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4-DIMENSIONAL LOCALLY CAT(0)-MANIFOLDS WITH NO RIEMANNIAN SMOOTHINGS

机译:4维局部CAT(0)-流形,无黎曼平滑

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摘要

We construct examples of 4-dimensional manifolds M supporting a locally CAT(0)-metric, whose universal covers M satisfy Hruska's isolated flats condition, and con-tain 2-dimensional flats F with the property that θ~∞ F ≌ S~1 → S~3 ≌θ~∞ M are nontrivial knots. As a consequence, we obtain that the group π_1(M) cannot be isomorphic to the fundamental group of any compact Riemannian manifold of nonpositive sectional curvature. In particular, if K is any compact locally CAT(0)-manifold, then M × K is a locally CAT(0)-manifold which does not support any Riemannian metric of nonpositive sectional curvature.
机译:我们构造支持局部CAT(0)度量的4维流形M的示例,其通用覆盖M满足Hruska的孤立平面条件,并包含2个平面F,其具有θ〜∞F≌S〜1的性质→S〜3≌θ〜∞M是非平凡的结。结果,我们得到了π_1(M)群不能与任何非正截面曲率的紧致黎曼流形的基本群同构。特别是,如果K是任何紧凑的局部CAT(0)流形,则M×K是不支持任何非正截面曲率的黎曼度量的局部CAT(0)流形。

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