Large and small group homology

TL;DR

This paper constructs non-large subspaces in the rational homology of finitely generated groups, demonstrating that certain largeness properties of manifolds depend only on their fundamental class images, and provides counterexamples to longstanding questions.

Contribution

It introduces a method to identify non-large homology subspaces and shows that largeness properties of manifolds depend solely on their fundamental class images, answering Gromov's question.

Findings

Constructed non-large vector subspaces in homology.

Showed largeness properties depend only on fundamental class images.

Provided examples of essential manifolds not having hyperspherical universal covers.

Abstract

For several instances of metric largeness like enlargeability or having hyperspherical universal covers, we construct non-large vector subspaces in the rational homology of finitely generated groups. The functorial properties of this construction imply that the corresponding largeness properties of closed manifolds depend only on the image of their fundamental classes under the classifying map. This is applied to construct examples of essential manifolds whose universal covers are not hyperspherical, thus answering a question of Gromov (1986), and, more generally, essential manifolds which are not enlargeable.