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

Kimberly Druschel

Algebraic & Geometric Topology 15 (2015) 169–190

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 .

orbifolds, cobordism, vertex groups, finite subgroups of SO(n)
Mathematical Subject Classification 2010
Primary: 57R18, 57R90
Secondary: 55N32, 57R65
Received: 1 November 2013
Revised: 23 May 2014
Accepted: 21 July 2014
Published: 23 March 2015
Kimberly Druschel
Department of Mathematics and Computer Science
Saint Louis University
221 N. Grand Blvd.
Saint Louis, MO