The Virtually Cyclic Classifying Space of the Heisenberg Group
Andrew Manion, Lisa Pham, and Jonathan Poelhuis
TL;DR
This paper constructs a minimal-dimension model for the virtually cyclic classifying space of the Heisenberg group, showing it has dimension 3, which matches its virtual cohomological dimension, and proving no lower dimension model exists.
Contribution
It provides the first explicit construction of a 3-dimensional model for the virtually cyclic classifying space of the Heisenberg group, establishing minimality.
Findings
The constructed model has dimension 3.
No model of lower dimension exists for the Heisenberg group.
The dimension matches the group's virtual cohomological dimension.
Abstract
We are interested in the relationship between the virtual cohomological dimension (or vcd) of a discrete group Gamma and the smallest possible dimension of a model for the classifying space of Gamma relative to its family of virtually cyclic subgroups. In this paper we construct a model for the virtually cyclic classifying space of the Heisenberg group. This model has dimension 3, which equals the vcd of the Heisenberg group. We also prove that there exists no model of dimension less than 3.
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Taxonomy
TopicsGeometric and Algebraic Topology · Topological and Geometric Data Analysis · Homotopy and Cohomology in Algebraic Topology