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

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For G=S3וs× S3, 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 ε0(X). This topological question is transformed into a matrix problem over Qotimes Z^, since ε0(BG) is the set of monomial matrices whose nonzero entries are odd squares. It will be shown that a submonoid of ε0(X), denoted by δ0(X), determines the decomposability of X. Namely, if Nodd denotes the monoid of odd natural numbers, Theorem 2 shows that the monoid δ0(X) is isomorphic to a direct sum of copies of Nodd. Moreover the space X splits into m indecomposable spaces if and only if δ0(X)_??_(Nodd)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 quaternionic projective spaces BG=HP∈ftyוs× HP∈fty.

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