= Fixed-point-free perturbation of a doubled-surface reflection
Double a compact orientable genus-zero <topological surface> with $g+1$ boundary circles. The resulting closed surface has genus $g$. The <involution> exchanging the halves fixes the seam circles and has local collar form $(s,t)\mapsto(s,-t)$. Compose it with a flow rotating each seam coordinate by a small nonzero amount, with speed supported in its collar. Equality with the initial point would force $t=0$, where the rotation is nonzero; off the collars the original reflection already has no fixed points. The new map is fixed-point-free and homotopic to the reflection through the flow. This gives a surface example in every genus where having fixed points is not a <homotopy> invariant.
Back to article page
