Rational homology 3-spheres and simply connected definite bounding

TL;DR

This paper constructs infinite families of rational homology 3-spheres that are homology cobordant to given spheres but do not bound simply connected definite 4-manifolds, revealing new obstructions in 4-manifold topology.

Contribution

It introduces a method to produce infinite families of rational homology 3-spheres with specific cobordism properties and bounding obstructions, expanding understanding of 4-manifold boundaries.

Findings

Constructed infinite families of rational homology 3-spheres with specific cobordism properties.

Showed these spheres cannot bound simply connected definite 4-manifolds.

Demonstrated the existence of spheres homology cobordant to lens spaces but not obtainable via knot surgery.

Abstract

For each rational homology 3-sphere $Y$ which bounds simply connected definite 4-manifolds of both signs, we construct an infinite family of irreducible rational homology 3-spheres which are homology cobordant to $Y$ but cannot bound any simply connected definite 4-manifold. As a corollary, for any coprime integers $p,q$, we obtain an infinite family of irreducible rational homology 3-spheres which are homology cobordant to the lens space $L(p,q)$ but cannot obtained by a knot surgery.