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Let $M$ be a connected closed oriented manifold with at least one orientation-reversing homeomorphism $M\to M$.

Let $S\subset M$ be a connected closed embedded submanifold of lower dimension. Let $f:M\to M$ be an orientation-preserving homeomorphism.

Is there an orientation-reversing homeomorphism $g:M\to M$ such that $f|_S=g|_S$?

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  • $\begingroup$ If you take the torus and three cohomologous nonintersecting loops? $\endgroup$
    – Ben McKay
    Apr 18, 2021 at 12:57
  • $\begingroup$ $CP^2$ does not admit any orientation reversing homeomorphism. $\endgroup$
    – Thomas Rot
    Apr 18, 2021 at 13:03
  • $\begingroup$ @ThomasRot then I suppose the question is not about that manifold $\endgroup$
    – malfat
    Apr 18, 2021 at 13:03
  • $\begingroup$ @BenMcKay I think your point is relevant. I added a condition. $\endgroup$
    – malfat
    Apr 18, 2021 at 13:06
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    $\begingroup$ Take $S$ the boundary of an embedded disk in a connected closed oriented surface $M$ of genus 1 or more. Let $f$ be the identity. The embedded disk must be preserved by $g$, with orientation on its boundary, so in its interior. $\endgroup$
    – Ben McKay
    Apr 18, 2021 at 13:12

1 Answer 1

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If I understand the question correctly, the lens space $L(5,2)$ provides a counterexample. Let $f$ be the identity, and let $S \subset L$ be a circle that generates the first homology. For concreteness sake, take $L$ to be the union of two solid tori of the form $S^1 \times D^2$, and let $S = S^1 \times pt$ in one of the solid tori. So the restriction of $f$ to $S$ is the identity.

From the classification of lens spaces, there is an orientation reversing self-homeomorphism $g:L \to L$. Any such $g$ acts by multiplication by $\pm 2$ on $H_1(L)$. In particular, its restriction to $S$ can't be the identity.

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