Fox reimbedding and Bing submanifolds

TL;DR

This paper introduces a new concept called amalgamated Heegaard genus to unify and generalize classical theorems by Fox and Bing, providing a criterion for embedding 3-manifolds with specific properties.

Contribution

It establishes an equivalence between two embedding conditions in 3-manifolds, generalizing key classical results and offering new insights into manifold embeddings.

Findings

Equivalence of embedding conditions in 3-manifolds

Generalization of Fox's reimbedding theorem and Bing's sphere characterization

Unified framework for manifold embedding and knot isotopy

Abstract

Let M be an orientable closed connected 3-manifold. We introduce the notion of amalgamated Heegaard genus of M with respect to a closed separating 2-manifold F, and use it to show that the following two statements are equivalent: (i) a compact connected 3-manifold Y can be embedded in M so that the exterior of the image of Y is a union of handlebodies; and (ii) a compact connected 3-manifold Y can be embedded in M so that every knot in M can be isotoped to lie within the image of Y . Our result can be regarded as a common generalization of the reimbedding theorem by Fox [Fox48] and the characterization of 3-sphere by Bing [Bin58], as well as more recent results of Hass and Thompson [HT89] and Kobayashi and Nishi [KN94].