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This question is about problem 2.C.5 from Allen Hatcher's Topology. The statement of the problem is as follows:

Let $M$ be a closed orientable surface embedded in $\mathbb{R}^3$ in such a way that reflection across a plane $P$ determines a homeomorphism $r:M\rightarrow M$ fixing $M\cap P$, a collection of circles. Is it possible to homotope $r$ to have no fixed points?

Progress I've made so far: The Euler characteristic of set of fixed points (ie $M\cap P$, a disjoint union of copies of $S^1$) is 0, which implies by some previous work (Hatcher 2.C.4) that the Lefshetz number of $r$ is 0. So we can't give a definitive no by using the Lefshetz theorem.

Issues I'm having: This question seems like there's so much room to do things that I'm getting a little freaked out. I think I might be able to do this homotoping by taking $[-\delta,\delta]\times S^1$, then for each circle fixed by $r$ rotating the $S^1$ a bit while fixing the ends- but this seems like the might be problems about actually doing this.

I'd appreciate any suggestions or comments about the problem or a means of solution.

EDIT: Turns out I can make this work by twisting the collars as described above. Apparently all I needed to do to figure this out was to write about it.

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If you've solved it, please mark it as solved. –  Ben Oct 23 '12 at 2:52

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