Geodesic
Oriented orbifold vertex groups and cobordism and an associated differential graded algebra
Druschel, Kimberly1
1Department of Mathematics and Computer Science, Saint Louis University, 221 N. Grand Blvd., Saint Louis, MO, USA
Algebraic and Geometric Topology, Tome 15 (2015) no. 1, pp. 169-190
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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 Sn−1∕G. This extends to a map dn: βn → βn−1 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, ℋ∗. 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 ℋ∗.

DOI : 10.2140/agt.2015.15.169
Classification : 57R18, 57R90, 55N32, 57R65
Keywords: orbifolds, cobordism, vertex groups, finite subgroups of SO(n)

Druschel, Kimberly  1

1 Department of Mathematics and Computer Science, Saint Louis University, 221 N. Grand Blvd., Saint Louis, MO, USA 
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Druschel, Kimberly. Oriented orbifold vertex groups and cobordism and an associated differential graded algebra. Algebraic and Geometric Topology, Tome 15 (2015) no. 1, pp. 169-190. doi: 10.2140/agt.2015.15.169

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