cobordism spectrum

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cobordism spectrum

Form the set ΩOn of cobordism classes of n-manifolds.

The O denotes that fact that any bundle over a manifold M admits a Riemannian metric, so embedding M into R yields a Reduction of structure group.md) to $O(N to O(N) for some N.

Define addition as disjoint union and multiplication by the Cartesian product to form a graded ring ΩO, and it turns out that there is a spectrum MO such that ΩOnGrpπnMO

Idea: any manifold is determined by its embedding into R, take the normal bundle ν, form the Thom space Mν by collapsing the complement of the normal bundle. This yields a map S?Mν. Now use the fact that ν is an N-dimensional bundle and is classified by a map MBO(N), the classifying space for O(N)-bundles with universal bundle γN,. We take Thom spaces (?) to get a map MνBO(N)γN, then take limN[SN+n,BO(N)γN]unstable? This is independent of the embedding and only depends on the cobordism class of M, so we define MO(n):=BO(N)γN.

Equivariant structure

See orthogonal spectra. attachments/Pasted%20image%2020220510185755.png attachments/Pasted%20image%2020220510185834.png attachments/Pasted%20image%2020220510185854.png

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