Homology cobordism for some Sol manifolds

TL;DR

This paper proves that for certain Sol manifolds with specific homology, being cobordant implies they are actually homeomorphic, establishing a precise classification criterion.

Contribution

It establishes a complete classification of Sol manifolds with first homology of order 16 up to integer homology cobordism, showing they are homeomorphic if cobordant.

Findings

Two Sol manifolds with first homology of order 16 are cobordant iff they are homeomorphic.

Provides a classification criterion for these Sol manifolds based on cobordism and homeomorphism.

Enhances understanding of the relationship between cobordism and topology in geometric 3-manifolds.

Abstract

We show that two 3-manifolds that admit Sol geometry and have first homology group of order 16 are integer homology cobordant if and only if they are homeomorphic.