The eta-invariant and parity conditions. (Q1432512)

Let \(A\) be an elliptic self-adjoint partial differential operator of order \(k\) on a closed smooth manifold \(M\) of dimension \(m\). Let \(\bar\eta(A)\) be the mod \(\mathbb{Z}\) reduction of the eta invariant. If \(m+k\) is odd, then \(\bar\eta(A)\) is a homotopy invariant determined by the principle symbol. An interesting example in even dimensions is the eta invariant of the Dirac operator on a non-orientable pin\({}^c\) manifold; \(\bar\eta\) is non-trivial for this operator and plays an important role in the study of the pin\({}^c\) bordism groups. It also plays an important role in the study of the Gromov-Lawson conjecture in the non-orientable context. However it remained an open problem to construct operators of even order with non-trivial \(\bar\eta\) on odd dimensional manifolds. In the present paper, the authors give a formula for \(\bar\eta\) if \(k+m\) is odd which is \(K\)-theoretic in nature. They show \(\bar\eta\) is \(2\)-torsion. They construct a second-order geometric operator on an odd dimensional manifold so that \(\bar\eta\) is non-trivial.