Equivariant Cohomology of Configuration Spaces Mod 2
Equivariant Cohomology of Configuration Spaces Mod 2
复制标题
配置空间的等变上同调 Mod 2
DOI：
10.1007/978-3-030-84138-6
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发表时间：
2021
影响因子：
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通讯作者：
G. Ziegler
中科院分区：
文献类型：
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作者：
Pavle V. M. Blagojević;F. Cohen;M. Crabb;W. Lück;G. Ziegler
F（M，n）：={（x1，...，xn）∈ Mn：对于流形M上不同点的所有有序n元组的所有1≤ i< j≤ n}，xi= xj始于1962年，由Fadell和Neujerth [47]以及Fox和Neujerth [52]开始，史前史可以追溯到Artin [7-9]的工作。不久之后，Arnold在他1969年的开创性工作[5]中给出了有序构形空间F（R2，n）的整上同调环的描述。从那时起，有序位形空间的拓扑被从许多方面进行了深入的研究，同时在各种问题，理论，甚至不同的数学领域和更远的领域，特别是在物理学中找到了应用。每个位形空间F（M，n）都配备了对称群在n个字母Sn上的自然自由作用，由点的排列给出。伴随轨道空间F（M，n）/Sn，称为无序位形空间，是一个重要而富有挑战性的研究对象。(The对称群的自由作用也是后面要讨论的小立方体操作数结构的基本成分。在他1970年的有影响力的论文[53]中，使用基本的新思想，Fuks给出了无序位形空间H <$（F（R2，n）/Sn; F2）的上同调代数作为上同调H <$（BO（n）; F2）的图像的描述。在研究无限循环空间和迭代循环空间的过程中，博德曼和沃格特[17]发明了与构形空间F（Rd，n）同伦类型相同的对象，并在May [76，Sec. 4]关于一个重要结构的定义，我们现在称之为小立方体运算符，见第二章。7. Frederick Cohen在他1976年的贡献[33]中给出了无序配置空间F（Rd，n）/Sn的上同调的第一个描述，其中n是素数，具有平凡系数（包括环结构）和扭曲系数（包括相关的模结构）[33，Thm. 5.2和Thm。5.3]。光滑流形M上的点的无序位形空间的同调在1989年由Bödigheimer等人确定。[20]在M是奇数维并且系数在任意域中的情况下，并且在
F (M, n):={(x1,..., xn)∈ Mn: xi= xj for all 1≤ i< j≤ n} of all ordered n-tuples of distinct points on a manifold M started in 1962 with the work of Fadell and Neuwirth [47] and Fox and Neuwirth [52], with prehistory going back to the work of Artin [7–9]. Soon after, Arnold, in his seminal work [5] from 1969, gave a description of the integral cohomology ring of the ordered configuration space F (R2, n). From that point on, the topology of the ordered configuration spaces was studied very intensively from many aspects, while finding applications in diverse problems, theories, and even different fields of mathematics and beyond, notably in physics.Each configuration space F (M, n) is equipped with a natural free action of the symmetric group on n letters Sn, given by the permutation of points. The associated orbit space F (M, n)/Sn, called the unordered configuration space, is an important and challenging object to study.(The free action of the symmetric group is also an essential ingredient of the little cubes operad structure to be discussed later.) In his influential 1970 paper [53] using fundamental new ideas, Fuks, gave a description of the cohomology algebra of the unordered configuration space H∗(F (R2, n)/Sn; F2) as an image of the cohomology H∗(BO (n); F2). In the course of study of infinite and iterated loop spaces, objects of the same homotopy type as the configuration space F (Rd, n) were invented by Boardman and Vogt [17] and adapted in a beautiful way by May [76, Sec. 4] for the definition of an important structure that we now call the little cubes operad; see Chap. 7. Frederick Cohen, in his 1976 contribution [33], gave the first descriptions of the cohomology of the unordered configuration space F (Rd, n)/Sn, for n a prime, with trivial coefficients (including the ring structure) and with twisted coefficients (including the relevant module structure)[33, Thm. 5.2 and Thm. 5.3]. The homology of the unordered configuration space for points on a smooth manifold M has been determined in 1989 by Bödigheimer et al.[20] in the case when M is odd-dimensional and coefficients are in an arbitrary field, and in the