Haefliger’s approach for spherical knots modulo immersions

$\def\Emb{\overline{Emb}}$We show that for the spaces of spherical embeddings modulo immersions $\Emb (S^n, S^{n+q})$ and long embeddings modulo immersions $\Emb_\partial (D^n, D^{n+q})$, the set of connected components is isomorphic to $\pi_{n+1} (SG, SG_q)$ for $q \geqslant 3$. 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.

Keywords

embeddings modulo immersions, spherical embedding, framed disked embedding

2010 Mathematics Subject Classification

57R40, 57R42, 58D10

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The author has benefited from a guest position at the Max Planck Institute for Mathematics in Bonn. Her travel was also partially supported by her advisor V. Turchin’s Simon Foundation grant, award ID: 519474.

Received 4 November 2021

Received revised 16 July 2022

Accepted 28 July 2022

Published 4 October 2023