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Exotic families of embeddings

Dave Auckly, Daniel Ruberman

TL;DR

The paper develops a framework to produce exotic embeddings of 3‑manifolds in 4‑manifolds that are topologically trivial but smoothly nontrivial, and it proves two main results: infinite non‑isotopic embeddings of certain homology 3‑spheres into $S^4$ with diffeomorphic complements, and infinite higher‑dimensional families of embeddings parameterized by spheres. The authors build these phenomena via cork twists (notably infinite order corks) and a stabilization technique that couples submanifold sums with families of diffeomorphisms, then detect nontriviality using gauge‑theoretic invariants (Donaldson and Konno’s Seiberg‑Witten‑based family invariants). They provide explicit constructions based on $E(2)$ and log transforms, and extend the analysis to families of submanifolds, introducing a marking framework and multiset invariants to distinguish different parameterizations. The work suggests a parameterized cork‑theoretic mechanism for embedding phenomena and broadens the scope of exotic smooth structures in dimension four with concrete, computable invariants and examples.

Abstract

We construct a number of topologically trivial but smoothly non-trivial families of embeddings of 3-manifolds in 4-manifolds. These include embeddings of homology spheres in $S^4$ that are not isotopic but have diffeomorphic complements, and families (parameterized by high-dimensional spheres) of embeddings of any 3-manifold that embeds in a blown-up K3 surface. In each case, the families are constructed so as to be topologically trivial in an appropriate sense. We also illustrate a general technique for converting a non-trivial family of embeddings into a non-trivial family of submanifolds.

Exotic families of embeddings

TL;DR

The paper develops a framework to produce exotic embeddings of 3‑manifolds in 4‑manifolds that are topologically trivial but smoothly nontrivial, and it proves two main results: infinite non‑isotopic embeddings of certain homology 3‑spheres into with diffeomorphic complements, and infinite higher‑dimensional families of embeddings parameterized by spheres. The authors build these phenomena via cork twists (notably infinite order corks) and a stabilization technique that couples submanifold sums with families of diffeomorphisms, then detect nontriviality using gauge‑theoretic invariants (Donaldson and Konno’s Seiberg‑Witten‑based family invariants). They provide explicit constructions based on and log transforms, and extend the analysis to families of submanifolds, introducing a marking framework and multiset invariants to distinguish different parameterizations. The work suggests a parameterized cork‑theoretic mechanism for embedding phenomena and broadens the scope of exotic smooth structures in dimension four with concrete, computable invariants and examples.

Abstract

We construct a number of topologically trivial but smoothly non-trivial families of embeddings of 3-manifolds in 4-manifolds. These include embeddings of homology spheres in that are not isotopic but have diffeomorphic complements, and families (parameterized by high-dimensional spheres) of embeddings of any 3-manifold that embeds in a blown-up K3 surface. In each case, the families are constructed so as to be topologically trivial in an appropriate sense. We also illustrate a general technique for converting a non-trivial family of embeddings into a non-trivial family of submanifolds.
Paper Structure (10 sections, 10 theorems, 52 equations, 9 figures)

This paper contains 10 sections, 10 theorems, 52 equations, 9 figures.

Table of Contents

  1. Introduction
  2. Embeddings into S4w
  3. Stabilization and sums
  4. Submanifold sums
  5. Higher parameter families
  6. Families and invariants
  7. Specific choices
  8. Proof of Theorem \ref{['T:B']}
  9. Families of submanifolds
  10. Examples

Key Result

Theorem 1.1

There exist an infinite collection of integer homology $3$-spheres, $Y_n$, each with an infinite number of smooth embeddings $j_{p,n}\colon Y_n \to S^4$ so that

Figures (9)

  • Figure 1: Infinite order corks $X_+(n)$
  • Figure 2: The homology spheres $Y_n$
  • Figure 3: The homology sphere $Y_n$ as a splice
  • Figure 4: Construction of the embedding $j_p$
  • Figure 5: Construction cartoon
  • ...and 4 more figures

Theorems & Definitions (30)

  • Theorem 1.1
  • Remark 1.2
  • Theorem 1.3
  • Remark 1.4
  • Proposition 2.1
  • proof : Proof of Proposition \ref{['P:cork']}
  • proof : Proof of Theorem \ref{['T:A']}
  • Remark 3.1
  • Definition 4.1
  • Lemma 4.2
  • ...and 20 more