The geometric Hopf invariant and double points
Algebraic Topology
2010-05-18 v2 Geometric Topology
Abstract
The geometric Hopf invariant of a stable map F is a stable Z_2-equivariant map h(F) such that the stable Z_2-equivariant homotopy class of h(F) is the primary obstruction to F being homotopic to an unstable map. In this paper we express the geometric Hopf invariant of the Umkehr map F of an immersion f:M^m \to N^n in terms of the double point set of f. We interpret the Smale-Hirsch-Haefliger regular homotopy classification of immersions f in the metastable dimension range 3m<2n-1 (when a generic f has no triple points) in terms of the geometric Hopf invariant.
Cite
@article{arxiv.1002.2907,
title = {The geometric Hopf invariant and double points},
author = {Michael Crabb and Andrew Ranicki},
journal= {arXiv preprint arXiv:1002.2907},
year = {2010}
}
Comments
25 pages, LATEX, accepted for publication in the Smale issue of the Journal of Fixed Point Theory and Applications. Minor revision of v1