TITLE: Orientation reversal and restriction to submanifold of lower dimension
QUESTION [5 upvotes]: 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 $f|_S=g|_S$?

REPLY [7 votes]: 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.