Cup-products for the polyhedral product functor
Cup-products for the polyhedral product functor
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DOI：
10.1017/s0305004112000230
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发表时间：
2012-06
影响因子：
0.8
通讯作者：
A. Bahri;M. Bendersky;F. Cohen;S. Gitler
中科院分区：
文献类型：
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作者：
A. Bahri;M. Bendersky;F. Cohen;S. Gitler
摘要 Davis-Januszkiewicz 引入了现在称为多面体上的矩角流形的流形 [6]。 Buchstaber–Panov 引入并广泛研究了为任何抽象单纯复形 K 定义的矩角复形 [4]。他们用Stanley-Reisner代数的Tor代数完整地描述了有理上同调环结构[4]。 Denham–Suciu [7] 和 Franz [9] 的工作给出了后续的发展，随后是 [1, 2]。也就是说，给定一系列基于 CW 对 X, A) = {(Xi, Ai)}mi=1 以及具有 m 个顶点的抽象单纯复形 K，则存在 Buchstaber-Panov 矩角复形的直接扩展。表示为 Z(K;(X,A)) 的扩展被称为多面体积函子，该术语由 Bill Browder 提出，并且与特殊情况 (X,A) = (D2, S1) 中的 Buchstaber-Panov 矩角复形一致 [1, 2]。证明了分解定理，该定理将 Z(K; (X, A)) 的悬置分解为由 K 的完整子复形确定的空间束。本文研究了 Z(K; (X, A)) 上同调环的杯积结构。本文的新结果是，上同调环的结构是根据 Z(K; (X, A)) 的“稳定”分解所产生的几何分解给出的[1, 2]。这里的方法给出了多面体积函子的许多新值的上同调环结构的确定，并检索了许多已知的结果。针对悬架对族以及 Xi 是 Ai 上的锥体的情况进行显式计算。这些结果补充和扩展了 Davis–Januszkiewicz [6]、Buchstaber–Panov [3, 4]、Panov [13]、Baskakov–Buchstaber–Panov, [3]、Franz, [8, 9] 以及 Hochster [12] 的结果。此外，在下述条件下（本质上是 Künneth 定理的强形式），这些定理也适用于任何上同调理论。
Abstract Davis–Januszkiewicz introduced manifolds which are now known as moment-angle manifolds over a polytope [6]. Buchstaber–Panov introduced and extensively studied moment-angle complexes defined for any abstract simplicial complex K [4]. They completely described the rational cohomology ring structure in terms of the Tor-algebra of the Stanley-Reisner algebra [4]. Subsequent developments were given in work of Denham–Suciu [7] and Franz [9] which were followed by [1, 2]. Namely, given a family of based CW-pairs X, A) = {(Xi, Ai)}mi=1 together with an abstract simplicial complex K with m vertices, there is a direct extension of the Buchstaber–Panov moment-angle complex. That extension denoted Z(K;(X,A)) is known as the polyhedral product functor, terminology due to Bill Browder, and agrees with the Buchstaber–Panov moment-angle complex in the special case (X,A) = (D2, S1) [1, 2]. A decomposition theorem was proven which splits the suspension of Z(K; (X, A)) into a bouquet of spaces determined by the full sub-complexes of K. This paper is a study of the cup-product structure for the cohomology ring of Z(K; (X, A)). The new result in the current paper is that the structure of the cohomology ring is given in terms of this geometric decomposition arising from the “stable” decomposition of Z(K; (X, A)) [1, 2]. The methods here give a determination of the cohomology ring structure for many new values of the polyhedral product functor as well as retrieve many known results. Explicit computations are made for families of suspension pairs and for the cases where Xi is the cone on Ai. These results complement and extend those of Davis–Januszkiewicz [6], Buchstaber–Panov [3, 4], Panov [13], Baskakov–Buchstaber–Panov, [3], Franz, [8, 9], as well as Hochster [12]. Furthermore, under the conditions stated below (essentially the strong form of the Künneth theorem), these theorems also apply to any cohomology theory.