Topological Entropy and Loop Space Homology

Abstract

The chapter begins with the definitions of rationally elliptic and rationally hyperbolic manifolds and with a summary of various properties and characterizations of rationally elliptic manifolds. Afterwards, we discuss results of J.P. Serre [Se1] and M. Gromov [Gr1] which allows us to relate the growth of n _T (x, y) with the topology of M via Morse theory. Using these ideas we show the result in [P3] that says that if M is a closed manifold that fibres over a closed simply connected rationally hyperbolic manifold, then for any C ^∞ Riemannian metric g on M, h _top (g) > 0. The more classical result of E. Dinaburg [D] which says that if π₁ (M) grows exponentially, then h _top (g) > 0 for any g, is also discussed here. The chapter finishes with various definitions of entropies of manifolds. We connect them with other notions, such as Gromov’s minimal volume and simplicial volume (see the chain of inequalities (5.7)), and we propose various related problems.