Haefliger's approach for spherical knots modulo immersions

TL;DR

This paper establishes a topological isomorphism between the set of connected components of certain spherical embedding spaces modulo immersions and specific homotopy groups, revealing geometric insights into these embeddings.

Contribution

It introduces a new isomorphism linking embedding spaces modulo immersions to homotopy groups, providing a geometric interpretation of the long exact sequence for these spaces.

Findings

Connected components of embedding spaces correspond to specific homotopy groups.

The long exact sequence of the triad has geometric interpretations related to embeddings and immersions.

Provides a framework for understanding spherical embeddings modulo immersions through homotopy theory.

Abstract

We show that for the spaces of spherical embeddings modulo immersions $\bar{Emb}(S^n,S^{n+q})$ and long embeddings modulo immersions $\bar{Emb}_{\partial}(D^n,D^{n+q})$, the set of connected components is isomorphic to $\pi_{n+1}(SG,SG_q)$ for $q\geq 3$. As a consequence, we show that all the terms of the long exact sequence of the triad $(SG;SO,SG_q)$ have a geometric meaning relating to spherical embeddings and immersions.