# Oriented Cobordism Cohomology

*Edit: When I say cobordism, I mean oriented cobordism unless stated otherwise. Also note that I accidentally flip-flopped \(\Omega^*\) and \(\Omega_*\) â€” \(\Omega^n\) should be cobordism classes of maps from manifolds of codimension \(n\) to \(X\), and \(\Omega_n\) is cobordism classes of maps from manifolds of dimension \(n\) to \(X\).*

Letâ€™s say \(M_1, M_2,\) and \(X\) are differentiable manifolds. We have a map \(f_1\), and a map \(f_2\).

Letâ€™s think of a movie: Our first frame is the map from \(M_1 \to X\) Our last frame is the map from \(M_2 \to X\) What is in between? The instructions for how to deform \(M_1 \to X\) to look like \(M_2 \to X\).

Each movie frame is a map from \(M \to X\), which we can stack up (like a CT-scan â€” such that the first frame is \(M_1 \to X\), and the last is \(M_2 \to X\)) to get another map, \(W \to X \times [0,1]\) which is â€œboundedâ€ by the first and last map: \([0,1]\) is our time interval.

If such a \(W\) exists, \(M_1 \to X\) and \(M_2 \to X\) are â€œcobordantâ€ (same boundary), and \(W\) is their â€œcobordismâ€ (the manifold that has them as its boundary). Itâ€™s just an equivalence relation.

Note that we can also play with cobordism *without* \(X\) by looking at cobordism classes of \(n\)-dimensional oriented manifolds.

*Note that the pair of pants is just an aesthetically pleasing example of a cobordism, and there are many other examples!*

Letâ€™s make something using the recipe:

- Take in a manifold \(X\)
- Look the collection of maps from \(n\)-dimensional oriented manifolds to \(X\) (up to cobordism of maps)
- Equip the collection with connected sum.
- Output: \(n\)-dimensional oriented cobordism group (wrt X).

\(\Omega^*(X)\) is the slice category \(\text{Man}/X\), up to cobordism of maps.

*Why are the non-trivial groups \(\Omega^{2n}(-)\)? Iâ€™ve been told that all manifolds with odd dimension are nilbordant, but Iâ€™m not sure I believe this.*

But this is just a group! Letâ€™s make it a ring! *This particular ring is called â€œThomâ€™s ringâ€ in the literature.*

- Take in a manifold \(X\)
- Look the collection of maps from \(n\)-dimensional oriented manifolds to \(X\) (up to cobordism of maps) for all \(n \in \mathbb{N}\)
- Equip the collection with connected sum and cartesian product.
- Output: oriented cobordism ring (wrt X).

The graded ring \(\Omega^*(-)\) is a cohomology theory (graded by dimension).

*Notational aside: \(MU^*(-)\) and \(\Omega^*_{U}\) are alternative notations for \(\Omega^*(-)\) used in the literature.*