Bustamante Mauricio, Krannich Manuel, Kupers Alexander
Departamento de Matemáticas, Universidad Católica de Chile, Santiago, Chile.
Department of Mathematics, Karlsruhe Institute of Technology, 76131 Karlsruhe, Germany.
Math Ann. 2024;388(4):3321-3371. doi: 10.1007/s00208-023-02594-x. Epub 2023 Mar 29.
Given a closed smooth manifold of even dimension with finite fundamental group, we show that the classifying space of the diffeomorphism group of 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 of arbitrary codimension into 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.
给定一个具有有限基本群的偶数维闭光滑流形，我们证明该流形的微分同胚群的分类空间是有限型的，并且在每个维度上都有有限生成的同伦群。我们还证明了这个结果对于带边界的流形的一个变体，并推断出任意余维数的紧致子流形到该流形的光滑嵌入空间在包含映射处具有有限生成的高阶同伦群，前提是补集的基本群是有限的。作为一个中间结果，我们表明具有有限基本群的有限CW复形的简单同伦自等价的同伦类群，在有限核的意义下与一个算术群可公度。
