---
date: 2022-02-23 18:45
modification date: Wednesday 23rd February 2022 18:45:08
title: "cobordism"
aliases: [cobordism, bordism, cobordism ring]

---

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


---

- Tags 
	- #todo/untagged
- Refs:
	- [Course notes](https://web.ma.utexas.edu/users/dafr/M392C-2012/Notes/) #resources/notes 
	- Proof of the [cobordism hypothesis](Unsorted/cobordism%20hypothesis.md) <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](homology%20of%20a%20point%20as%20a%20coefficient%20ring.md)
	- [ring spectra](Unsorted/ring%20spectra.md)
	- [TQFT](TQFT.md)

---

# cobordism

![](attachments/Pasted%20image%2020220403214242.png)
![](attachments/Pasted%20image%2020220403214317.png)

# Bordism category

![](attachments/Pasted%20image%2020220408100528.png)
![](attachments/Pasted%20image%2020220408100536.png)

See [TQFT](Unsorted/TQFT.md).

# Motivation

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

There is a sequence of spaces, $\MO(n)$, and maps $\Sigma \MO(n)\to \MO(n+1)$ such that the $k$th stable homotopy groups of this spectrum are isomorphic to the [stable framing](stable%20framing) you get the [stable homotopy groups of spheres](stable homotopy groups of spheres.md).

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

- [framed](framed.md) bordism classes of manifolds
$$
\Omega^{fr}_n \cong \pi_n^S
$$


# Invariants

![](attachments/Pasted%20image%2020220403214815.png)
![](attachments/Pasted%20image%2020220403214914.png)

![](attachments/Pasted%20image%2020220403214929.png)

# Theorems

![](attachments/Pasted%20image%2020220403215002.png)
![](attachments/Pasted%20image%2020220403215026.png)
![](attachments/Pasted%20image%2020220403215038.png)

# Ring spectra structure

![](attachments/Pasted%20image%2020220404090128.png)
![](attachments/Pasted%20image%2020220404092953.png)