Almost complex structures on spheres

TL;DR

This review paper discusses the well-known fact that only spheres S^2 and S^6 admit almost complex structures, using characteristic classes and K-theory, and aims to provide a clear, self-contained exposition of the proof.

Contribution

It offers a comprehensive, accessible review of the proof that only S^2 and S^6 admit almost complex structures, emphasizing clarity and motivation.

Findings

S^2 and S^6 are the only spheres with almost complex structures

The proof uses characteristic classes and Bott periodicity in K-theory

The paper provides a clear, self-contained exposition

Abstract

In this paper we review the well-known fact that the only spheres admitting an almost complex structure are S^2 and S^6. The proof described here uses characteristic classes and the Bott periodicity theorem in topological K-theory. This paper originates from the talk "Almost Complex Structures on Spheres" given by the second author at the MAM1 workshop "(Non)-existence of complex structures on S^6", held in Marburg from March 27th to March 30th, 2017. It is a review paper, and as such no result is intended to be original. We tried to produce a clear, motivated and as much as possible self-contained exposition.