Strongly chiral rational homology spheres with hyperbolic fundamental groups

TL;DR

This paper constructs high-dimensional, strongly chiral rational homology spheres with hyperbolic fundamental groups, extending known 3-dimensional examples and exploring their self-map degree properties.

Contribution

It introduces new high-dimensional examples of strongly chiral hyperbolic rational homology spheres with specific homology and fundamental group properties.

Findings

Existence of strongly chiral rational homology spheres in high dimensions.

Construction of hyperbolic fundamental groups with specified homology.

Analysis of self-map degree sets and their relation to $r$-spins.

Abstract

For each $m\geq0$ and any prime $p\equiv3\ \mathrm{(mod \ 4)}$, we construct strongly chiral rational homology $(4m+3)$-spheres, which have real hyperbolic fundamental groups and only non-zero integral intermediate homology groups isomorphic to $\mathbb{Z}_{2p}$ in degrees $1,2m+1$ and $4m+1$. This gives group theoretic analogues in high dimensions of the existence of strongly chiral hyperbolic rational homology $3$-spheres, as well as of the existence of strongly chiral hyperbolic manifolds of any dimension that are not rational homology spheres, which was shown by Weinberger. One of our tools will be $r$-spins. We thus investigate the relationship between the sets of degrees of self-maps of a given manifold and its $r$-spins, and give classes of manifolds for which the sets are equal.