Volume, Cheeger and Gromov

On Invariants of Hirzebruch and Cheeger–Gromov

We prove that, if M is a compact oriented manifold of dimension 4k + 3, where k > 0, such that π1(M) is not torsion-free, then there are infinitely many manifolds that are homotopic equivalent to M but not homeomorphic to it. To show the infinite size of the structure set of M , we construct a secondary invariant τ(2) : S(M) → R that coincides with the ρ–invariant of Cheeger–Gromov. In particul...

Cheeger-gromov Theory and Applications to General Relativity

This paper surveys aspects of the convergence and degeneration of Riemannian metrics on a given manifold M , and some recent applications of this theory to general relativity. The basic point of view of convergence/degeneration described here originates in the work of Gromov, cf. [27]-[29], with important prior work of Cheeger [14], leading to the joint work of [16]. This Cheeger-Gromov theory ...

Lectures on Cheeger-Gromov Theory of Riemannian manifolds

Cheeger Isoperimetric Constants of Gromov-hyperbolic Spaces with Quasi-poles

Let X be a non-compact complete manifold (or a graph) which admits a quasi-pole and has bounded local geometry. Suppose that X is Gromov-hyperbolic and the diameters (for a fixed Gromov metric) of the connected components of X(∞) have a positive lower bound. Under these assumptions we show that X has positive Cheeger isoperimetric constant. Examples are also constructed to show that the Cheeger...

Gromov ’ s invariant and the volume of a hyperbolic manifold

This is an electronic edition of the 1980 notes distributed by Princeton University. The text was typed in T E X by Sheila Newbery, who also scanned the figures. Typos have been corrected (and probably others introduced), but otherwise no attempt has been made to update the contents. Numbers on the right margin correspond to the original edition's page numbers. Thurston's Three-Dimensional Geom...