Finite part of operator K-theory for groups finitely embeddable into Hilbert space and the degree of non-rigidity of manifolds
Finite part of operator K-theory for groups finitely embeddable into Hilbert space and the degree of non-rigidity of manifolds
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DOI：
10.2140/gt.2015.19.2767
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发表时间：
2013-08
期刊：
影响因子：
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通讯作者：
S. Weinberger;Guoliang Yu
中科院分区：
文献类型：
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作者：
S. Weinberger;Guoliang Yu
本文根据离散群的挠量，研究了离散群的极大$C^*$-代数的K-理论的下界。我们称之为有限部分的运营商K-理论，并给出了一个下界，是有效的一大类群体，称为“可嵌入群”。可嵌入群的类包括所有剩余有限群、顺从群、Gromov怪物群、虚挠自由群（例如$Out（F_n）$）以及固定给定点的解析连通流形的任何解析自同构群。是否每个可数群都可嵌入是一个开放问题。我们应用这个结果来衡量的程度非刚性的任何紧定向流形$M$的维数为$4k-1$（k>1）$。在这种情况下，我们得到了结构群$S（M）$的秩的下界。对于维数大于或等于5且具有正数量曲率度量的紧致黎曼流形M，存在一个阿贝尔群P（M）度量M上所有正数量曲率度量空间的大小.当紧致光滑自旋流形M的维数为2k-1（k>2）且M的基本群是可嵌入的时，得到了阿贝尔群P（M）的秩的一个下界.
In this paper, we study lower bounds on the K-theory of the maximal $C^*$-algebra of a discrete group based on the amount of torsion it contains. We call this the finite part of the operator K-theory and give a lower bound that is valid for a large class of groups, called the "finitely embeddable groups". The class of finitely embeddable groups includes all residually finite groups, amenable groups, Gromov's monster groups, virtually torsion free groups (e.g. $Out(F_n)$), and any group of analytic diffeomorphisms of an analytic connected manifold fixing a given point. It is an open question if every countable group is finitely embeddable. We apply this result to measure the degree of non-rigidity for any compact oriented manifold $M$ with dimension $4k-1$ $(k>1)$. We derive a lower bound on the rank of the structure group $S(M)$ in this case. For a compact Riemannian manifold $M$ with dimension greater than or equal to 5 and positive scalar curvature metric, there is an abelian group $P(M)$ that measures the size of the space of all positive scalar curvature metrics on $M$. We obtain a lower bound on the rank of the abelian group $P(M)$ when the compact smooth spin manifold $M$ has dimension $2k-1$ $(k>2)$ and the fundamental group of $M$ is finitely embeddable.