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 th e monoid of rational equivalences of X; denoted by epsilon(0)(X) This topol ogical question is transformed into a matrix problem over Q x Z<^>, since e psilon(0)(BG) is the set of monomial matrices whose nonzero entries are odd squares. It will be shown that a submonoid of epsilon(0)(X), denoted by de lta(0)(X), determines the decomposability of X. Namely, if Nodd 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 spl its into m indecomposable spaces if and only if delta(0)(X) congruent to (N -odd)(m). 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 q uaternionic projective spaces BG = HPinfinity x ... x HPinfinity.