Principal bundles over tori and maps which induce the identity on homotopy (Q690278)

The authors study principal bundles over tori. These manifolds and others constructed according to a method of George Cooke, the authors show, answer in the negative a question of the reviewer. They provide finite complexes \(X\) such that \(\pi_ 1(X)\) acts trivially on \(\pi_ n(X)\) for \(n>0\) and yet \(G_ 1(X)=0\). The methods employed are rational homotopy theory. Also the group of homotopy equivalences of some of these spaces are calculated. Finally these spaces possess self-homomorphisms which are not based isotopic to the identity, yet induce the identity on all the homotopy groups.