A Bott periodicity theorem for ℓp-spaces and the coarse Novikov conjecture at infinity

Abstract

We formulate and prove a Bott periodicity theorem for an ℓp-space (1≤p<∞). For a proper metric space X with bounded geometry, especially for a coarsely connected space, we introduce a version of K-homology at infinity and the Roe algebra at infinity and show that to prove the coarse Novikov conjecture, it suffices to prove the coarse assembly map at infinity is an injection. As a result, we show that the coarse Novikov conjecture holds for any metric space with bounded geometry which admits a fibred coarse embedding into an ℓp-space. These include all box spaces of a residually finite hyperbolic group, and a large class of warped cones of a compact metric space with an action by a hyperbolic group.