Smith Theory for algebraic varieties

We show how an approach to Smith Theory about group actions on CW-complexes using Bredon cohomology can be adapted to work for algebraic varieties.Comment: Published by Algebraic and Geometric Topology at http://www.maths.warwick.ac.uk/agt/AGTVol4/agt-4-8.abs.htm

A new description of equivariant cohomology for totally disconnected groups

We consider smooth actions of totally disconnected groups on simplicial complexes and compare different equivariant cohomology groups associated to such actions. Our main result is that the bivariant equivariant cohomology theory introduced by Baum and Schneider can be described using equivariant periodic cyclic homology. This provides a new approach to the construction of Baum and Schneider as well as a computation of equivariant periodic cyclic homology for a natural class of examples. In addition we discuss the relation between cosheaf homology and equivariant Bredon homology. Since the theory of Baum and Schneider generalizes cosheaf homology we finally see that all these approaches to equivariant cohomology for totally disconnected groups are closely related

On the domain of the assembly map in algebraic K-theory

We compare the domain of the assembly map in algebraic K-theory with respect to the family of finite subgroups with the domain of the assembly map with respect to the family of virtually cyclic subgroups and prove that the former is a direct summand of the later.Comment: Published by Algebraic and Geometric Topology at http://www.maths.warwick.ac.uk/agt/AGTVol3/agt-3-35.abs.htm

Dehn Twists in Heegaard Floer Homology

We derive a new exact sequence in the hat-version of Heegaard Floer homology. As a consequence we see a functorial connection between the invariant of Legendrian knots and the contact element. As an application we derive two vanishing results of the contact element making it possible to easily read off its vanishing out of a surgery presentation in suitable situations.Comment: 61 pages, 39 figures; added details to several proofs. Version published by Algebr. Geom. Topol. 10 (2010), 465--52

Fundamental groups of topological stacks with slice property

The main result of the paper is a formula for the fundamental group of the coarse moduli space of a topological stack. As an application, we find simple general formulas for the fundamental group of the coarse quotient of a group action on a topological space in terms of the fixed point data. The formulas seem, surprisingly, to be new. In particular, we recover, and vastly generalize, results of Armstrong, Bass, Higgins, Rhodes

Topological geodesics and virtual rigidity

We introduce the notion of a topological geodesic in a 3-manifold. Under suitable hypotheses on the fundamental group, for instance word-hyperbolicity, topological geodesics are shown to have the useful properties of, and play the same role in several applications as, geodesics in negatively curved spaces. This permits us to obtain virtual rigidity results for 3-manifolds.Comment: Published by Algebraic and Geometric Topology at http://www.maths.warwick.ac.uk/agt/AGTVol1/agt-1-18.abs.htm

Poincare duality for K-theory of equivariant complex projective spaces

We make explicit Poincare duality for the equivariant K-theory of equivariant complex projective spaces. The case of the trivial group provides a new approach to the K-theory orientation

Concordance of Zp×ZpZ_p\times\Z_p actions on S4S^4

We consider locally linear Z_p x Z_p actions on the four-sphere. We present simple constructions of interesting examples, and then prove that a given action is concordant to its linear model if and only if a single surgery obstruction taking to form of an Arf invariant vanishes. We discuss the behavior of this invariant under various connected-sum operations, and conclude with a brief discussion of the existence of actions which are not concordant to their linear models

A homotopy-theoretic view of Bott-Taubes integrals and knot spaces

We construct cohomology classes in the space of knots by considering a bundle over this space and "integrating along the fiber" classes coming from the cohomology of configuration spaces using a Pontrjagin-Thom construction. The bundle we consider is essentially the one considered by Bott and Taubes, who integrated differential forms along the fiber to get knot invariants. By doing this "integration" homotopy-theoretically, we are able to produce integral cohomology classes. We then show how this integration is compatible with the homology operations on the space of long knots, as studied by Budney and Cohen. In particular we derive a product formula for evaluations of cohomology classes on homology classes, with respect to connect-sum of knots.Comment: 32 page