正数量曲率的余维双指标障碍
Codimension two index obstructions to positive scalar curvature
摘要：
利用余维为2的超曲面给出了闭自旋流形上正数量曲率黎曼度量存在的一般性障碍。证明是基于Dirac算子的粗指标理论与希尔伯特C*-模丛扭曲。在此基础上，我们给出了完备黎曼流形上与Hilbert C ~*-模丛扭在一起的Dirac型算子的极小闭包是这个流形的L ~ 2-截面的Hilbert C ~*-模上的正则自伴算子的完备且完备的证明.此外，对于完备黎曼流形上的Dirac算子的粗指标，给出了Roe消灭定理的一个新的证明，该流形的数量曲率在紧致子集外是一致正的.这个证明立即推广到Dirac算子扭曲希尔伯特C*-模丛。
英文摘要：
We derive a general obstruction to the existence of Riemannian metrics of positive scalar curvature on closed spin manifolds in terms of hypersurfaces of codimension two. The proof is based on coarse index theory for Dirac operators that are twisted with Hilbert C*-module bundles. Along the way we give a complete and self-contained proof that the minimal closure of a Dirac type operator twisted with a Hilbert C*-module bundle on a complete Riemannian manifold is a regular and self-adjoint operator on the Hilbert C*-module of L2-sections of this bundle. Moreover, we give a new proof of Roe's vanishing theorem for the coarse index of the Dirac operator on a complete Riemannian manifold whose scalar curvature is uniformly positive outside of a compact subset. This proof immediately generalizes to Dirac operators twisted with Hilbert C*-module bundles.
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