AGT 5 (2005) Paper 44 (Abstract)

Discrete Morse theory and graph braid groups

Daniel Farley, Lucas Sabalka

Abstract. If Gamma is any finite graph, then the unlabelled configuration space of n points on Gamma, denoted UC^n(Gamma), is the space of n-element subsets of Gamma. The braid group of Gamma on n strands is the fundamental group of UC^n(Gamma). We apply a discrete version of Morse theory to these UC^n(Gamma), for any n and any Gamma, and provide a clear description of the critical cells in every case. As a result, we can calculate a presentation for the braid group of any tree, for any number of strands. We also give a simple proof of a theorem due to Ghrist: the space UC^n(Gamma) strong deformation retracts onto a CW complex of dimension at most k, where k is the number of vertices in Gamma of degree at least 3 (and k is thus independent of n).

## Wednesday, August 31, 2005

### Subelliptic spinc Dirac operators

Charlie Epstein kindly gave me permission to publish this paper on index theory for generalizations of the d-bar Neumann problem and its relationship to index theory for Fourier integral operators.

## Tuesday, August 30, 2005

## Tuesday, August 09, 2005

### [math/0508135] On the generalized Nielsen realization problem

[math/0508135] On the generalized Nielsen realization problem

Authors: Jonathan Block, Shmuel Weinberger

Subj-class: Geometric Topology

MSC-class: 57N

The main goal of this paper is to give the first examples of equivariant aspherical Poincare complexes, that are not realized by group actions on closed aspherical manifolds $M$. These will also provide new counterexamples to the Nielsen realization problem about lifting homotopy actions of finite groups to honest group actions. Our examples show that one cannot guarantee that a given action of a finitely generated group $\pi$ on Euclidean space extends to an action of $\Pi$, a group containing $\pi$ as a subgroup of finite index, even when all the torsion of $\Pi$ lives in $\pi$.

Authors: Jonathan Block, Shmuel Weinberger

Subj-class: Geometric Topology

MSC-class: 57N

The main goal of this paper is to give the first examples of equivariant aspherical Poincare complexes, that are not realized by group actions on closed aspherical manifolds $M$. These will also provide new counterexamples to the Nielsen realization problem about lifting homotopy actions of finite groups to honest group actions. Our examples show that one cannot guarantee that a given action of a finitely generated group $\pi$ on Euclidean space extends to an action of $\Pi$, a group containing $\pi$ as a subgroup of finite index, even when all the torsion of $\Pi$ lives in $\pi$.

## Monday, August 08, 2005

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