发表机构
Nankai University（南开大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究分类空间的有理实现问题，利用Sullivan极小模型导子李代数中Gottlieb元素的结构结果，证明偶维球面及特定Eilenberg-MacLane空间乘积无法作为π-有限空间的第一类自同态分类空间，给出K(Q^r,n)实现为分类空间时X的有理同伦型。
AI 中文摘要
本文研究分类空间$\boldsymbol{B}(X)$的有理实现问题。我们证明：若偶维球面$S^{2n}$可实现为单连通空间$X$的$\boldsymbol{B}(X)$，则$X$是$\boldsymbol{\text{π}}$-无穷的，且次数高于$2n-1$的有理Gottlieb元素均为零。特别地，偶维球面无法作为任何单连通$\boldsymbol{\text{π}}$-有限空间$X$的$\boldsymbol{B}\text{aut}_1(X)$。我们还证明：对所有$n\boldsymbol{\text{≥}}2$及$s,t\boldsymbol{\text{≥}}1$，Eilenberg–MacLane空间的乘积$K(\boldsymbol{\text{Q}}^s,n)\times K(\boldsymbol{\text{Q}}^t,n+1)$无法作为任何单连通$\boldsymbol{\text{π}}$-有限空间$X$的$\boldsymbol{B}\text{aut}_1(X)$。此外，我们证明：若$r\boldsymbol{\text{≥}}2$、$n\boldsymbol{\text{≥}}3$且$K(\boldsymbol{\text{Q}}^r,n)$可实现为$\boldsymbol{\text{π}}$-有限空间$X$的$\boldsymbol{B}(X)$，则$X\boldsymbol{\text{≃}}_{\boldsymbol{\text{Q}}}K(\boldsymbol{\text{Q}}^r,n-1)$。证明基于Sullivan极小模型的导子李代数中Gottlieb元素的两个结构结果，为这些实现问题提供了统一方法。
英文摘要
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.
Comments10 pages
