A Bott periodicity theorem for ${\ell }^p$-spaces and the coarse Novikov conjecture at infinity

Abstract

We formulate and prove a Bott periodicity theorem for an ${\ell }^p$-space ($1\le p<\infty$). For a proper metric space $X$ with bounded geometry, we introduce a version of $K$-homology at infinity, denoted by $K_{\ast }^{\infty }(X)$, and the Roe algebra at infinity, denoted by $C_{\infty }^{\ast }(X)$. Then the coarse assembly map descents to a map from ${\lim }_{d\to \infty }{K}_{\ast }^{\infty }(P_d(X))$ to $K_{\ast }(C_{\infty }^{\ast }(X))$, called the coarse assembly map at infinity. We 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 ${\ell }^p$-space. These include all box spaces of a residually finite hyperbolic group and a large class of warped cones of a compact space with an action by a hyperbolic group.

55 pages