Non-cobordant hyperbolic manifolds

TL;DR

The paper demonstrates that in most dimensions, there exist closed hyperbolic manifolds that are not boundaries of higher-dimensional manifolds, using cobordism and involution techniques.

Contribution

It establishes non-cobordism results for hyperbolic manifolds in all dimensions except those of the form 4m+3, expanding understanding of their boundary properties.

Findings

Existence of non-cobordant hyperbolic manifolds in dimensions ≥4, excluding 4m+3.

Use of involution fixed points and geodesic embeddings in proofs.

Outline of approaches for remaining dimensions 4m+3 ≠ 2^k-1.

Abstract

In all dimensions $n \ge 4$ not of the form $4m+3$, we show that there exists a closed hyperbolic $n$-manifold which is not the boundary of a compact $(n+1)$-manifold. The proof relies on the relationship between the cobordism class and the fixed point set of an involution on the manifold, together with a geodesic embedding of Kolpakov, Reid and Slavich. We also outline a possible approach to cover the dimensions $4m+3 \ne 2^k-1$.