Approximations of delocalized eta invariants by their finite analogues

Abstract

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 $\tilde{M}$ is a $\hat{I}“$-regular covering space of $M$. Let $⟨\hat{I}\pm ⟩$ be the conjugacy class of a non-identity element $\hat{I}\pm \in \hat{I}“$. Suppose $\{\hat{I}{“}_i\}$ is a sequence of finite-index normal subgroups of $\hat{I}“$ that distinguishes $⟨\hat{I}\pm ⟩$. Let $\ddot{I}{€}_{\hat{I}{“}_i}$ be the quotient map from $\hat{I}“$ to $\hat{I}“/\hat{I}{“}_i$ and $⟨\ddot{I}{€}_{\hat{I}{“}_i}(\hat{I}\pm )⟩$ the conjugacy class of $\ddot{I}{€}_{\hat{I}{“}_i}(\hat{I}\pm )$ in $\hat{I}“/\hat{I}{“}_i$. 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 $\tilde{M}$ at the conjugacy class $⟨\hat{I}\pm ⟩$ is equal to the limit of the delocalized eta invariants for the Dirac operators of $M_{\hat{I}{“}_i}$ at the conjugacy class $⟨\ddot{I}{€}_{\hat{I}{“}_i}(\hat{I}\pm )⟩$, where $M_{\hat{I}{“}_i}=\tilde{M}/\hat{I}{“}_i$ is the finite-sheeted covering space of $M$ determined by $\hat{I}{“}_i$. In another main result of the paper, we prove that the limit of the delocalized eta invariants for the Dirac operators of $M_{\hat{I}{“}_i}$ at the conjugacy class $⟨\ddot{I}{€}_{\hat{I}{“}_i}(\hat{I}\pm )⟩$ converges, under the assumption that the rational maximal Baum-Connes conjecture holds for $\hat{I}“$.

41 pages. Minor revision and updated reference