Embedding and knotting of manifolds in Euclidean spaces

TL;DR

This survey explains the topology of higher-dimensional manifolds, focusing on embedding and knotting in Euclidean spaces, highlighting classical and modern results, and illustrating the use of algebraic and geometric methods.

Contribution

It provides an accessible overview of embedding and knotting results, including recent classifications and generalizations of the Haefliger-Weber theorem, with emphasis on intuitive understanding.

Findings

Explicit classification of knotted tori

Generalization of the Haefliger-Weber embedding theorem below the metastable range

Counterexamples outside the metastable range

Abstract

A clear understanding of topology of higher-dimensional objects is important in many branches of both pure and applied mathematics. In this survey we attempt to present some results of higher-dimensional topology in a way which makes clear the visual and intuitive part of the constructions and the arguments. In particular, we show how abstract algebraic constructions appear naturally in the study of geometric problems. Before giving a general construction, we illustrate the main ideas in simple but important particular cases, in which the essence is not veiled by technicalities. More specifically, we present several classical and modern results on the embedding and knotting of manifolds in Euclidean space. We state many concrete results (in particular, recent explicit classification of knotted tori). Their statements (but not proofs!) are simple and accessible to non-specialists. We outline a general approach to embeddings via the classical van Kampen-Shapiro-Wu-Haefliger-Weber 'deleted product' obstruction. This approach reduces the isotopy classification of embeddings to the homotopy classification of equivariant maps, and so implies the above concrete results. We describe the revival of interest in this beautiful branch of topology, by presenting new results in this area (of Freedman, Krushkal, Teichner, Segal, Spiez and the author): a generalization the Haefliger-Weber embedding theorem below the metastable dimension range and examples showing that other analogues of this theorem are false outside the metastable dimension range.