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 $f_2$.

Screenshot from 2015-01-27 15:55:55

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.

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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:

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  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.

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  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. 

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