---
date: 2022-04-03 21:43
modification date: Sunday 3rd April 2022 21:43:43
title: "Pontrayagin-Thom"
aliases: [Pontrayagin-Thom, Thom spectra, MO]

---

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


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- Tags 
	- #todo/untagged
- Refs:
	- #todo/add-references
- Links:
	- [surgery](surgery.md)

---

# Pontrayagin-Thom

See [immersion](immersion), [normal bundle](Unsorted/normal%20bundle.md), [embedding](Unsorted/embedding.md), [tubular neighborhood](tubular%20neighborhood)
![](attachments/Pasted%20image%2020220403214345.png)
![](attachments/Pasted%20image%2020220403214424.png)
![](attachments/Pasted%20image%2020220403214443.png)

![](attachments/Pasted%20image%2020220403214518.png)
![](attachments/Pasted%20image%2020220403214531.png)


# Thom spectra

First example of [spectra](Unsorted/spectra.md):
![](attachments/Pasted%20image%2020220403214613.png)

Relation to [bordism](Unsorted/cobordism.md):
![](attachments/Pasted%20image%2020220403214742.png)

$H^{*}(M O)$ is free as module over the [Steenrod algebra](Steenrod%20algebra.md) $A^{*} .$

# MU and BP

See [Brown-Peterson spectra](Unsorted/Brown-Peterson%20spectra.md).
![](attachments/Pasted%20image%2020220404093607.png)

![](attachments/Pasted%20image%2020220404093539.png)