We prove that for any group π with cohomological dimension at least n the nth power of the Berstein class of π is nontrivial. This allows us to prove the following Berstein–Svarc theorem for all n:
Theorem. For a connected complex X with dim X = cat X = n, we have ≠ 0 whereis the Berstein class of X.
Previously it was known for n ≥ 3.
We also prove that, for every map f: M → N of degree ±1 of closed orientable manifolds, the fundamental group of N is free provided that the fundamental group of M is.