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On the rho invariant for manifolds with boundary. (English) Zbl 1032.58013

Let \(M\) be a closed odd dimensional Riemannian manifold and let \(D\) be the tangential operator of the signature complex. Let \(\alpha:\pi_1(M)\rightarrow U(n)\) be a representation of the fundamental group. One sets \[ \rho(M,\alpha):=\eta(D_\rho)-k\eta(D) \] to be the relative eta invariant; this is independent of the Riemannian metric on \(M\).
If, more generally, \(M\) is a compact odd dimensional Riemannian manifold with boundary, the authors impose APS (i.e. spectral) boundary conditions with respect to quite general Lagrangian subspaces to define the eta invariant. The relative eta invariant now depends on the choice of Riemannian metric \(g\) on the boundary and Lagrangian subspace and the authors derive an appropriate ‘cut and paste’ formula for the \(\eta\) and \(\rho\) invariants in this setting. The relative eta invariant \(\rho\) is shown to depend only on the pseudo-isotopy class of the metric on the boundary; specific examples to illustrate this dependence are given. Relations with spectral flow are given and the machinery of determinant bundles is invoked to study the variation mod \(Z\).
The article concludes with a discussion of the relations to the program of constructing topological quantum field theories.

MSC:

58J28 Eta-invariants, Chern-Simons invariants
57M27 Invariants of knots and \(3\)-manifolds (MSC2010)
58J32 Boundary value problems on manifolds
58J30 Spectral flows
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