关于Gorenstein空间的一点注记
A note on Gorenstein spaces
摘要：
与增广微分分次代数$R= R^{\geq 0}$相关联的是同伦不变量${\mathcal T}(R)$。这是一个分次向量空间，如果$H^0(R)$是基场，$H^{>N}(R)= 0$则dim $\, {\mathcal T}(R)= 1$当且仅当$H(R)$是一个庞加莱对偶代数。在苏利文扩展$\land W\to \land W\otimes \land Z\to \land Z$的情况下，其中暗淡$\, H(\land Z)<\infty$，我们表明，$${\mathcal T}(\land W\otimes \land Z)= {\mathcal T}(\land W)\otimes {\mathcal T}(\land Z).$$这是适用于有限维CW复合物$X$的基本组$G$的行为幂零的上同调$H(\widetilde{X};\mathbb Q)$的通用覆盖空间。若$H(X;\mathbb Q)$是Poincaré对偶代数，且$H(\widetilde{X};\mathbb Q)$和$H(BG;\mathbb Q)$是有限维的，则它们也是Poincaré对偶代数.
英文摘要：
Associated with an augmented differential graded algebra $R= R^{\geq 0}$ is a homotopy invariant ${\mathcal T}(R)$. This is a graded vector space, and if $H^0(R)$ is the ground field and $H^{>N}(R)= 0$ then dim$\, {\mathcal T}(R)= 1$ if and only if $H(R)$ is a Poincaré duality algebra. In the case of Sullivan extensions $\land W\to \land W\otimes \land Z\to \land Z$ in which dim$\, H(\land Z)<\infty$ we show that $${\mathcal T}(\land W\otimes \land Z)= {\mathcal T}(\land W)\otimes {\mathcal T}(\land Z).$$ This is applied to finite dimensional CW complexes $X$ where the fundamental group $G$ acts nilpotently in the cohomology $H(\widetilde{X};\mathbb Q)$ of the universal covering space. If $H(X;\mathbb Q)$ is a Poincaré duality algebra and $H(\widetilde{X};\mathbb Q)$ and $H(BG;\mathbb Q)$ are finite dimensional then they are also Poincaré duality algebras.
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