Embedding 3-manifolds in spin 4-manifolds

TL;DR

This paper introduces an invariant called the embedding number for orientable 3-manifolds based on their embedding in connected sums of S^2 x S^2, explores its properties, computes it for specific families, and discusses its growth and decomposition implications.

Contribution

It defines the embedding number invariant, calculates it for lens spaces and Brieskorn spheres, and shows how to construct spheres with arbitrarily large embedding numbers, assuming the 11/8-Conjecture.

Findings

Embedding number is well-defined and has specific properties.

Calculations for lens spaces and Brieskorn spheres.

Construction of spheres with arbitrarily large embedding numbers.

Abstract

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 embedding number of the 3-manifold. We give some general properties of this invariant, and make calculations for families of lens spaces and Brieskorn spheres. We show how to construct rational and integral homology spheres whose embedding numbers grow arbitrarily large, and which can be calculated exactly if we assume the 11/8-Conjecture. In a different direction we show that any simply connected 4-manifold can be split along a rational homology sphere into a positive definite piece and a negative definite piece.