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The eta-invariant and parity conditions. (English. Russian original) Zbl 1044.58027
Dokl. Math. 63, No. 2, 189-193 (2001); translation from Dokl. Akad. Nauk, Ross. Akad. Nauk 377, No. 3, 305-309 (2001).
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.

##### MSC:
 58J28 Eta-invariants, Chern-Simons invariants
##### Keywords:
eta invariant; Dirac operator