The geometric Hopf invariant and double points

Michael Crabb, Andrew Ranicki

Research output: Contribution to journalArticlepeer-review

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 -> 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.

Original languageEnglish
Pages (from-to)325-350
Number of pages26
JournalJournal of Fixed Point Theory and Applications
Volume7
Issue number2
DOIs
Publication statusPublished - Oct 2010

Keywords

  • Geometric Hopf invariant
  • immersion
  • double point
  • INTERSECTIONS
  • IMMERSIONS
  • BORDISM
  • Primary 55Q25
  • Secondary 57R42

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