Theorem 36.5:

The degree satisfies the following properties.

(i)

The degree of the identity map $S^n\longrightarrow S^n$ is $+1$.

(ii)

If $f$ and $g$ are two continuous maps from $S^n$ to itself then deg$\;(g\circ f) = ($deg$\;g)($deg$\;f)$.

(iii)

Homotopic maps from $S^n$ into itself have the same degree.

(iv)

If $f:S^n\longrightarrow S^n$ is a homotopy equivalence then degree of $f$ is $\pm 1$.

(v)

Any map homotopic to the constant map has degree zero.

(vi)

Any two maps $S^n\longrightarrow S^n$ having the same degree are homotopic (Theorem of H. Hopf).