nLab moduli space of framed manifolds

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.

(Randal-Williams 10, section 1.1, 2.2)

Related concepts

• cobordism category, cobordism hypothesis

moduli spaces

• moduli space of complex structures

• moduli space of conformal structures

• moduli space of curves

• moduli space of Riemannian metrics

• moduli stack of elliptic curves

• moduli space of Calabi-Yau spaces

• moduli stack of tori

• moduli stack of formal groups

• moduli space of connections, moduli space of flat connections

• Hitchin moduli space

• moduli space of monopoles

• moduli space of framed manifolds

• moduli space of bundles

• moduli space of (higher) line bundles

• quiver variety

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)