Embedding of 3-manifolds in spin 4-manifolds

An invariant of orientable 3-manifolds is defined by taking the minimum $n$ such that a given 3-manifold embeds in the connected sum of $n$ copies of $S^2\times S^2$, and we call this $n$ the \emph{embedding number} of the 3-manifold. We discuss some general properties of this invariant, and discuss some calculations for families of lens spaces and Brieskorn spheres. We can construct rational and integral homology spheres whose embedding numbers grow arbitrarily large, and which can be calculated by exactly if we assume the $11/8$-Conjecture. This is joint work with Paolo Aceto and Marco Golla.