Seminar on low-dimensional mathematics
March 10, 2006

Higher-dimensional analog of Goussarov's knot theory
V.M.Nezhinskij

     Let p and n be natural numbers. An r-component link is a sequence of
smooth embeddings of r copies of p-sphere in n-sphere with pairwise
disjoint images. A 1-component link is also called a knot. A link is
a Brunnian one if any proper sublink of the link is isotopically
trivial.
     We assume that n-p>2. In this case the classes of isotopic Brunnian
r-component links form a group C_r with respect to the componentwise
connected summation. The operation of a connected summation of all
components of a link induces a homomorphism f_r: C_r \ arrow C_1.  We
set G_r = C_1/f_r (C_r).  At first, we'll formulate properties of
groups G_r, which are analogues to the corresponding properties of
Goussarov's groups of classes of r-equivalent one-dimentional knots.
Further, we'll formulate results, concerning the calculation of these
groups for some natural numbers r, n and p. Finally, we'll formulate
theorem on the reduction the problem of the calculation of our groups
to some problem of homotopy theory for any natural numbers r, n and
p; the reduction is a higher-dimensional analogue of the universal
Goussarov-Vassiljev finite type invariant of one-dimensional knots.
     There is a generalization of groups G_r to spherical links with any
(finite) number of components, the dimensions of different components
may differ each other. (We assume that the dimension of any component
of a spherical link is greater than two.) The theory of these groups
will be the subject of the talk too.

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