On self H-equivalences of an H-space with respect to any multiplication
An H-space usually supports many H-multiplications. The authors study the group HE(X) of self-homotopy equivalences which are H-maps for all possible multiplications. In particular it is shown that if X is U(n), SU(n) or Sp(n), then HE(X) is a finite nilpotent group and any member induces the identity on \(\pi_*(X)_{(p)}\) for all sufficiently large primes p. The proofs are not obvious and demonstrate considerable technical skill. Roughly speaking, the strategy is to replace the calculation of HE(X) by that of the group of H-equivalences for a particular multiplication and other groups depending only on the space without reference to H- structures, then to analyse the situation for a direct product of H- spaces and finally to localize at a sufficiently large prime so that X becomes a product of spheres and convenient multiplications can be chosen on each factor. The proof of nilpotency now depends upon earlier resuls quoted from the literature. In addition the paper includes other theorems and examples of interest.
scientific article; zbMATH DE number 3971759