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- W1985080848 abstract "Given a manifold N and a number m, we study the following question: is the set of isotopy classes of embeddingsN → S m 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) ⊂ ℤ 2 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 ofC 1 -smooth embeddingsS p × S q → S m is 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 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." @default.
- W1985080848 created "2016-06-24" @default.
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- W1985080848 date "2015-06-01" @default.
- W1985080848 modified "2023-09-26" @default.
- W1985080848 title "When is the set of embeddings finite up to isotopy?" @default.
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- W1985080848 doi "https://doi.org/10.1142/s0129167x15500512" @default.
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