Arnold's seminar,
December 15th, 1998

Victor A. Vassiliev
"Immersions and their intersections (after T.Ekholm)"

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\centerline{Immersions and their intersections (after T.~Ekholm)}
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T.~Ekholm studies generic immersions
$S^k \to R^{2k-r},$ $r=0,1,2,$ up to two equivalence relations:

a) regular diffeotopies, i.e. pairs consisting of smooth isotopies
of both $S^k$ and $R^{2k-r}$;

b) regular homotopies, i.e. paths in the space of immersions.

For $k \ge 4$  the complete system of invariants of all these classes
is obtained in terms of the (global) geometry of the selfintersection set
of the immersion.

For immersions $S^3 \to R^5$ it is
known (Smale's theorem) that the group of regular homotopy
classes of immersions is free cyclic. It is proved that the
classes representable by embeddings form a subgroup of
index $24$, and complete system of invariants of the quotient group
(i.e. of obstructions to such a representation) also is constructed.

Some more key notions (which will be explained
together with their relations to the above problems): Smale invariant,
spin structure, Arf invariant, Seifert surface.

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