Self homotopy equivalences of virtually nilpotent spaces

Abstract

The aim of this paper is to prove Theorem 1.1 below, a generalization to virtually nilpotent spaces of a result of Wilkerson [13] and Sullivan [12]. We recall that a CW complex Y is virtually nilpotent if (i) Y is connected, (ii) 7r~ Y is virtually nilpotent (i.e. has a nilpotent subgroup of finite index) and (iii) for every integer n > 1, zr~Y has a subgroup of finite index which acts nilpotently on 7r, Y. The class of virtually nilpotent spaces is much larger than the class of nilpotent spaces. For instance such non-nilpotent spaces as the Klein bottle and the real projective spaces are virtually nilpotent, and so is, of course, any connected space with a finite fundamental group.