Collapsing and Dirac-type Operators

We analyze the limit of the spectrum of a geometric Dirac-type operator under a collapse with bounded diameter and bounded sectional curvature. In the case of a smooth limit space B, we show that the limit of the spectrum is given by the spectrum of a certain first-order differential operator on B, which can be constructed using superconnections. In the case of a general limit space X , we express the limit operator in terms of a transversally elliptic operator on a G-space X̌ with X = X̌/G. As an application, we give a characterization of manifolds which do not admit uniform upper bounds, in terms of diameter and sectional curvature, on the k-th eigenvalue of the square of a Dirac-type operator. We also give a formula for the essential spectrum of the square of a Dirac-type operator on a finite-volume manifold with pinched negative sectional curvature.