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

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

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

Orbit Spaces of Gradient Vector Fields

We study orbit spaces of generalized gradient vector fields for Morse functions. Typically, these orbit spaces are non-Hausdorff. Nevertheless, they are quite structured topologically and are amenable to study. We show that these orbit spaces are locally contractible. We also show that the quotient map associated to each such orbit space is a weak homotopy equivalence and has the path lifting property.Comment: 16 pages, 4 figures; strengthened a main result (Corollary 3.5) and updated the introduction and the conclusio

The probability that xx and yy commute in a compact group

We show that a compact group $G$ has finite conjugacy classes, i.e., is an FC-group if and only if its center $Z(G)$ is open if and only if its commutator subgroup $G'$ is finite. Let $d(G)$ denote the Haar measure of the set of all pairs $(x,y)$ in $G \times G$ for which $[x,y] = 1$; this, formally, is the probability that two randomly picked elements commute. We prove that $d(G)$ is always rational and that it is positive if and only if $G$ is an extension of an FC-group by a finite group. This entails that $G$ is abelian by finite. The proofs involve measure theory, transformation groups, Lie theory of arbitrary compact groups, and representation theory of compact groups. Examples and references to the history of the discussion are given at the end of the paper.Comment: 17 pages; we have cut some points ; to appear in Math. Proc. Cambridge Phil. So

Abels's groups revisited

We generalize a class of groups introduced by Herbert Abels to produce examples of virtually torsion free groups that have Bredon-finiteness length m-1 and classical finiteness length n-1 for all 0 < m <= n. The proof illustrates how Bredon-finiteness properties can be verified using geometric methods and a version of Brown's criterion due to Martin Fluch and the author.Comment: 17 pages, 2 figures, v2 more detaile

Cohomology of the space of commuting n-tuples in a compact Lie group

Consider the space Hom(Z^n,G) of pairwise commuting n-tuples of elements in a compact Lie group G. This forms a real algebraic variety, which is generally singular. In this paper, we construct a desingularization of the generic component of Hom(Z^n,G), which allows us to derive formulas for its ordinary and equivariant cohomology in terms of the Lie algebra of a maximal torus in G and the action of the Weyl group. This is an application of a general theorem concerning G-spaces for which every element is fixed by a maximal torus.Comment: 11 pages Changes made: Implemented referee recommendations, in particular to use the Vietoris mapping theorem to generalize results and simplify argument