## Algebraic & Geometric Topology

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

Kimberly Druschel

#### 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 $Sn−1∕G$. 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 $ℋ∗$.

#### Article information

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

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

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

#### 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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