Smale invariants of homotopy spheres

ANDRÁS SZUCS
Department of Analyisis,
Eötvös University, Budapest

Abstract

There are embeddings of the 4k-1-sphere in R^6k-1non regularly homotopic to the standard embedding. This shows that it is impossible to read off the Smale invariant (i.e. the regular homotopy class) of an immersion $S^{4k-1} \to R^{4k+1}$ like in the case of immersions $S^n \to R^{2n}$ just by looking at the multiple points. Still we solve this unsolvable problem, we read off the regular homotopy class of an immersion by looking at the singularities of any generic map the immersion bounds. We prove three formulas for the smale invariant. These formulas have the following consequences:

1) If two immersions $S^{4k-1} \to R^{4k+1}$ are not regularly homotopic, then they are not regularly homotopic in R^6k-1 either. In particular there are non-regularly homotopic embeddings $S^{4k-1} \subset R^{6k-1}$ showing that Kervaire's theorem is sharp. (By this theorem any embedding $S^n \subset R^q$ is regularly homotopic to the standard one if 2q > 3n+1. )

2) Any homotopy joining two non-regularly homotopic immersions $S^{4k-1} \to R^{4k+1}$ will have at least a_k(2k-1)! cusp points when we make it generic in R^6k-1. Here a_k = 2 if k is odd, and a_k = 1 if k is even.

We show the simplest non-trivial embedding $S^{4k-1} \subset R^{4k+1}.$For this purpose we extend the formulas for the Smale invariants to the immersions of the homotopy spheres. We compute also the group of immersions of homotopy 4k-1-spheres in R^4k+1.