Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

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


Author: Gabriel Katz
Journal: Trans. Amer. Math. Soc. 357 (2005), 1177-1224
MSC (2000): Primary 57M15, 57R45
Published electronically: October 5, 2004
MathSciNet review: 2110437
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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

\begin{displaymath}\chi_-(\Sigma) \geq \chi_-(F_{best}) - \rho^\circ(\Sigma, f)\cdot Var_{\chi_-}(f).\end{displaymath}

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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Additional Information

Gabriel Katz
Affiliation: Department of Mathematics, Bennington College, Bennington, Vermont 05201-6001
Address at time of publication: Department of Mathematics, Brandeis University, Waltham, Massachusetts 02454
Email: gabrielkatz@rcn.com, gkatz@bennington.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-04-03577-9
Received by editor(s): May 15, 2002
Received by editor(s) in revised form: October 10, 2003
Published electronically: October 5, 2004
Article copyright: © Copyright 2004 American Mathematical Society