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

DOI:
https://doi.org/10.1090/S0002-9947-04-03577-9

Published electronically:
October 5, 2004

MathSciNet review:
2110437

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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 .

**[C]**Calabi, E.,*An Intrinsic Characterization of Harmonic 1-Forms*, in*Global Analysis, Papers in Honor of K. Kodaira*, D.C. Spencer and S. Ianga, Eds., (1969), pp. 101-117. MR**40:6585****[Cr]**Crowell R. H.,*Genus of alternating link types*, Annals of Math., 69 (1959), pp. 258-275. MR**20:6103b****[FKL]**Farber, M., Katz, G., and Levine J.,*Morse Theory of Harmonic Forms,*Topology, vol. 37, No. 3 (1998), pp. 469-483. MR**99i:58026****[FHS]**Freedman, M., Hass, H., Scott, P.,*Least Area Incompressible Surfaces in 3-Manifolds*, Inventiones Mathematicae, No. 71 (1983), pp. 609-642. MR**85e:57012****[G]**Gabai, D.,*Foliations and the Topology of 3-Manifolds*, J. Differential Geometry, No. 18 (1983), pp. 445-503. MR**86a:57009****[H]**Hass, J.,*Surfaces Minimizing Area in Their Homology Class and Group Actions on 3-Manifolds*, Math. Zeitschrift, Vol. 199, (1988), pp. 501-509.MR**90d:57017****[HS]**Hass, J., Scott, P.,*The Existence of Least Area Surfaces in 3-Manifolds*, Trans. of AMS, vol. 310, No. 1, (1988), pp. 87-114. MR**90c:53022****[Ho]**Honda, K.,*A Note on Morse Theory of Harmonic 1-Forms*, Topology No 38 (1) (1999), pp. 223-233. MR**99h:58030****[K]**Katz, G.,*Harmonic Forms and Near-minimal Singular Foliations*, Comment. Math. Helvetici 77 (2002), 39-77. MR**2003b:57042****[Mc]**McMullen, C. T.,*The Alexander polynomial of a 3-manifold and the Thurston norm on cohomology*, Ann. Sci. École Norm. Sup. (4) 35 (2002), 153-171. MR**2003d:57044****[MY]**Meeks, W.H., Yau, S.T.,*The Existence of Embedded Minimal Surfaces and the Problem of Uniqueness*, Math. Z., No 179 (1982), pp. 151-168.MR**83j:53060****[M]**Milnor, J.,*Lectures on the**-cobordism Theorem*, Princeton University Press, 1965. MR**32:8352****[M1]**Milnor, J.,*Singular points of complex hypersurfaces*, Princeton University Press, 1968. MR**39:969****[Mur]**Murasugi K.,*On the genus of the alternating knot, I, II*, J. Math. Soc. Japan 10 (1958), pp. 94-105, 235-248. MR**20:6103a****[Su]**Sullivan, D.,*A Cohomological Characterization of Foliations Consisting of Minimal Surfaces*, Comment. Math. Helvetici, No.54 (1979), pp. 218-223. MR**80m:57022****[T]**Thurston, W.P.,*A Norm for the Homology of 3-Manifolds*, Memoirs of AMS, Vol. 59, No. 339 (1986), pp.100-130. MR**88h:57014****[W]**Wall, C.T.C.,*Surgery on Compact Manifolds*, Academic Press, London & New York, 1970. MR**55:4217**

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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:
https://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