Smooth (non)rigidity of piecewise rank one locally symmetric manifolds

We define \emph{piecewise rank 1} manifolds, which are aspherical manifolds that generally do not admit a nonpositively curved metric but can be decomposed into pieces that are diffeomorphic to finite volume, irreducible, locally symmetric, nonpositively curved manifolds with $\pi_1$-injective cusps. We prove smooth (self) rigidity for this class of manifolds in the case where the gluing preserves the cusps' homogeneous structure. We compute the group of self homotopy equivalences of such a manifold and show that it can contain a normal free abelian subgroup and thus, can be infinite. Elements of this abelian subgroup are twists along elements in the center of the fundamental group of a cusp.Comment: 20 pages, 1 figur

An algorithm to classify the asymptotic set associated to a polynomial mapping

We provide an algorithm to classify the asymptotic sets of the dominant polynomial mappings F: \C^3 \to \C^3 of degree 2, using the definition of the so-called "{\it fa\c{c}ons}" in \cite{Thuy}. We obtain a classification theorem for the asymptotic sets of dominant polynomial mappings F: \C^3 \to \C^3 of degree 2. This algorithm can be generalized for the dominant polynomial mappings F: \C^n \to \C^n of degree $d$, with any $(n, d) \in {(\N^*)}^2$