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The eta invariant for even dimensional \(PIN_ c\) manifolds. (English) Zbl 0602.58041
This paper is devoted to studying the eta invariant for the tangential \(PIN_ c\) operator on a compact even dimensional manifold M without boundary. The eta invariant is an \({\mathbb{R}} mod {\mathbb{Z}}\) valued invariant which measures the spectral asymmetry of a self-adjoint elliptic pseudodifferential operator defined on a compact manifold. The author shows that the eta invariant corresponding to an operator whose leading symbol is given by Clifford multiplication belongs to \({\mathbb{Z}}[2^{-\ell -1}]\), where \(2\ell =\dim (M)\); the eta invariant is applied to compute the reduced complex K-group of real projective space.
The author studies functorial constructions related to taking the product of two even dimensional manifolds and the twisted product of two odd- dimensional manifolds. These constructions are applied to obtain other manifolds where the eta invariant is nontrivial by taking the twisted product of a circle and a lensspace.
In an earlier paper [Invent. Math. 76, 421-453 (1984; Zbl 0547.58032)] the author used the eta invariant of the tangential \(SPIN_ c\) complex to calculate the complex K-theory of odd dimensional spherical space forms.
Reviewer: V.Deundyak

MSC:
58J10 Differential complexes
55N15 Topological \(K\)-theory
58J20 Index theory and related fixed-point theorems on manifolds
58J50 Spectral problems; spectral geometry; scattering theory on manifolds
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