Algebraic & Geometric Topology

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

Kimberly Druschel

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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 .

Article information

Source
Algebr. Geom. Topol., Volume 15, Number 1 (2015), 169-190.

Dates
Received: 1 November 2013
Revised: 23 May 2014
Accepted: 21 July 2014
First available in Project Euclid: 16 November 2017

Permanent link to this document
https://projecteuclid.org/euclid.agt/1510840910

Digital Object Identifier
doi:10.2140/agt.2015.15.169

Mathematical Reviews number (MathSciNet)
MR3325735

Zentralblatt MATH identifier
1321.57036

Subjects
Primary: 57R18: Topology and geometry of orbifolds 57R90: Other types of cobordism [See also 55N22]
Secondary: 55N32: Orbifold cohomology 57R65: Surgery and handlebodies

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

Citation

Druschel, Kimberly. Oriented orbifold vertex groups and cobordism and an associated differential graded algebra. Algebr. Geom. Topol. 15 (2015), no. 1, 169--190. doi:10.2140/agt.2015.15.169. https://projecteuclid.org/euclid.agt/1510840910


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