The Mayer-Vietoris Sequence for the Analytic Structure Group

The source of the coarse assembly map is the K-homology of the space $X$; we now review a C*-algebraic definition of K-homology.

We will often use the notation “$\sim$” to denote equality modulo compact operators; thus $T\in\mathfrak{D}^{*}_{\rho}(X)$ if and only if $[T,\rho(f)]\sim 0$ for every $f\in C_{0}(X)$ and $T\in\mathfrak{C}^{*}_{\rho}(X)$ if and only if $T\rho(f)\sim\rho(f)T\sim 0$ for every $f\in C_{0}(X)$.

Note that the dual algebra, the locally compact algebra, and their quotient each depend on the representation $\rho$. However, their K-theory groups are independent of $\rho$ so long as $C_{0}(X)$ is separable (equivalently, $X$ is second countable), $H$ is separable, and $\rho$ is ample (meaning $\rho$ is nondegenerate and $\rho(f)$ is a compact operator only if $f=0$ for $f\in C_{0}(X)$). See chapter 5 of [3] for further details. Because of this we will often abuse notation and write $\mathfrak{D}^{*}(X)$, $\mathfrak{C}^{*}(X)$ and $\mathfrak{Q}^{*}(X)$ with the understanding that the representations used to define these C*-algebras are ample and the Hilbert spaces are separable.

The K-homology groups form a generalized homology theory (in the sense of the Eilenberg-Steenrod axioms) which is naturally dual to topological K-theory. In particular they enjoy a certain functoriality property which can be understood at the level of dual algebras as follows. To begin, let $\phi\colon\mathcal{U}\to Y$ be a continuous proper map from an open subset $\mathcal{U}$ of $X$ to $Y$. Extend $\phi$ to a map $\phi^{+}\colon X^{+}\to Y^{+}$ between one-point compactifications by sending the complement of $\mathcal{U}$ in $X$ to the point at infinity in $Y^{+}$. Then $\phi^{+}$ induces a $*$-homomorphism $C(Y^{+})\to C(U^{+})\subseteq C(X^{+})$ given by $f\mapsto f\circ\phi^{+}$, and we define $\phi^{*}$ to be the restriction of this map to $C_{0}(Y)$.

By Voiculescu’s theorem, every map $\phi$ of the sort described in the definition is topologically covered by some isometry. If $V$ is a topological covering isometry for $\phi$ then the $*$-homomorphism $\text{Ad}_{V}\colon\mathbb{B}(H_{X})\to\mathbb{B}(H_{Y})$ maps $\mathfrak{D}^{*}(X)$ into $\mathfrak{D}^{*}(Y)$ and $\mathfrak{C}^{*}(X)$ into $\mathfrak{C}^{*}(Y)$, and the induced maps on K-theory depend only on $\phi$ and not on the chosen covering isometry ([3], chapter 5). Thus every continuous proper map $\phi\colon\mathcal{U}\to Y$ on an open subset of $X$ induces a map $\phi_{*}=(\text{Ad}_{V})_{*}\colon K_{p}(X)\to K_{p}(Y)$ on K-homology defined using a topological covering isometry.

The target of the assembly map is the K-theory of the coarse algebra of the space $X$, so we now review the definition and basic properties of this C*-algebra. Along the way we will also define the structure algebra.

For much of what follows we will be interested in proper metric spaces, i.e. a metric spaces whose closed and bounded subsets are all compact. Any complete Riemannian manifold is a proper metric space by the Hopf-Rinow theorem; indeed, this is one our main sources of examples.

The set of all controlled operators forms a $*$-subalgebra, and we will be interested in the intersection of this $*$-subalgebra with the dual algebra and the locally compact algebra. Most of the results of this paper generalize easily to account for free and proper group actions, so we will include them at the outset (though we will not need to consider equivariant dual algebras or equivariant K-homology). For simplicity, the reader is invited to ignore the group action during the first reading.

So let $X$ be a proper metric space equipped with a free and proper action of a countable discrete group $G$ of isometries of $X$. We will need to consider representations of $C_{0}(X)$ which are compatible with the $G$-action; the motivating example is the representation of $C_{0}(X)$ on $L^{2}(X,\mu)$ by multiplication operators where $\mu$ is a $G$-invariant Borel measure. More abstractly:

As usual if $H$ carries a unitary representation $U$ of $G$ then we say that an operator $T\in\mathbb{B}(H)$ is $G$-equivariant if $U_{\gamma}TU_{\gamma}^{*}=T$ for every $\gamma\in G$.

As before, both $D_{G}^{*}(X)$ and $C_{G}^{*}(X)$ depend on the representation used to define them, but the ambiguity disappears at the level of K-theory under mild additional hypotheses. For the coarse algebra it is enough to assume that the Hilbert space is separable and the representation is ample, but for the structure algebra we must assume that the Hilbert space is separable and the representation is the countable direct sum of a fixed ample representation. Following [3], we refer to such representations as very ample.

There is a considerable amount of literature on $C_{G}^{*}(X)$ due in part to its relationship with index theory. Its K-theory groups are functorial for a class of maps between metric spaces called coarse maps which are compatible with large scale geometry (in the same way that continuous maps are compatible with small scale geometry). There is a general theory of abstract coarse structures, but for our present purposes it will suffice to review the basic features of the coarse structure of a metric space.

If $S$ is any set and $\alpha_{1},\alpha_{2}\colon S\to X$ are maps, we say that $\alpha_{1}$ and $\alpha_{2}$ are close if there is a constant $C$ such that $d(\alpha_{1}(s),\alpha_{2}(s))\leq C$ for every $s\in S$. A map $\phi\colon X\to Y$ between metric spaces is said to be coarse if $\phi^{-1}(B)$ is a bounded subset of $X$ whenever $B$ is a bounded subset of $Y$ and $\phi\circ\alpha_{1}$ and $\phi\circ\alpha_{2}$ are close whenever $\alpha_{1}$ and $\alpha_{2}$ are close. $\phi$ is said to be a coarse equivalence if there is another coarse map $\psi\colon Y\to X$ such that $\phi\circ\psi$ and $\psi\circ\phi$ are close to the identity maps on $Y$ and $X$, respectively. For instance, the standard embedding of $\mathbb{Z}$ into $\mathbb{R}$ is a coarse equivalence.

As with K-homology, the functoriality of coarse K-theory can be implemented at the level of the coarse algebra using another notion of covering isometry.

According to chapter 6 of [3], every coarse map $\phi$ admits a coarse covering isometry $V$ so long as $\rho_{X}$ is nondegenerate and $\rho_{Y}$ is ample; a straightforward variation on that argument shows that if a discrete group $G$ acts freely and properly on $X$ and $Y$ by isometries and $\phi$ is $G$-invariant then $V$ can be taken to be a $G$-equivariant coarse covering isometry. Moreover $\text{Ad}_{V}$ maps $C_{G}^{*}(X)$ into $C_{G}^{*}(Y)$ and the induced map on K-theory is independent of the $G$-equivariant coarse covering isometry chosen. Also note that if $V$ coarsely covers $\phi$ then it also coarsely covers any map which is close to $\phi$, so $\phi$ induces a map $\phi_{*}=(\text{Ad}_{V})_{*}\colon K_{p}(C_{G}^{*}(X))\to K_{p}(C_{G}^{*}(Y))$ which depends only on the closeness equivalence class of $\phi$.

Recall that we defined the structure algebra of a proper $G$-space $X$ to be the C*-algebra $D_{G}^{*}(X)$ obtained by taking the norm closure of the space of $G$-invariant pseudolocal controlled operators associated to a representation of $C_{0}(X)$.

While $K_{p}(X)$ is an invariant of the small scale geometry of $X$ and $K_{p}(C_{G}^{*}(X))$ is an invariant of the large scale geometry of $X$, the analytic structure group depends simultaneously on both large and small scale behavior. As usual this will be clarified by understanding its functorial properties: we will prove that $S_{p}(X,G)$ is functorial for uniform maps, i.e. maps which are both continuous and coarse. This functoriality is once again implemented using covering isometries.

If $X$ and $Y$ carry $G$-actions and $H_{X}$ and $H_{Y}$ carry $G$-equivariant representations then one can define $G$-equivariant uniform covering isometries in the same way as in our discussion of $G$-equivariant coarse covering isometries. $G$-equivariant uniform covering isometries always exist, but the proof of this fact does not appear in the literature and thus we provide it here. Our strategy is to build ”local” covering isometries and patch them together using a partition of unity.

To construct $G$-equivariant uniform covering isometries, we break $X$ and $Y$ up into pieces for which the lemma applies. To make the argument work it is important that the $(Y,G)$-module be very ample, meaning it is the countable direct sum of a fixed ample representation.

If $V$ is a $G$-equivariant covering isometry for a $G$-equivariant uniform map $\phi$ then $\text{Ad}_{V}$ maps $D_{G}^{*}(X)$ into $D_{G}^{*}(Y)$ and the induced map on K-theory is independent of the $G$-equivariant covering isometry chosen. These observations follow immediately by combining the corresponding arguments for K-homology and coarse K-theory. The conclusion is that every $G$-equivariant uniform map $\phi\colon X\to Y$ induces a map $\phi_{*}=(\text{Ad}_{V})_{*}\colon S_{p}(X,G)\to S_{p}(Y,G)$.

We are now ready to construct the (equivariant) coarse assembly map described in the introduction. As usual let $X$ be a proper metric space on which $G$ acts freely and properly by isometries. Let $X_{G}$ denote the quotient space of $X$ by $G$. There is an isomorphism

(2.1)

$$Q_{G}^{*}(X)\cong\mathfrak{Q}^{*}(X_{G})$$

and hence the long exact sequence in K-theory associated to the short exact sequence

$$0\to C_{G}^{*}(X)\to D_{G}^{*}(X)\to Q_{G}^{*}(X)\to 0$$

takes the form:

(2.2)

$$\to K_{p+1}(C_{G}^{*}(X))\to S_{p}(X,G)\to K_{p}(X_{G})\to K_{p}(C_{G}^{*}(X))\to$$

In the case where $X_{G}$ is a compact manifold and $X$ is its universal cover, this is the analytic surgery exact sequence investigated by Higson and Roe in [4], [5], and [6]. In any event, the boundary map $K_{p}(X_{G})\to K_{p}(C_{G}^{*}(X))$ is the desired coarse assembly map.

The key is the isomorphism (2.1). It is proved in [3] and in [9] (using sheaf-theoretic techniques). We will give a brief outline of the argument in [3] so that we can adapt it to the relative setting in the next section. The main tool is a certain truncation operator defined as follows:

Here are the properties of the truncation construction which we will need:

The isomorphism 2.1 can be constructed using truncation operators in two steps. The first step is to pass from equivariant operators associated to $X$ to ordinary operators on $X_{G}$. This is more or less straightforward for operators supported in an open subset of $X$ of the form $\mathcal{U}=\mathcal{U}_{G}\times G$ where $\mathcal{U}_{G}$ is an open subset of $X_{G}$, so we cover $X$ by countably many evenly covered open sets of uniformly bounded diameter, lift to an open cover $\{\mathcal{U}_{n}\}$ of $X$, and truncate a general $G$-equivariant operator $T$ along this open cover using a $G$-invariant partition of unity. $\mathfrak{T}(T)$ descends to a pseudolocal controlled operator associated to $X_{G}$ and it differs from $T$ by a locally compact $G$-equivariant controlled operator by Proposition 2.15. This yields an isomorphism $Q_{G}^{*}(X)\cong Q^{*}(X_{G})$.

The second step is to pass from a general pseudolocal operator $T$ associated to $X_{G}$ to a pseudolocal controlled operator. Once again, the strategy is to truncate $T$ along a uniformly bounded open cover of $X_{G}$; Proposition 2.15 guarantees that $\mathfrak{T}(T)$ is pseudolocal and controlled and that it differs from $T$ by a locally compact operator. This yields an isomorphism $Q^{*}(X_{G})\cong\mathfrak{Q}^{*}(X_{G})$. The fact that $\mathfrak{T}$ is continuous is used in both steps to handle the fact that a general operator in the coarse algebra or the structure algebra is the norm limit of controlled operators but may not itself be controlled.

Let $A$ be a C*-algebra and let $I_{1}$ and $I_{2}$ be C*-ideals in $A$ such that $I_{1}+I_{2}=A$. There is a Mayer-Vietoris sequence in K-theory associated to these data which takes the form:

(3.1)

$$\to K_{p+1}(A)\to K_{p}(I_{1}\cap I_{2})\to K_{p}(I_{1})\oplus K_{p}(I_{2})\to K_{p}(A)\to$$

This is apparently a folklore result among operator algebraists; it appears in [7] but it is probably older. To construct it, consider the C*-algebra $\Omega(A,I_{1},I_{2})$ defined to be the set of all continuous paths $f\colon[0,1]\to A$ such that $f(0)\in I_{1}$ and $f(1)\in I_{2}$. There is a short exact sequence

(3.2)

$$0\to SA\to\Omega(A,I_{2},I_{2})\to I_{1}\oplus I_{2}\to 0$$

where $SA$ is the suspension of $A$, the first map is inclusion, and the second map is evaluation at $0$ and $1$.

The Mayer Vietoris sequence (3.1) can thus be obtained as the long exact sequence in K-theory associated to the short exact sequence (3.2)

We conclude by providing formulas for the boundary maps in the Mayer-Vietoris sequence. These formulas use the following device:

Partitions of unity exist and are unique up to homotopy by a straightforward functional calculus argument. Let us use a partition of unity to calculate the Mayer-Vietoris boundary map $\partial_{MV}\colon K_{1}(A)\to K_{0}(I_{1}\cap I_{2})$. This begins with the following lemma.

Our formula for $\partial_{MV}$ is as follows:

The proof uses an explicit formula for the K-theory boundary map $K_{1}(A/(I_{1}\cap I_{2}))\to K_{0}(I_{1}\cap I_{2})$ and some elementary homotopies. Further detail is available in [10].

There is a similar formula for the Mayer-Vietoris boundary map in the other degree, but in this case we must assume that $A=I_{1}+I_{2}$ admits a partition of unity $(a_{1},a_{2})$ such that $a_{1}$ and $a_{2}$ are projections; this is required to ensure that $a_{1}\textbf{p}+a_{2}$ is a projection if p is a projection. Partitions of unity of this sort are available in the examples relevant to this paper.

This formula can be reduced to the formula in the other degree using suspensions and Bott periodicity. Again, the details are provided in [10].

Let $X$ be a proper metric space and let $Y_{1}$ and $Y_{2}$ be subspaces such that $X=Y_{1}\cup Y_{2}$. In this section we will associate to $Y_{1}$ and $Y_{2}$ ideals $\mathfrak{Q}^{*}(Y_{1}\subseteq X)$ and $\mathfrak{Q}^{*}(Y_{2}\subseteq X)$ in $\mathfrak{D}^{*}(X)$ and show that $\mathfrak{D}^{*}(X)=\mathfrak{D}^{*}(Y_{1}\subseteq X)+\mathfrak{D}^{*}(Y_{2}\subseteq X)$. Combined with some excision results, the abstract Mayer-Vietoris sequence of the previous section will yield a Mayer-Vietoris sequence in K-homology:

$$\to K_{p+1}(X)\to K_{p}(Y_{1}\cap Y_{2})\to K_{p}(Y_{1})\oplus K_{p}(Y_{2})\to K_{p}(X)\to$$

We will also carry out similar constructions for the coarse K-theory group and the analytic structure group of $X$ and conclude by showing that the three Mayer-Vietoris sequences can be interwoven with the analytic surgery exact sequence to form the braid diagram which appears in the introduction.

To begin, fix a representation $\rho$ of $C_{0}(X)$ on a separable Hilbert space $H$ and let $Y\subseteq X$ be a subspace. Say that an operator $T\in\mathbb{B}(H)$ is locally compact for $X-Y$ if $\rho(f)T\sim T\rho(f)\sim 0$ for every $f\in C_{0}(X-Y)$, and say that $T$ is supported near $Y$ if $\text{Supp}(T)\subseteq B_{R}(Y)\times B_{R}(Y)$ for some $R$-neighborhood $B_{R}(Y)$.

Defining $Q_{G}^{*}(Y\subseteq X)=D_{G}^{*}(Y\subseteq X)/C_{G}^{*}(Y\subseteq X)$, there is still an isomorphism

$$Q_{G}^{*}(Y\subseteq X)\cong\mathfrak{Q}^{*}(Y_{G}\subseteq X_{G})$$

The proof proceeds exactly as before, only now we build the required truncation operator using a collection of open sets $\{\mathcal{U}_{n}\}$ in $X$ such that $Y\subseteq\bigcup_{n}\mathcal{U}_{n}\subseteq B_{R}(Y)$ for some $R>0$. This ensures that $\mathfrak{T}(T)$ is supported near $Y$ for any operator $T$.

An crucial fact about these ideals is that their K-theory depends only on $Y$.