When is the set of embeddings finite up to isotopy?

Abstract

Given a manifold N and a number m, we study the following question: is the set of isotopy classes of embeddingsN → Smfinite? 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 embeddingsSp × Sq → Smis infinite if and only if eitherq + 1orp + q + 1is 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 ⊔ Sq → Sm and Dp × Sq → Sm. The latter classification problems are reduced to homotopy ones, which are solved rationally.