# Oriented Cobordism Cohomology

Edit: When I say cobordism, I mean oriented cobordism unless stated otherwise. Also note that I accidentally flip-flopped $$\Omega^*$$ and $$\Omega*$$ — $$\Omega^n$$ 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 $$X$$._

Let’s say $$M_1, M_2,$$ and $$X$$ are differentiable manifolds. We have a map $$f_1$$, and a map $$f_2$$. Let’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 — 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 “bounded” 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 “cobordant” (same boundary), and $$W$$ is their “cobordism” (the manifold that has them as its boundary). It’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!

Let’s make something using the recipe:

1. Take in a manifold $$X$$
2. Look the collection of maps from $$n$$-dimensional oriented manifolds to $$X$$ (up to cobordism of maps)
3. Equip the collection with connected sum.
4. 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 $$\Omega^{2n}(-)$$? I’ve been told that all manifolds with odd dimension are nilbordant, but I’m not sure I believe this.

But this is just a group! Let’s make it a ring! This particular ring is called “Thom’s ring” in the literature.

1. Take in a manifold $$X$$
2. Look the collection of maps from $$n$$-dimensional oriented manifolds to $$X$$ (up to cobordism of maps) for all $$n \in \mathbb{N}$$
3. Equip the collection with connected sum and cartesian product.
4. 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._

Written on January 28, 2015