Sur les structures affines homotopes à zéro des groupes de Lie. (On nullhomotopic affine structures on Lie groups) (Q920417)

This paper is an important study of left invariant locally affine structures on nilpotent Lie groups, particularly of the problem of extending such a stucture from connected maximal subgroups to the whole group. The author defines a class of structures which he calls homotopic to zero (which, morally, is a vanishing cohomology property) and \({\mathcal P}_ F\) (which is a set of technical conditions also of a cochain cohomological nature), and shows that the extension of such structures is related to killing certain cohomology classes. As a result, the author proves the existence of a left invariant locally affine structure on a connected nilpotent Lie group G which induces complete affine structures on a Jordan-Hölder sequence of subgroups, and which can be extended to a codimension 1 central extension of G. Many examples and consequences are also given.