高维流形的Torelli空间
Torelli spaces of high-dimensional manifolds
摘要：
一个流形的Torelli群是所有作为该流形的同调单位元的复同态的群。本文计算了某些高维、高连通流形的Torelli群的分类空间的上同调的不变部分（在同调的自同构作用下不变），这些流形具有有理系数，在一定的次数范围内.这是基于加拉修斯-兰德尔-威廉姆斯的工作的aromorphism群体的这些流形，博雷尔的经典成果算术群体，和方法从手术理论和pseudositopy理论。作为一个推论，我们发现，所有的米勒-森田-芒福德特征类是非平凡的上同调的分类空间的Torelli组，除了那些与Hirzebruch类，其消失是被迫的家庭指数定理。
英文摘要：
The Torelli group of a manifold is the group of all diffeomorphisms which act as the identity on the homology of the manifold. In this paper, we calculate the invariant part (invariant under the action of the automorphisms of the homology) of the cohomology of the classifying space of the Torelli group of certain high-dimensional, highly connected manifolds, with rational coefficients and in a certain range of degrees. This is based on Galatius--Randal-Williams' work on the diffeomorphism groups of these manifolds, Borel's classical results on arithmetic groups, and methods from surgery theory and pseudoisotopy theory. As a corollary, we find that all Miller--Morita--Mumford characteristic classes are nontrivial in the cohomology of the classifying space of the Torelli group, except for those associated with the Hirzebruch class, whose vanishing is forced by the family index theorem.
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