Lens spaces which are realizable as closures of homology cobordisms over planar surfaces

TL;DR

This paper characterizes which lens spaces can be realized as closures of homology cobordisms over planar surfaces, showing all lens spaces can be represented with three boundary components, using number theory tools.

Contribution

It provides a criterion for realizing lens spaces as closures of homology cobordisms over planar surfaces with specified boundary components, and proves all lens spaces can be represented with three boundaries.

Findings

Every lens space can be realized as a closure over a planar surface with three boundary components.

A specific condition on lens spaces determines their realizability as such closures.

The Chebotarev density theorem is used in the proof.

Abstract

We determine the condition on a given lens space having a realization as a closure of homology cobordism over a planar surface with a given number of boundary components. As a corollary, we see that every lens space is represented as a closure of homology cobordism over a planar surface with three boundary components. In the proof of this corollary, we use Chebotarev density theorem.