Geometry and Topology Seminar

Abstract: An affine manifold (M,C) is a differentiable manifold M endowed with a connection which curvature and torsion forms vanish identically. The connection defines on M an atlas which transition functions are affine maps. The affine manifold is said complete if and only if the connection is complete. 40 years ago, L. Auslander has proposed the following conjecture: The fundamental group of a compact and complete affine manifold is polycyclic. In [1] I have proposed the following conjecture. Let (M,C) be a compact and complete affine manifold, there exists a finite galoisian cover of M which is the source space of a non trivial affine map. This last conjecture implies the Auslander conjecture and leads to the following problem. Given two affine manifolds (B,C_1) and (F,C_2) classify every affine bundle which basis space is (B,C_1) and which typical fiber is diffeomorphic to F and is endowed with an affine structure which linear holonomy is the one of (F,C_2). The purpose of my talk is to present this problem and to show how I used stack theory to solve it.