Cut numbers of manifolds
Author:
Adam S. Sikora
Journal:
Trans. Amer. Math. Soc. 357 (2005), 20072020
MSC (2000):
Primary 57M05, 57M27, 20F34, 11E76
Published electronically:
October 7, 2004
MathSciNet review:
2115088
Fulltext PDF Free Access
Abstract 
References 
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Additional Information
Abstract: We investigate the relations between the cut number, and the first Betti number, of manifolds We prove that the cut number of a ``generic'' manifold is at most This is a rather unexpected result since specific examples of manifolds with large and are hard to construct. We also prove that for any complex semisimple Lie algebra there exists a manifold with and Such manifolds can be explicitly constructed.
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 K. S. Brown, Cohomology of groups, Graduate Texts in Mathematics, SpringerVerlag, 1982. MR 83k:20002
 [CLO]
 D. Cox, J. Little, D. O'Shea, Ideals, Varieties, and Algorithms, An Introduction to Computational Algebraic Geometry and Commutative Algebra, 2nd edition, Springer, 1997. MR 97h:13024
 [CM]
 T. D. Cochran, P. Melvin, Quantum cyclotomic orders of 3manifolds, Topology 40 (2001), no. 1, 95125. MR 2002f:57022
 [Dw]
 W. G. Dwyer, Homology, Massey products and maps between groups, J. Pure Appl. Alg 6 (1975), 177190. MR 52:6710
 [Fe]
 R. A. Fenn, Techniques of Geometric Topology, Cambridge Univ. Press, 1983. MR 87a:57002
 [Gi]
 P. Gilmer, Integrality for TQFTs, preprint, www.arxiv.org/abs/math.QA/0105059, to appear in Duke Math. Journal.
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 G. B. Gurevich, Foundations of the theory of algebraic invariants, P. Noordhoff Ltd  Groningen, The Netherlands, 1964. MR 32:1211
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 R. Hartshorne, Algebraic Geometry, Graduate Texts in Mathematics, SpringerVerlag, 1977. MR 57:3116
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 S. Harvey, On the Cut Number of a manifold, Geom. Topol. 6 (2002) 409424. MR 2003g:57017
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 J. Hempel, manifolds, Annals of Mathematical Studies 86, Princeton Univ. Press, 1976. MR 54:3702
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 J. E. Humphreys, Introduction to Lie Algebras and Representation Theory, Graduate Texts in Mathematics, SpringerVerlag, 1972. MR 48:2197
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 C. J. Leininger, A. W. Reid, The corank conjecture for manifold groups, Algebr. Geom. Topol. 2 (2002), 9491000. MR 2002m:57019
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 W. Magnus, A. Karrass, D. Solitar, Combinatorial Group Theory, Dover Publications, Inc. 1976. MR 54:10423
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 A. A. Razborov, On systems of equations in a free group, Izv. Akad. Nauk SSSR Ser. Mat. 48 (1984), no. 4, 779832. English translation: Math. USSR Izvestiya 25 (1985), no. 1, 115162. MR 86c:20033
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 H. Samelson, Notes on Lie Algebras, Van Nostrand Reinhold Co., 1969. MR 40:7322
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 J. R. Stallings, Problems about free quotients of groups, Geometric group theory (Columbus, OH, 1992), 165182, Ohio State Univ. Math. Res. Inst. Publ. 3, de Gruyter, Berlin, 1995. MR 97b:20028
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 D. Sullivan, On the intersection ring of compact three manifolds, Topology 14 (1975), 275277. MR 52:4296
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 V. Turaev, Milnor invariants and Massey products, J. Soviet Math. 12 (1979), 128137.
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 V. Turaev, Cohomology rings, linking forms and invariants of spin structures of threedimensional manifolds, Mat. Sb. (N.S.) 120 (162) (1983), no. 1, 6883, 143. English translation from: Math. USSR Sbornik 48 no 1 (1984), 6579. MR 84g:57009
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 B. L. van der Waerden, Modern Algebra, Frederick Ungar Pub. Co., vol II, 1950.
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Additional Information
Adam S. Sikora
Affiliation:
Department of Mathematics, 244 Mathematics Building, SUNY at Buffalo, Buffalo, New York 14260
Email:
asikora@buffalo.edu
DOI:
http://dx.doi.org/10.1090/S0002994704035810
PII:
S 00029947(04)035810
Keywords:
Cut number,
3manifold,
corank,
skewsymmetric form,
cohomology ring
Received by editor(s):
October 28, 2002
Received by editor(s) in revised form:
December 2, 2003
Published electronically:
October 7, 2004
Article copyright:
© Copyright 2004
American Mathematical Society
