K-Theory and Dynamics. I

A history and survey of the Novikov conjecture

“…Consequently, a number of authors have proven vanishing theorems for Whitehead groups of fundamental groups of aspherical manifolds and polyhedra, beginning with the proof in [Hi] that Wh(Z) = 0 and in [BHS] that Wh(Z n ) = 0. Other efforts along these lines include [FarHs2], [FarHs9], [FJ3], [FJ12], [Ni2], [Ni3], [Q4] and [Wald1]. A notable recent effort is the paper [Hu] in which Hu proves that Wh(Γ) = 0 when Γ is the fundamental group of a finite nonpositively curved (≡ CAT(0)) polyhedron.…”

A PL geometric study of algebraic 𝐾 theory

“…In this section we shall establish a long thin stratified decomposition and prove a topological foliated control result, that is the topological version of a smooth foliated control result of Farrell and Jones in [FJ1,[6][7][8]. This result can handle the geodesic flow G of any PL manifold with A < 0.…”

“…This completes the proof. See [FJ1,§2], and [Hu,6.3]. Clearly, W = ÍV+ u W° U W~ , ÍV+uW0 is an .4 together imply that there is / > 0 such that for any e > 0 there are x > 0 and homeomorphism C7+ U G° A R+ U R° such that the «-cobordism xg-x(W+ U W°) over C7+ U C7° has diameter < (I, e).…”

“…Our study originates from an attempt to carry a very successful strategy created by F. T. Farrell and L. E. Jones in [FJ1] into M. Gromov's theory of hyperbolic groups [Gl], to "say something about the Whitehead groups of hyperbolic groups. "…”

“…To describe the theory of Farrell-Jones in [FJ1], first recall the 5-cobordism theorem: Let W be an /i-cobordism over Mn , n > 5 . Then W is a product if and only if x(W) £ Wh(7TiM) vanishes; any element of Wh^i-M) is the Whitehead torsion of an A-cobordism over M. Next there is the control theory: This theory measures an /i-cobordism by a real number, the diameter, and tells that the /z-cobordism is a product when the diameter is small enough.…”