Harmonic maps 𝐌³→𝐒¹ and 2-cycles, realizing the Thurston norm

Katz, Gabriel

doi:10.1090/S0002-9947-04-03577-9

Harmonic maps $\mathbf {M^3 \rightarrow S^1}$ and 2-cycles, realizing the Thurston norm
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by Gabriel Katz PDF

Trans. Amer. Math. Soc. 357 (2005), 1177-1224 Request permission

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 harmonic map $f: M^3 \rightarrow S^1$ with Morse-type singularities delivers the Thurston norm $\chi _-([F_{best}])$ of its homology class $[F_{best}] \in H_2(M^3; \mathbb {Z})$. 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 $\tilde M^3 \rightarrow M^3$. Based on these invariants, for any Morse map $f$, we introduce the notion of its twist $\rho _{\chi _-}(f)$. We prove that, for a harmonic $f$, $\chi _-([F_{best}]) = \chi _-(F_{best})$ if and only if $\rho _{\chi _-}(f) = 0$.

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