On the fundamental group and triple Massey's product

TL;DR

This paper investigates how certain algebraic invariants, like Massey's product, relate to the fundamental group of topological spaces under pseudo-homeomorphisms, revealing their potential to distinguish fundamental groups.

Contribution

It establishes that invariants preserved under pseudo-homeomorphisms depend only on the fundamental group and identifies triple Massey's product as such an invariant.

Findings

Invariants under pseudo-homeomorphisms depend solely on the fundamental group.

Triple Massey's product can distinguish fundamental groups of arrangements.

Necessary conditions are provided for invariants to differentiate fundamental groups.

Abstract

Let us say that a map of arcwise connected topological spaces (having the homotopy type of CW-complexes) is a pseudo-homeomorphism if it induces an isomorphism of the first integer homology groups and an epimorphism of the second integer homology groups. We prove that any invariant of a topological space w.r.t. pseudo-homeomorphisms is an invariant of the fundamental group of this space. We also describe a necessary condition for the fundamental groups to be distinguished by such invariants. As an example we show that the invariant used in math.AG/9805056 to distinguish the fundamental groups of combinatorially equivalent arrangements is, in fact, a form of triple Massey's product on the first integer homology group.