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

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 $f: M^3 \to S^1$withMorse-typesingularitiesdeliverstheThurstonnorm$\chi_-([F_{best}])$ofitshomologyclass$[F_{best}] \in H_2(M^3; \Z)$. In particular, for a map $f$ with connected fibers and any well-positioned oriented surface $\mathrm{\Sigma }\subset M$ in the homology class of a fiber, we show that the Thurston number ${\chi }_{-}(\mathrm{\Sigma })$ satisfies an inequality $${\chi }_{-}(\mathrm{\Sigma })\ge {\chi }_{-}(F_{best})-{\rho }^{\circ }(\mathrm{\Sigma },f)\cdot Va{r}_{{\chi }_{-}}(f)\mathrm{.}$$ Here the variation $Va{r}_{{\chi }_{-}}(f)$ is can be expressed in terms of the ${\chi }_{-}$-invariants of the fiber components, and the twist ${\rho }^{\circ }(\mathrm{\Sigma },f)$ measures the complexity of the intersection of $\mathrm{\Sigma }$ with a particular set $F_R$ of “bad” fiber components. This complexity is tightly linked with the optimal “$\tilde{f}$-height” of $\mathrm{\Sigma }$, 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, $\rho_{\chi_-}(f) = 0$.