Harmonic maps 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 | References | Similar Articles | Additional Information

Abstract: Let be an oriented 3-manifold. We investigate when one of the fibers or a combination of fiber components, , of a *harmonic* map with Morse-type singularities delivers the Thurston norm of its homology class .

In particular, for a map with connected fibers and any well-positioned oriented surface in the homology class of a fiber, we show that the Thurston number satisfies an inequality

Here the variation is can be expressed in terms of the -invariants of the fiber components, and the twist measures the complexity of the intersection of with a particular set of ``bad" fiber components. This complexity is tightly linked with the optimal ``-height" of , being lifted to the -induced cyclic cover .

Based on these invariants, for any Morse map , we introduce the notion of its *twist* . We prove that, for a harmonic , if and only if .

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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