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Constructing equivariant maps for representations

Stefano Francaviglia (2009)

Annales de l’institut Fourier

We show that if Γ is a discrete subgroup of the group of the isometries of ℍ k , and if ρ is a representation of Γ into the group of the isometries of ℍ n , then any ρ -equivariant map F : ℍ k → ℍ n extends to the boundary in a weak sense in the setting of Borel measures. As a consequence of this fact, we obtain an extension of a result of Besson, Courtois and Gallot about the existence of volume non-increasing, equivariant maps. Then, we show that the weak extension we obtain is actually a measurable ρ -equivariant...

Constructing generic smooth maps of a manifold into a surface with prescribed singular loci

Osamu Saeki (1995)

Annales de l'institut Fourier

It is known that the singular set S ( f ) of a generic smooth map f : M → N of an n -dimensional manifold into a surface is a closed 1-dimensional submanifold of M and that it has a natural stratification induced by the absolute index. In this paper, we give a complete characterization of those 1-dimensional (stratified) submanifolds which arise as the singular set of a generic map in terms of the homology class they represent.

Constructing manifolds by homotopy equivalences I. An obstruction to constructing PL-manifolds from homology manifolds

Hajime Sato (1972)

Annales de l'institut Fourier

We aim at constructing a PL-manifold which is cellularly equivalent to a given homology manifold M n . The main theorem says that there is a unique obstruction element in H n - 4 ( M , ℋ 3 ) , where ℋ 3 is the group of 3-dimensional PL-homology spheres modulo those which are the boundary of an acyclic PL-manifold. If the obstruction is zero and M is compact, we obtain a PL-manifold which is simple homotopy equivalent to M .

Construction of Einstein metrics by generalized Dehn filling

Richard H. Bamler (2012)

Journal of the European Mathematical Society

In this paper, we present a new approach to the construction of Einstein metrics by a generalization of Thurston's Dehn filling. In particular in dimension 3, we will obtain an analytic proof of Thurston's result.

Contact 3-manifolds twenty years since J. Martinet's work

Yakov Eliashberg (1992)

Annales de l'institut Fourier

The paper gives an account of the recent development in 3-dimensional contact geometry. The central result of the paper states that there exists a unique tight contact structure on S 3 . Together with the earlier classification of overtwisted contact structures on 3-manifolds this result completes the classification of contact structures on S 3 .

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