Elliptic operators in subspaces and the eta invariant (Q1868424)

Let \(A\) be a self-adjoint elliptic operator on a closed manifold \(M\) of dimension \(m\). One defines \(\eta(s,A):=\sum_\nu sign(\lambda_\nu)|\lambda_\nu|^{-s}\) as a measure of the spectral asymmetry of \(A\). This has an analytic continuation to \(C\); \(s=0\) is a regular value and one sets \(\eta(A)=\eta(0,A)/2\). If \(A_t\) is a smooth \(1\) parameter family of partial differential operators of order \(r\) and \(r+m\) is odd, then the mod \(Z\) reduction of \(\eta(A_t)\) is independent of the parameter \(t\). The authors derive a formula for this fractional part of the eta invariant in topological terms using the index theorem for elliptic operators on subspaces. The authors also study the \(K\)-theory of \(M\) with coefficients in \(Z_n\). The authors have used the results of this paper in subsequent work to construct some new second order even order operators in odd dimensions with non-trivial eta invariant.