moduli space of framed manifolds in nLab

Contents

Contents

Idea

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

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

|ℳ Σ fr| ϕ≃BΓ fr(Σ,ϕ), {\vert \mathcal{M}^{fr}_{\Sigma} \vert}_\phi \simeq B \Gamma^{fr}(\Sigma,\phi) \,,

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

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

|ℳ Σ fr|=∐[ϕ]|ℳ Σ fr| ϕ {\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)

• cobordism category, cobordism hypothesis

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)