Embeddings in high dimension

Abstract

The spaces of smooth high dimensional spherical and long embeddings attracted a lot of attention starting mid-20th century. In 1966, Andre Haefliger proved that the group of isotopy classes of spherical (framed) embeddings of Sn in Sn+q for q ≥ 3 can be described by means of homotopy groups of spheres and orthogonal groups. We show in the second chapter of this dissertation that Haefliger’s result can be adapted and extended to spherical and long embeddings modulo immersions of codimension at least three. The spaces of such embeddings have been actively studied in the last 20 years. Moreover, we prove that a partial result is true for concordance classes of spherical and long embeddings modulo immersions of codimension two. Our motivation for third and fourth chapters is to utilize the combinatorial techniques developed in 1990’s and further on for classical knots and links in R3, to gain a geometric intuition about high dimensional knots and links. In particular, we are interested in generalizing the work of Polyak and Viro in the classical knot theory, where they provided explicit formulas for finite type knot invariants in terms of Gauss diagrams. We obtained Polyak-Viro type formulas for 2- and 3-component spherical and long links of dimension (2ℓ − 1) in R3ℓ, ℓ ≥ 2, as discussed in chapter three. Although, the generalization to knots (one-component links) in these (co)dimensions turns out to be a more difficult problem, we give a conjectural formula for an invariant distinguishing such knots in the final fourth chapter.