#### Volume 15, issue 1 (2015)

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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
##### Abstract

We develop a homology of vertex groups as a tool for studying orbifolds and orbifold cobordism and its torsion. To a pair $\left(G,H\right)$ of conjugacy classes of degree-$n$ and degree-$\left(n-1\right)$ finite subgroups of $SO\left(n\right)$ and $SO\left(n-1\right)$ we associate the parity with which $H$ occurs up to $O\left(n\right)$ conjugacy as a vertex group in the orbifold ${S}^{n-1}∕G$. This extends to a map ${d}_{n}:{\beta }_{n}\to {\beta }_{n-1}$ between the ${Z}_{2}$ vector spaces whose bases are all such conjugacy classes in $SO\left(n\right)$ and then $SO\left(n-1\right)$. Using orbifold graphs, we prove $d:\beta \to \beta$ is a differential and defines a homology, ${\mathsc{ℋ}}_{\ast }$. We develop a map $s:{\beta }_{\ast }^{-}\to {\beta }_{\ast +1}^{-}$ 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 $\psi$ between the set of diffeomorphism classes of closed, locally oriented $n$–orbifolds and ${\beta }_{n}$ maps into $ker{d}_{n}$ and that this map is onto $ker{d}_{n}$ for $n\le 4$. We relate $d$ to orbifold cobordism and surgery and show that $\psi$ quotients to a map between oriented orbifold cobordism and ${\mathsc{ℋ}}_{\ast }$.

##### Keywords
orbifolds, cobordism, vertex groups, finite subgroups of SO(n)
##### Mathematical Subject Classification 2010
Primary: 57R18, 57R90
Secondary: 55N32, 57R65