Fundamental classes not representable by products

TL;DR

This paper demonstrates that certain rationally essential manifolds with large fundamental groups cannot be obtained via non-zero degree maps from product manifolds, highlighting restrictions on their topological structure.

Contribution

It establishes new obstructions to representing specific classes of manifolds as images of product manifolds under maps of non-zero degree.

Findings

Manifolds with large fundamental groups do not admit non-zero degree maps from products.

Examples include non-positively curved rank one manifolds and irreducible locally symmetric spaces.

Certain manifolds admit such maps only if they are virtually products.

Abstract

We prove that rationally essential manifolds with suitably large fundamental groups do not admit any maps of non-zero degree from products of closed manifolds of positive dimension. Particular examples include all manifolds of non-positive sectional curvature of rank one and all irreducible locally symmetric spaces of non-compact type. For closed manifolds from certain classes, say non-positively curved ones, or certain surface bundles over surfaces, we show that they do admit maps of non-zero degree from non-trivial products if and only if they are virtually diffeomorphic to products.