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When is the set of embeddings finite up to isotopy?

Mikhail Skopenkov

National Research University Higher School of Economics, Moscow, Russia

Institute for Information Transmission Problems, Russian Academy of Sciences, Moscow, Russia

https://doi.org/10.1142/S0129167X15500512Cited by:2 (Source: Crossref)

Abstract

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.

Keywords:

AMSC: 57R52, 57R40, 57R65

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