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

Full-text PDF Free Access

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 .

**[C]**Eugenio Calabi,*An intrinsic characterization of harmonic one-forms*, Global Analysis (Papers in Honor of K. Kodaira), Univ. Tokyo Press, Tokyo, 1969, pp. 101–117. MR**0253370****[Cr]**Richard Crowell,*Genus of alternating link types*, Ann. of Math. (2)**69**(1959), 258–275. MR**0099665****[FKL]**Michael Farber, Gabriel Katz, and Jerome Levine,*Morse theory of harmonic forms*, Topology**37**(1998), no. 3, 469–483. MR**1604870**, 10.1016/S0040-9383(97)82730-9**[FHS]**Michael Freedman, Joel Hass, and Peter Scott,*Least area incompressible surfaces in 3-manifolds*, Invent. Math.**71**(1983), no. 3, 609–642. MR**695910**, 10.1007/BF02095997**[G]**David Gabai,*Foliations and the topology of 3-manifolds*, J. Differential Geom.**18**(1983), no. 3, 445–503. MR**723813****[H]**Joel Hass,*Surfaces minimizing area in their homology class and group actions on 3-manifolds*, Math. Z.**199**(1988), no. 4, 501–509. MR**968316**, 10.1007/BF01161639**[HS]**Joel Hass and Peter Scott,*The existence of least area surfaces in 3-manifolds*, Trans. Amer. Math. Soc.**310**(1988), no. 1, 87–114. MR**965747**, 10.1090/S0002-9947-1988-0965747-6**[Ho]**Ko Honda,*A note on Morse theory of harmonic 1-forms*, Topology**38**(1999), no. 1, 223–233. MR**1644028**, 10.1016/S0040-9383(98)00018-4**[K]**Gabriel Katz,*Harmonic forms and near-minimal singular foliations*, Comment. Math. Helv.**77**(2002), no. 1, 39–77. MR**1898393**, 10.1007/s00014-002-8331-5**[Mc]**Curtis T. McMullen,*The Alexander polynomial of a 3-manifold and the Thurston norm on cohomology*, Ann. Sci. École Norm. Sup. (4)**35**(2002), no. 2, 153–171 (English, with English and French summaries). MR**1914929**, 10.1016/S0012-9593(02)01086-8**[MY]**William W. Meeks III and Shing Tung Yau,*The existence of embedded minimal surfaces and the problem of uniqueness*, Math. Z.**179**(1982), no. 2, 151–168. MR**645492**, 10.1007/BF01214308**[M]**John Milnor,*Lectures on the ℎ-cobordism theorem*, Notes by L. Siebenmann and J. Sondow, Princeton University Press, Princeton, N.J., 1965. MR**0190942****[M1]**John Milnor,*Singular points of complex hypersurfaces*, Annals of Mathematics Studies, No. 61, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1968. MR**0239612****[Mur]**Kunio Murasugi,*On the genus of the alternating knot. I, II*, J. Math. Soc. Japan**10**(1958), 94–105, 235–248. MR**0099664****[Su]**Dennis Sullivan,*A homological characterization of foliations consisting of minimal surfaces*, Comment. Math. Helv.**54**(1979), no. 2, 218–223. MR**535056**, 10.1007/BF02566269**[T]**William P. Thurston,*A norm for the homology of 3-manifolds*, Mem. Amer. Math. Soc.**59**(1986), no. 339, i–vi and 99–130. MR**823443****[W]**C. T. C. Wall,*Surgery on compact manifolds*, Academic Press, London-New York, 1970. London Mathematical Society Monographs, No. 1. MR**0431216**

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC (2000):
57M15,
57R45

Retrieve articles in all journals with MSC (2000): 57M15, 57R45

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