Rational homology $3$–spheres and simply connected definite bounding

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 $p$ and $q$, we obtain an infinite family of irreducible rational homology $3$–spheres which are homology cobordant to the lens space $L\left(p,q\right)$ but cannot be obtained by a knot surgery.