# Rank of a maximal subgroup in $H^1(M,Z)$ with trivial cup-product

**Author:** Irina Gelbukh

Fourth International Conference of Applied Mathematics and Computing (FICAMC);
Plovdiv, Bulgaria, August 12–18, 2007, p. 167.

**Abstract:** Let $M$ be a smooth closed oriented manifold, $h^{max}(M)$ be the maximal rank
of a maximal subgroup in $H^1(M,Z)$ with trivial cup-product, and $h^{min}(M)$
the minimal rank of such a subgroup. It has been shown that the value of $h(M)$
characterizes the topology of Morse form foliations on $M$: e.g., if $rk \omega
> h(M)$, where $\omega$ is a Morse form on $M$, then its foliation has a
minimal component. We give upper and lower bounds on $h{max}(M)$ and $h^{min}(M)$
in terms of the first and second Betti numbers. In addition, we calculate
these values for a connected sum and direct product of manifolds.

**Keywords:** Manifold topology; Morse form; foliations (The paper provides no keywords; these keywords are provided by a Zentralblatt review.)

**PDF:** Rank of a maximal subgroup in $H^1(M,Z)$ with trivial cup-product

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