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

Irina Gelbukh

doi:10.1515/ms-2016-0298

Published by De Gruyter June 5, 2017

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

Irina Gelbukh

From the journal Mathematica Slovaca

https://doi.org/10.1515/ms-2016-0298

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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

(Communicated by Július Korbaš)

References

[1] Arnoux, P.—Levitt, G.: Sur l’unique ergodicité des 1-formes fermées singulières, Invent. Math. 84 (1986), 141–156.10.1007/BF01388736Search in Google Scholar

[2] Babalic, E. M.—Lazaroiu, C. I.: Foliated eight-manifolds for M-theory compactification, Journal of High Energy Physics 1 (2015), 140.10.1007/JHEP01(2015)140Search in Google Scholar

[3] Babalic, E. M.—Lazaroiu, C. I.: Singular foliations for M-theory compactification, Journal of High Energy Physics 3 (2015), 116.10.1007/JHEP03(2015)116Search in Google Scholar

[4] Chen, B-L.—LeFloch, PH. G.: Local foliations and optimal regularity of Einstein spacetimes, J. Geom. Phys. 59 (2009), 913–941.10.1016/j.geomphys.2009.04.002Search in Google Scholar

[5] Dimca, A.—Papadima, S.—Suciu, A.: Quasi-Kähler groups, 3-manifold groups, and formality, Math. Z. 268 (2011), 169–186.10.1007/s00209-010-0664-ySearch in Google Scholar

[6] Gelbukh, I.: Co-rank and Betti number of a group, Czechoslovak Math. J. 65 (2015), 565–567.10.1007/s10587-015-0195-0Search in Google Scholar

[7] Gelbukh, I.: Presence of minimal components in a Morse form foliation, Differ. Geom. Appl. 22 (2005), 189–198.10.1016/j.difgeo.2004.10.006Search in Google Scholar

[8] Gelbukh, I.: Number of minimal components and homologically independent compact leaves for a Morse form foliation, Stud. Sci. Math. Hung. 46 (2009), 547–557.10.1556/sscmath.2009.1108Search in Google Scholar

[9] Gelbukh, I.: On the structure of a Morse form foliation, Czechoslovak Math. J. 59 (2009), 207–220.10.1007/s10587-009-0015-5Search in Google Scholar

[10] Gelbukh, I.: Close cohomologous Morse forms with compact leaves, Czechoslovak Math. J. 63 (2013), 515–528.10.1007/s10587-013-0034-0Search in Google Scholar

[11] Gilmer, P.: Heegaard genus, cut number, weak p-congruence, and quantum invariants, J. Knot Theory Ramifications 18 (2009), 1359–1368.10.1142/S021821650900749XSearch in Google Scholar

[12] Harvey, S.: On the cut number of a 3-manifold, Geom. Topol. 6 (2002), 409–424.10.2140/gt.2002.6.409Search in Google Scholar

[13] Imanishi, H.: On codimension one foliations defined by closed one forms with singularities, J. Math. Kyoto Univ. 19 (1979), 285–291.10.1215/kjm/1250522432Search in Google Scholar

[14] Jaco, W.: Heegaard splittings and splitting homomorphisms, Trans. Amer. Math. Soc. 146 (1969), 365–375.10.1090/S0002-9947-1969-0253340-0Search in Google Scholar

[15] Jaco, W.: Geometric realizations for free quotients, J. Austral. Math. Soc. 14 (1972), 411–418.10.1017/S1446788700011034Search in Google Scholar

[16] Katz, M.—Rudyak, Y.—Sabourau, S.: Systoles of 2-complexes, Reeb graph, and Grushko decomposition, Int. Math. Res. Not. IMRN 2006 (2006), 1–30.10.1155/IMRN/2006/54936Search in Google Scholar

[17] Leininger, C. J.—Reid, A. W.: The co-rank conjecture for 3-manifold groups, Algebraic and Geometric Topology 2 (2002), 37–50.10.2140/agt.2002.2.37Search in Google Scholar

[18] Levitt, G.: 1-formes fermées singulières et groupe fondamental, Invent. Math. 88 (1987), 635–667.10.1007/BF01391835Search in Google Scholar

[19] Levitt, G.: Groupe fondamental de l’espace des feuilles dans les feuilletages sans holonomie, J. Diff. Geom. 31 (1990), 711–761.10.4310/jdg/1214444632Search in Google Scholar

[20] Lyndon, R. C.: The equation a²b²=c² in free groups, Mich. Math. J. 6 (1959), 89–95.10.1307/mmj/1028998143Search in Google Scholar

[21] Lyndon, R. C.: Dependence in groups, Colloq. Math. XIV (1966), 275–283.10.4064/cm-14-1-275-283Search in Google Scholar

[22] Lyndon, R. C.—Schupp, P. E.: Combinatorial Group Theory. Mathematics, Springer, Berlin, 2001.10.1007/978-3-642-61896-3Search in Google Scholar

[23] Makanin, G. S.: Equations in a free group, Math. USSR Izvestiya 21 (1983), 483–546.10.1070/IM1983v021n03ABEH001803Search in Google Scholar

[24] Mel’nikova, I. A.: A test for non-compactness of the foliation of a Morse form, Russ. Math. Surveys 50 (1995), 444–445.10.1070/RM1995v050n02ABEH002092Search in Google Scholar

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

[26] Razborov, A. A.: On systems of equations in a free group, Math. USSR Izvestiya 25 (1985), 115–162.10.1017/CBO9780511566073.022Search in Google Scholar

[27] Sikora, A.: Cut numbers of 3-manifolds, Trans. Amer. Math. Soc. 357 (2005), 2007–2020.10.1090/S0002-9947-04-03581-0Search in Google Scholar

Received: 2015-3-29

Accepted: 2015-5-31

Published Online: 2017-6-5

Published in Print: 2017-6-27

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

Abstract

References

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