The geometric Hopf invariant and double points

Michael Crabb; Andrew Ranicki

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arXiv · 1002.2907

The geometric Hopf invariant and double points

Michael Crabb·Andrew Ranicki

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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.

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Michael Crabb, Andrew Ranicki. 2010-05-17. The geometric Hopf invariant and double points. https://arxiv.org/abs/1002.2907

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