How to show that every continuous function $f:S^1\to S^1$ without fixed points is homotopic to the identity? (without using homology nor the concept of degree).
[Math] If $f:S^1\to S^1$ doesn’t have any fixed point then it is homotopic to the identity
algebraic-topologygeneral-topologyhomotopy-theory
Best Answer
$f : S^1 \to S^1$ be a map with no fixed points. Consider the projection of the straightline homotopy $$H(s, t) = \frac{(1-t)f(s) - ts}{\left \lVert(1-t)f(s)-ts\right \rVert}$$ between $f$ and the antipodal map $-\text{id}$, which is well-defined since $f(x) \neq x$ for all $x$. Compose this with the homotopy $$H(s, t) = e^{i\pi (1-t)} s$$ which rotates $-\text{id}$ to the identity map $\text{id}$. Thus, by transitivity, $f \sim \text{id}$.