On $[X, U(n)]$ when $\mathrm{dim} X$ is $2n+1$

Abstract

In this paper we investigate the homotopy set $[X,U(n)]$, where $X$ is a finite CW-complex with its dimension $2n + 1$ and $U(n)$ is the unitary group. This homotopy set has the group structure and can be described as a central extension using ordinary cohomology and $K$-theory.