A self-homeomorphism of $L_{p,q}$ is isotopic to one which preserves heegaard splitting

Question

Consider the lens space $L_{p,q}$, which we can describe using its standard heegaard splitting, i.e. define $L_{p,q}$ as a quotient of two solid tori, identifying meridians on the boundary of one with $(p,q)$-curves on the other. Call $H_1$ and $H_2$ the two solid tori, and $T$ the torus with identified points. I am working on a proof that $L_{p,q}$ has no orientation reversing self-homeomorphism if $q$ is a quadratic residue mod $p$. A proof can be found here, but it can be greatly simplified if we assume that any homeomorphism $L_{p,q} \to L_{p,q}$ is isotopic to one which maps $T$ to $T$. In this textbook the author claims that this might always be the case, although he doesn´t give any reasons for this assumption, not even intuitive ones.

Does anybody know if and why this might be the case? It seems to me that we need to exclude cases in which, for example, one torus ends un knotted inside the other, but unfortunately I am very far from being able to justify anything rigorously.

Thanks for any insight.

Answer 1

Suppose that $L = L(p,q)$ is a lens space. Let $T$ be the standard genus one Heegaard splitting. Then $T$ is unique up to isotopy. It follows that every self-homeomorphism of $L$ preserves $T$ up to isotopy.

The uniqueness result was first proved by Bonahon and Otal in their paper "Scindements de Heegaard des espaces lenticulaires". A proof can also be found in Theorem 2.5 of Hatcher's three-manifold notes, available here: http://www.math.cornell.edu/~hatcher/3M/3Mdownloads.html