Bordism and regular homotopy of low-dimensional immersions

John Forbes Hughes

Vol. 156 (1992), No. 1, 155–184

DOI: 10.2140/pjm.1992.156.155

Abstract

In this paper we study the geometric characteristics of low-dimensional immersions. Smale asked, in his paper on immersions of the k-sphere in Rⁿ, what are explicit generators for the groups of regular homotopy classes of immersions? We answer this for the 3-sphere in R⁴ and R⁵. For S³ in R⁴, the answer is:

Theorem The standard (Froissart-Morin) eversion of S² in R³ has, as a track, an immersion of S² × I in R⁴ whose ends are embedded S²s. Each of these bounds a 3-ball in R⁴. Capping off the track with these 3-balls yields an immersion K : S³ → R⁴. Performing the eversion twice and capping off gives an immersion E : S³ → R⁴. The immersions E and K generate the group of regular homotopy classes of immersions of S³ in R⁴.

We also relate the invariants of an immersion which bounds an immersion of a manifold of one higher dimension to the characteristic classes of that manifold.

Mathematical Subject Classification 2000

Primary: 57R42

Secondary: 55Q99

Milestones

Received: 5 December 1989

Revised: 6 May 1991

Published: 1 November 1992