论文标题
其有限类似物的离域ETA不变性近似
Approximations of delocalized eta invariants by their finite analogues
论文作者
论文摘要
对于给定的自动伴侣一阶椭圆差差分运算符，我们证明了与常规覆盖空间相关的离域ETA不变式何时可以通过与有限片覆盖空间相关的Delocalized ETA不变性近似。我们的主要结果之一是以下内容。假设$ m $是封闭的平滑旋转歧管，$ \ widetilde m $是$γ$的覆盖空间$ m $。令$ \langleα\ rangle $为γ$中的非身份元素$α\的共轭类别。假设$ \ {γ_i\} $是$γ$的一系列有限索引正常子组，该子组区分了$ \langleα\ rangle $。令$π_{γ_I} $为商图，从$γ$到$γ/γ_i$和$ \langleπ_{γ_I}（α）\ rangle $ $π_{γ_I}（α）$gumγ__i$ $γ_i$ $。如果$ m $上的标量曲率在下面有足够大的正数，那么在$ \ wideTilde m $的Dirac运算符中，ETA不变级别的共轭类别$ \langleα\ rangle $等于$ m m m m m m im______________________________________________________________________________________ $ \langleπ_{γ_i}（α）\ rangle $，其中$ m_ {γ_I} = \ widetilde m/γ_i$是$ m $由$γ_i$确定的$ M $的有限空间。在本文的另一个主要结果中，我们证明了$ m_ {γ_i} $的Dirac Operators的限制在偶联类$ \langleπ_{γ_i}（α）\ rangle $ commation committion committion $ maximal baum-conund $ conture $ conture $ contruce pred $ contiment $ consection $ \ langleπ_{γ_I}（α）\ langleπ_{γ_i}（α）
For a given self-adjoint first order elliptic differential operator on a closed smooth manifold, we prove a list of results on when the delocalized eta invariant associated to a regular covering space can be approximated by the delocalized eta invariants associated to finite-sheeted covering spaces. One of our main results is the following. Suppose $M$ is a closed smooth spin manifold and $\widetilde M$ is a $Γ$-regular covering space of $M$. Let $\langle α\rangle$ be the conjugacy class of a non-identity element $α\in Γ$. Suppose $\{Γ_i\}$ is a sequence of finite-index normal subgroups of $Γ$ that distinguishes $\langle α\rangle$. Let $π_{Γ_i}$ be the quotient map from $Γ$ to $Γ/Γ_i$ and $\langle π_{Γ_i}(α) \rangle$ the conjugacy class of $π_{Γ_i}(α)$ in $Γ/Γ_i$. If the scalar curvature on $M$ is everywhere bounded below by a sufficiently large positive number, then the delocalized eta invariant for the Dirac operator of $\widetilde M$ at the conjugacy class $\langle α\rangle$ is equal to the limit of the delocalized eta invariants for the Dirac operators of $M_{Γ_i}$ at the conjugacy class $\langle π_{Γ_i}(α) \rangle$, where $M_{Γ_i}= \widetilde M/Γ_i$ is the finite-sheeted covering space of $M$ determined by $Γ_i$. In another main result of the paper, we prove that the limit of the delocalized eta invariants for the Dirac operators of $M_{Γ_i}$ at the conjugacy class $\langle π_{Γ_i}(α) \rangle$ converges, under the assumption that the rational maximal Baum-Connes conjecture holds for $Γ$.
