1 Introduction↩︎

To any proper metric space \(M\), one has a canonically associated “boundary at infinity”, its Higson corona \(\partial_h M\) (see Definition [defn95higson95corona]). There exists an “index pairing”, which pairs classes in the \(K\)-theory group \(K^{p-1}(\partial_h M)\) with classes in the \(K\)-theory group \(K_p^{\mathrm{top}}(C^*M)\) of the Roe algebra of \(M\), to obtain an integer EmeMey?. This is the pairing on the right hand side of 1 below.

On the other hand, given a class \(x\) in \(K^{p-1}(\partial_h M)\), we may form the Chern character \(\operatorname{ch}(x)\) in Alexander–Spanier cohomology of \(\partial_h M\) gorokhovsky? and then transgress to obtain a coarse cohomology class \(T\operatorname{ch}(x)\) in \(H \! X^p(M)\). Coarse cohomology does not directly pair with \(K\)-theory classes in the Roe algebra, but it does pair (via the Chern character in cyclic homology) with algebraic \(K\)-theory classes over the dense subalgebra \(\mathcal{B}M \subset C^*M\), the algebra of locally trace-class, finite propagation operators on \(M\) roe_coarse_cohomology?. This is the pairing on the left hand side of 1 .

In total, we have described two different ways to obtain a complex number from a \(K\)-theory class \(\xi\) in \(K^{\mathrm{alg}}_p(\mathcal{B}M)\) and a class \(x\) in \(K^{p-1}(\partial_h M)\), and we may ask whether these two numbers are equal: \[\label{PairingEquation} \big\langle \chi \operatorname{ch}(\xi), T\operatorname{ch}(x)\big\rangle ~\stackrel{?}{=}~ \langle \iota_* \xi, x \rangle.\tag{1}\] In the term \(\chi \operatorname{ch}(\xi)\) on the left hand side, \(\operatorname{ch}\) denotes the Chern character from algebraic \(K\)-theory to periodic cyclic homology and \(\chi\) denotes the so-called coarse character map essentially introduced by Roe roe_coarse_cohomology?, which maps (periodic) cyclic homology of \(\mathcal{B}M\) to coarse homology. On the right hand side, \[\iota_* \colon K_*^{\mathrm{alg}}(\mathcal{B}M) \longrightarrow K_*^{\mathrm{top}}(C^*M)\] denotes the canonical comparison map. The following result was proved by Roe in roe_coarse_cohomology?.

Roe posed the question whether 1 holds in general roe_coarse_cohomology?, but to our knowledge, no answer to this question is available in the literature. In concrete examples, the left hand side of 1 is given as an operator trace and a priori takes values in complex numbers, while the right hand side of 1 can be written as the index of an operator, which takes integer values. The equality of both is a “quantization result” for the trace-pairing; see LudewigThiangI?, where equality of 1 is proved for general proper metric spaces and arbitrary classes in \(K_*^\mathrm{alg}(\mathcal{B}M)\), but only specific coarse cohomology classes.

The purpose of this note is to show the following results. The first one is a variant of the above mentioned result of John Roe and introduces one of the main players of this note, the algebraic assembly map, which is defined in §4.

Our second theorem complements the mentioned results from roe_coarse_cohomology?, LudewigThiangI? by giving conditions on the space \(M\) such that 1 holds.

Applying a recent result of Bunke–Engel coarse_KH?, we obtain that the above conditions are satisfied, for example, for \(M = \mathbb{R}^n\).

Note that Theorem B (and consequently Theorem C) also hold more generally for the algebra \(\mathcal{B}^p M\) of finite propagation, locally \(p\)-summable operators (Remark [rem95nilinvariance]).

Outline of the argument.↩︎

Let \((M_N, N)\) be a finite CW-pair and \(N\) be Higson dominated. Then in Section 3.1 we reduce 1 to commutativity of the diagram \[\label{eq95big95diag95intro} \begin{tikzcd}[column sep=3cm] K^{\mathrm{top}}_*(C^*M) \ar[r, "\tau_N"] & \tilde{K}_{*-1}^\mathrm{an}(N) \\ K^{\mathrm{alg}}_*(\mathcal{B}M) \ar[u, "\iota_*"] \ar[d, "\operatorname{ch}"'] & \tilde{K}_{*-1}(N) \ar[dd, "\operatorname{ch}"] \ar[u, "\cong"] \\ H \! P_*(\mathcal{B}M) \ar[d, "\chi"'] \\ H \! X^{\mathrm{per}}_*(M) \ar[r, "T_N"'] & \tilde{H}_{*-1}^{\mathrm{per}}(N) \end{tikzcd}\tag{2}\] stating that the two maps from \(K^{\mathrm{alg}}_*(\mathcal{B}M)\) to \(\tilde{H}_{*-1}^{\mathrm{per}}(N)\) in the diagram coincide.

Unfortunately, we cannot establish commutativity of the diagram 2 unconditionally. Instead, in the Section 3.2, under the additional assumption that \(M\) is a smooth manifold, we make a further reduction to \(1\)-summable Fredholm modules by using the algebraic assembly map which we discuss in Section 4.1. This leads to Theorem A, which appears as Theorem [ThmMain1] in the main text.

Again, unfortunately, it is not clear if the assumptions of this intermediate theorem (that all classes in \(K^\mathrm{alg}_*(\mathcal{B}M)\) come from \(1\)-summable Fredholm modules) are always satisfied, hence we introduce Weibel’s \(K \! H\)-theory as a further tool. This step is based on the following diagram which can be grafted into the above one: \[\label{eq95big95diag95295intro} \begin{tikzcd} & K^{\mathrm{top}}_*(C^*M) & & K^{\mathrm{an},\mathrm{lf}}_*(M) \ar[ll, "A"'] \ar[ddd, "\substack{\text{Connes--} \\ \text{Chern} \\ \text{character}}"] \\ K \! H_*(\mathcal{B}M) \ar[ur] \ar[dr] & K^{\mathrm{alg}}_*(\mathcal{B}M) \ar[u, "\iota_*"] \ar[l] \ar[d, "\operatorname{ch}"'] & \left\{\substack{\text{1-summable}\\ \text{even/odd}\\\text{Fredholm} \\ \text{modules}}\right\} \ar[ur, "{[-]}"'] \ar[l, "A^{\mathrm{alg}}"'] \\ & H \! P_*(\mathcal{B}M) \ar[d, "\chi"'] \\ & H \! X^{\mathrm{per}}_*(M) & & H_*^{\mathrm{dR}, \mathrm{lf}, \mathrm{per}}(M), \ar[ll, "c"'] \end{tikzcd}\tag{3}\] This leads to Theorem B and consequently Theorem C.

1.0.0.1 Acknowledgements

The first named author acknowledges financial support by the Deutsche Forschungsgemeinschaft DFG through Priority Programme SPP 2026 “Geometry at Infinity” (EN 1163/5-1, project number 441426261, Macroscopic invariants of manifolds). The second named author would like to thank SFB 1085 “Higher Invariants” for financial support. We thank Bernhard Hanke for a careful reading of the first version of this article.

2 Preliminaries↩︎

2.1 The index pairing and the analytic transgression map↩︎

Let \(M\) be a proper metric space. We start by discussing the index pairing \[\label{IndexPairing} K_*^{\mathrm{top}}(C^*M) \times \tilde{K}^{*-1}(\partial_hM) \longrightarrow \mathbb{Z},\tag{4}\] which is, equivalently, a map \[K^{\mathrm{top}}_*(C^*M) \longrightarrow \mathop{\mathrm{Hom}}\big(\tilde{K}^{*-1}(\partial_h M), \mathbb{Z}\big).\] Here \(\partial_h M\) is the Higson corona of \(M\), defined as follows:

\(C^*M\) in 4 is the Roe algebra of \(M\), which is the norm closure of the algebra \(\mathcal{B}M\), consisting of all finite propagation, locally trace-class operators, acting on a separable Hilbert space \(H\) with an ample representation of \(C_0(M)\). In the following we will use that by Borel functional calculus we can extend this representation to a representation of the \(C^*\)-algebra \(C_b(M)\) of all continuous, bounded functions on \(M\).

The pairing 4 may be defined as follows. Consider the map \[C^*M \otimes C(\partial_h M) \longrightarrow C^*M/\mathbb{K}(H), \qquad T \otimes [f] \longmapsto [T\tilde{f}],\] where we lift \(f \in C(\partial_h M) = C_h(M)/C_0(M)\) to an element \(\tilde{f}\) of \(C_h(M)\), which is a well-defined \(*\)-homomorphism by the fact that elements of \(C_h(M)\) have compact commutator with elements of \(C^*M\) qiao_roe?, bunke2024coronascalliastypeoperators?. The index pairing 4 is then obtained by applying the \(K_1\)-functor to this \(*\)-homomorphism, precomposing with the exterior product map and postcomposing with the \(K\)-theory boundary map to \(K_0(\mathbb{K}(H)) \cong \mathbb{Z}\). This explains the right hand side of 1 .

Suppose now that \(N\) is a metrizable, Higson dominated corona of \(M\), which means that there exists a continuous surjection \(\partial_h M \to N\). Write \(M_N \mathrel{\vcenter{:}}= M \sqcup_{\partial_h M} N\) for the corresponding compactification of \(M\) by \(N\). The ample representation of \(C_0(M)\) on \(H\) may then be extended to a representation of \(C(M_N)\). By \(C^*_N M\) we denote the \(C^*\)-algebra generated by the locally compact operators on \(H\) with controlled propagation for the coarse structure \(\mathcal{C}_N\) induced by the compactification \(M_N\), see higson_roe?. Because the corona \(N\) is Higson dominated, the coarse structure \(\mathcal{C}_N\) is coarser than the metric one and hence we have an inclusion \(\iota_{\mathcal{C}}\colon C^* M \subset C^*_N M\). This provides a map \(K_*^{\mathrm{top}}(\iota_{\mathcal{C}})\colon K_*^{\mathrm{top}}(C^* M) \to K_*^{\mathrm{top}}(C^*_N M)\).

The significance of \(C^*_N M\) is that it coincides with the relative dual algebra \(\mathfrak{D}(C(M_N)/\!\!/C_0(M))\) and the \(K\)-theory of the latter is isomorphic to the reduced analytic \(K\)-homology \(\tilde{K}^{\mathrm{an}}_{*-1}(N)\) of the corona \(N\), see higson_roe?. (In roe_coarse_cohomology? this result is attributed to the never published preprint higson_never? of Higson.) Hence we have an isomorphism \(\tilde{\tau}_N\colon K_*^{\mathrm{top}}(C^*_N M) \cong \tilde{K}_{*-1}^{\mathrm{an}}(N)\). Note that this result uses metrizability of \(N\) (at least, the argument of Higson–Roe needs it). Combining this with the previous map, we obtain a map \[\tau_N \colon K_*^{\mathrm{top}}(C^*M) \xrightarrow{K_*^{\mathrm{top}}(\iota_{\mathcal{C}})} K_*^{\mathrm{top}}(C^*_NM) \stackrel{\tilde{\tau}_N}\cong \tilde{K}_{*-1}^{\mathrm{an}}(N) \cong \tilde{K}_{*-1}(N).\]

Analytic \(K\)-homology satisfies strong excision, which results in an isomorphism \(K^\mathrm{an}_*(M_N,N) = K^{\mathrm{an},\mathrm{lf}}_*(M)\). The boundary map in the long exact sequence for the pair \((M_N,N)\) is therefore a map \(\partial\colon K_*^{\mathrm{an},\mathrm{lf}}(M) \to K^\mathrm{an}_{*-1}(N)\) whose image is contained in the reduced \(K\)-homology \(\tilde{K}_{*-1}^\mathrm{an}(N)\). One then has the following result:

2.2 The algebraic Chern character and the coarse character map↩︎

Let \(M\) be still a proper metric space. The algebraic Chern character is abstractly defined as a natural transformation \[\operatorname{ch}\colon K^{\mathrm{alg}}_*(\mathcal{B}M) \longrightarrow H \! P_*(\mathcal{B}M)\] from algebraic \(K\)-theory to periodic cyclic homology (see loday_cyclic_hom?). Recall that there are natural comparison maps \[H \! P_*(\mathcal{B}M) \longrightarrow H \! C_{*+2k}(\mathcal{B}M), \rlap{\qquad k \in \mathbb{Z}.}\] In degrees \(* \in \{0, 1\}\), the postcomposition of the Chern character with these maps allows explicit formulas; they are \[\begin{align} \operatorname{ch}([P]) &= \frac{(2k)!}{k!}\cdot \big[\mathrm{Tr}(\underbrace{P\otimes \cdots \otimes P}_{2k+1})\big] \in H \! C_{2k}(\mathcal{B}M) \\ \operatorname{ch}([U])&= k!\cdot \big[\mathrm{Tr}(\underbrace{(U-1) \otimes (U^{-1} - 1)\otimes \cdots }_{k+1~\text{pairs}})\big] \in H \! C_{1 + 2k}(\mathcal{B}M). \end{align}\]

We will use the following picture of coarse homology: \(n\)-chains are given by locally finite, signed Borel measures on \(M^{n+1}\) that are supported in an \(r\)-neighborhood of the diagonal, endowed with the Alexander–Spanier differential; this yields the chain complex \(C\!X_*(M)\) of coarse cochains. One then has coarse character maps \[\label{CoarseCharacterMap} \chi_n \colon H \! C_n(\mathcal{B}M) \longrightarrow H \! X_n(M),\tag{5}\] see LudewigThiangI?. On the level of cyclic chains, \(\chi_n(T_0 \otimes \cdots \otimes T_n)\) is the unique signed measure on \(M^{n+1}\) with the property that for \(f_0, \dots, f_n \in C_c(M)\), one has the formula \[\int_M \cdots \int_M f_0(x_0) \cdots f_n(x_n) d\chi(T_0 \otimes \cdots \otimes T_n)(x_0, \dots x_n) = \mathrm{Tr}(f_0 T_0 \cdots f_n T_n).\]

We define periodic version of the coarse character map \[\chi \colon H \! P_*(\mathcal{B}M) \longrightarrow H \! X^{\mathrm{per}}_*(M) := \prod_{k \in \mathbb{Z}} H \! X_{* + 2k}(M)\] which involves powers of \(2 \pi i\); it is given as the product of the compositions \[\begin{tikzcd} H \! P_*(\mathcal{B}M) \ar[r] & H \! C_{*+2k}(\mathcal{B}M) \ar[r, "(2 \pi i)^{-k}\cdot \chi_{*+2k}"] &[2cm] H \! X_{*+2k}(M). \end{tikzcd}\] We conclude that for \(*\in \{0, 1\}\) the composition \[\begin{tikzcd} K^{\mathrm{alg}}_*(\mathcal{B}M) \ar[r] & H \! P_*(\mathcal{B}M) \ar[r, "\chi"] & H \! X_{*}^{\mathrm{per}}(M) \end{tikzcd}\] in the factor labelled by \(k \in \mathbb{Z}\) is given by \[\int_{M^{2k+1}} f_0 \otimes \cdots \otimes f_{2k}\, d\operatorname{ch}([P]) = \frac{(2k)!}{(2 \pi i)^k k!} \cdot \chi(f_0 P \cdots f_{2k} P)\] in the even case and by \[\int_{M^{2k+2}} f_0 \otimes \cdots \otimes f_{2k+1}\, d\operatorname{ch}([U]) = \frac{k!}{(2 \pi i)^k} \cdot \chi\big(f_0 (U-1) f_1 (U^{-1} - 1) \cdots f_{2k+1} (U^{-1} - 1)\big)\] in the odd case.

2.3 Analytic and topological Chern characters, and boundary maps↩︎

In this section the assumption will come in that \(M\) is a smooth manifold and that it is the interior of a finite CW-pair \((M_N,N)\), where \(M_N\) is any compactification of \(M\) by a corona \(N\). This finiteness assumption is due to baum_higson_schick? which we are going to use.

Let us discuss the different pictures for \(K\)-homology and the respective Chern characters on them. The necessity for changing pictures is seen in Diagram 2 : The Chern character on the left hand side is defined analytically by computing traces of operators and we have to compare it with a Chern character on the corona \(N\) of \(M\). Assuming \(N\) to be a (compact) manifold we could use the analytic Chern character on \(\tilde{K}^\mathrm{an}_{*-1}(N)\) and therefore stay completely in the analytic world to prove commutativity of 2 . Since we do not want to restrict ourselves even further by assuming that \(N\) is a smooth manifold, we are forced to use a different picture for the \(K\)-homology of \(N\) and the respective Chern character.

There are two other pictures of \(K\)-homology: The topological one, which is the homology theory associated to the \(K\)-theory spectrum, and the geometric one, introduced by Baum–Douglas baum_douglas?. Each comes with a comparison map to analytic \(K\)-homology which is known to be an isomorphism whenever \((M_N, N)\) is a finite CW-pair baum_higson_schick?. Each of these two pictures of \(K\)-homology also comes with a naturally defined Chern character on it to the periodized homology of \(N\). Since for our purposes both of these two pictures of \(K\)-homology do the job, we just pick one once and for all and denote it by \(\tilde{K}_{*-1}(N)\). This explains the upper right vertical isomorphism in Diagram 2 and our necessity for assuming that \((M_N, N)\) is a finite CW-pair.

Let us discuss the Connes–Chern character which is the Chern character on \(K\)-homology in the analytic picture connes_noncomm_diff_geo? (from here on we need in this discussion that \(M\) is a smooth manifold): Analytic \(K\)-homology is defined in terms of Fredholm modules, but the character is only defined for finitely summable ones (i.e., \(p\)-summable for some \(p\in [1,\infty)\)), i.e. we have a map \(K^{\mathrm{an},\mathrm{lf},\mathrm{fin}}_*(M) \to H^{\mathrm{dR},\mathrm{lf},\mathrm{per}}_*(M)\) to locally finite, periodized de Rham homology of \(M\). By the localized index formula of Connes–Moscovici connes_moscovici?, one concludes that the Connes–Chern character of the finitely summable Fredholm module defined by a graded generalized Dirac operator on \(M\) (in the sense of gromov_lawson_complete?) coincides with the classical Chern character of the \(K\)-homology class of that operator connes_moscovici?. By this together with the fact that (rationally) every \(K\)-homology class of \(M\) may be represented by the class of a graded generalized Dirac operator on \(M\) (cf. footnote 2), we infer that the Connes–Chern character descends to \(K^{\mathrm{an},\mathrm{lf}}_*(M)\) and we have a commutative diagram \[\label{eq95comp95Chern} \begin{tikzcd} K^{\mathrm{an}, \mathrm{lf}}_*(M) \ar[dd, "\substack{\text{Connes--} \\ \text{Chern} \\ \text{character}}"'] &[-0.3cm] K^{\mathrm{lf}}_*(M) \ar[dd, "\operatorname{ch}"] \ar[l] \\ \\ H_*^{\mathrm{dR}, \mathrm{lf}, \mathrm{per}}(M) & H_*^{\mathrm{lf}, \mathrm{per}}(M) \ar[l] \end{tikzcd}\tag{6}\]

Let \(M_N\) be any compactification of \(M\) by a corona \(N \mathrel{\vcenter{:}}= M_N \setminus M\). As discussed in Section 2.1 that analytic \(K\)-homology satisfies strong excision, and therefore we have an isomorphism \(K_*^\mathrm{an}(M_N,N) \cong K_*^{\mathrm{an},\mathrm{lf}}(M)\), the boundary map in the long exact sequence in analytic \(K\)-homology for the pair \((M_N,N)\) identifies with a map \(\partial\colon K_*^{\mathrm{an},\mathrm{lf}}(M) \to \tilde{K}_{*-1}^\mathrm{an}(N)\). Under the assumption that \((M_N,N)\) is a finite CW-pair we can then switch pictures and get a boundary map \(\partial\colon K_*^{\mathrm{lf}}(M) \to \tilde{K}_{*-1}(N)\). Now we use the fact that the Chern character is a natural transformation of cohomology theories to get a commutative diagram \[\label{BigDiagram95rightpartsmall} \begin{tikzcd} K^{\mathrm{lf}}_*(M) \ar[r, "\partial"] \ar[dd, "\operatorname{ch}"] &[-0.3cm] \tilde{K}_{*-1}(N) \ar[dd, "\operatorname{ch}"] \\ \\ H_*^{\mathrm{lf}, \mathrm{per}}(M) \ar[r, "\partial"] & \tilde{H}_{*-1}^{\mathrm{per}}(N), \end{tikzcd}\tag{7}\] where the bottom map is the boundary map in periodized homology.

Combining all the above results in the following:

3 Main argument↩︎

In this section, we explain the main diagram used for the proof.

3.1 Transgression and the main diagram↩︎

We start our discussion here with the assumption that \(M\) is a general proper metric space.

The Higson corona supports transgression maps \[\label{TransgressionMapsHigsonCorona} T \colon \tilde{H}^{*-1}(\partial_h M) \longrightarrow H \! X^{*}(M), \qquad T \colon H \! X_{*}(M) \longrightarrow \tilde{H}_{*-1}(\partial_h M),\tag{9}\] relating Alexander–Spanier (co)homology of the Higson corona \(\partial_h M\) with the coarse (co)homology of \(M\), see engel_wulff?; we remark here that it is important to use Alexander–Spanier (co)homology as \(\partial_h M\) is usually not homotopy equivalent to a CW-complex.

The idea to define these transgression maps is as follows: Let us first note that \(H \! X^*(M) \cong H^*(\mathcal{P}(M))\) and \(H \! X_*(M) \cong H_*^\mathrm{lf}(\mathcal{P}(M))\), where \(\mathcal{P}(M)\) is the total Rips complex of \(M\) considered as a \(\sigma\)-space, which means that we remember the sequence of Rips complexes \(P_R(M)\) for \(R \in \mathbb{N}\) assembling to \(\mathcal{P}(M)\), and the functors \(H^*(-)\) and \(H_*^\mathrm{lf}(-)\) are correspondingly extended to \(\sigma\)-spaces. Because Alexander–Spanier (co)homology satisfies strong excision, we have \(H^*(\mathcal{P}(M)) \cong H^*(\overline{\mathcal{P}(M)},\partial_h M)\) and \(H_*^\mathrm{lf}(\mathcal{P}(M)) \cong H_*(\overline{\mathcal{P}(M)},\partial_h M)\), where \(\overline{\mathcal{P}(M)}\) denotes the Higson compactification of \(\mathcal{P}(M)\).³ The transgression maps are then the boundary maps in the long exact sequences of the pair \((\overline{\mathcal{P}(M)},\partial_h M)\) together with the observation that they factor through (resp. have image contained in) the corresponding reduced groups.

The canonical map \(M \to \mathcal{P}(M)\) yields a canonical comparison map \[c \colon H^{\mathrm{lf}}_*(M) \longrightarrow H \! X_*(M).\] Naturality of boundary maps shows that the transgression map fits into the commutative diagram \[\begin{tikzcd} \label{BoundaryComparisonMap} H^{\mathrm{lf}}_*(M) \ar[d, "\partial"'] \ar[r, "c"] & H \! X_*(M) \ar[d, "T"] \\ \tilde{H}_{*-1}(\partial_h M) \ar[r, equal] & \tilde{H}_{*-1}(\partial_h M), \end{tikzcd}\tag{10}\] where the left hand side is the boundary map for the pair \((\overline{M}, \partial_h M)\) with \(\overline{M}\) being the Higson compactification of \(M\).

The transgression maps satisfy the duality \[\langle Tx, \xi\rangle = \langle x, T\xi\rangle\] for chains \(x \in H \! X_p(M)\) and cochains \(\xi \in \tilde{H}^{p-1}(\partial_hM)\), where the left hand side denotes the pairing in Alexander–Spanier (co)homology of \(\partial_hM\) and the right hand side denotes the pairing in coarse (co)homology of \(M\). Using this, we may move the transgression on the left hand side of 1 to obtain \[\label{ShiftTransgression} \langle \chi \operatorname{ch}(\xi), T \operatorname{ch}(x) \rangle = \langle T\chi \operatorname{ch}(\xi), \operatorname{ch}(x) \rangle .\tag{11}\]

If now \(N\) is a Higson dominated corona, there are transgression maps (which we denote by \(T_N\)) for \(N\) that factor through 9 and analogously to the diagram 10 , we have a commutative diagram \[\begin{tikzcd} \label{BoundaryComparisonMap95N} H^{\mathrm{lf}}_*(M) \ar[d, "\partial"'] \ar[r, "c"] & H \! X_*(M) \ar[d, "T_N"] \\ \tilde{H}_{*-1}(N) \ar[r, equal] & \tilde{H}_{*-1}(N), \end{tikzcd}\tag{12}\]

Provided \((M_N, N)\) is a finite CW-pair (so that we have the isomorphism in the upper right corner of 13 ), we may now ask whether the diagram \[\label{SmallerDiagram} \begin{tikzcd}[column sep=3cm] K^{\mathrm{top}}_*(C^*M) \ar[r, "\tau_N"] & \tilde{K}_{*-1}^\mathrm{an}(N) \\ K^{\mathrm{alg}}_*(\mathcal{B}M) \ar[u, "\iota_*"] \ar[d, "\operatorname{ch}"'] & \tilde{K}_{*-1}(N) \ar[dd, "\operatorname{ch}"] \ar[u, "\cong"] \\ H \! P_*(\mathcal{B}M) \ar[d, "\chi"'] \\ H \! X^{\mathrm{per}}_*(M) \ar[r, "T_N"'] & \tilde{H}_{*-1}^{\mathrm{per}}(N) \end{tikzcd}\tag{13}\] commutes, i.e., if the two maps from \(K^{\mathrm{alg}}_*(\mathcal{B}M)\) to \(\tilde{H}_{*-1}^{\mathrm{per}}(N)\) that we see in the diagram coincide.

3.2 Reduction to \(1\)-summable Fredholm modules↩︎

We will not be able to establish the commutativity of 13 unconditionally. Our discussion is based on the following inflation of 13 , where we have to assume that \(N\) is Higson dominated, and \((M_N, N)\) is a finite CW-pair and \(M\) is a smooth manifold (since we will use the discussion in Section 2.3): \[\label{BigDiagram} \begin{tikzcd} &[-0.1cm] &[-1.1cm] &[-0.4cm] &[-0.4cm] \tilde{K}_{*-1}^\mathrm{an}(N) \\ K^{\mathrm{top}}_*(C^*M) \ar[rrrru, bend left=10, "\tau_N"] & & K^{\mathrm{an}, \mathrm{lf}}_*(M) \ar[ll, "A"'] \ar[ddd, "\substack{\text{Connes--} \\ \text{Chern} \\ \text{character}}"] & K^{\mathrm{lf}}_*(M) \ar[r, "\partial"] \ar[ddd, "\operatorname{ch}"] \ar[l, "\cong"'] & \tilde{K}_{*-1}(N) \ar[ddd, "\operatorname{ch}"] \ar[u, "\cong"'] \\ K^{\mathrm{alg}}_*(\mathcal{B}M) \ar[u, "\iota_*"] \ar[d, "\operatorname{ch}"'] & \left\{\substack{\text{1-summable}\\ \text{even/odd}\\\text{Fredholm} \\ \text{modules}}\right\} \ar[ur, "{[-]}"'] \ar[l, "A^{\mathrm{alg}}"'] \\ H \! P_*(\mathcal{B}M) \ar[d, "\chi"'] \\ H \! X^{\mathrm{per}}_*(M) \ar[rrrr, bend right=15, "T_N"] & & H_*^{\mathrm{dR}, \mathrm{lf}, \mathrm{per}}(M) \ar[ll, "c"'] & H_*^{\mathrm{lf}, \mathrm{per}}(M) \ar[r, "\partial"] \ar[l, "\cong"'] & \tilde{H}_{*-1}^{\mathrm{per}}(N) \end{tikzcd}\tag{16}\] Here we consider the set of \(1\)-summable Fredholm modules, which on the one hand define elements in the analytic \(K\)-homology group \(K_*^{\mathrm{an}, \mathrm{lf}}(M)\). On the other hand, we will show that (in the case that \(* \in \{0, 1\}\)) there is an “algebraic assembly map” that yields elements in \(K_*^{\mathrm{alg}}(\mathcal{B}M)\), in such a way that both subdiagrams of the large left rectangle commute.

Commutativity of the diagram involving \(\tau_N\) is Lemma [lem95boundary95factors1] (together with a change of pictures for \(K\)-homology) and commutativity of the right two squares is Proposition [prop95different95cherns]. The diagram involving \(T_N\) is 12 . Regarding the left square, the following result will be proved in Sections 4.1 and 4.2:

Notice, however, that the arrows in 16 are oriented in such a way that the commutativity of all subdiagrams above does not yield the commutativity of 13 . However, we may record the following consequence of Proposition [prop95both95subdiag95commute]:

Note that it is not clear (and we actually do not really expect it to be true) that all classes in \(K_*^{\mathrm{alg}}(\mathcal{B}M)\) lie in the image of the algebraic assembly map \(A^{\mathrm{alg}}\).

To move on with our argument, we consider the extended diagram \[\label{BigDiagram2} \begin{tikzcd} & K^{\mathrm{top}}_*(C^*M) & & K^{\mathrm{an},\mathrm{lf}}_*(M) \ar[ll, "A"'] \ar[ddd, "\substack{\text{Connes--} \\ \text{Chern} \\ \text{character}}"] \\ K \! H_*(\mathcal{B}M) \ar[ur, "\eta_{\mathrm{top}}"] \ar[dr] & K^{\mathrm{alg}}_*(\mathcal{B}M) \ar[u, "\iota_*"] \ar[l, "\eta_{K \! H}"'] \ar[d, "\operatorname{ch}"'] & \left\{\substack{\text{1-summable}\\ \text{even/odd}\\\text{Fredholm} \\ \text{modules}}\right\} \ar[ur, "{[-]}"'] \ar[l, "A^{\mathrm{alg}}"'] \\ & H \! P_*(\mathcal{B}M) \ar[d, "\chi"'] \\ & H \! X^{\mathrm{per}}_*(M) & & H_*^{\mathrm{dR}, \mathrm{lf}, \mathrm{per}}(M), \ar[ll, "c"'] \end{tikzcd}\tag{17}\] where \(K \! H_*(\mathcal{B}M)\) denotes Weibel’s homotopy \(K\)-theory of the algebra \(\mathcal{B} M\). For our purposes, the important property of it is that both the comparison map \(\iota_*\) and the Chern character \(\operatorname{ch}\) factor through it. An immediate consequence is the following:

A case where both assumptions of the above theorem are satisfied is the following. It is well-known that the coarse assembly map is an isomorphism for \(M = \mathbb{R}^n\), and a recent result of Bunke–Engel coarse_KH? implies that the comparison map \(K \! H_*(\mathcal{B}M) \to K^{\mathrm{top}}_*(C^*M)\) is also an isomorphism for \(M = \mathbb{R}^n\). Briefly put, the reason for this latter claim is that both sides are coarse homology theories, so that the comparison map begin an isomorphism follows from the fact that it is an isomorphism on a point by results of Cortiñas–Thom CT?.

The above corollary is, even though \(\mathbb{R}^n\) seems to be an extremely special case of the very general Theorem [ThmMainThm2], a new result: We don’t expect that every class in \(K^{\mathrm{alg}}_*(\mathcal{B}\mathbb{R}^n)\) is representable by a graded generalized Dirac operator on \(\mathbb{R}^n\) and therefore Roe’s original result does not cover all cases here.

4 Algebraic assembly↩︎

In this section, we define our algebraic assembly map, which allows to compare the algebraic Chern character with the Connes–Chern character.

4.1 Definition↩︎

Let \(M\) be a proper metric space. In this section, we define a map \[A^{\mathrm{alg}} \colon \left\{\substack{\text{1-summable}\\ \text{even/odd}\\\text{Fredholm} \\ \text{modules}}\right\} \longrightarrow K^{\mathrm{alg}}_*(\mathcal{B}M),\] for \(* \in \{0, 1\}\), which makes the diagram \[\label{DiagramSection6} \begin{tikzcd} K^{\mathrm{top}}_*(C^*M) & & K^{\mathrm{an},\mathrm{lf}}_*(M) \ar[ll, "A"'] \\ K^{\mathrm{alg}}_*(\mathcal{B}M) \ar[u, "\iota_*"] & \left\{\substack{\text{1-summable}\\ \text{even/odd}\\\text{Fredholm} \\ \text{modules}}\right\} \ar[ur, "{[-]}"'] \ar[l, "A^{\mathrm{alg}}"'] \end{tikzcd}\tag{19}\] commutative. We do not attempt to show that \(A^{\mathrm{alg}}\) respects any equivalence relations one might put on the set of \(1\)-summable Fredholm modules. Throughout, we fix an ample⁶ \(M\)-module \(H\), on which the algebra \(\mathcal{B}M\) of finite propagation, locally trace-class operators acts, and we will use the notation \(\mathcal{A}M\) for the algebra of finite propagation operators on \(M\). Note that \(\mathcal{A}M\) contains \(\mathcal{B}M\) as an ideal.

We fix a dense *-subalgebra \(\mathcal{A}\subset C_c(M)\) containing partitions of unity subordinate to any finite open cover, as well as square roots of these functions. If \(M\) is a manifold, then \(\mathcal{A}= C_c^\infty(M)\) is a valid choice.

4.1.0.1 The even case.

An even, \(1\)-summable Fredholm module is a bounded, self-adjoint operator \(T\) on \(H \oplus H\) having the form \[T = \begin{pmatrix} 0 & T_- \\ T_+ & 0 \end{pmatrix},\] such that for each \(f \in \mathcal{A}\) the operators \(f(T^2 - 1)\), \((T^2 - 1)f\) and \([T, f]\) are trace-class.

We choose a uniformly bounded and locally finite open cover \((U_i)_{i \in I}\) of \(M\) and let \((\chi_i)_{i \in I}\) be a subordinate partition of unity from \(\mathcal{A}\). Then we set \[\tilde{T}_+ \mathrel{\vcenter{:}}= \sum_{i \in I} \chi_i^{1/2} T_+ \chi_i^{1/2}.\] This operator has finite propagation by uniform boundedness of the cover and is invertible modulo \(\mathcal{B}M\), hence we obtain an element \([\tilde{T}_+]\) in \(K_1^{\mathrm{alg}}(\mathcal{A}M/\mathcal{B}M)\). Then we set \[A^{\mathrm{alg}}(T) \mathrel{\vcenter{:}}= \partial([\tilde{T}_+]) \in K_0^{\mathrm{alg}}(\mathcal{B}M),\] where \(\partial\) is the boundary map in algebraic \(K\)-theory.⁷ Another choice of cover \((U_i)_{i \in I}\) and subordinate partition of unity \((\chi_i)_{i \in I}\) leads to an operator \(\tilde{T}^\prime_+\) which differs from \(\tilde{T}_+\) by a finite propagation, locally trace-class operator and hence defines the same element in \(K_1^{\mathrm{alg}}(\mathcal{A}M/\mathcal{B}M)\). Consequently, \(A^{\mathrm{alg}}(T)\) is well-defined.

The diagram 19 commutes, because the usual assembly map \(A\) in topological \(K\)-theory is defined by exactly the same procedure.

4.1.0.2 The odd case.

An odd, \(1\)-summable Fredholm module is a bounded, self-adjoint operator \(T\) on \(H\) such that for each \(f \in \mathcal{A}\) the operators \(f(T^2 - 1)\), \((T^2 - 1)f\) and \([T, f]\) are trace-class.

Let us first recall the definition of the usual assembly map \(A\) in this case. First we replace \(T\) by the finite propagation operator \[\label{eq95truncation} \tilde{T} \mathrel{\vcenter{:}}= \sum_{i \in I} \chi_i^{1/2} T \chi_i^{1/2}\tag{20}\] as in the case of even Fredholm modules. Then \(P = (1-\tilde{T})/2\) is a projection in \(\mathcal{A}M / \mathcal{B}M\) and we obtain a class \([P] \in K_0^\mathrm{alg}(\mathcal{A}M / \mathcal{B}M)\). Then we define \[\label{eq95defnA95odd} A(T) \mathrel{\vcenter{:}}= \partial\left[P\right] \in K_1^{\mathrm{top}}(C^*M),\tag{21}\] where we pass to the completions of the algebras and \(\partial\) denotes the boundary map in the \(6\)-term exact sequence of \(K\)-theory of \(C^*\)-algebras.

Note that \(\partial\left[P\right]\) is usually represented by \([\exp(2 \pi i \cdot P)]\), but we have to deviate from this to define \(A^\mathrm{alg}\), because in general \(\exp(2 \pi i \cdot P)\) has infinite propagation. But since homotopic invertibles over \(C^*M\) define the same class in \(K_1^{\mathrm{top}}(C^*M)\), we have some flexibility here to use a different function \(\varphi\) instead of \(\exp(2 \pi i \cdot -)\). The rough idea is as follows: If \(\varphi\) does not take the value \(0\), then we will get by functional calculus an invertible element \(\varphi(P)\), and if \(\varphi\) is homotopic to \(\exp(2 \pi i \cdot -)\) through functions which all do not take the value \(0\), then \(\varphi(P)\) will represent the same class in \(K_1^{\mathrm{top}}(C^*M)\) as \([\exp(2 \pi i \cdot P)]\). Taking care of the details in this argument, especially taking care of the non-unitality of \(C^*M\), we arrive at the following:

We consider continuous functions \(\varphi \colon I \to \mathbb{C}\), where \(I \subset \mathbb{R}\) is a sufficiently large interval containing \([0,1]\) and the spectrum of \(P\), with

1. \(\varphi(0) = \varphi(1) = 1\);

2. \(\varphi\) does not have any zeros; and

3. the restriction of \(\varphi\) to \([0, 1]\) has winding number \(+1\) around \(0\).⁸

By (i) and (ii) we infer that \(\varphi(P) - 1\) is in \(C^* M\) and is invertible; we obtain an element \([\varphi(P)] \in K_1^\mathrm{top}(C^* M)\). We observe that the function \(\varphi_0(\lambda) = \exp(2 \pi i \lambda)\) satisfies all these conditions and that any function \(\varphi\) satisfying (i), (ii) and (iii) may by homotoped to \(\varphi_0\) through functions also satisfying (i), (ii) and (iii). Hence \(\varphi(P)\) represents the class \(A(T)\) for any \(\varphi\) with these properties.

It is a consequence of the Weierstraß approximation theorem that there in fact exists a polynomial function \(\varphi\) satisfying (i), (ii) and (iii). One explicit choice is \[\varphi(\lambda) = 1 + (\lambda^2 - \lambda)\psi(\lambda), \qquad \text{with}\qquad \psi(\lambda) = 2i\left(t - \tfrac{1}{2}\right) + 5.\] For such a function \(\varphi\), the operator \(\varphi(P)\) has finite propagation. Since \(P^2 - P \in \mathcal{B}M\), we conclude that \(\varphi(P) - 1 = (P^2 - P)\in \mathcal{B}M\). Hence we may set \[A^{\mathrm{alg}}(T) \mathrel{\vcenter{:}}= [\varphi(P)] \in K_1^{\mathrm{alg}}(\mathcal{B}M),\] which agrees with \(A(T)\) after passing to topological \(K\)-theory by the above discussion.

4.2 Comparison with the Connes–Chern character↩︎

Here, \(M\) has to be a smooth manifold. In this section, we discuss commutativity of the diagram \[\label{eq95moscovici95wu} \begin{tikzcd} & & K^{\mathrm{an}, \mathrm{lf}}_*(M) \ar[ddd, "\substack{\text{Connes--} \\ \text{Chern} \\ \text{character}}"] \\ K^{\mathrm{alg}}_*(\mathcal{B}M) \ar[d, "\operatorname{ch}"'] & \left\{\substack{\text{1-summable}\\ \text{even/odd}\\\text{Fredholm} \\ \text{modules}}\right\} \ar[ru, "{[-]}"] \ar[l, "A^{\mathrm{alg}}"'] \ar[ddr, "{\text{Moscovici--Wu}}"', dashed, bend left=10] & \\ H \! P_*(\mathcal{B}M) \ar[d, "\chi"'] \\ H \! X^{\mathrm{per}}_*(M) & & H_*^{\mathrm{dR}, \mathrm{lf}, \mathrm{per}}(M). \ar[ll, "c"'] \end{tikzcd}\tag{22}\] which is (without the dashed arrow that we will explain below) the lower left part of 16 . Commutativity of this diagram is essentially proved by Moscovici–Wu moscovici_wu?, but unfortunately, they use a rather different language to state their results. Below we discuss how to nevertheless extract the desired statement from their paper.

They start with a finitely summable (i.e., \(p\)-summable for some \(p\in [1,\infty)\)) Fredholm module and first apply the truncation procedure from 20 with a sequence of covers \(\smash{(U_i^{(k)})_{i \in I_k}^{k \in \mathbb{N}}}\) and corresponding sequence of subordinate partitions of unity satisfying that the diameters of the covers converge (uniformly in \(i\)) to \(0\) as \(k \to \infty\). This results in a sequence of operators \((\tilde{T}^{(k)})_{k \in \mathbb{N}}\) with the property that the propagation of these operators converges to \(0\) as \(k \to \infty\). From these operators they then construct cycles in homology akin to the ones we get when applying the composition \(\chi \circ \operatorname{ch}\circ A^\mathrm{alg}\); and since the propagation of the operators goes to \(0\), they can then interpret the whole sequence of these homological cycles as a single class in Alexander–Spanier homology of \(M\). Identifying the latter with de Rham homology, we get the dashed arrow in the above diagram. Basically by construction, the left part of 22 commutes, and that the right part commutes is proven by Moscovici–Wu.

Notice that the construction of Moscovici–Wu is essentially a precursor to the localization algebra idea of Guoliang Yu yu_localization?.

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1. Universität Greifswald, Walther–Rathenau–Straße 47, 17489 Greifswald, Germany
   alexander.engel@uni-greifswald.de↩︎

2. Universität Greifswald, Walther–Rathenau–Straße 47, 17489 Greifswald, Germany
   matthias.ludewig@uni-greifswald.de↩︎

3. This is abuse of terminology: The space \(\mathcal{P}(M)\) is usually not locally compact and hence \(\overline{\mathcal{P}(M)} = \mathcal{P}(M) \sqcup \partial_h M\) is not compact.↩︎

4. This is because every locally finite \(K\)-homology class may be represented by the class of a graded, generalized Dirac operator and they define summable classes. The argument in the compact case is outlined in, e.g., lawson_michelsohn?. This argument extends to the locally finite case.↩︎

5. Unfortunately, there is a discrepancy here between the two occurrences of the word “summable”: The representation result of \(K\)-homology classes allows us only to represent classes by \(p\)-summable modules with \(p\) depending on the dimension of the manifold. But we define the algebraic assembly map \(A^\mathrm{alg}\) only for \(1\)-summable modules. To resolve this issue, we observe that \(A^\mathrm{alg}\) may also be defined on \(p\)-summable modules and maps them then to \(K_*^\mathrm{alg}(\mathcal{B}^p M)\), where \(\mathcal{B}^p M\) is the algebra of finite propagation, locally \(p\)-summable operators. Then we just exploit that the inclusion \(\mathcal{B}M \to \mathcal{B}^p M\) induces an isomorphism on \(K \! H\)-theory, see Remark [rem95nilinvariance].↩︎

6. Recall that this means that the representation of \(C_0(M)\) on \(H\) is non-degenerate and that no non-zero function from \(C_0(M)\) acts as a compact operator.↩︎

7. The ideal inclusion \(\mathcal{B}(M) \subset \mathcal{A}(M)\) does probably not satisfy excision in algebraic \(K\)-theory, so the boundary map generally sends \(K_j(\mathcal{A}(M)/\mathcal{B}(M))\) to the relative \(K_{j-1}(\mathcal{A}(M), \mathcal{B}(M))\). However, in degree zero, one always has \(K_0(\mathcal{A}(M), \mathcal{B}(M)) \cong K_0(\mathcal{B}(M))\) rosenberg_algebraic_K?. Moreover, there exists an explicit description of the boundary map from \(K_1\) to \(K_0\) analogous to the index map in topological \(K\)-theory, which lands directly in \(K_0(\mathcal{B}(M))\).↩︎

8. This is a convenient way of phrasing that \(\varphi\) is homotopic to \(\exp(2 \pi i \cdot -)\) through functions which all do not take the value \(0\).↩︎