Embedding Seifert manifolds in S^4

TL;DR

This paper investigates conditions under which Seifert manifolds and certain 3-manifolds embed smoothly in the 4-sphere, using Donaldson's theorem and invariants to identify obstructions and construct new embeddings.

Contribution

It introduces new obstructions based on Donaldson's theorem for embedding Seifert manifolds in S^4 and constructs explicit embeddings, advancing understanding of 3- and 4-manifold embeddings.

Findings

Connected sums of lens spaces with certain properties embed in S^4

Constraints on Seifert invariants for embeddings with non-orientable base or odd Betti number

New explicit embeddings constructed and analyzed using d and mu-bar invariants

Abstract

Using an obstruction based on Donaldson's theorem on the intersection forms of definite 4-manifolds, we determine which connected sums of lens spaces smoothly embed in S^4. We also find constraints on the Seifert invariants of Seifert 3-manifolds which embed in S^4 when either the base orbifold is non-orientable or the first Betti number is odd. In addition we construct some new embeddings and use these, along with the d and mu-bar invariants, to examine the question of when the double branched cover of a 3 or 4 strand pretzel link embeds.