The geometric Hopf invariant and double points

Details The geometric Hopf invariant and double points The geometric Hopf invariant and double points

2010-02-15

arxiv.org/abs/1002.2907v2

Mathematics

Algebraic Topology

25 pages, LATEX, accepted for publication in the Smale issue of the Journal of Fixed Point Theory and Applications. Minor revi

Scientific paper

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.

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