---
date: 2022-03-23 17:07
modification date: Wednesday 23rd March 2022 17:07:10
title: topological K theory
aliases: [topological K theory, "p-adic etale K theory", KO]

---

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


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- Tags 
	- #todo/untagged
- Refs:
	- #todo/add-references
- Links:
	- [prismatic](Unsorted/prismatic%20cohomology.md)

---

# topological K theory

![](attachments/Pasted%20image%2020220527232439.png)
![](attachments/Pasted%20image%2020220527232427.png)

Relation to [Bott Periodicity](Unsorted/Bott%20Periodicity.md):

![](attachments/Pasted%20image%2020220527232807.png)

Relation to [algebraic K theory](Unsorted/K-theory.md) via the [Serre-Swan](Unsorted/Serre-Swan.md) theorem:
![](attachments/Pasted%20image%2020220527233028.png)

![](attachments/Pasted%20image%2020220527233100.png)

## Infty cat definition

![](attachments/Pasted%20image%2020220510161326.png)

Some properties:
![](attachments/Pasted%20image%2020220510161336.png)

## Classical

See [HH](Unsorted/HH.md) and [Topological cyclic homology](Unsorted/Topological%20cyclic%20homology.md), [cyclotomic trace](cyclotomic%20trace), [coherent cohomoloy](coherent%20cohomoloy).
![](attachments/Pasted%20image%2020220323170712.png)

![](attachments/Pasted%20image%2020220323170842.png)

![](attachments/Pasted%20image%2020220503123853.png)
![](attachments/Pasted%20image%2020220503123952.png)

# Classifying space

![](attachments/Pasted%20image%2020220503124646.png)

# Stability

![](attachments/Pasted%20image%2020220503124743.png)

# Products

![](attachments/Pasted%20image%2020220503124818.png)

# E_infty space structure

![](attachments/Pasted%20image%2020220503130140.png)

# Adjoint bundles

![](attachments/Pasted%20image%2020220503131146.png)

# Examples

Setup for [Bott Periodicity](Unsorted/Bott%20Periodicity.md):

![](attachments/Pasted%20image%2020220508184630.png)
![](attachments/Pasted%20image%2020220508184706.png)
![](attachments/Pasted%20image%2020220508184720.png)

# Problems

See [Hopf invariant one](Unsorted/Hopf%20invariant%20one.md) and [[hairy ball theorem]].
![](attachments/Pasted%20image%2020220527232539.png)