On the Berstein-Svarc Theorem in dimension 2

TL;DR

This paper proves that the nth power of the Berstein class is nontrivial for groups of cohomological dimension n, extending the Berstein-Svarc theorem to all dimensions and exploring implications for manifold maps.

Contribution

It generalizes the Berstein-Svarc theorem to all dimensions and establishes new results on the fundamental groups of manifolds related by degree ±1 maps.

Findings

The nth power of the Berstein class is nontrivial for groups of cohomological dimension n.

For connected complexes with dimension equal to category, the nth power of the Berstein class is nontrivial.

If a degree ±1 map exists between closed orientable manifolds, the fundamental group of the target is free if the source's is.

Abstract

We prove that for any group of the cohomological dimension $n$ the $n$th power of the Berstein class of the group is nontrivial. This allows to prove the following Berstein-Svarc theorem for all $n$: Theorem. For a connected complex $X$ with $\dim X=\cat X=n$, the $n$th power of the Berstein class of $X$ is nontrivial. Previously it was known for $n\ge 3$. We also prove that, for every map $f: M \to N$ of degree $\pm 1$ of closed orientable manifolds, the fundamental group of $N$ is free provided that the fundamental group of $M$ is.