Harmonic maps M^3 --> S^1 and 2-cycles, realizing the Thurston norm

Abstract

Let $M^3$ be an oriented 3-manifold. We investigate when one of the fibers or a combination of fiber components, $F_{best}$, of a \emph{harmonic} map $f: M^3 \to S^1$ with Morse-type singularities delivers the Thurston norm $\chi_-([F_{best}])$ of its homology class $[F_{best}] \in H_2(M^3; \Z)$. In particular, for a map $f$ with connected fibers and any well-positioned oriented surface $\Sigma \subset M$ in the homology class of a fiber, we show that the Thurston number $\chi_-(\Sigma)$ satisfies an inequality $$\chi_-(\Sigma) \geq \chi_-(F_{best}) - \rho^\circ(\Sigma, f)\cdot Var_{\chi_-}(f).$$ Here the variation $Var_{\chi_-}(f)$ is can be expressed in terms of the $\chi_-$-invariants of the fiber components, and the twist $\rho^\circ(\Sigma, f)$ measures the complexity of the intersection of $\Sigma$ with a particular set $F_R$ of "bad" fiber components. This complexity is tightly linked with the optimal "$\tilde f$-height" of $\Sigma$, being lifted to the $f$-induced cyclic cover $\tilde M^3 \to M^3$. Based on these invariants, for any Morse map $f$, we introduce the notion of its \emph{twist} $\rho_{\chi_-}(f)$. We prove that, for a harmonic $f$, $\chi_-([F_{best}]) = \chi_-(F_{best})$, if and only if, $\rho_{\chi_-}(f) = 0$.