Contents

# Contents

## Idea

The moduli space $\mathcal{M}^{fr}_{\Sigma}$ of framings on a given manifold $\Sigma$.

The homotopy type of the connected component of any fixed framing $\phi$ is

${\vert \mathcal{M}^{fr}_{\Sigma} \vert}_\phi \simeq B \Gamma^{fr}(\Sigma,\phi) \,,$

where $\Gamma^{fr}(\Sigma,\phi)\hookrightarrow \Gamma(\Sigma, or(\phi))$ is the subgroup of the mapping class group of $\Sigma$ (for the given orientation) which fixes the isotopy class of the framing $[\phi]$.

In particular this is a homotopy 1-type for every $\phi$, and so the whole

${\vert \mathcal{M}^{fr}_{\Sigma} \vert} = \underset{[\phi]}{\coprod} {\vert \mathcal{M}^{fr}_{\Sigma} \vert}_\phi$

is a homotopy 1-type.

## References

### General

• Schumacher, Tsuji, section 8 of Quasi-projectivity of moduli spaces of polarized varieties pdf

### Moduli space of framed surfaces

• Oscar Randal-Williams, Homology of the moduli spaces and mapping class groups of framed, r-Spin and Pin surfaces (arXiv:1001.5366)

• Daan Krammer, A Garside like structure on the framed mapping class group, 2007 (pdf)

Last revised on October 14, 2019 at 12:52:07. See the history of this page for a list of all contributions to it.