Rational self-equivalences of spaces in the genus of a product of quaternionic projective spaces

For G = S(3) x ... x S(3), let X be a space such that the (P)-completion (X)(P)^ is homotopy equivalent to (BG)(P)^ for any prime p. We investigate the monoid of rational equivalences of X, denoted by (E0) (X). This topological question is transformed into a matrix problem over QxZ^, since (E0){BG) is the set of monomial matrices whose nonzero entries are odd squares. It will be shown that a submonoid (E0)(X) denoted by (delta 0 )(X), determines the decomposability of X. Namely, if N( odd) denotes the monoid of odd natural numbers, Theorem 2 shows that the monoid (delta 0)(X) is isomorphic to a direct sum of copies of N odd- Moreover the space X splits into m indecomposable spaces if and only if (delta 0 )(X) When such a space X is indecomposable, the relationship between {X. X] and {BG, BG] is discussed. Our results indicate that the homotopy set (X, X) contains less maps if X is not homotopy equivalent to the product of quaternionic projective spaces BG = HP(benahayat) x... x HP(benahayat)