Aspherical Manifolds with Virtually 3-Step Nilpotent Fundamental Group

Abstract

A closed aspherical manifold of dimension ? 3, 4 with virtually 3-step nilpotent fundamental group supports a complete affinely flat structure. Topological and Seifert Bundle classifications are given for a special case called Seifert relatives of flat Riemannian manifolds. We use the Seifert fiber space construction as a main tool. 0. Introduction. It is well-known that given a torsion-free virtually poly-Z group ir, there exists a closed smooth K(-,r, 1) manifold (eg. a double coset space of a Lie group). A question we are interested in is whether such a manifold is homeomorphic to a quotient of R' by an action of a discrete group of affine motions. The best known result seems to be J. Scheuneman's [18], stating that the answer is yes if ir is 3-step nilpotent. In this paper we generalize this to virtually 3-step nilpotent groups (in dimension ? 3, 4). The main idea is to refine the Seifert fiber space construction of P. Conner and F. Raymond [4]. This gives rise to an embedding of such ir into the group of affine motions so that Rm/Ir is compact. Then we apply T. Farrell and W. C. Hsiang's theorem [6] stating that any two closed aspherical manifolds with isomorphic fundamental groups which are virtually nilpotent are homeomorphic in dimension ?3, 4. This paper is organized as follows. In section 1, we refine the Seifert fiber space construction by restricting the coefficient module to some submodules of M(W, Rk). The main result is that, if the first coboundary homomorphism 6 of certain cohomology exact sequence is surjective, then one can convert the algebraic object (group) into a geometric one (group action). Manuscript received November 23, 1981. Manuscript revised May 14, 1981. *This is a part of the author's Ph.D. Thesis at the University of Michigan, 1981. The second part (geometric realization) will appear as a separate article elsewhere. The author wishes to express his sincere appreciation to Professor Frank Raymond for his guidance in this work.