Complex K-theory of 4-complexes

TL;DR

This paper reviews key properties of the complex K-theory ring for 4-dimensional CW-complexes, highlighting unique features that distinguish this dimension from higher ones, especially regarding vector bundles and cohomology.

Contribution

It summarizes specific facts about the structure of K^0(X) for 4-complexes, emphasizing the dimension's special features and how they differ from higher dimensions.

Findings

Every complex vector bundle is determined by Chern classes.

All even cohomology classes are realizable as Chern classes.

K^0(X) is fully determined by the even cohomology ring.

Abstract

This short note summarizes a number of facts about the ring $K^0(X)$ for $X$ a $4$-dimensional CW-complex. Unusual features of this dimension are that every complex vector bundle is determined up to stable isomorphism by its Chern classes, that every even cohomology class arises as a Chern class of a vector bundle, and that $K^0(X)$ is completely determined as a ring by knowledge of the even-dimensional cohomology ring $H^{\text{even}}(X; \mathbb Z)$. (All of these fail in high dimensions.)