On lens spaces bounding smooth 4-manifolds with $\boldsymbol{b_2=1}$

TL;DR

This paper investigates which lens spaces can bound smooth 4-manifolds with second Betti number one, revealing infinite families with specific topological constraints and constructing examples related to rational homology projective planes.

Contribution

It identifies infinite families of lens spaces that bound certain smooth 4-manifolds with second Betti number one, under various topological conditions, and constructs explicit examples.

Findings

Infinite families of lens spaces bound smooth 4-manifolds with b_2=1.

Some lens spaces cannot bound simple 4-manifolds with a single 0- and 2-handle.

Existence of lens spaces bounding 4-manifolds with specific Betti number properties.

Abstract

We study which lens spaces can bound smooth 4-manifolds with second Betti number one under various topological conditions. Specifically, we show that there are infinite families of lens spaces that bound compact, simply-connected, smooth 4-manifolds with second Betti number one, yet cannot bound a 4-manifold consisting of a single 0-handle and 2-handle. Additionally, we establish the existence of infinite families of lens spaces that bound compact, smooth 4-manifolds with first Betti number zero and second Betti number one, but cannot bound simply-connected 4-manifolds with second Betti number one. The construction of such 4-manifolds with lens space boundaries is motivated by the study of rational homology projective planes with cyclic quotient singularities.