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Commutators of \(C^{\infty }\)-diffeomorphisms preserving a submanifold. (English) Zbl 1169.57029

J. Math. Soc. Japan 61, No. 2, 427-436 (2009); erratum and addendum ibid. 65, No. 4, 1329-1336 (2013).
Let \((M^m,N^n)\) be a C\(^\infty\)-manifold pair, where \(N\) is a proper submanifold and let \(D^\infty_c(M,N)\) be the group of all C\(^\infty\) diffeomorphisms of \(M\) isotopic to the identity through C\(^\infty\) diffeomorphisms preserving \(N\) and having compact support. Then \(D_c^\infty(M,N)\) is a perfect group for \(n\geq 1\). If \(D_c^\infty(\text{int}M)\) and \(D^\infty(\partial M)\) are uniformly perfect then so is \(D_c^\infty(M,\partial M)\) when \(m\geq 2\).

MSC:

57R50 Differential topological aspects of diffeomorphisms
58D05 Groups of diffeomorphisms and homeomorphisms as manifolds
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References:

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