Self-homotopy equivalences of \(SO(4)\) (Q1572927)

The author is specifically interested in the group of homotopy self-equivalences for rank 2 H-spaces which are not simply connected. In this paper, the case of \(SO(4)\) is considered in detail. The principal result maps \({\mathcal E} (SO(4))\) onto the invertible \(2\times 2\) matrices with integers on the diagonal and other entries integer multiples of \(\sqrt{2}\). The kernel is identified as a subgroup of order \(2^8\cdot 3^2\). The methods consist of very careful examination of \(SO(4)\) and related complexes from the point of view of compositions, suspensions and Whitehead products.