The Novikov conjecture, the group of volume preserving diffeomorphisms and Hilbert-Hadamard spaces

Abstract

In this paper, we prove the Novikov conjecture for a class of highly non-linear groups, namely discrete subgroups of the diffeomorphism group of a compact smooth manifold. This removes the volume-preserving condition in a previous work. This result is proved by studying operator K-theory and group actions on continuous fields of infinite dimensional non-positively curved spaces.Proof. It follows from [AAS20, Section 5] that the above homomorphism gives rise to an isomorphism between KK Γ R, * (EΓ, A) and KK Γ R, * (EΓ, A) τ , which is a subgroup of KK Γ R, * (EΓ, A) called its τ -part. Lemma 2.9. Let Γ be a discrete group, let X be a free and proper Γ-space, let A and B be two Γ-C * -algebras, and let ϕ : A → B be a *homomorphism that is Γ-equivariant and a homotopy equivalence, i.e., there exists a (possibly non-equivariant) * -homomorphism ψ : B → A such that ψ • ϕ and ϕ • ψ are homotopic to the identity maps on A and B, respectively. Then the homomorphismProof. This uses a cutting-and-pasting argument similar to [GWY21, Lemma 8.3].2.2. Hilbert-Hadamard spaces. In this section, we review some basics of Hilbert-Hadamard spaces (cf. [GWY21, Section 3]).Definition 2.10. A metric space (X, d) is CAT(0) if for any p, q, r, m ∈ X satisfying d(q, m) = d(r, m) = 1 2 d(q, r), the following CN inequality

The coarse Novikov conjecture for extensions of coarsely embeddable groups

“…A fundamental idea underlining the approach in [33] is that the index of a Dirac operator is more computable if the Dirac operator is twisted by a family of "almost flat Bott bundles". This approach inspires several later progresses on the coarse Novikov conjecture for coarse embeddings into certain Banach spaces [19,6] or non-positively curved manifolds [12,27]. See also [9,11,20,24,32] for closely related developments.…”

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

“…We will show that the Bott map (β x 0 ) * : K * (S) → K * (A(B)) is an isomorphism for any x 0 ∈ B. One can find a proof in [10,Remark 7.7] for the case when p = 2. We remark that the Bott periodicity theorem holds for any Banach space which is spherical equivalent to ℓ 2 by using a similar argument.…”

“…However, Q((A n ⊗K) n∈N ) is built by using a dense subspace of B which may not be invarinat under t. To solve this problem, we construct a new algebra A(B) for an ℓ p -space B = ℓ p (N, R) in Section 3. Our construction is inspired from the paper of S. Gong, J. Wu and G. Yu [10] for the case when p = 2. We construct a Bott homomorphism β x 0 : S → A(B) associated with a base point x 0 ∈ B by using the p/2-H ölder extension of the Mazur map introduced by E. Odell and T. Schlumprecht in [15] and extended by Q. Cheng in [6].…”

The coarse Baum-Connes conjecture for certain relative expanders

“…the fundamental group of a closed manifold) equipped with a word length metric, and the Gromov's zero-in-the-spectrum conjecture and the positive scalar curvature conjecture when X is a Riemannian manifold. See [55] for a comprehensive survey for the coarse Baum-Connes conjecture, and [5,9,18,19,20,21,22,24,41,48,49,50,51,53] for some recent developments.…”