Last modified date: <%+ tp.file.last_modified_date() %>

• Tags
  • #todo/untagged
• Refs:
  • Course notes #resources/notes
  • Proof of the cobordism hypothesis https://arxiv.org/pdf/0905.0465.pdf#page=1
  • https://arxiv.org/pdf/2111.01095.pdf#page=2
• Links:
  • homology of a point as a coefficient ring
  • ring spectra
  • TQFT

cobordism

Bordism category

See TQFT.

Motivation

• Can I find a list of all manifolds up to diffeomorphism? No. 
  • Weaken to homeomorphism? Still no.
  • Weaken to homotopy equivalence. Still pretty much no.
  • A really weak notion: cobordism.

There is a sequence of spaces, \({\operatorname{MO}}(n)\), and maps \(\Sigma {\operatorname{MO}}(n)\to {\operatorname{MO}}(n+1)\) such that the \(k\)th stable homotopy groups of this spectrum are isomorphic to the stable framing you get the stable homotopy groups of spheres.

Note: bordism is one of the coarsest equivalence relations we can put on manifolds. Hope to understand completely!

• framed bordism classes of manifolds \begin{align*} \Omega^{fr}_n \cong \pi_n^S \end{align*}

Invariants

Theorems

Ring spectra structure