# Orientation reversal and restriction to submanifold of lower dimension

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 $$f|_S=g|_S$$?

• If you take the torus and three cohomologous nonintersecting loops? – Ben McKay Apr 18 at 12:57
• $CP^2$ does not admit any orientation reversing homeomorphism. – Thomas Rot Apr 18 at 13:03
• @ThomasRot then I suppose the question is not about that manifold – malfat Apr 18 at 13:03
• @BenMcKay I think your point is relevant. I added a condition. – malfat Apr 18 at 13:06
• 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. – Ben McKay Apr 18 at 13:12

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.