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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

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.


(Communicated by Július Korbaš)


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Received: 2015-3-29
Accepted: 2015-5-31
Published Online: 2017-6-5
Published in Print: 2017-6-27

© 2017 Mathematical Institute Slovak Academy of Sciences

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