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Smale invariants of homotopy spheres % latex2html id marker 105

Department of Analyisis,
Eötvös University, Budapest


There are embeddings of the 4k-1-sphere in R6k-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 R6k-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 ak(2k-1)! cusp points when we make it generic in R6k-1. Here ak = 2 if k is odd, and ak = 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 R4k+1.

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Tohru Okuzono