= Injective self-map of a circle
Every continuous injective map $g:S^1\to S^1$ is a <homeomorphism>. After normalizing $g(1)=1$ by a rotation, it is <homotopy>[homotopic] either to the identity or to complex conjugation, and has <degree of a continuous mapping>[degree] $1$ or $-1$.
To see this without assuming surjectivity, lift $t\mapsto g(e^{2\pi it})$ on $[0,1]$ through the <covering map> $u\mapsto e^{2\pi iu}$ to a path $G$ with $G(0)=0$. Injectivity makes $G$ injective on $[0,1)$, hence strictly monotone. Its endpoint $d=G(1)$ is an integer. Strict monotonicity excludes $d=0$; if $|d|>1$, the <intermediate value theorem> gives an interior integer value, contradicting injectivity of $g$. Thus $d=\pm1$, and $\exp(2\pi i((1-s)G(t)+sdt))$ descends to a <homotopy> from $g$ to $z\mapsto z^d$. If $g$ omitted a point, a continuous branch of the <complex argument> on the punctured circle would make it <null-homotopic>, contradicting this nonzero <degree of a continuous mapping>[degree]. Hence it is surjective. A continuous bijection from a <compact space> to a <Hausdorff space> is a <homeomorphism>.
Back to article page
