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Статья

When is the set of embeddings finite up to isotopy?

International Journal of Mathematics. 2015. Vol. 26. No. 7, Article number 1550051. P. 1-28.

Given a manifold N and a number m, we study the following question: is the set of isotopy classes of embeddings N → Sm finite? 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 that m > 2p + q + 2 and m < p + 3q/2 + 2. Then the set of isotopy classes of C1-smooth embeddings Sp × Sq → Sm is infinite if and only if either q + 1 or p + q + 1 is divisible by 4, or there exists a point (x, y) in the set FCS(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 Sp × Sq→ Sm to the classification of embeddings Sp+q {square cup} Sq → Sm and Dp × Sq → Sm. The latter classification problems are reduced to homotopy ones, which are solved rationally. © 2015 World Scientific Publishing Company.