---
date:  2021-04-26
modification date: Monday 25th October 2021 23:36:47
title: Why study K theory
aliases: [Why study K theory]

---

Tags: 
#todo #higher-algebra/K-theory 
Refs:
[K-theory](K-theory.md)

# Why study K theory

Tags: #higher-algebra/K-theory 

# Why study K-theory?

See [K-theory](K-theory.md). Examples of results gleaned from the [Adams operations](Adams%20operations).

It played a major role in the second proof of the [Aatiyah-Singer Index Theorem](Aatiyah-Singer%20Index%20Theorem.md) (circa 1962).
In 1955, Jean-Pierre Serre had used the analogy of vector bundles with [Serre's conjecture on vector bundles](Serre's%20conjecture%20on%20vector%20bundles.md),

Geometrically

- Finitely generated [projective (modules)](Unsorted/projective%20(modules).md) modules $\mapstofrom$ [vector bundles](vector%20bundles.md) over $\AA^N$, 
- Free modules $\mapstofrom$ trivial vector bundles. 
  
See also [Serre-Swan](Unsorted/Serre-Swan.md).
 
Affine space is topologically contractible, so It admits no non-trivial topological vector bundles. It also admits no non-trivial holomorphic vector bundles. Jean-Pierre Serre remarked that the corresponding question was not known for [algebraic vector bundle](algebraic%20vector%20bundle): 

> "It is not known whether there exist [projective (modules)](Unsorted/projective%20(modules).md) $A\dash$modules of [finite type](finite%20type.md), where $A = k[x_1, ..., x_n]$ is a polynomial ring over a field.