Narrow Results

Given a manifold N and a number m, we study the following question: is the set of isotopy classes of embeddingsN → S^mfinite? In case when the manifold N is a sphere the answer was given by A. Haefliger in 1966. In case when the manifold N is a disjoint union of spheres the answer was given by D. Crowley, S. Ferry and the author in 2011. We consider the next natural case when N is a product of two spheres. In the following theorem, FCS(i, j) ⊂ ℤ² is a specific set depending only on the parity of i and j which is defined in the paper.

Theorem.Assume thatm > 2p + q + 2andm < p + 3q/2 + 2. Then the set of isotopy classes ofC¹-smooth embeddingsS^p × S^q → S^mis infinite if and only if eitherq + 1orp + q + 1is divisible by 4, or there exists a point(x, y)in the setFCS(m - p - q, m - q)such that(m - p - q - 2)x + (m - q - 2)y = m - 3.

Our approach is based on a group structure on the set of embeddings and a new exact sequence, which in some sense reduces the classification of embeddings S^p × S^q → S^m to the classification of embeddings S^p+q ⊔ S^q → S^m and D^p × S^q → S^m. The latter classification problems are reduced to homotopy ones, which are solved rationally.

A method of construction of the local approximations in the case of functions defined on n-dimensional (n ≥ 1) smooth manifold with boundary is proposed. In particular, spline and finite-element methods on manifold are discussed. Nondegenerate simplicial subdivision of the manifold is introduced and a simple method for evaluations of approach is examined (the evaluations are optimal as to N-width of corresponding compact set).