Harmonic maps <IMG BORDER="0" SRC="/tran/2005-357-03/S0002-9947-04-03577-9/gif-title0/img1.gif" > and 2-cycles, realizing the Thurston norm期刊界 All Journals 搜尽天下杂志传播学术成果专业期刊搜索期刊信息化学术搜索

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Harmonic maps and 2-cycles, realizing the Thurston norm

Authors:	Gabriel Katz

Institution:	Department of Mathematics, Bennington College, Bennington, Vermont 05201-6001

Abstract:	Let 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 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 of ``bad" fiber components. This complexity is tightly linked with the optimal `` $\tilde f$ -height" of $\Sigma$ , being lifted to the -induced cyclic cover $\tilde M^3 \rightarrow M^3$ . Based on these invariants, for any Morse map , we introduce the notion of its twist $\rho_{\chi_-}(f)$ . We prove that, for a harmonic , $\chi_-(F_{best}]) = \, \chi_-(F_{best})$ if and only if $\rho_{\chi_-}(f) = 0$ .

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