Embedding and knotting of manifolds in Euclidean spaces

Abstract

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, Spież 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.

A new invariant and parametric connected sum of embeddings

“…(By [Bry72] for m ≥ n + 3 the classification of embeddings of PL manifolds is the same in the PL and TOP categories. Analogously to [ These theorems have many specific corollaries [Sk07]. In this paper we study the case one dimension lower than in the Isotopy Theorem (b).…”

“…For recent surveys see [ReSk99,Sk07]; whenever possible we refer to these surveys and not to original papers.…”

Hardness of embedding simplicial complexes in $\mathbb{R}^d$

“…See Haefliger [Hae64] and Weber [Web67] for original sources and, e.g., Skopenkov [Sko06] for a modern overview, proof sketch, and extensions. Thus, for example, for (k, d) = (3, 6), (4, 8), (5, 9), etc., the embeddability of a given k-dimensional simplicial complex into R d is equivalent to the existence of an equivariant map of the space |K| 2 into S d−1 .…”

Embeddings of $k$-connected $n$-manifolds into $\mathbb {R}^{2n-k-1}$

“…A reader who is bothered by new terms can replace in this paper the p-parallelizability by the almost parallelizability. 4 The embeddability into R 2n−k−1 is equivalent to W n−k−1 (N ) = 0, where W n−k−1 (N ) is the normal Stiefel-Whitney class [Sk08,§2,Pr07,11.3].…”

“…For recent surveys, see [RS99], [Sk08], [HCEC]; whenever possible we refer to these surveys, not to original papers.…”