Flexible exponent of geometric 3-manifolds and Legendrian maps of Seifert spaces

Abstract

A classical question in quantitative topology is to bound the mapping degree $\mathrm{deg}(f)$ in terms of its Lipchitz constant $\mathrm{Lip}(f)$. For a closed, oriented manifold $M$, the flexible exponent $\alpha (M)$ is the infimum of $\alpha \ge 0$ such that $|\mathrm{deg}f|\le C(\mathrm{Lip}f{)}^{\alpha }$ holds for all differentiable map $f:M\to M$. The flexible exponent measures how effectively a manifold can wrap itself through self-maps. For geometric 3-manifolds $M$ in the sense of Thurston, we give the complete result for $\alpha (M)$: \[ α(M)= \begin{cases} 3 & M \text{ modeled on } \mathbb S^3,\mathbb E^3,\mathbb S^2\times\mathbb E^1,\\ \frac83 & M \text{ modeled on Nil},\\ 2 & M \text{ modeled on Sol},\\ 1 & M \text{ modeled on }\mathbb H^2\times\mathbb E^1,\\ 0 & M \text{ modeled on } \mathbb H^3,\widetilde{\rm SL_2}. \end{cases} \] To prove $\alpha (M)=8/3$ for Nil 3-manifold $M$, we construct the so-called Legendrian map: a smooth self-map $f:M\to M$ such that $f$ is homotopic to the identity and $f$ maps all $S^1$-fibers into the orthogonal contact plane field simultaneously. Moreover, we prove that any Legendrian map must not be a diffeomorphism.

34 pages, comments welcomed