Embedding 3-manifolds in spin 4-manifolds

Aceto, Paolo and Golla, Marco and Larson, Kyle (2017) Embedding 3-manifolds in spin 4-manifolds. JOURNAL OF TOPOLOGY, 10 (2). pp. 301-323. ISSN 1753-8416

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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 S2 ×S2, 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. © 2017 London Mathematical Society.

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