Journal of the Mathematical Society of Japan

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

Kenshi ISHIGURO, Jesper MØLLER, and Dietrich NOTBOHM

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For G=S3×...×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 QZ, 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 aspace 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×...×HP.

Article information

J. Math. Soc. Japan, Volume 51, Number 1 (1999), 45-61.

First available in Project Euclid: 10 June 2008

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 55P10: Homotopy equivalences 55P60: Localization and completion
Secondary: 55R35: Classifying spaces of groups and $H$-spaces

genus classifying space $p$-completion Lie groups self-maps


ISHIGURO, Kenshi; MØLLER, Jesper; NOTBOHM, Dietrich. Rational self-equivalences of spaces in the genus of a product of quaternionic projective spaces. J. Math. Soc. Japan 51 (1999), no. 1, 45--61. doi:10.2969/jmsj/05110045.

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