For , let be a space such that the -completion is homotopy equivalent to (BG) for any prime . We investigate the monoid of rational equivalences of , denoted by . This topological question is transformed into a matrix problem over , since is the set of monomial matrices whose nonzero entries are odd squares. It will be shown that a submonoid of , denoted by , determines the decomposability of . Namely, if denotes the monoid of odd natural numbers, Theorem 2 shows that the monoid is isomorphic to a direct sum of copies of . Moreover the space splits into indecomposable spaces if and only if . When such aspace is indecomposable, the relationship between and is discussed. Our results indicate that the homotopy set contains less maps if is not homotopy equivalent to the product of quaternionic projective spaces .
Kenshi ISHIGURO. Jesper MØLLER. Dietrich NOTBOHM. "Rational self-equivalences of spaces in the genus of a product of quaternionic projective spaces." J. Math. Soc. Japan 51 (1) 45 - 61, January, 1999. https://doi.org/10.2969/jmsj/05110045