ANDRÁS SZUCS

Department of Analyisis,

Eötvös University, Budapest

Department of Analyisis,

Eötvös University, Budapest

Abstract

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

2) Any homotopy joining two non-regularly homotopic immersions
will have at least
*a*_{k}(2*k*-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
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 4*k*-1-spheres
in *R*^{4k+1}.