The Hopf invariant of a Haefliger knot
Geometric Topology
2007-05-23 v1 Algebraic Topology
Abstract
We show that Haefliger's differentiable (6,3)-knot bounds, in 6-space, a 4-manifold (a Seifert surface) of arbitrarily prescribed signature. This implies, according to our previous paper, that the Seifert surface has been prolonged in a prescribed direction near its boundary. This aspect enables us to understand a resemblance between Ekholm-Szucs' formula for the Smale invariant and Boechat-Haefliger's formula for Haefliger knots. As a consequence, we show that an immersion of the 3-sphere in 5-space can be regularly homotoped to the projection of an embedding in 6-space if and only if its Smale invariant is even. We also correct a sign error in our previous paper: "A geometric formula for Haefliger knots" [Topology 43 (2004) 1425-1447].
Keywords
Cite
@article{arxiv.math/0506356,
title = {The Hopf invariant of a Haefliger knot},
author = {Masamichi Takase},
journal= {arXiv preprint arXiv:math/0506356},
year = {2007}
}