arXiv subjects

Courtney Thatcher

Publications and source records attributed to Courtney Thatcher.

3 recordsLinked to original sources

arXiv2022

Linear actions of $\mathbb{Z}/p\times\mathbb{Z}/p$ on $S^{2n-1}\times S^{2n-1}$

For an odd prime $p$, we consider free actions of $(\mathbb{Z}/p)^2$ on $S^{2n-1}\times S^{2n-1}$ given by linear actions of $(\mathbb{Z}/p)^2$ on $\mathbb{R}^{4n}$. Simple examples include a lens space cross a lens space, but $k$-invariant calculations show that other quotients exist. Using the tools of Postnikov towers and surgery theory, the quotients are classified up to homotopy by the $k$-invariants and up to homeomorphism by the Pontrjagin classes. We will present these results and demonstrate how to calculate the $k$-invariants and the Pontrjagin classes from the rotation numbers.

math.GT↗

arXiv2019

$\mathbb Z_{/p}\times \mathbb Z_{/p}$ actions on $S^n\times S^n$

We determine the homotopy type of quotients of $S^n \times S^n$ by free actions of $\mathbb Z_{/p} \times \mathbb Z_{/p}$ where $2p>n+3$. Much like free $\mathbb Z_{/p}$ actions, they can be classified via the first $p$-localized $k$-invariant, but there are restrictions on the possibilities, and these restrictions are sufficient to determine every possibility in the $n=3$ case. We use this to complete the classification of free $\mathbb Z_{/p} \times \mathbb Z_{/p}$ actions on $S^3 \times S^3$, for $p>3$, by reducing the problem to the simultaneous classification of pairs of binary quadratic forms. Although the restrictions are not sufficient to determine which $k$-invariants are realizable in general, they can sometimes be used to rule out free actions by groups that contain $\mathbb Z_{/p}\times\mathbb Z_{/p}$ as a normal Abelian subgroup.

math.GT↗

arXiv2014

Orbispaces and their Mapping Spaces via Groupoids: A Categorical Approach

In this paper, we give an accessible introduction to the theory of orbispaces via groupoids. We define a certain class of topological groupoids, which we call orbigroupoids. Each orbigroupoid represents an orbispace, but just as with orbifolds and Lie groupoids, this representation is not unique: orbispaces are Morita equivalence classes of orbigroupoids. We show how to formalize this equivalence by defining the category of orbispaces as a bicatecory of fractions from the category of orbigroupoids. We focus particularly on laying the groundwork for future work in creating mapping objects for orbispaces which are themselves orbispaces, and providing a concrete description of how this mapping space construction will get its orbispace structure. Throughout this paper, we illustrate our definitions and results with numerous examples which we hope will be useful in seeing how the categorical point of view is used to study these spaces.

math.CT↗