A Novikov fundamental group

TL;DR

This paper introduces a Novikov fundamental group associated with a cohomology class on a closed manifold, extending the classical fundamental group concept and providing new lower bounds for critical points of Morse closed 1-forms.

Contribution

It defines a new Novikov fundamental group that generalizes the classical fundamental group in the context of Novikov homology.

Findings

Establishes a new algebraic invariant for closed manifolds.

Provides lower bounds for critical points of Morse closed 1-forms.

Differentiates from bounds derived via Novikov homology.

Abstract

Given a $1$-cohomology class $u$ on a closed manifold $M$, we define a Novikov fundamental group associated to $u$, generalizing the usual fundamental group in the same spirit as Novikov homology generalizes Morse homology to the case of non exact $1$-forms. As an application, lower bounds for the minimal number of index $1$ and $2$ critical points of Morse closed $1$-forms are obtained, that are different in nature from those derived from the Novikov homology.