On The Rational Realization of Even-dimensional Spheres and Products of Eilenberg--MacLane Spaces as Classifying Spaces

In this paper, we study the rational realization problem for the classifying space $\B(X)$. We prove that if $S^{2n}$ is realized as $\B(X)$ for a simply-connected space $X$, then $X$ is $π$-infinite and has vanishing rational Gottlieb elements above degree $2n-1$. In particular, even-dimensional spheres cannot be realized as $B\mathrm{aut}_1(X)$ for any simply-connected $π$-finite space $X$. We also prove that, for all $n\geq 2$ and $s,t\geq 1$, the product of Eilenberg--MacLane spaces $K(\Q^s,n)\times K(\Q^t,n+1)$ cannot be realized as $B\mathrm{aut}_1(X)$ for any simply-connected $π$-finite space $X$. Moreover, we prove that if $r\geq 2$ and $n\geq 3$ and $K(\Q^r,n)$ is realized as $\B(X)$ for a $π$-finite space $X$, then $X\simeq_{\Q}K(\Q^r,n-1)$. The proofs are based on two structural results for Gottlieb elements in the derivation Lie algebra of a Sullivan minimal model, which provide a uniform method for these realization problems.