Oriented orbifold vertex groups and cobordism and an associated differential graded algebra

Druschel, Kimberly

doi:10.2140/agt.2015.15.169

Download this article
	For screen
	For printing

Recent Issues

The Journal

About the Journal

Editorial Board

Subscriptions

Submission Guidelines

Submission Page

Policies for Authors

Ethics Statement

ISSN 1472-2739 (online)

ISSN 1472-2747 (print)

Author Index

To Appear

Other MSP Journals

Oriented orbifold vertex groups and cobordism and an associated differential graded algebra

Kimberly Druschel

Algebraic & Geometric Topology 15 (2015) 169–190

DOI: 10.2140/agt.2015.15.169

Abstract

We develop a homology of vertex groups as a tool for studying orbifolds and orbifold cobordism and its torsion. To a pair $(G, H)$ of conjugacy classes of degree- $n$ and degree- $(n - 1)$ finite subgroups of $SO (n)$ and $SO (n - 1)$ we associate the parity with which $H$ occurs up to $O (n)$ conjugacy as a vertex group in the orbifold $S^{n - 1} ∕ G$ . This extends to a map $d_{n} : β_{n} \to β_{n - 1}$ between the $Z_{2}$ vector spaces whose bases are all such conjugacy classes in $SO (n)$ and then $SO (n - 1)$ . Using orbifold graphs, we prove $d : β \to β$ is a differential and defines a homology, $ℋ_{*}$ . We develop a map $s : β_{*}^{-} \to β_{* + 1}^{-}$ for a subcomplex of groups which admit orientation-reversing automorphisms. We then look at examples and algebraic properties of $d$ and $s$ , including that $d$ is a derivation. We prove that the natural map $ψ$ between the set of diffeomorphism classes of closed, locally oriented $n$ –orbifolds and $β_{n}$ maps into $ker d_{n}$ and that this map is onto $ker d_{n}$ for $n \leq 4$ . We relate $d$ to orbifold cobordism and surgery and show that $ψ$ quotients to a map between oriented orbifold cobordism and $ℋ_{*}$ .

Keywords

orbifolds, cobordism, vertex groups, finite subgroups of SO(n)

Mathematical Subject Classification 2010

Primary: 57R18, 57R90

Secondary: 55N32, 57R65

References

Publication

Received: 1 November 2013

Revised: 23 May 2014

Accepted: 21 July 2014

Published: 23 March 2015

Authors

Kimberly Druschel
Department of Mathematics and Computer Science Saint Louis University 221 N. Grand Blvd. Saint Louis, MO USA

Volume 15, issue 1 (2015)