G$ on a closed topological manifold the equivariant cohomology equipped with its canonical $H^*(BG)$-module structure is Cohen-Macaulay. The proof…

In this article we provide a version of the Leray-Serre spectral sequence for equidimensional (i.e. smooth with all orbits of the same dimension) actions of compact connected Lie groups on compact manifolds. The main part of this article…

We compute the equivariant K-homology of the groups PSL_2 of imaginary quadratic integers with trivial and non-trivial class-group. This was done before only for cases of trivial class number. We rely on reduction theory in the form of the…

This text is an introduction to equivariant cohomology, a classical tool for topological transformation groups, and to equivariant intersection theory, a much more recent topic initiated by D. Edidin and W. Graham. It is based on lectures…

We use a variation of a classical construction of A. Hatcher to construct virtually all stable exotic smooth structures on compact smooth manifold bundles whose fibers have sufficiently large odd dimension (at least twice the base dimension…

Let $B$ denote the upper triangular subgroup of $SL_2(C)$, $T$ its diagonal torus and $U$ its unipotent radical. A complex projective variety $Y$ endowed with an algebraic action of $B$ such that the fixed point set $Y^U$ is a single point,…

For any compact Lie group $G$ and any $n$ we construct a smooth $G$-manifold $U_n(G)$ such that any smooth $n$-dimensional $G$-manifold can be embedded in $U_n(G)$ with a trivial normal bundle. Furthermore, we show that such embeddings are…

We show by studying the symplectic geometry of the extended moduli space that the intersection cohomology of the representation space $Hom(\pi_1(\Sigma),G)/G$ for a simply connected compact Lie group $G$ is naturally embedded into the $G$…

Following Hopkins and Singer, we give a definition for the differential equivariant K-theory of a smooth manifold acted upon by a finite group. The ring structure for differential equivariant K-theory is developed explicitly. We also…