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Mathematics > Geometric Topology

arXiv:math/0002076 (math)
[Submitted on 10 Feb 2000]

Title:Geometric formulas for Smale invariants of codimension two immersions

Authors:Tobias Ekholm, Andras Szucs
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Abstract: We give three formulas expressing the Smale invariant of an immersion f of a (4k-1)-sphere into (4k+1)-space. The terms of the formulas are geometric characteristics of any generic smooth map g of any oriented 4k-dimensional manifold, where g restricted to the boundary is an immersion regularly homotopic to f in (6k-1)-space.
The formulas imply that if f and g are two non-regularly homotopic immersions of a (4k-1)-sphere into (4k+1)-space then they are also non-regularly homotopic as immersions into (6k-1)-space. Moreover, any generic homotopy in (6k-1)-space connecting f to g must have at least a_k(2k-1)! cusps, where a_k=2 if k is odd and a_k=1 if k is even.
Comments: 22 pages, LaTeX2e
Subjects: Geometric Topology (math.GT)
MSC classes: 57R42; 57R45
Cite as: arXiv:math/0002076 [math.GT]
  (or arXiv:math/0002076v1 [math.GT] for this version)
  https://doi.org/10.48550/arXiv.math/0002076

Submission history

From: Tobias Ekholm [view email]
[v1] Thu, 10 Feb 2000 18:02:22 UTC (22 KB)
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