Trivial unit conjecture and homotopy theory. (Q2253033)

Let \(\mathbb ZG\) be the integral group ring of a torsion-free group \(G\). There is the following trivial unit conjecture for \(G\): any unit of \(\mathbb ZG\) is of the form \(\pm g\) for some \(g\in G\). In this note the author gives the following homotopy theoretic answer to the above conjecture. Theorem 1. Let \(G\) be a torsion-free group. The trivial unit conjecture for \(G\) is true if and only if for an Eilenberg-Mac Lane space \(X=BG\), the element \([f]\in\pi_d(X\vee S^d,S^d)\) (the relative homotopy group of the universal covering space) vanishes for some lifting of \(S^d\) whenever the inclusion \(i_f\colon X\to Y_f\) is a homotopy equivalence.