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

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, ${\mathcal{\mathscr{H}}}_{\ast }$. 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 ${\mathcal{\mathscr{H}}}_{\ast }$.