The Hopf invariant of a Haefliger knot

Details The Hopf invariant of a Haefliger knot The Hopf invariant of a Haefliger knot

2005-06-17

arxiv.org/abs/math/0506356v1

Mathematische Zeitschrift 256 (2007) pp.35-44

Mathematics

Geometric Topology

Scientific paper

10.1007/s00209-006-0053-8

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].

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