Some Applications of K-Theory

Abstract

Let α: S ^{n - 1} → S ^{n - 1} be a continuous map. Then $\tilde K_\mathbb{C}^{n - 1}\left( {{S^{n - 1}}} \right) \approx \mathbb{Z}$, and α induces an endomorphism of ℤ of the form x → λx where λ ∈ ℤ. The integer λ is called the degree of α, and is denoted by deg(α). In particular, if α is a homeomorphism, we have deg(α) = ± 1. It is possible to prove (cf. Hu [1]) that π _{n - 1}(S ^{n - 1}) ≈ ℤ, where the isomorphism is given by the degree. However, we do not require this result in this section.