Slowly increasing cohomology for discrete metric spaces with polynomial growth (Q1928187)

The authors associate to any discrete metric space \(X\) of polynomial growth the subcomplex \((CS^*(X),\partial)\) of the Alexander-Spanier complex of \(X\) consisting of slowly increasing cochains. The resulting cohomology groups are referred to as the slowly increasing cohomology of \(X\) (with real or complex coefficients). Alternatively (see Theorem 2.1), this cohomology may be computed as the cohomology of the subcomplex of polynomially bounded cochains. These cohomology groups are a coarse isometric invariant (Theorem 3.2). In an earlier work [J. Funct. Anal. 197, No. 1, 228--246 (2003; Zbl 1023.46060)], the authors have shown that, if \(X\) has polynomial growth, then the uniform Roe algebra \(B^*(X)\) contains a smooth Fréchet subalgebra \(S(X)\) of Haagerup type. The main result, shown in Theorem 4.1, is that slowly increasing cocycles may be extended to continuous cyclic cocycles on the Fréchet algebra \(S(X)\). An immediate consequence is a pairing \(K^t_n(B^*(X))\otimes HS^*(X)\to \mathbb{C}\) which, combined with the earlier work of \textit{G.-L. Yu} [J. Funct. Anal. 133, No. 2, 442--473 (1995; Zbl 0849.58066)], is applied to prove the following result. Theorem 6.3. Let \(M\) be a uniformly contractible complete Riemannian manifold with polynomial growth and polynomial radius growth. Let \(D\) be the generalized Dirac operator on the Clifford bundle over \(M\). Then \(0\leq[\text{Ind}(D)]\in K_0(B^*(M))\).