Volume 15, issue 1 (2015)

Download this article
Download this article For screen
For printing
Recent Issues

Volume 17
Issue 6, 3213–3852
Issue 5, 2565–3212
Issue 4, 1917–2564
Issue 3, 1283–1916
Issue 2, 645–1281
Issue 1, 1–643

Volume 16, 6 issues

Volume 15, 6 issues

Volume 14, 6 issues

Volume 13, 6 issues

Volume 12, 4 issues

Volume 11, 5 issues

Volume 10, 4 issues

Volume 9, 4 issues

Volume 8, 4 issues

Volume 7, 4 issues

Volume 6, 5 issues

Volume 5, 4 issues

Volume 4, 2 issues

Volume 3, 2 issues

Volume 2, 2 issues

Volume 1, 2 issues

The Journal
About the Journal
Subscriptions
Editorial Board
Editorial Interests
Editorial Procedure
Submission Guidelines
Submission Page
Author Index
To Appear
ISSN (electronic): 1472-2739
ISSN (print): 1472-2747
Oriented orbifold vertex groups and cobordism and an associated differential graded algebra

Kimberly Druschel

Algebraic & Geometric Topology 15 (2015) 169–190
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 Sn1G. This extends to a map dn: βn βn1 between the Z2 vector spaces whose bases are all such conjugacy classes in SO(n) and then SO(n 1). Using orbifold graphs, we prove d: β β is a differential and defines a homology, . We develop a map s: β 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 βn maps into kerdn and that this map is onto kerdn for n 4. We relate d to orbifold cobordism and surgery and show that ψ quotients to a map between oriented orbifold cobordism and .

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