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

Weinberger, Shmuel

Proc. Natl. Acad. Sci. USA 85, No. 15, 5362-5363 (1988).

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

Keywords:

higher signature; representation of fundamental group; smooth manifold; eta-invariant; signature operator; homotopy invariant; intersection homology; Novikov conjecture