Notes

Notes will be posted occasionally to supplement the material in lectures and references.

• Disc Retraction

  Thursday, Aug 24, 2023.

  We sketch here some more details of the construction of a retraction $\rho: D^2 \to S^1$ from the disc $D^2$ to the circle $S^1$ given a map $f: D^2 \to D^2$ without fixed points. Geometric construction For a point $x\in D^2$, let $\lambda_x: [0, \infty) \to \mathbb{R}^2$ be the ray from $f(x)$ through $x$, with $\lambda_x(1)= 1$. By convexity, for $\lambda_x(s)\in (0, 1)$, $\lambda(s)$ is in interior of the disc $D^2$.

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