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# Mathematica Slovaca

Editor-in-Chief: Pulmannová, Sylvia

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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

## Abstract

We study $\begin{array}{}{b}_{1}^{\prime }\end{array}$ (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 $\begin{array}{}{b}_{1}^{\prime }\end{array}$ (M) and the first Betti number b1(M) by explicitly constructing manifolds with any possible combination of $\begin{array}{}{b}_{1}^{\prime }\end{array}$ (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 $\begin{array}{}{b}_{1}^{\prime }\end{array}$ (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

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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,

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