Vol. 110, No. 2, 1984

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Numerical invariants of homotopies into spheres

Jerrold Norman Siegel and Frank Williams

Vol. 110 (1984), No. 2, 417–428

A classical problem in the calculus of variations may be expressed: Given two points in the same component of a Riemannian manifold M, what is the length of the shortest path connecting them? In this paper we discuss parametric analogs to this problem. The width of a homotopy is the supremum of the lengths of the paths traced by the points of X. Work of Allan Calder and the first author on the topology of Stone-Cech compactifications led to the study of the question: Given a space X and two homotopic maps f,g : X M, what is the width of the shortest homotopy between them?

In this paper we obtain bounds bq(M) that depend only on the dimension q of X for the answer to this question in case M is a sphere or a projective space (with the standard metric). We also introduce a related sequence of invariants Bq(M) and compute these numbers for spheres and projective spaces. Of particular interest is the fact that b2n2(Sn) detects elements of Hopf invariant one while B2n2(Sn) does not.

Mathematical Subject Classification 2000
Primary: 55R65
Secondary: 53C20
Received: 7 December 1982
Published: 1 February 1984
Jerrold Norman Siegel
Frank Williams
Department of Mathematics, MSC 3MB
New Mexico State University
PO Box 30001
Las Cruces NM 88003-8001
United States