Surfaces in a background space and the homology of mapping class groups
Surfaces in a background space and the homology of mapping class groups
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背景空间中的表面和映射类群的同源性
DOI：
10.1090/pspum/080.1/2483932
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发表时间：
2006
期刊：
影响因子：
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通讯作者：
I. Madsen
中科院分区：
文献类型：
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作者：
R. Cohen;I. Madsen
本文研究了单连通空间X， Sg,n（X,）中黎曼曲面空间的拓扑结构。这是由三元组（Fg,n, f）组成的空间，其中Fg，n是g属和n个边界分量的黎曼曲面，是边界的参数化，@Fg,n，和f: Fg，n！X是一个满足边界条件的连续映射。我们证明了关于这些空间的三个定理。我们的主要定理是空间S1，n（X;）的稳定同调型的识别，它被定义为空间Sg，n（X;）在g格变大时的极限。我们关于这个稳定拓扑的结果是Madsen和Weiss定理的一个参数化版本，证明了关于映射类群稳定上同调的Mumford猜想的推广。我们的第二个结果描述了一个稳定范围，其中Sg，n（X;）的同构与稳定同构。最后证明了具有一定扭转系数族的映射类群的同调性的一个稳定性定理。第二个和第三个定理是对Harer和Ivanov稳定性定理的推广。
In this paper we study the topology of the space of Riemann surfaces in a simply connected space X, Sg,n(X,). This is the space consisting of triples, (Fg,n,�,f), where Fg,n is a Riemann surface of genus g and n-boundary components, � is a parameterization of the boundary, @Fg,n, and f : Fg,n! X is a continuous map that satisfies a boundary condition . We prove three theorems about these spaces. Our main theorem is the identification of the stable homology type of the space S1,n(X;), defined to be the limit as the genus g gets large, of the spaces Sg,n(X;). Our result about this stable topology is a parameterized version of the theorem of Madsen and Weiss proving a generalization of the Mumford conjecture on the stable cohomology of mapping class groups. Our second result describes a stable range in which the homology of Sg,n(X;) is isomorphic to the stable homology. Finally we prove a stability theorem about the homology of mapping class groups with certain families of twisted coefficients. The second and third theorems are generalizations of stability theorems of Harer and Ivanov.