Jump to ContentJump to Main Navigation
Show Summary Details
More options …

Mathematica Slovaca

Editor-in-Chief: Pulmannová, Sylvia

6 Issues per year


IMPACT FACTOR 2016: 0.346
5-year IMPACT FACTOR: 0.412

CiteScore 2016: 0.42

SCImago Journal Rank (SJR) 2016: 0.489
Source Normalized Impact per Paper (SNIP) 2016: 0.745

Mathematical Citation Quotient (MCQ) 2016: 0.24

Online
ISSN
1337-2211
See all formats and pricing
More options …
Volume 67, Issue 3 (Jun 2017)

Issues

The co-rank of the fundamental group: The direct product, the first Betti number, and the topology of foliations

Irina Gelbukh
Published Online: 2017-06-05 | DOI: https://doi.org/10.1515/ms-2016-0298

Abstract

We study b1 (M), the co-rank of the fundamental group of a smooth closed connected manifold M. We calculate this value for the direct product of manifolds. We characterize the set of all possible combinations of b1 (M) and the first Betti number b1(M) by explicitly constructing manifolds with any possible combination of b1 (M) and b1(M) in any given dimension. Finally, we apply our results to the topology of Morse form foliations. In particular, we construct a manifold M and a Morse form ω on it for any possible combination of b1 (M), b1(M), m(ω), and c(ω), where m(ω) is the number of minimal components and c(ω) is the maximum number of homologically independent compact leaves of ω.

MSC 2010: Primary 14F35; 57N65; 57R30

Keywords: co-rank; inner rank; manifold; fundamental group; direct product; Morse form foliation

References

  • [1]

    Arnoux, P.—Levitt, G.: Sur l’unique ergodicité des 1-formes fermées singulières, Invent. Math. 84 (1986), 141–156.CrossrefGoogle Scholar

  • [2]

    Babalic, E. M.—Lazaroiu, C. I.: Foliated eight-manifolds for M-theory compactification, Journal of High Energy Physics 1 (2015), 140.CrossrefGoogle Scholar

  • [3]

    Babalic, E. M.—Lazaroiu, C. I.: Singular foliations for M-theory compactification, Journal of High Energy Physics 3 (2015), 116.CrossrefGoogle Scholar

  • [4]

    Chen, B-L.—LeFloch, PH. G.: Local foliations and optimal regularity of Einstein spacetimes, J. Geom. Phys. 59 (2009), 913–941.CrossrefWeb of ScienceGoogle Scholar

  • [5]

    Dimca, A.—Papadima, S.—Suciu, A.: Quasi-Kähler groups, 3-manifold groups, and formality, Math. Z. 268 (2011), 169–186.CrossrefGoogle Scholar

  • [6]

    Gelbukh, I.: Co-rank and Betti number of a group, Czechoslovak Math. J. 65 (2015), 565–567.Web of ScienceGoogle Scholar

  • [7]

    Gelbukh, I.: Presence of minimal components in a Morse form foliation, Differ. Geom. Appl. 22 (2005), 189–198.CrossrefGoogle Scholar

  • [8]

    Gelbukh, I.: Number of minimal components and homologically independent compact leaves for a Morse form foliation, Stud. Sci. Math. Hung. 46 (2009), 547–557.Web of ScienceGoogle Scholar

  • [9]

    Gelbukh, I.: On the structure of a Morse form foliation, Czechoslovak Math. J. 59 (2009), 207–220.Web of ScienceGoogle Scholar

  • [10]

    Gelbukh, I.: Close cohomologous Morse forms with compact leaves, Czechoslovak Math. J. 63 (2013), 515–528.Web of ScienceGoogle Scholar

  • [11]

    Gilmer, P.: Heegaard genus, cut number, weak p-congruence, and quantum invariants, J. Knot Theory Ramifications 18 (2009), 1359–1368.Google Scholar

  • [12]

    Harvey, S.: On the cut number of a 3-manifold, Geom. Topol. 6 (2002), 409–424.CrossrefGoogle Scholar

  • [13]

    Imanishi, H.: On codimension one foliations defined by closed one forms with singularities, J. Math. Kyoto Univ. 19 (1979), 285–291.CrossrefGoogle Scholar

  • [14]

    Jaco, W.: Heegaard splittings and splitting homomorphisms, Trans. Amer. Math. Soc. 146 (1969), 365–375.CrossrefGoogle Scholar

  • [15]

    Jaco, W.: Geometric realizations for free quotients, J. Austral. Math. Soc. 14 (1972), 411–418.CrossrefGoogle Scholar

  • [16]

    Katz, M.—Rudyak, Y.—Sabourau, S.: Systoles of 2-complexes, Reeb graph, and Grushko decomposition, Int. Math. Res. Not. IMRN 2006 (2006), 1–30.Google Scholar

  • [17]

    Leininger, C. J.—Reid, A. W.: The co-rank conjecture for 3-manifold groups, Algebraic and Geometric Topology 2 (2002), 37–50.CrossrefGoogle Scholar

  • [18]

    Levitt, G.: 1-formes fermées singulières et groupe fondamental, Invent. Math. 88 (1987), 635–667.CrossrefGoogle Scholar

  • [19]

    Levitt, G.: Groupe fondamental de l’espace des feuilles dans les feuilletages sans holonomie, J. Diff. Geom. 31 (1990), 711–761.CrossrefGoogle Scholar

  • [20]

    Lyndon, R. C.: The equation a2b2=c2 in free groups, Mich. Math. J. 6 (1959), 89–95.Google Scholar

  • [21]

    Lyndon, R. C.: Dependence in groups, Colloq. Math. XIV (1966), 275–283.Google Scholar

  • [22]

    Lyndon, R. C.—Schupp, P. E.: Combinatorial Group Theory. Mathematics, Springer, Berlin, 2001.Google Scholar

  • [23]

    Makanin, G. S.: Equations in a free group, Math. USSR Izvestiya 21 (1983), 483–546.Google Scholar

  • [24]

    Mel’nikova, I. A.: A test for non-compactness of the foliation of a Morse form, Russ. Math. Surveys 50 (1995), 444–445.Google Scholar

  • [25]

    Mel’nikova, I. A.: Maximal isotropic subspaces of skew-symmetric bilinear mapping, Mosc. Univ. Math. Bull. 54 (1999), 1–3.Google Scholar

  • [26]

    Razborov, A. A.: On systems of equations in a free group, Math. USSR Izvestiya 25 (1985), 115–162.Google Scholar

  • [27]

    Sikora, A.: Cut numbers of 3-manifolds, Trans. Amer. Math. Soc. 357 (2005), 2007–2020.Google Scholar

About the article


Received: 2015-03-29

Accepted: 2015-05-31

Published Online: 2017-06-05

Published in Print: 2017-06-27


Citation Information: Mathematica Slovaca, ISSN (Online) 1337-2211, ISSN (Print) 0139-9918, DOI: https://doi.org/10.1515/ms-2016-0298.

Export Citation

© 2017 Mathematical Institute Slovak Academy of Sciences. Copyright Clearance Center

Comments (0)

Please log in or register to comment.
Log in