On the homotopy classification of maps

TL;DR

This paper provides conditions under which a continuous map between topological spaces is null homotopic, focusing on the algebraic properties of cohomology and homotopy groups.

Contribution

It introduces specific criteria involving torsion-free cohomology and polynomial cohomology rings that guarantee null homotopy of maps.

Findings

Trivial induced cohomology homomorphism implies null homotopy under certain conditions.

Spaces with torsion-free cohomology and polynomial cohomology rings satisfy the null homotopy criterion.

The results connect algebraic properties of spaces to their homotopy classification.

Abstract

We establish certain conditions which imply that a map $f:X\to Y$ of topological spaces is null homotopic when the induced integral cohomology homomorphism is trivial; one of them is: $H^*(X)$ and $\pi_*(Y)$ have no torsion and $H^*(Y)$ is polynomial.