多项式增长与可缩空间上的粗糙指标理论
微分几何
2018-04-11 v4 K理论与同调
摘要
我们将证明，对于具有有界几何和多项式体积增长的多项式可缩流形，每个粗（coarse）与粗糙（rough）上同调类都与一致 Roe 代数的 K-理论连续配对。作为一个应用，我们将讨论此类流形上 Dirac 算子的粗糙指标类的非零性，并进一步得到对具有几乎幂零基本群的闭流形上正数量曲率度量的高余维指标阻碍。我们将给出多项式体积增长流形的一致 Roe 代数（的稠密光滑子代数）的同调的计算。
引用
@article{arxiv.1505.03988,
title = {Rough index theory on spaces of polynomial growth and contractibility},
author = {Alexander Engel},
journal= {arXiv preprint arXiv:1505.03988},
year = {2018}
}
备注
v4: final version, to appear in J. Noncommut. Geom. v3: added a computation of the homology of (a smooth subalgebra of) the uniform Roe algebra. v2: added as corollaries to the main theorem the multi-partitioned manifold index theorem and the higher-codimensional index obstructions against psc-metrics, added a proof of the strong Novikov conjecture for virtually nilpotent groups, changed the title