Question about the definition of the degree of a continuous map $f : S^n \rightarrow S^n$.

Question

From Rotman's algebraic Topology:

A continuous map $f : S^n \rightarrow S^n$ (where $n \gt 0$) has degree $m$, denoted by $d(f) = m$, if $f_*:H_n(S^n) \rightarrow H_n(S^n)$ is multiplication by $m$.

What does "is multiplication by $m$" mean for n-th homology groups?

Answer 1

$H_n(S^n)\simeq\mathbb{Z}$ and so every homomorphism

$$f:H_n(S^n)\to H_n(S^n)$$

is actually a homomorphism

$$f:\mathbb{Z}\to \mathbb{Z}$$

and every such homomorphism is given by

$$f(x)=mx$$

for some unique $m\in\mathbb{Z}$.