## Algebraic & Geometric Topology

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

### 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 ${S}^{n-1}\u2215G$. 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 ${\mathcal{\mathscr{H}}}_{\ast}$.

#### Article information

**Source**

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

**Dates**

Received: 1 November 2013

Revised: 23 May 2014

Accepted: 21 July 2014

First available in Project Euclid: 16 November 2017

**Permanent link to this document**

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

**Subjects**

Primary: 57R18: Topology and geometry of orbifolds 57R90: Other types of cobordism [See also 55N22]

Secondary: 55N32: Orbifold cohomology 57R65: Surgery and handlebodies

**Keywords**

orbifolds cobordism vertex groups finite subgroups of SO(n)

#### 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