Timothy A. Schroeder
$\ell^2$-homology and planar graphs.
Fri, 10/07/2011 - 15:01 — SomNambulAuthors: Timothy A. Schroeder
Subjects: Geometric Topology
AbstractIn his 1930 paper, Kuratowksi categorized planar graphs, proving that a
finite graph $\Gamma$ is planar if and only if it does not contain a subgraph
that is homeomorphic to $K_5$, the complete graph on 5 vertices, or $K_{3,3}$,
the complete bipartite graph on six vertices. In their 2001 paper, Davis and
Okun point out that the $K_{3,3}$ graph can be understood as the nerve of a
right-angled Coxeter system and prove that this graph is not planar using
results from $\ell^2$-homology. In this paper, we employ a similar method
proving $K_5$ is not planar.On the three-dimensional Singer Conjecture for Coxeter groups.
Wed, 09/02/2009 - 12:02 — SomNambulAuthors: Timothy A. Schroeder
Subjects: Geometric Topology
AbstractWe give a proof of the Singer conjecture (on the vanishing of reduced
$\ell^2$-homology except in the middle dimension) for the Davis Complex
$\Sigma$ associated to a Coxeter system $(W,S)$ whose nerve $L$ is a
triangulation of $\mathbb{S}^2$. We show that it follows from a theorem of
Andreev, which gives the necessary and sufficient conditions for a classical
reflection group to act on $\mathbb{H}^3$.
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