On the range of the relative higher index and the higher rho-invariant for positive scalar curvature
On the range of the relative higher index and the higher rho-invariant for positive scalar curvature
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DOI：
10.1016/j.aim.2021.107897
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发表时间：
2017-12
影响因子：
1.7
通讯作者：
Zhizhang Xie;Guoliang Yu;Rudolf Zeidler
中科院分区：
文献类型：
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作者：
Zhizhang Xie;Guoliang Yu;Rudolf Zeidler
设M是一个闭的自旋流形，它支持一个正的标量曲率度量。M上的正数量曲率度量的协调类集合在固定一个正数量曲率度量后构成一个阿贝尔群P（M）。群P（M）度量M上正数量曲率度量空间的大小。Weinberger和Yu用π 1（M）的挠元个数给出了P（M）的秩的一个下界.本文通过研究群C_n-代数Cr_n（π 1（M））的从P（M）到真实的K-理论的相对高指数映射的象，给出了P（M）的秩的一个较尖锐的下界.我们证明了它在一定的同调程度上合理地包含Baum-Connes组装映射的像，这取决于M的维数。同时利用高阶rho-不变量得到了正数量曲率边群的下界。
Let M be a closed spin manifold which supports a positive scalar curvature metric. The set of concordance classes of positive scalar curvature metrics on M forms an abelian group P (M) after fixing a positive scalar curvature metric. The group P (M) measures the size of the space of positive scalar curvature metrics on M. Weinberger and Yu gave a lower bound of the rank of P (M) in terms of the number of torsion elements of π 1 (M). In this paper, we give a sharper lower bound of the rank of P (M) by studying the image of the relative higher index map from P (M) to the real K-theory of the group C⁎-algebra C r⁎(π 1 (M)). We show that it rationally contains the image of the Baum–Connes assembly map up to a certain homological degree depending on the dimension of M. At the same time we obtain lower bounds for the positive scalar curvature bordism group by applying the higher rho-invariant.