论文标题
在具有边界的流形上，正标曲率指标的相对扭转和界体类别
Relative torsion and bordism classes of positive scalar curvature metrics on manifolds with boundary
论文作者
论文摘要
在本文中，我们定义了一个相对的$ l^2 $ - $ρ$ -Invariant，用于零二维的旋转歧管上的零度旋转歧管，并表明它们是在边界附近串制的正标曲率指标的Bordism类别的不变性。作为应用程序，我们表明，如果$ 4K+3 $维旋转歧管和边界的旋转歧管承认了一个指标，并且大致说明，在歧管及其边界的基本组的差异中存在一个扭曲元素，那么给定歧管上的此类PSC指标的许多bord bord class类别。反过来，该结果意味着PSC指标在此类歧管上的模量空间具有无限的许多路径成分。我们还指出了如何定义带有边界的奇数旋转歧管的离域$η$ invariants，然后可以将其用于获得相似的结果，以$ 4K+1 $ 1 $维的歧管。
In this paper, we define a relative $L^2$-$ρ$-invariant for Dirac operators on odd-dimensional spin manifolds with boundary and show that they are invariants of the bordism classes of positive scalar curvature metrics which are collared near the boundary. As an application, we show that if a $4k+3$-dimensional spin manifold with boundary admits such a metric and if, roughly speaking, there exists a torsion element in the difference of the fundamental groups of the manifold and its boundary, then there are infinitely many bordism classes of such psc metrics on the given manifold. This result in turn implies that the moduli-space of psc metrics on such manifolds has infinitely many path components. We also indicate how to define delocalised $η$-invariants for odd-dimensional spin manifolds with boundary, which could then be used to obtain similar results for $4k+1$-dimensional manifolds.
