具有有限基本群的流形的自同构空间的有限性
Finiteness properties of automorphism spaces of manifolds with finite fundamental group
摘要：
给定一个具有有限基本群的偶数维闭光滑流形$2n\ge6$，证明了$M$的同胚群的分类空间${\rm BDiff}(M)$是有限型的，并且在每一次上都有同胚生成的同伦群.我们还证明了这个结果的一个变种的流形的边界和推导出的空间的光滑嵌入的一个紧凑的子流形$N\subset M$的任意余维到$M$已连续生成更高的同伦群的基础上的包含，提供的基本群的补充是有限的。作为中间结果，我们证明了有限CW复形的简单同伦自等价的同伦类群与有限基本群的有限核可分解为一个算术群。
英文摘要：
Given a closed smooth manifold $M$ of even dimension $2n\ge6$ with finite fundamental group, we show that the classifying space ${\rm BDiff}(M)$ of the diffeomorphism group of $M$ is of finite type and has finitely generated homotopy groups in every degree. We also prove a variant of this result for manifolds with boundary and deduce that the space of smooth embeddings of a compact submanifold $N\subset M$ of arbitrary codimension into $M$ has finitely generated higher homotopy groups based at the inclusion, provided the fundamental group of the complement is finite. As an intermediate result, we show that the group of homotopy classes of simple homotopy self-equivalences of a finite CW complex with finite fundamental group is up to finite kernel commensurable to an arithmetic group.
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