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Homotopy invariance of \(\eta\)-invariants. (English) Zbl 0659.57016

Given a smooth manifold M of odd dimension and a representation \(\rho\) : \(\pi_ 1(M)\to U(n)\) of its fundamental group, Atiyah, Patodi, and Singer introduced the eta-invariant \(\eta_{\rho}(M)\) by choosing a Riemannian metric and extending the signature operator to differential forms with coefficients in the flat vector bundle defined by \(\rho\). They showed tha \(\eta_{\rho}(M)\) is independent of the metric; however, it is not a homotopy invariant in general.
The author announces that, for manifolds with certain types of fundamental groups, \(\eta_{\rho}(M)\) is indeed an oriented homotopy invariant, generalizing an earlier result by W. Neumann. The proof will use deep tools such as intersection homology and work on the Novikov conjecture.
Reviewer: W.Singhof

MSC:

57R19 Algebraic topology on manifolds and differential topology
58J50 Spectral problems; spectral geometry; scattering theory on manifolds
57R20 Characteristic classes and numbers in differential topology
58J20 Index theory and related fixed-point theorems on manifolds
57M05 Fundamental group, presentations, free differential calculus
53C99 Global differential geometry
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