Algebraic Topology · Problem 2C.5

Cohomology — Allen Hatcher

Mathematics — Topology · reflection-surface-fixed-points

Problem

Let

M

be a closed orientable surface embedded in

\mathbb{R}^3

in such a way that reflection across a plane

P

defines a homeomorphism

r: M \to M

fixing

M \cap P

, a collection of circles. Is it possible to homotope

r

to have no fixed points?