Manifolds with a unique embedding

• Autorzy: Zbigniew Jelonek
• Języki publikacji: EN

Abstrakty

EN

We show that if X, Y are smooth, compact k-dimensional submanifolds of ℝⁿ and 2k+2 ≤ n, then each diffeomorphism ϕ: X → Y can be extended to a diffeomorphism Φ: ℝⁿ → ℝⁿ which is tame (to be defined in this paper). Moreover, if X, Y are real analytic manifolds and the mapping ϕ is analytic, then we can choose Φ to be also analytic.
We extend this result to some interesting categories of closed (not necessarily compact) subsets of ℝⁿ, namely, to the category of Nash submanifolds (with Nash, real-analytic and smooth morphisms) and to the category of closed semi-algebraic subsets of ℝⁿ (with morphisms being semi-algebraic continuous mappings). In each case we assume that X, Y are k-dimensional and ϕ is an isomorphism, and under the same dimension restriction 2k+2 ≤ n we assert that there exists an extension Φ :ℝⁿ → ℝⁿ which is an isomorphism and it is tame.
The same is true in the category of smooth algebraic subvarieties of ℂⁿ, with morphisms being holomorphic mappings and with morphisms being polynomial mappings.

Kategorie tematyczne

• 32Q28: Stein manifolds
• 32H02: Holomorphic mappings, (holomorphic) embeddings and related questions
• 14R10: Affine spaces (automorphisms, embeddings, exotic structures, cancellation problem)
• 14P15: Real analytic and semianalytic sets
• 57R40: Embeddings
• 14P10: Semialgebraic sets and related spaces

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Colloquium Mathematicum
• Rocznik: 2009
• Tom: 117
• Numer: 2
• Strony: 299-317

Daty

wydano

2009

Twórcy

autor

Zbigniew Jelonek

• Instytut Matematyczny, Polska Akademia Nauk, Św. Tomasza 30, 31-027 Kraków, Poland

Identyfikatory

DOI

10.4064/cm117-2-13