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Volume 67, Issue 3


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


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


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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, Volume 67, Issue 3, Pages 645–656, ISSN (Online) 1337-2211, ISSN (Print) 0139-9918, DOI: https://doi.org/10.1515/ms-2016-0298.

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Irina Gelbukh
Discrete & Computational Geometry, 2017

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