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On the generalized Massey–Rolfsen invariant for link maps

A. Skopenkov (2000)

Fundamenta Mathematicae

For K = K 1 . . . K s and a link map f : K m let K = i < j K i × K j , define a map f : K S m - 1 by f ( x , y ) = ( f x - f y ) / | f x - f y | and a (generalized) Massey-Rolfsen invariant α ( f ) π m - 1 ( K ) to be the homotopy class of f . We prove that for a polyhedron K of dimension ≤ m - 2 under certain (weakened metastable) dimension restrictions, α is an onto or a 1 - 1 map from the set of link maps f : K m up to link concordance to π m - 1 ( K ) . If K 1 , . . . , K s are closed highly homologically connected manifolds of dimension p 1 , . . . , p s (in particular, homology spheres), then π m - 1 ( K ) i < j π p i + p j - m + 1 S .

On the group of real analytic diffeomorphisms

Takashi Tsuboi (2009)

Annales scientifiques de l'École Normale Supérieure

The group of real analytic diffeomorphisms of a real analytic manifold is a rich group. It is dense in the group of smooth diffeomorphisms. Herman showed that for the n -dimensional torus, its identity component is a simple group. For U ( 1 ) fibered manifolds, for manifolds admitting special semi-free U ( 1 ) actions and for 2- or 3-dimensional manifolds with nontrivial U ( 1 ) actions, we show that the identity component of the group of real analytic diffeomorphisms is a perfect group.

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