A Bott periodicity theorem for $\ell^{p}$-spaces and the coarse Novikov conjecture at infinity¹¹1Supported in part by NSFC (No. 11831006, 12171156).

The Novikov conjecture claims that the higher signatures of closed oriented smooth manifolds are invariant under orientation preserving homotopy equivalences. It is implied by the injectivity of the Baum-Connes assembly map [2]. The coarse Novikov conjecture, as a geometric analogue of the Novikov conjecture, is the injectivity part of the coarse Baum–Connes conjecture [12, 18]. It offers a technique to determine when an elliptic operators higher index on a noncompact complete Riemannian manifold is nonzero. A detailed survey on the coarse Novikov conjecture can be found in [31].

In [4], X. Chen, Q. Wang and G. Yu generalized Gromov’s notion of coarse embedding into Hilbert space and introduced the notion of fibred coarse embedding (see Definition 2.4) and showed that if a discrete metric space $X$ with bounded geometry admits a fibred coarse embedding into Hilbert space, then the maximal coarse Baum–Connes conjecture holds for $X$. Later in [8], M. Finn-Sell showed that the coarse Novikov conjecture also holds for $X$ in such a case.

Our main theorem in this paper states as follows:

Warped cones over actions of hyperbolic groups and box spaces of residually finite hyperbolic groups are typical examples to which our main result appies. A sequence of nested normal subgroups $\{\Gamma_{i}\}$ of a residually finite group $\Gamma$ is said to be a filtration if the sequence has trivial intersection. The box space of $\Gamma$ associated to a filtration $\{\Gamma_{i}\}$, denoted by ${\bf Box}_{\{\Gamma_{i}\}}\Gamma$, is the coarse disjoint union $\bigsqcup\Gamma/\Gamma_{i}$, where each quotient is equipped with the word length metric. Warped cones are metric spaces introduced by J. Roe [20] from discrete group actions on compact metric spaces. Typically, let $\Gamma$ be a finite generated group, $M$ a compact Riemannian manifold. The warped cone for $(\Gamma,M)$, denoted by $\mathcal{O}_{\Gamma}(M)$, is constructed from the action $\Gamma$ on $M$ by diffeomorphisms. It is a metric space whose underlying topological space is the open cone $\mathcal{O}M$ on $M$, but whose coarse geometry produces large groups of translations. The second section will have a comprehensive introduction for warped cones.

In [22], D. Sawicki and J. Wu found a relationship between fibred coarse embeddability of warped cones and existence of proper affine isometric actions of discrete groups on Banach spaces. They proved that if $(\Gamma,M)$ is free and admits a linearization by unitary operators in a Hilbert space $\mathcal{H}$ and $\Gamma$ admits a proper affine isometric action on $\mathcal{H}$ (i.e., $\Gamma$ is a-T-menable), then $\mathcal{O}_{\Gamma}(M)$ admits a fibred coarse embedding into Hilbert space. The result still holds if we replace the Hilbert space with an $\ell^{p}$-space. The same question is also considered by Q. Wang and Z. Wang in [24]. They provided a different proof for the case of $\ell^{p}$ spaces and replace the condition $\Gamma\curvearrowright M$ is free by that $M$ is required to contain a dense and free orbit of $\Gamma$ in their paper.

As corollaries, we have the following results:

Combining the result in [30, 21, 3], we also have

We now give a short explaination about serveral key ingredients in our proof. We briefly recall the proof of the maximal coarse Baum-Connes conjecture for a metric space which admits a fibred coarse embedding into a Hilbert space in [4]. The authors firstly reduced the issue to proving the maximal coarse Baum-Connes conjecture at infinity for the coarse disjoint unions of finite metric spaces. By using a geometric Dirac-dual-Dirac argument, they further reduced the problem to a twisted version of coarse Baum-Connes conjecture at infinity by using a $C^{*}$-algebra constructed in [11]. Then the theorem follows by using a cutting and pasting argument of G. Yu introduced in the Section 6 of [29]. We try to follow the outline of their proof, however there are servarl problems.

For the first step, we need to use a Mayer-Vietoris argument and Five Lemma to reduce the problem to ’infinity’. First of all, a coarse Mayer-Vietoris argument in [13] enables us to just take into consideration of the coarse Baum-Connes conjecture for coarse disjoint unions of finite subsets. For the case when $X=\bigsqcup_{n\in\mathbb{N}}X_{n}$ where each $X_{n}$ is finite, one can use the following diagram, see [4]:

The vertical sequence on the right side is exact only when we take the maximal norm and the top horizantal map is an isomorphism. Then it suffices to prove the assembly map at infinity $\mu_{\infty}$ is an isomorphism.

However, when we consider the coarse Novikov conjecture for certain spaces, the coarse Mayer-Vietories argument above does not work anymore as the Five Lemma fails if we only have the injectivity part of the assembly map $\mu$. The argument holds only for a coarse disjoint union of a sequence of finite metric spaces $X=\bigsqcup_{n\in\mathbb{N}}X_{n}$. One can still reduce the coarse Novikov conjecture for $X$ to ’infinity’ by using the following diagram, see [25, 26].

Notice that $i_{*}:K_{*}(\mathcal{K})\to K_{*}(C^{*}(X))$ is injective only when $X$ is a coarse disjoint union of finite spaces, see [16]. By using a diagram chasing argument, to prove $\mu$ is injective, it suffices to prove $\mu_{\infty}$ is injective.

For the general case (especially the case when $X$ can not be seen as a coarse disjoint union of finite metric spaces), in order to make the whole progress work well, we introduce a notion of $K$-homology at infinity for a locally compact Hausdorff space and a notion of Roe algebra at infinity for a proper metric space in Section 4 and construct the following diagram:

A metric space with bounded geometry is said to admit an infinite coarse component if there exists $R>0$ such that the Rips complex $P_{R}(X)$ has an unbounded component. It is an opposite statement to the coarse disjoint union (see Proposition 5.5). We prove that $\pi_{*}$ is an injection if $X$ admits an infinite coarse component. Thus we can still reduce the Novikov conjecture for $X$ to proving $\mu_{\infty}$ is injective in this case. Moreover, we show in Section 5 that our construction coincides with the construction in [4] for the coarse disjoint union case. This means that we can always reduce the coarse Novikov conjecture to ’infinity’ by using our construction (see Theorem 5.7).

For the second step, we need to construct a twisted version of the coarse Baum-Connes conjecture by using an algebra associated with $B=\ell^{p}(\mathbb{N},\mathbb{R})$ for $p\in[1,\infty)$. As $B$ is of property (H) introduced by G. Kasparov and G. Yu in [14], we wanted to use the $C^{*}$-algebra $\mathcal{Q}((\mathcal{A}_{n}\widehat{\otimes}\mathcal{K})_{n\in\mathbb{N}})$ introduced in [14, 5] to solve this problem. For any $a\in\mathcal{Q}((\mathcal{A}_{n}\widehat{\otimes}\mathcal{K})_{n\in\mathbb{N}})$, we shall need a $C^{*}$-isomorphism $t^{*}:\mathcal{Q}((\mathcal{A}_{n}\widehat{\otimes}\mathcal{K})_{n\in\mathbb{N}})\to\mathcal{Q}((\mathcal{A}_{n}\widehat{\otimes}\mathcal{K})_{n\in\mathbb{N}})$ associated to an affine isometry $t$ on $B$ to translate the support of $a$. However, $\mathcal{Q}((\mathcal{A}_{n}\widehat{\otimes}\mathcal{K})_{n\in\mathbb{N}})$ 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 $\mathcal{A}(B)$ for an $\ell^{p}$-space $B=\ell^{p}(\mathbb{N},\mathbb{R})$ 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 $\beta_{x_{0}}:\mathcal{S}\to\mathcal{A}(B)$ associated with a base point $x_{0}\in B$ by using the $p/2$-Hölder extension of the Mazur map introduced by E. Odell and T. Schlumprecht in [15] and extended by Q. Cheng in [6]. We show that the Bott homomorphism induces an isomorphism on $K$-theory which does not depend on the choice of the base point, i.e., $(\beta_{x_{0}})_{*}:K_{*}(\mathcal{S})\to K_{*}(\mathcal{A}(B))$ is an isomorphism for any $x_{0}\in B$ (see Theorem 3.15).

Actually, it has been pointed out in G. Kasparov and G. Yu’s paper [14] that the Bott periodicity theorem holds for $\ell^{p}$-spaces. Indeed, the Mazur map can be extended to a homeomorphism between $\ell^{p}(\mathbb{N},\mathbb{R})$ and $\ell^{2}(\mathbb{N},\mathbb{R})$ and the Bott periodicity for $\ell^{2}(\mathbb{N},\mathbb{R})$ has been proved in [11]. However, the constructions in [11] and [14] also rely on a dense subspace of $\ell^{p}(\mathbb{N},\mathbb{R})$ which is sightly different from ours.

This paper is organized as follows. In Section 2, we briefly recall the Roe algebra and Yu’s localization algebra, and states the coarse Novikov conjecture. After that we go over the concept of warped metric and warped cones introduced by John Roe [20]. In Section 3, we introduce an algebra $\mathcal{A}(B)$ associated with $B=\ell^{p}(\mathbb{N},\mathbb{R})$ and calculate its $K$-theory. It will play an important role in the following proof. In Section 4, we introduce $K$-homology at infinity, Roe algebra at infinity and the coarse Novikov conjecture at infinity for a proper metric space. In Section 5. we compare the case of coarse disjoint union and the case of spaces which admit an infinite coarse component, and reduce both cases to the coarse Novikov conjecture at infinity. In Section 6, we define twisted algebras at infinity and compute their $K$-theories by using a cutting and pasting technique introduced by G. Yu in [29]. In Section 7, we define the Bott map and complete the proof.

First of all, let us recall the definition of the Roe algebra and the coarse Baum-Connes assembly map for a proper metric space (cf. [2, 18, 19]) in this section.

Assume that $(X,d)$ is a proper metric space. For $\delta>0$, a $\delta$-net of $X$ is a discrete subset of $X_{\delta}\subset X$ such that there exists $r>0$ such that $d(x,x^{\prime})>r$ for all distinct $x,x^{\prime}\in X_{\delta}$ and $\bigcup_{x\in X_{\delta}}B(x,\delta)=X$. Moreover, $X$ is said to have bounded geometry if $X$ contains a net $X_{\delta}$ with bounded geometry for some $\delta>0$, i.e., for any $r>0$ there exists $N>0$ such that any ball of radius $r$ in $X_{\delta}$ contains at most $N$ elements.

We denote $C_{0}(X)$ to be the $C^{*}$-algebra of all continuous functions on $X$ which vanish at infinity. A separable infinite-dimensional Hilbert space $H_{X}$ is said to be an $X$-module if $H_{X}$ is equipped with a non-degenerate $*$-representation $\pi:C_{0}(X)\to\mathcal{B}(H_{X})$. Moreover, $H_{X}$ is said to be ample if no non-zero element of $C_{0}(X)$ acts as a compact operator.

Note that $C^{*}(X,H_{X})$ does not depend on the choice of $X$-module $H_{X}$ up to a non-canonical $*$-isomorphism [13]. Hence, it is convenient to discuss the following specific $X$-module. Let $Z\subseteq X$ be a countable dense subset of $X$ and $H$ be a fixed infinite dimensional Hilbert space. We denote $\mathcal{K}(H)$ to be the set of all compact operators on $H$. We define $H_{X}=\ell^{2}(Z)\otimes H$ to be an ample $X$-module equipped with the pointwise multiplication representation of $C_{0}(X)$ on $\ell^{2}(Z)$. It is easy to check that $H_{X}$ is ample.

Notice that $\mathbb{C}_{f}[X]$ can be viewed as a dense $*$-subalgebra of $\mathbb{C}[X,H_{X}]$ in $C^{*}(X)$, where the algebraic operation of $\mathbb{C}_{f}[X]$ is given by viewing $T\in\mathbb{C}_{f}[X]$ as $Z$-by-$Z$ matrix. In this sequel, we will use $\mathbb{C}_{f}[X]$ to replace $\mathbb{C}[X]$ to define the Roe algebra of $X$.

We next recall the assembly map $\mu$ for the Roe algebras. Let $X$ be a proper metric space. Recall that the $K$-homology groups $K_{i}(X)=KK_{i}(C_{0}(X),\mathbb{C})$ $(i=0,1)$ are generated by certain cycles modulo certain equivalence relations

• (1)

  a cycle for $K_{0}(X)$ is a pair $(H_{X},F)$, where $H_{X}$ is an $X$-module and $F$ is a bounded linear operator acting on $H_{X}$ such that $F^{*}F-I$ and $FF^{*}-I$ are locally compact, and $\varphi F-F\varphi$ is compact for all $\varphi\in C_{0}(X)$;

• (2)

  a cycle for $K_{1}(X)$ is a pair $(H_{X},F)$, where $H_{X}$ is an $X$-module and $F$ is a self-adjoint operator acting on $H_{X}$ such that $F^{2}-I$ is locally compact, and $\varphi F-F\varphi$ is compact for all $\varphi\in C_{0}(X)$.

Let $(H_{X},F)$ represent a cycle in $K_{0}(X)$. For any $R>0$, one can always take a locally finite, uniformly bounded open cover $\{U_{i}\}_{i\in I}$ of $X$ such that the diameter of each $U_{i}$ is no more than $R$. Let $\{\phi_{i}\}_{i\in I}$ be a continuous partition of unity subordinate to the open cover $\{U_{i}\}_{i\in I}$. Define

where the sum converges in the strong operator topology. It is not hard to see that $(H_{X},F)$ and $(H_{X},F^{\prime})$ are equivalent via $(H_{X},(1-t)F+tF^{\prime})$, where $t\in[0,1]$. Note that both $F^{\prime}$ and $F^{\prime 2}-1$ have finite propagation, so $F^{\prime}$ is a multiplier of $C^{*}(X)$ and $F^{\prime}$ is invertible modulo $C^{*}(X)$. Hence $F^{\prime}$ gives rise to an element, denoted by $\partial([F^{\prime}])$ in $K_{0}(C^{*}(X))$, where

is the boundary map of $K$-theory, and $M(C^{*}(X))$ is the multiplier algebra of $C^{*}(X)$. We define the index of $(H_{X},F)$ to be $\partial([F^{\prime}])$. Similarly, we can define the index map from $K_{1}(X)$ to $K_{1}(C^{*}(X))$.

Now we are ready to introduce the coarse Baum–Connes assembly map for a metric space with bounded geometry.

We will next discuss the semi-simplicial metric of the Rips complex of $X$. One can find more details of Rips complex in the section 7.2 of [27]. We shall first define the spherical metric $d_{S}$ on $P_{d}(X)$. On each path connected component of $P_{d}(X)$, the spherical metric is the maximal metric whose restriction to each simplex $\{\sum_{i=0}^{n}t_{i}x_{i}\mid t_{i}\geq 0,\sum_{i}t_{i}=1\}$ is the metric obtained by identifying the simplex with $S^{n}_{+}$ via the map

where $S^{n}_{+}=\{(x_{0},\cdots,x_{n})\in\mathbb{R}^{n+1}\mid x_{i}\geq 0,\sum_{i=0}^{n}x_{i}^{2}=1\}$ endowed with the standard Riemannian metric on the unit $n$-sphere.

For any $x,y\in P_{d}(Z)$, a semi-simplicial path $\delta$ between $x$ and $y$ is a finite sequence of points

where $x_{i}$ and $y_{i}$ are in $X$ for each $i=\{1,\cdots,n\}$. The length of $\delta$ is defined to be

We define the semi-spherical metric $d_{P_{d}}$ on $P_{d}(X)$ by

One can check that $(P_{0}(X),d_{P_{0}})$ identifies isometrically with $(X,d)$. Moreover, it has been proved in [27, Proposition 7.2.11] that the canonical inclusion $i_{d}:X\to P_{d}(X)$ is a coarse equivalence for each $d\geq 0$.

If $d<d^{\prime}$, then $P_{d}(X)$ is included in $P_{d^{\prime}}(X)$ as a subcomplex via a simplicial map. Passing to inductive limit, we obtain the assembly map

To state our main theorem, we will also need some preparations of coarse geometry and group actions. Recall that a metric space $(X,d)$ is said to admits a coarse embedding into a Hilbert space if there exists a map $f:X\to\mathcal{H}$ and two non-decreasing unbounded functions $\rho_{-},\rho_{+}:\mathbb{R}_{+}\to\mathbb{R}_{+}$ such that

The concept of fibred coarse embedding is introduced by X. Chen, Q. Wang and G. Yu in [4] as a generalization of coarse embedding. In this paper, we will only focus on the case when a space admits a fibred coarse embedding into a real $\ell^{p}$-space as following:

Notice that for $x\in X$ and $R^{\prime}>R>0$, if there exists two trivilizations $t_{x,R}$ and $t_{x,R^{\prime}}$ for $B(x,R)$ and $B(x,R^{\prime})$, respectively. Without loss of generality, we may assume that

For the notational convenience, we denote $t_{x}$ and $t_{xy}$ for all $t_{x,R}$ and $t_{xy,R}$, respectively.

In the rest of this paper, we will prove the following theorem:

In this section, we will introduce a $C^{*}$-algebra $\mathcal{A}(B)$ associated to a given real Banach space $B=\ell^{p}(\mathbb{N},\mathbb{R})$ and calculate its $K$-theory.

In [14], G. Kasparov and G. Yu introduced a $C^{*}$-algebra for Banach space with Property (H). They pointed out that its $K$-theory can be calculated if the base space is an $\ell^{p}$-space. We provide a detailed proof in this section. We would like to mention that our construction is slightly different from theirs. The $C^{*}$-algebra $\mathcal{A}(X)$ in [14] is generated by some certain functions on a dense subspace $V$ of $B$. To build the twisted algebras in Section 6, for any affine isometry in Definition 2.4, we shall need a $C^{*}$-isomorphism associated to the affine isometry to translate the support of a function in $\mathcal{A}(X)$. However, the affine isometries in Definition 2.4 may not preserve the dense subspace in [14]. Our construction $\mathcal{A}(B)$ will circumvent this problem.