## Journal of the Mathematical Society of Japan

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

#### Abstract

For $G=S^{3}\times\cdots\times S^{3}$, let $X$ be a space such that the $p$-completion $(X)_{p}^{\wedge}$ is homotopy equivalent to (BG)$)_{p}^{\wedge}$ for any prime $p$. We investigate the monoid of rational equivalences of $X$, denoted by $\epsilon_{0}(X)$. This topological question is transformed into a matrix problem over $Q\otimes Z^{\wedge}$, since $\epsilon_{0}(BG)$ 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 $\delta_{0}(X)$, determines the decomposability of $X$. Namely, if $N_{odd}$ 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 $X$ splits into $m$ indecomposable spaces if and only if $\delta_{0}(X)\cong(N_{odd})^{m}$. 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 $BG=HP^{\infty}\times\cdots\times HP^{\infty}$.

#### Article information

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

Dates
First available in Project Euclid: 10 June 2008

https://projecteuclid.org/euclid.jmsj/1213108350

Digital Object Identifier
doi:10.2969/jmsj/05110045

Mathematical Reviews number (MathSciNet)
MR1661000

Zentralblatt MATH identifier
0924.55008

#### Citation

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. https://projecteuclid.org/euclid.jmsj/1213108350