On intersection forms of definite 4-manifolds bounded by a rational homology 3-sphere

Abstract

We show that, if a rational homology 3-sphere $Y$ bounds a positive definite smooth 4-manifold, then there are finitely many negative definite lattices, up to the stable-equivalence, which can be realized as the intersection form of a smooth 4-manifold bounded by $Y$. To this end, we make use of constraints on definite forms bounded by $Y$ induced from Donaldson's diagonalization theorem, and correction term invariants due to Frøyshov, and Ozsváth and Szabó. In particular, we prove that all spherical 3-manifolds satisfy such finiteness property.

17 pages, 5 figures; Typos fixed. We added more results including properties on spherical 3-manifolds. The version to appear in Topology and its Applications