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

Abstract

For $G=S^3\times \text{...}\times S^3$, 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 ${ϵ}_0\left(X\right)$. This topological question is transformed into a matrix problem over $Q\otimes Z^{\wedge }$, since ${ϵ}_0\left(BG\right)$ is the set of monomial matrices whose nonzero entries are odd squares. It will be shown that a submonoid of ${ϵ}_0\left(X\right)$, denoted by ${\delta }_0\left(X\right)$, determines the decomposability of $X$. Namely, if $N_{odd}$ denotes the monoid of odd natural numbers, Theorem 2 shows that the monoid ${\delta }_0\left(X\right)$ is isomorphic to a direct sum of copies of $N_{odd}$. Moreover the space $X$ splits into $m$ indecomposable spaces if and only if ${\delta }_0\left(X\right)\cong (N_{odd}{)}^m$. 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 $BG=H{P}^{\infty }\times \text{...}\times H{P}^{\infty }$.