3-manifolds that bound no definite 4-manifolds

Abstract

We produce a rational homology 3‑sphere that does not smoothly bound either a positive or negative definite 4‑manifold. Such a 3‑manifold necessarily cannot be rational homology cobordant to a Seifert fibered space or any 3‑manifold obtained by Dehn surgery on a knot. The proof requires an analysis of short characteristic covectors in bimodular lattices.