Rational homotopy theory of automorphisms of manifolds

IF 4.9 1区 数学 Q1 MATHEMATICS
Alexander Berglund, I. Madsen
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引用次数: 29

Abstract

We study the rational homotopy types of classifying spaces of automorphism groups of smooth simply connected manifolds of dimension at least five. We give dg Lie algebra models for the homotopy automorphisms and the block diffeomorphisms of such manifolds. Moreover, we use these models to calculate the rational cohomology of the classifying spaces of the homotopy automorphisms and block diffeomorphisms of the manifold #^g S^d x S^d relative to an embedded disk as g tends to infinity. The answer is expressed in terms of stable cohomology of arithmetic groups and invariant Lie algebra cohomology. Through an extension of Kontsevich's work on graph complexes, we relate our results to the (unstable) homology of automorphisms of free groups with boundaries.
流形自同构的有理同伦理论
研究了至少五维光滑单连通流形自同构群的分类空间的有理同伦类型。给出了这类流形的同伦自同构和块微分同构的dg李代数模型。此外,我们利用这些模型计算了流形#^g S^d x S^d的同伦自同构和块微分同构的分类空间在g趋于无穷时相对于嵌入盘的有理上同调。用算式群的稳定上同调和不变李代数上同调来表示答案。通过推广Kontsevich在图复合体上的工作,我们将我们的结果与有边界的自由群的自同构的(不稳定)同调联系起来。
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来源期刊
Acta Mathematica
Acta Mathematica 数学-数学
CiteScore
6.00
自引率
2.70%
发文量
6
审稿时长
>12 weeks
期刊介绍: Publishes original research papers of the highest quality in all fields of mathematics.
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