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January, 1999 Rational self-equivalences of spaces in the genus of a product of quaternionic projective spaces
Kenshi ISHIGURO, Jesper MØLLER, Dietrich NOTBOHM
J. Math. Soc. Japan 51(1): 45-61 (January, 1999). DOI: 10.2969/jmsj/05110045


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.


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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.


Published: January, 1999
First available in Project Euclid: 10 June 2008

zbMATH: 0924.55008
MathSciNet: MR1661000
Digital Object Identifier: 10.2969/jmsj/05110045

Primary: 55P10 , 55P60
Secondary: 55R35

Keywords: $p$-completion , classifying space , genus , Lie groups , self-maps

Rights: Copyright © 1999 Mathematical Society of Japan


Vol.51 • No. 1 • January, 1999
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