================================================================================ FAILURE OF RATIONAL INJECTIVITY FOR MAXIMAL COARSE ASSEMBLY A complete, friendly, self-contained proof tour ================================================================================ AUTHORS: OpenAI DATE: 2026-10-05 THIS FILE: coarse_assembly_rational_injectivity_v2.txt SOURCE PAPER: coarse_assembly_rational_injectivity_original.txt COMPANION: "A counterexample to the coarse Novikov conjecture", OpenAI 2026 WHAT THIS FILE IS Every statement, every lemma, and every proof step of the paper is here. The mathematics is intact. What is different is the writing: it starts from scratch, uses plain words around every symbol, and never leaves a definition to the reader's imagination. If the original paper says "let T be a locally compact finite-propagation operator", this version first reminds you what each of those words means. WHAT YOU NEED TO KNOW * sets: subsets, unions, Cartesian products, functions * groups: groups, subgroups, quotient groups, homomorphisms * matrices: multiplication, inverses. Note "E_12" is the 3-by-3 matrix with a 1 in row 1 column 2 and 0 everywhere else. * graphs: vertices and edges, path distance, connectedness No prior exposure to coarse geometry, K-theory, or C*-algebras is required. Those are built from scratch, in order, with examples. WHAT THE THEOREM SAYS, IN WORDS Build a metric space X by gluing together a countable family X_1, X_2, X_3, ... of finite connected graphs. Use the graph path metric inside each X_j, place different components at growing distances so that every bounded ball touches only finitely many components, and require a single vertex-degree bound D that works for all X_j. On such a space there is an "infinite-order" topological class alpha in the K-homology KX_1(X). "Infinite order" means no nonzero integer multiple of alpha is zero. The theorem says one can choose the graphs X_j so that the natural map mu_X^max : KX_1(X) -> K_1(C^*_max(X)) (the maximal coarse assembly map) sends alpha to the ZERO class. Because alpha has infinite order, its class in KX_1(X) tensor Q is still nonzero. So the rationalized assembly map kills a nonzero element. Therefore mu_X^max tensor Q is NOT injective. ================================================================================ THE THEOREM AND ITS CONSEQUENCE (formal) ================================================================================ Theorem (main). There exist finite connected graphs X_j (j = 1, 2, ...), a common vertex-degree bound D independent of j, a coarse disjoint union X = disjoint union of the X_j (section 1.3), and an infinite-order element alpha of KX_1(X), such that mu_X^max(alpha) = 0 in K_1(C^*_max(X)). Consequence. mu_X^max tensor Q is not injective. The rest of this file proves this, in seven parts. The proof is synthetic: part 2 builds the space, part 3 shows alpha exists and survives every scale, parts 4 and 5 give the analytic machinery, part 6 slides alpha's analytic image from one end of X to the other, and part 7 combines the two. ================================================================================ PART 1. COARSE GEOMETRY, K-THEORY, AND THE ASSEMBLY MAP ================================================================================ Coarse geometry (also "large-scale geometry") studies a metric space through its behaviour as we zoom farther and farther out. No single point matters; only the pattern of distances above any fixed cutoff. The theorem lives in this world, so this part sets up the dictionary. If you already know what a Rips complex, a Roe algebra, and a coarse assembly map are, skim to Part 2. 1.1. Bounded geometry, uniform discreteness, properness ------------------------------------------------------- Definition. A metric space Z is * uniformly discrete if some r > 0 exists with d(x,y) >= r for all distinct x, y in Z; * proper if every closed ball B(z, R) = {w : d(z,w) <= R} of finite radius is compact; * of bounded geometry if in addition, for each R > 0, the quantity sup over z in Z of |B(z, R)| (number of points in the radius-R ball about z) is finite. For a connected graph with the path metric (each edge length 1): uniform discreteness holds with r = 1, properness is automatic when the graph is locally finite (each vertex has finitely many neighbors), and bounded geometry follows from a uniform vertex-degree bound. In a graph of maximum vertex degree D, the number of vertices in a radius-R ball is at most 1 + D + D(D-1) + ... + D(D-1)^{R-1}; finite for each R, uniformly in the vertex. Examples: * Infinite path with the usual edge metric: bounded geometry (each radius-R ball has at most 2R+1 points). * Infinite binary tree: bounded geometry (each radius-R ball has at most 2^{R+1} - 1 points). * Wedge of countably many circles sharing a single center: NOT bounded geometry. The ball of radius 1 around the center is infinite. 1.2. Finite-propagation operators and the Roe algebra ----------------------------------------------------- Let H_Z = l^2(Z) tensor l^2(N) be the "ample module" of Z. l^2(Z) has orthonormal basis {delta_z : z in Z}; l^2(N) is the standard separable Hilbert space. Any bounded linear operator T on H_Z can be written as a matrix of blocks T_{xy} : l^2(N) -> l^2(N), for x, y in Z. Definition. * T has finite propagation R if T_{xy} = 0 whenever d(x,y) > R. Concrete meaning: T only "feels" nearby vertices. * T is locally compact if aT and Ta are compact operators for every a in C_0(Z) (continuous functions vanishing at infinity). For T of finite propagation this holds iff every vertex entry T_{xy} is a compact operator on l^2(N). Write C[Z] for the set of bounded, finite-propagation, locally compact operators. It is a *-algebra (closed under sums, products, adjoints). The REDUCED Roe algebra C^*(Z) is the completion of C[Z] in the operator norm on H_Z. The MAXIMAL Roe algebra C^*_max(Z) is the completion of C[Z] in the largest possible norm over all *- representations. For each t in C[Z], its max norm is ||t||_max = sup over pi of ||pi(t)||, the supremum being over all *-homomorphisms pi from C[Z] to the algebra of bounded operators on some Hilbert space. The reduced norm is ||t||_op, the concrete operator norm. Always ||t||_op <= ||t||_max. The theorem is interesting because in the relevant situation, the maximal norm is STRICTLY LARGER than the operator norm for some elements, and this is exactly what will let a class vanish in one norm but not the other. 1.3. Coarse disjoint unions ---------------------------- Let (X_j)_{j >= 1} be finite metric spaces. A coarse disjoint union is a metric on the set X = disjoint union over j of X_j such that (i) on each X_j, the restricted metric is the original one; (ii) for each R > 0, at most finitely many PAIRS (X_i, X_j) with i != j contain any two points at distance <= R. Condition (ii) says different components are farther and farther apart as j grows. If all X_j share a bounded-geometry constant, then X is a bounded-geometry space. A concrete construction. Pick one point c_j in each X_j. Join c_j and c_{j+1} by a new "auxiliary" edge of length L_j = 100 + j^2. * L_j -> infinity, so components are pairwise farther apart as j grows. * A path that leaves a component and returns must traverse at least two auxiliary edges of total length at least 2 L_j, so it cannot shorten an intrinsic path inside that component. * Hence the restriction of the new path metric to X_j is still the old intrinsic metric. * Bounded balls touch only finitely many components. The set X carries no new points; the auxiliary edges define a metric on the same set. 1.4. The Rips complex ---------------------- Definition. For a metric space Z and r > 0, the Rips complex P_r(Z) is the simplicial complex with * vertices: the points of Z, * an n-simplex for each (n+1)-point subset {z_0, ..., z_n} of Z whose diameter is at most r. "Simplicial complex" means a topological space built by gluing together triangles, tetrahedra, and their higher-dimensional cousins, one face at a time. Points of Z are the 0-simplices; each pair at distance <= r spans an edge; each triple pairwise close spans a triangle, and so on. For bounded-geometry Z, dim P_r(Z) is finite for each fixed r: an n-simplex lies in a single radius-r ball, which has bounded size. Examples: * Z a discrete set with pairwise distances > r: P_r(Z) = Z as isolated points. * Z = R^d with the usual metric: P_r(Z) is a "nerve" of the cover by r-balls and is contractible. 1.5. Coarse K-homology ----------------------- Definition. KX_q(Z) = colim over r -> infinity of K_q(C_0(P_r(Z)), C). Here K_q(-) is analytic K-homology, i.e., Kasparov's KK_q(C_0(Y), C) for Y a compact or locally compact space. We need only three facts: * Functoriality. A continuous map h : Z -> Y (or more generally a coarse bornologous proper map) induces a homomorphism h_* : KX_q(Z) -> KX_q(Y). * Local computation. For a finite simplicial complex M, KX_q(M) = K_q(M), the usual topological K-homology of M. * Countable additivity. For a countable disjoint union Y = disjoint union of finite complexes M_j, the natural map KX_q(Y) -> PRODUCT over j of KX_q(M_j) is an isomorphism. This is countable additivity in the first variable of Kasparov's KK-theory. The third fact is what makes "coarse disjoint union of graphs" the right setup. A coarse K-homology class on X is a SEQUENCE of classes, one per component, not a finite sum. The setup allows infinitely many components to contribute, one generator per component, and the resulting product is a huge group with plenty of infinite-order elements. For compact spaces, K-homology is small and computable in low degrees. For the 3-torus T^3 = (S^1)^3 we will use K_1(T^3) = Z^4, with four generators: three "coordinate-loop" classes and one "product of two loops". We do not need the elliptic-operator representatives of these generators, only the pairing with K-theory of the torus. 1.6. Topological K-theory of a C*-algebra ------------------------------------------ For a unital C*-algebra A: * K_0(A) is the Grothendieck group of projections in matrix algebras over A. A projection is p with p^2 = p = p^*, of some size n. Equivalence is unitary equivalence after stabilising by augmenting the size. * K_1(A) is the group of unitaries in matrix algebras over A, modulo the subgroup of unitaries that sit in the identity component of the unitary group. We use ONE fact from K-theory: a short exact sequence of C*- algebras 0 -> I -> A -> A/I -> 0 gives a six-term cyclic exact sequence K_0(I) -> K_0(A) -> K_0(A/I) ^ | | v K_1(A/I) <- K_1(A) <- K_1(I) The maps alternate between "induced", "quotient", and "boundary". Exactness is what will deny existence of a lifting element in Part 3 (the Mayer-Vietoris argument). We also use: for K the algebra of compact operators on a separable infinite-dimensional Hilbert space, K_1(K) = 0. So the six-term sequence applied to 0 -> K -> A -> A/K -> 0 gives injectivity of K_1(A) -> K_1(A/K). This is used in Part 6. 1.7. The maximal coarse assembly map ------------------------------------- There is a natural group homomorphism mu_Z^max : KX_q(Z) -> K_q(C^*_max(Z)), the maximal coarse assembly map. It takes a K-homology class (which lives in the topology of the space) to its analytic K-theory index (which lives in the K-theory of the Roe algebra). Two properties are used in this paper. * Naturality. For a coarse bornologous proper map h : Z -> Z' giving a *-homomorphism h_A : C^*_max(Z) -> C^*_max(Z'), the natural square KX_q(Z) --mu_Z^max--> K_q(C^*_max(Z)) |h_* |h_{A*}_* v v KX_q(Z') --mu_Z'^max-> K_q(C^*_max(Z')) commutes. * Boundary formula. For a K-homology class [F] in KX_1(Z) represented by a bounded self-adjoint "pseudolocal" operator F that is a multiplier of C^*_max(Z), with F^2 - 1 belonging to the ideal C[Z], the assembly map is mu_Z^max([F]) = [exp(2 pi i p)] in K_1(C^*_max(Z)), where p = (1+F)/2. Both properties are standard in maximal coarse index theory: see Gong-Wang-Yu 2008, Oyono-Oyono and Yu 2009, Willett and Yu 2012, Schick and Seyedhosseini 2021. Part 4 reconstructs the specific facts used, in self-contained form. For now, take them as a description of mu_Z^max. 1.8. What the theorem says about mu_X^max ------------------------------------------ Injectivity of mu_X^max would mean: no nonzero K-homology class gets sent to zero. The theorem builds a bounded-geometry X of the form "coarse disjoint union of finite graphs" and an infinite-order class alpha with mu_X^max(alpha) = 0. Injectivity therefore fails for such X. And because alpha is infinite order, the failure survives after tensoring with Q, i.e., the map after rationalization is not injective either. NOTATION for the rest of the paper. We write Z, Q, R, C, N for integers, rationals, reals, complex numbers, positive integers. Fixed symbols: C[Z], C^*_max(Z), C^*(Z), KX_q(Z), mu_Z^max, P_r(Z), K_q(-). All K-theory and K-homology in this paper use COMPLEX coefficients (C, not Z or Z/p) unless explicitly "with integral coefficients". ================================================================================ PART 2. THE SPACE X BUILT FROM FINITE GRAPHS ================================================================================ This part constructs X as a coarse disjoint union of finite graphs X_j of uniformly bounded vertex degree. Each X_j has two "torus ends" glued to a long central "collar". The whole point of the collar is to buy room to move analytic operators around (Part 6). The whole point of the ends is to supply a topological class (Part 3) at each end. The whole point of the varying fiber quotient is to make the class survive at every Rips scale while its analytic index is zero (Parts 3 and 5). 2.1. The picture ----------------- Fix approximately n = 6j + 1 "heights" labelled -3j, ..., 3j. At each height, place a single "slice" graph. Join consecutive slices by imperceptible edges. This creates a finite graph X_j arranged like a long sausage: base torus B_j | (reduction tower, shrinking fiber quotient step by step) | central collar = Y_j x [-j, ..., j] | (reduction tower again) | base torus B_j Both ends are copies of the SAME finite graph B_j. The middle is the product of Y_j (a graph with large finite fibers) with an interval. Between the ends and the middle, the fiber quotient shrinks by one congruence level per step. All graphs have uniformly bounded vertex degree; only the sizes explode (super-exponentially in j). 2.2. Base tori B_j ------------------- For each j >= 1, set l_j = 100(j+1). Let B_j = (Z / l_j Z)^3, the 3-dimensional discrete torus of side length l_j. As a graph: * vertices: triples (x, y, z) with x, y, z in {0, ..., l_j - 1}; * edges: change one coordinate by +/- 1 modulo l_j. Vertex degree is 6. Graph distance is sum of the three circular distances. 2.3. Fiber groups Q_k ---------------------- Let Gamma = SL_3(Z), the group of 3-by-3 integer matrices with determinant 1. Fix a finite symmetric generating set S of Gamma (say all elementary matrices + their inverses up to symmetrization). For k >= 1, let Q_k = image of Gamma under entrywise reduction modulo 2^k. Set Q_0 = {1}. Lemma geom:quotients (congruence tower): (a) The natural maps Q_{k+1} -> Q_k are surjective. (b) The kernel of Q_{k+1} -> Q_k has size at most 2^9 = 512. (c) For every fixed word radius R, the reduction map Gamma -> Q_k is injective on the S-word ball of radius R, for all sufficiently large k. Proof. (a) is immediate from definition. For (b), if M mod 2^k is fixed, its lifts to Q_{k+1} are M + 2^k D mod 2^{k+1}, with D an arbitrary 3-by-3 matrix mod 2. That is 2^9 possibilities. For (c), the S-word ball of radius R is a finite set of integer matrices. If g, h in the ball differ, some entry of g - h is a nonzero integer. Every nonzero integer is divisible by 2^k only finitely often. Take k larger than max v_2 for all nonzero entry-differences in the ball. */ 2.4. The twist that disappears at one congruence level --------------------------------------------------- Set m_j = 2^j. In Gamma = SL_3(Z), define U_j = 1 + m_j E_12, V_j = 1 + m_j E_23, where E_ab is the matrix unit (a "1" in row a, column b, and "0" elsewhere). Both are in SL_3(Z) with determinant 1. Direct matrix multiplication gives U_j V_j U_j^{-1} V_j^{-1} = 1 + m_j^2 E_13. (*) Read (*) in two congruence levels at once. * Modulo 2^{2j} = m_j^2, the right-hand side is 1. So U_j and V_j COMMUTE in Q_{2j}. * Modulo 2^{3j}, the right-hand side is 1 + 2^{2j} E_13, which is NOT the identity (its (1,3) entry is 2^{2j} != 0 mod 2^{3j}). So U_j and V_j FAIL to commute in Q_{3j}. That's the entire magic. The twist becomes visible only one level higher. The whole point of the height profile below is to create one family of slices (near the middle, in Q_{2j}) where U_j and V_j commute, and another family (near the ends, in Q_{3j}) where they do not. 2.5. Transport matchings and seam numbers ------------------------------------------ For an oriented edge e : u -> v in B_j, and a coordinate i in {1, 2, 3}, choose representatives u_i, v_i in {0, ..., l_j - 1} of the two endpoints. The displacement delta_i(u,v) = v_i - u_i might be as small as -l_j + 1 or as large as l_j - 1. Let N_i(u,v) be the UNIQUE integer in the range (-l_j/2, l_j/2] with N_i(u,v) = v_i - u_i + l_j * a_i(u,v). Equivalently, a_i(u,v) counts the number of "positive seam crossings" of e in coordinate i, minus the number of "negative" ones. This is an integral, well-defined count. The three INTEGERS (a_1, a_2, a_3) are the SEAM NUMBERS of e. For transport around the first two circle coordinates, and for a congruence level k <= 2j, define the right-hand "transport" element g_{j,k}(a, b) = V_j^b U_j^a in Q_k. For k = 0, take g_{j,0} = 1. For k between 1 and 2j, use the reduction of V_j^b U_j^a modulo 2^k. 2.6. The fiber graph Y_{j,k} ----------------------------- Definition. For 0 <= k <= 2j, let Y_{j,k} be the graph with * vertex set B_j x Q_k; * horizontal "fiber" edges: over each fixed u in B_j, connect (u, q) to (u, s q) for each generator s in S (of Q_k); * horizontal "base" edges: for an oriented base edge u -> v with seam numbers (a, b, c), connect (u, q) to (v, q g_{j,k}(a, b)). That is: * Fiber direction: LEFT multiplication by generators of Q_k. * Base direction: RIGHT multiplication by the transport element. * Left and right multiplications commute, so the two edge systems are consistent. * Reversing a base edge reverses the matching, because U_j and V_j commute in Q_{2j} (hence in Q_k for k <= 2j). The map g_{j,k}(a, b) is invertible with inverse g_{j,k}(-a, -b). * Discard self-loops and repeated edges. Each fiber over u is the full Cayley graph of Q_k with left multiplication. The transport matchings realize the twisted bundle over B_j. Since B_j and Q_k are finite, so is Y_{j,k}; and since both are connected (B_j is, Q_k is, and the transport connects distinct base vertices), Y_{j,k} is connected. Degree is uniformly bounded in j,k by |S| + 6 (plus consistency corrections). Write Y_j = Y_{j, 2j}. 2.7. The height profile and the graph X_j ------------------------------------------ For each integer t with -3j <= t <= 3j, define k_j(t) = 2j for |t| <= j, k_j(t) = 3j - |t| for j <= |t| <= 3j. So k_j(-3j) = k_j(3j) = 0, k_j(-j) = k_j(j) = 2j, and on the middle strip |t| <= j the level is CONSTANT at 2j. Definition. The graph X_j has vertex set disjoint union over t = -3j, ..., 3j of (B_j x Q_{k_j(t)}). Edges: * Within a slice at height t: exactly the edges of Y_{j, k_j(t)}. * Between consecutive heights t and t+1: for each u in B_j and each q in Q_{k_j(t+1)}, add an edge between the vertex (u, reduction of q to Q_{k_j(t)}) in the lower slice and (u, q) in the upper slice. Or, if k_j(t) = k_j(t+1), connect the same (u, q) across the two slices. Vertices have degree bounded by |S| + 6 + 2^9 (the reduction kernel size) independent of j. The graph X_j is finite and connected. 2.8. The middle is a product; the ends are tori ------------------------------------------------ At heights -j, ..., j, k_j(t) = 2j throughout, so the corresponding subgraph is precisely Y_j x [-j, ..., j]. We call this the CENTRAL COLLAR. In this range the graph has a clean product structure: an edge either changes the base or the fiber or the height by one step, and no reduction maps are involved. For j <= |t| <= 3j, k_j(t) decreases by one at each step outward. At height +3j (or -3j) we have k_j(3j) = 0, so Q = Q_0 = {1} is trivial. The slice at height +3j is just B_j. Hence the leftmost slice (t = -3j) and the rightmost slice (t = +3j) are both copies of B_j. Everything else fills the interior with a varying fiber quotient. Two consequences we will use repeatedly: * The projections f_j : X_j -> B_j (forget (q, t)) are 1-Lipschitz. * The end inclusions i_{+j}, i_{-j} : B_j -> X_j (send u to (u, 1) at height +3j or -3j) are isometric embeddings. 2.9. Coarse disjoint union of the X_j -------------------------------------- Let B = disjoint union of B_j, Y = disjoint union of Y_j, X = disjoint union of X_j, each with the coarse-disjoint-union metric from section 1.3 (auxiliary edges of length L_j = 100 + j^2 connecting marked vertices (0,1) in successive components). The marked vertex of B_j is u = 0. The marked vertex of Y_j is (u,q) = (0,1). The marked vertex of X_j is (u,q,t) = (0,1,0). Since all X_j, Y_j, B_j have vertex-degree bound D (independent of j), all three unions have bounded geometry. 2.10. The two maps f : Y -> B and i_pm : B -> X ------------------------------------------------ Three coarse maps: * f : Y -> B. On component Y_j, project (u, q) -> u. This is a "forget the fiber" map. 1-Lipschitz on each component. * i_+ : B -> X. On component B_j, send u to (u, 1, +3j) in X_j (the +3j end). 1-Lipschitz on each component. * i_- : B -> X. On component B_j, send u to (u, 1, -3j) in X_j (the -3j end). 1-Lipschitz on each component. All three are bornologous (bounded distance change) and proper (preimages of bounded sets are bounded). They are the maps through which the topological class and the analytic index will meet. At scale r = 3, none of the Rips complexes P_3(B), P_3(Y), P_3(X) has a simplex spanning two different components (because the auxiliary edges are longer than 3). Consequently, * i_{+3} : P_3(B) -> P_3(X) is a proper simplicial inclusion, and * i_{-3} : P_3(B) -> P_3(X) is likewise a proper simplicial inclusion. This is the "topological side" of the construction: we can lift a K-homology class of B to either end of X, and the two lifts will turn out to be different. ================================================================================ PART 3. THE TOPOLOGICAL CLASS ALPHA SURVIVES AT EVERY SCALE ================================================================================ This part builds the class alpha in KX_1(X) and proves it has INFINITE ORDER. This is entirely a topological statement, independent of any Roe algebra or C*-algebra. The point of the construction is that at every fixed Rips scale, a certain integral K-theory class on P_r(Y) becomes divisible by a huge power of two, forcing any equality between the two endpoints of X to contradict divisibility at a large enough scale. 3.1. K-theory pairings and naturality -------------------------------------- For a finite complex M there is a bilinear pairing < , > : K^1(M) x K_1(M) -> Z, the Kasparov index pairing. It is a "counting" operation: for a K-theory class (e.g. a line bundle) and a K-homology class (e.g. an elliptic operator representative), the pairing gives the index, an integer. Naturality. For a continuous map h : M -> N of finite complexes, the induced maps on K-theory reverse direction (h^* : K^1(N) -> K^1(M)) and induce forward on K-homology (h_* : K_1(M) -> K_1(N)): = for eta in K^1(N), xi in K_1(M). This is the property that lets the same integer pairing be computed either "downstairs" (on B) or "upstairs" (on X). Whenever a K-theory class fails to exist on X (due to a divisibility obstruction), it must also fail downstairs. Mayer-Vietoris sequence. For a finite complex M written as the union of two subcomplexes A, B, with intersection I = A intersect B, there is a long exact sequence on K_1: K_1(I) -> K_1(A) + K_1(B) -> K_1(M). The exactness will be used as a "no lift" statement: if a class in K_1(M) has both restrictions liftable from A and from B in a compatible way, then it must be zero on the connecting image. The divisibility argument will contradict that. 3.2. The torus and its K-theory --------------------------------- For each j >= 1, let T_j = (R / l_j Z)^3 be the smooth 3-torus of side length l_j. Triangulate T_j with unit cubes: for each vertex of the lattice (Z/l_j Z)^3, glue its 8 surrounding unit cubes. Since the vertices of the resulting triangulation are the points of B_j, and every simplex has vertex diameter at most 3, the triangulated T_j lies canonically as a subcomplex of P_3(B_j): kappa_j : T_j -> P_3(B_j). Coordinate classes. Let u_1, u_2, u_3 in H^1(T_j; Z) = Z^3 be the "first, second, third circle" classes. Their cup products generate H^2 and H^3. K-theory of a torus is standard: K^0(T^3) = Z^4, K^1(T^3) = Z^4. The generators of K^1 come from three coordinate-loop classes (one for each S^1 factor) plus a "product" of two coordinate loops with the third, in the exterior-algebra pattern. Specifically, K^1(T^3) = H^0 oplus H^2 (odd degree part of exterior algebra) = Z oplus Z^3, interpreted through the usual K-theory of tori. What matters is that certain explicit classes are PRIMITIVE: they are not multiples of any integer > 1. 3.3. The K-theory class theta_j -------------------------------- Define three classical classes on T_j. * L_j: a line bundle pulled back from the first two circles, with first Chern class c_1(L_j) = u_1 cup u_2. * [L_j] - 1 in K^0(T_j): the reduced class. This is a generator of the (2,2)-part of K^0 (Kunneth). Equivalently, we can identify it via c_1 with a primitive element of H^2. * e_3 in K^1(T_j): the pullback of the POSITIVE generator of K^1(S^1) = Z under projection to the third circle. Define the combined class theta_j = ([L_j] - 1) * e_3 in K^1(T_j). Here "*" is the topological K-theory product (the exterior algebra product pulled back through K^0 tensor K^1 -> K^1). Lemma (primitiveness of theta_j). theta_j is a primitive element of K^1(T_j): there is a class tau_j in K_1(T_j) with = 1. Proof idea. The exterior-algebra description of K^*(T^3) identifies [L_j] - 1 with a generator of the H^2-part and e_3 with a generator of the H^1-part, so their product theta_j is the "product" of three circle generators, i.e., a primitive generator of a free direct summand of K^1(T_j). Because K^*(T^3) is torsion-free, the universal coefficient theorem identifies K_1(T_j) with Hom(K^1(T_j), Z), and a primitive element of a free abelian group admits an integer-valued functional taking value 1 on it. That functional is tau_j. 3.4. The K-homology class beta on B ------------------------------------ Recall that P_3(B) = disjoint union over j of P_3(B_j), because no simplex of P_3(B) spans two components (auxiliary edges exceed 3). By countable additivity of K-homology in the first variable of KK, KX_1(B) at scale 3 = PRODUCT over j of K_1(P_3(B_j)). The sequence ((kappa_j)_* tau_j)_{j >= 1} in PRODUCT over j of K_1(P_3(B_j)) defines a class beta_3 in KX_1(B) at scale 3. Let beta be its image in KX_1(B) (the direct limit over Rips scales). 3.5. The K-homology class alpha on X ------------------------------------- The two end maps i_{+3}, i_{-3} : P_3(B) -> P_3(X) are proper simplicial inclusions (section 2.10). Define alpha_3 = (i_{+3})_* beta_3 - (i_{-3})_* beta_3 in KX_1(X) at scale 3, and let alpha in KX_1(X) be its image in the direct limit. Then alpha = (i_+)_* beta - (i_-)_* beta in KX_1(X). The rest of Part 3 shows alpha has infinite order. 3.6. Retractions from Rips to the torus ---------------------------------------- Fix a Rips scale r >= 3. Choose j large enough that l_j > 4r. We will construct a continuous map p_{B,j} : P_r(B_j) -> T_j that restricts to the identity on the embedded torus T_j subset P_3(B_j) subset P_r(B_j). Lifting lemma. Let {u, v, w} be the vertices of a simplex of P_r(B_j); then d(u,v), d(v,w), d(u,w) are all <= r. Choose lifts u', v', w' in Z^3 (each coordinate in {0, ..., l_j - 1}) and integer displacements delta'(u', v') whose sum around the triangle is an integer multiple of l_j. That sum equals delta'(u', v') + delta'(v', w') - delta'(u', w'), so its coordinate entries are each between -2r and 2r. Since l_j > 4r, the only multiple of l_j in [-2r, 2r] is 0. So the sum is exactly zero. That means the "small lift" construction on simplices is COHERENT: we can interpolate linearly on each simplex after lifting to Z^3, and this interpolation agrees on faces. So a continuous map P_r(B_j) -> T_j exists, extending the identity on the triangulated torus. Geometry. The map p_{B,j} sends a simplex of P_r(B_j) to a point of T_j at most r away (in the torus metric) from each of the simplex's vertices. It is r-Lipschitz. Functorial lifts. Because the projection Y_j -> B_j and X_j -> B_j are 1-Lipschitz (section 2.10), they induce simplicial maps on the corresponding Rips complexes. Composing with p_{B,j} yields p_{Y,j} : P_r(Y_j) -> T_j, p_{X,j} : P_r(X_j) -> T_j. Because both projections agree with the base projection (in the sense (u,q) |-> u and (u,q,t) |-> u), the two maps satisfy the factorization p_{X,j} | family near middle = p_{Y,j} o (forget height). Precisely: there is a simplicial map h : P_r(X_j "middle part") -> P_r(Y_j) that forgets the height, and p_{X,j} restricts to p_{Y,j} o h on the "collar" subcomplex I (defined in section 3.9 below). 3.7. The congruence calculation: divisibility of the pulled-back Chern class -------------------------------------------------------------- Key divisibility statement. For each fixed r >= 3 and every sufficiently large j, the class p_{Y,j}^*(u_1 cup u_2) in H^2(P_r(Y_j); Z) is divisible by 2^j: p_{Y,j}^*(u_1 cup u_2) in 2^j * H^2(P_r(Y_j); Z). Equivalently, there is a line bundle L'_{j,r} on P_r(Y_j) with p_{Y,j}^* L_j = (L'_{j,r})^{tensor 2^j}. Proof. Take j large enough so l_j > 4r and the reduction Gamma -> Q_{2j} is injective on the S-word ball of radius 2r (from Lemma 2.3(c)). We construct an explicit integral 2-cocycle representing p_{Y,j}^*(u_1 cup u_2), then show that modulo 2^j it is a coboundary. That establishes divisibility. Representing u_1 and u_2 by seam cocycles. For each oriented Rips edge e : x -> y in P_r(Y_j), project to base vertices u, v in B_j, and write u,v coordinate-wise as representatives in {0, ..., l_j - 1}. Let a_{xy}, b_{xy} in Z be the seam numbers in the first two coordinates (from section 2.5). These are the cocycles: [a] = p_{Y,j}^* u_1, [b] = p_{Y,j}^* u_2. The cup product [a cup b] = p_{Y,j}^*(u_1 cup u_2) is a 2-cocycle assigning to an oriented 2-simplex (x_0 < x_1 < x_2), using a vertex ordering, the integer a_{01} b_{12} (where a_{ij} = a_{x_i x_j} and b_{ij} = b_{x_i x_j}). Lifting to Q_{3j}. For each vertex x in P_r(Y_j), write x = (u_x, q_x) with q_x in Q_{2j}, and choose a lift q_tilde_x in Q_{3j} (the reduction Q_{3j} -> Q_{2j} is surjective by Lemma 2.3(a)). For each oriented Rips edge e : x -> y with d(x,y) <= r, choose a graph path from x to y in Y_j of length at most r. Along this path, the base projection winds by total seam numbers a_{xy} and b_{xy}, and the left Cayley steps give a word w_{xy} in Gamma of length at most r. Then q_y = overline{w_{xy}} q_x g_{2j}(a_{xy}, b_{xy}) in Q_{2j}, where overline{w} denotes the reduction of w to Q_{2j}. Now consider an ordered triangle (x_0 < x_1 < x_2) in P_r(Y_j). Two facts combine. * Commutativity in Q_{2j}. Since U_j and V_j commute in Q_{2j}, the transport elements multiply: g_{2j}(a_{01},b_{01}) * g_{2j}(a_{12},b_{12}) = g_{2j}(a_{02},b_{02}). * Word consistency. Substituting the edge equations into the triangle equation for x_0, x_1, x_2 gives overline{w_{02}} = overline{w_{12}} overline{w_{01}}, in Q_{2j}. But w_{02} and w_{12} w_{01} both have length at most 2r (as words in Gamma), so by the injectivity of Gamma -> Q_{2j} on the radius-2r ball (Lemma 2.3(c)), they agree in Gamma: w_{02} = w_{12} w_{01}. (**) Lifting to Q_{3j}. Now work modulo 2^{3j}. Let N = kernel(Q_{3j} -> Q_{2j}). Every element of N has the form 1 + m_j^2 D mod m_j^3, with D a 3-by-3 matrix mod m_j (m_j = 2^j). Multiplication in N is addition mod m_j of D (since m_j^4 is divisible by m_j^3). So N is an abelian group. Define a homomorphism d : N -> Z/m_jZ, d(1 + m_j^2 D) = D_{13} mod m_j. This is well-defined by the structural analysis of elementary matrices above. For each vertex x, choose a lift q_tilde_x in Q_{3j} of q_x. For each oriented edge e : x -> y, define n_{xy} in N by q_tilde_y = w_tilde_{xy} q_tilde_x g_tilde(a_{xy}, b_{xy}) n_{xy}, where w_tilde is the reduction of the Gamma-word w to Q_{3j} and g_tilde is the lift of g to Q_{3j} (g_tilde(a,b) = V_j^b U_j^a mod m_j^3). Triangle identity. Substituting twice along the triangle (x_0, x_1, x_2) and using (**) to substitute w_{12} w_{01} = w_{02} gives q_tilde_2 = w_tilde_{02} q_tilde_0 g_tilde(a_{01}, b_{01}) g_tilde(a_{12}, b_{12}) n_{01} n_{12} = w_tilde_{02} q_tilde_0 g_tilde(a_{02}, b_{02}) C^{a_{01} b_{12}} n_{01} n_{12}. Here C = 1 + m_j^2 E_13 = the lifted commutator of U and V, and the power C^{a_{01} b_{12}} accounts for the fact that g_tilde(a,b) g_tilde(a',b') = g_tilde(a+a', b+b') C^{a b'} (proved by direct multiplication of elementary matrices). Comparing with the direct edge x_0 -> x_2 gives n_{02} = C^{a_{01} b_{12}} n_{01} n_{12}. Apply d. Since d(C) = 1 (mod m_j), this becomes c_{02} = a_{01} b_{12} + c_{01} + c_{12} mod m_j, where c_{xy} = d(n_{xy}). Rearranged with the standard simplicial coboundary convention (delta c)_{012} = c_{12} - c_{02} + c_{01}, this reads (a cup b)_{012} = a_{01} b_{12} = -(delta c)_{012} mod m_j. Hence [a cup b] is equal to -[delta c] = 0 mod m_j in H^2, i.e., p_{Y,j}^*(u_1 cup u_2) = 0 in H^2(P_r(Y_j); Z/2^j Z). Torsion-free step. Divisibility in cohomology mod m_j lifts to divisibility in integral cohomology by the long exact sequence of 0 -> Z --x m_j--> Z -> Z/m_j Z -> 0. The kernel of "reduce mod m_j" on H^2(-; Z) is m_j * H^2(-; Z). Since we just showed the class is in the kernel, it is divisible by m_j = 2^j. Line-bundle interpretation. Line bundles on a finite complex are classified by their first Chern class in H^2(-; Z). The first Chern class of a tensor power is the integer multiple. Since the first Chern class c_1(p_{Y,j}^* L_j) = p_{Y,j}^*(u_1 cup u_2) is divisible by 2^j, there is a line bundle L'_{j,r} on P_r(Y_j) with p_{Y,j}^* L_j = (L'_{j,r})^{tensor 2^j}. (QED) 3.8. From line-bundle roots to K-theory divisibility ------------------------------------------------------ The dimension bound. For fixed r, bounded geometry of Y (uniform vertex-degree bound) supplies an integer D_r >= 1 with dim P_r(Y_j) <= D_r for all j >= 1. Indeed, a Rips simplex is contained in a single radius-r ball, which contains at most D_r points. K-theory divisibility lemma. For each fixed r >= 3 and all sufficiently large j, p_{Y,j}^* theta_j is divisible by 2^{j - floor(log_2 D_r)} in K^1(P_r(Y_j)): p_{Y,j}^* theta_j in 2^{j - floor(log_2 D_r)} K^1(P_r(Y_j)). Proof. Take j satisfying Lemma 3.7 and also 2^j > D_r. Define z = [L'_{j,r}] - 1 in K^0(P_r(Y_j)). This is the reduced K-theory class of the line-bundle root. Its classifying map can be taken to land in complex projective space CP^{D_r} of dimension D_r (cellular approximation, using the dimension bound). The relation for the tautological line bundle H on CP^d, ([H] - 1)^{d+1} = 0 in K^0(CP^d), implies z^{D_r + 1} = 0 in K^0(P_r(Y_j)). Then, from p_{Y,j}^* L_j = (L'_{j,r})^{tensor 2^j}, we have p_{Y,j}^* [L_j] - 1 = (1 + z)^{2^j} - 1 = sum_{h=1}^{D_r} binomial(2^j, h) z^h. Divisibility of binomial coefficients. For 1 <= h <= D_r, write v_2(h) for the exponent of 2 in the binary expansion of h. The identity h * binomial(2^j, h) = 2^j * binomial(2^j - 1, h - 1) shows that binomial(2^j, h) is divisible by 2^{j - v_2(h)}. Since h <= D_r, v_2(h) <= floor(log_2 D_r). So binomial(2^j, h) is divisible by 2^{j - floor(log_2 D_r)} for all h in [1, D_r]. In particular, every coefficient in the sum is divisible by that integer. Consequence. p_{Y,j}^* [L_j] - 1 is divisible by 2^{j - floor(log_2 D_r)} in K^0(P_r(Y_j)). Multiplying by p_{Y,j}^* e_3 preserves divisibility, so p_{Y,j}^* theta_j in 2^{j - floor(log_2 D_r)} * K^1(P_r(Y_j)). QED. 3.9. Mayer-Vietoris detection ------------------------------ We want: for any nonzero integer n and any fixed r >= 3, we can find j large enough that n times (the two-end-point-difference) is nonzero in K_1(P_r(X_j)). Setup. Let t_{+,j,r}, t_{-,j,r} : T_j -> P_r(X_j) be the two torus inclusions, at heights +3j and -3j. Assume additionally j > 2r. Mayer-Vietoris cover. Cover P_r(X_j) by M^+ = P_r(X_j) restricted to {heights >= -r}, M^- = P_r(X_j) restricted to {heights <= +r}, I = M^+ intersect M^- = P_r(X_j) restricted to {-r <= t <= r}. Both sets are open subcomplexes, and they cover: a simplex not contained in either would have a vertex below -r and a vertex above +r, so its diameter is > r, contradiction. The intersection I is the subcomplex induced on the "middle" heights [-r, r]. A map h : I -> P_r(Y_j) forgetting height. Since j > 2r, every vertex of a simplex of I has heights between -r and +r, and every graph path of length <= r joining two such vertices stays within heights [-2r, 2r] sub h >= -j, i.e., inside the central collar (product region). In the central collar, X_j = Y_j x interval. So forgetting height gives a simplicial map I -> P_r(Y_j) whose images of simplex vertices have diameter <= r. This h satisfies the factorization p_{X,j} | I = p_{Y,j} o h. (QED) The Mayer-Vietoris sequence. Write a_± : I -> M^± for the inclusions. The relevant piece of the sequence on K-homology is K_1(I) --((a_+)_*,-(a_-)_*)--> K_1(M^+) + K_1(M^-) --(b_+)_* + (b_-)_*--> K_1(P_r(X_j)), where b_± : M^± -> P_r(X_j) are the inclusions. The middle combining map is "sum of the two restrictions". Exactness at the middle: the kernel of the second map is the image of the first. Now suppose, for contradiction, that n * (t_{+,j,r})_* tau_j = n * (t_{-,j,r})_* tau_j in K_1(P_r(X_j)). Since M^+ contains the +3j end and M^- contains the -3j end, and both containments come from the two sides, exactness gives some eta in K_1(I) with (a_+)_* eta = n * (t^+)_* tau_j in K_1(M^+), and (a_-)_* eta = n * (t^-)_* tau_j in K_1(M^-), where t^± : T_j -> M^± are the endpoint inclusions ("+" end inside M^+ at height +3j, "-" end inside M^- at height -3j). Pairing with the theta class. Pair the first equation with (p_{X,j} restricted to M^+)^* theta_j. On the right: < p_{X,j}^* theta_j, n (t^+)_* tau_j > = n < (t^+)^* p_{X,j}^* theta_j, tau_j > = n < (p_{X,j} o t^+)^* theta_j, tau_j > = n < theta_j, tau_j > (identity on T_j) = n. So the pairing of eta with (p_{X,j} | M^+)^* theta_j equals n. On the left, using a_+ and the factorization p_{X,j} | I = p_{Y,j} o h: n = < (p_{X,j} |_I)^* theta_j, eta > = < (p_{Y,j} o h)^* theta_j, eta > = < h^* p_{Y,j}^* theta_j, eta > = < p_{Y,j}^* theta_j, h_* eta >. Divisibility contradicts integer equality. Since p_{Y,j}^* theta_j is divisible by 2^{j - floor(log_2 D_r)} (Lemma 3.8), the last integer is a multiple of 2^{j - floor(log_2 D_r)}. Taking j large enough that j - floor(log_2 D_r) > v_2(|n|), the last integer is divisible by 2^{v_2(|n|)+1}. But n has v_2(|n|) and is not divisible by 2^{v_2(|n|)+1}. Contradiction. So for every nonzero integer n, and every r >= 3, there exists j = j(n, r) such that n * (t_{+,j,r})_* tau_j != n * (t_{-,j,r})_* tau_j in K_1(P_r(X_j)). 3.10. Alpha has infinite order ------------------------------- Proposition (infinite order of alpha). The class alpha in KX_1(X) defined in section 3.5 has infinite order. In particular alpha tensor 1 in KX_1(X) tensor Q is nonzero. Proof. Fix a nonzero integer n and a Rips scale r >= 3. Let alpha_r be the image of alpha_3 under the map KX_1 at scale 3 -> KX_1 at scale r. Concretely, alpha_r is the sum over j of the difference in K_1(P_r(X_j)) determined by the two ends. For j large enough (say j > r + 100 so no simplex of P_r(X) spans two components), the component P_r(X_j) is a clopen subcomplex of P_r(X). Extension by zero on functions gives a *-homomorphism C(P_r(X_j)) -> C_0(P_r(X)). Contravariance of Kasparov's KK in its first variable then defines a restriction map KX_1(P_r(X)) at scale r -> K_1(P_r(X_j)). The image of n * alpha_r under this restriction is exactly the difference n * ((t_{+,j,r})_* tau_j) - n * ((t_{-,j,r})_* tau_j). By Lemma 3.9, choosing j also large enough so that the divisibility contradiction applies, this difference is nonzero in K_1(P_r(X_j)). Hence n * alpha_r != 0 in KX_1(P_r(X)). Conclusion. We have proved: for every nonzero integer n and every r >= 3, the image of n * alpha_3 in KX_1 at scale r is nonzero. If n * alpha vanished in the direct limit KX_1(X), it would have to vanish at some finite scale r (direct limits of abelian groups detect elements this way). That is impossible. So alpha has infinite order. Tensor with Q. The kernel of the natural map from an abelian group A to A tensor Q is exactly the torsion subgroup of A. Since alpha has infinite order, its image in KX_1(X) tensor Q is nonzero. QED. ================================================================================ PART 4. THE MAXIMAL ROE ALGEBRA AND THE CANONICAL INDEX ================================================================================ We have an integral class alpha in KX_1(X) of infinite order. The rest of the paper shows mu_X^max(alpha) = 0. This requires working carefully in the MAXIMAL norm (not just the operator norm). Three fixed- propagation lemmas, one functoriality proposition, and one naturality proposition. All are standard in maximal coarse index theory (Oyono- Oyono and Yu 2009; Schick and Seyedhosseini 2021), restated here in full. Throughout, Z is a bounded-geometry metric space. H_Z = l^2(Z) tensor l^2(N). C[Z] is the *-algebra of bounded finite-propagation locally compact operators (section 1.2). A_Z = C^*_max(Z). 4.1. Fixed-propagation norm control ------------------------------------ Lemma (fixed-propagation control). For every R >= 0 there is a constant c_Z(R) < infinity such that, for every T in C[Z] with propagation at most R, ||T||_op <= ||T||_max <= c_Z(R) ||T||_op. Consequently, operator-norm convergence of Roe operators with a common propagation bound implies maximal-norm convergence. Proof. Let N_R = sup over z in Z of |B_Z(z, R)|, finite by bounded geometry. The matrix support supp(T) = {(x,y) : T_{xy} != 0} is a bipartite graph with row copy and column copy of Z, and each vertex has degree <= N_R (because T has propagation R). A bipartite graph of max degree <= N_R has an edge-color partition into at most 2 N_R - 1 matchings (greedy: assign each edge to the first color not used by an adjacent edge). Therefore T = T^{(1)} + T^{(2)} + ... + T^{(2 N_R - 1)}, where each T^{(a)} has at most one nonzero entry in each row and each column, and ||T^{(a)}||_op <= ||T||_op. The diagonal algebra D = PRODUCT over z in Z of K(H_0) is a C*-subalgebra of C[Z] (elements diag(k_z) : k_z in K(H_0), sup_z ||k_z|| finite). For a T^{(a)} as above, (T^{(a)})^* T^{(a)} is diagonal. Apply any *-representation pi of C[Z]. Restriction to D is contractive. So ||pi(T^{(a)})||^2 = ||pi((T^{(a)})^* T^{(a)})|| <= ||(T^{(a)})^* T^{(a)}||_op <= ||T||_op^2. Thus ||pi(T^{(a)})|| <= ||T||_op for every a, and ||pi(T)|| <= (2 N_R - 1) ||T||_op. Taking the sup over pi: ||T||_max <= (2 N_R - 1) ||T||_op, giving c_Z(R) = 2 N_R - 1. The lower bound ||T||_op <= ||T||_max is the concrete representation being one of the *-representations. QED. 4.2. The compact ideal ----------------------- Lemma (compact ideal K_Z). An operator in C[Z] supported on F x F, for a finite F subset Z, has the same maximal and concrete norms. The closure in A_Z of all such operators is an ideal K_Z, canonically isomorphic to the compact operators K(H_Z) on H_Z. In particular K_1(K_Z) = 0. Proof. Operators supported on F x F form exactly the C*-algebra K(l^2(F) tensor H_0). Any *-representation of C[Z] restricts to a contractive representation of this finite-type algebra. The concrete representation on H_Z has the correct (operator) norm, so it represents the same norm. The closure of the union over finite F of these algebras is K(H_Z) (finite-rank plus limit), giving K_Z = K(H_Z). To see that K_Z is an ideal: let T in C[Z] have finite propagation R. Then T * (compact on F x F) * T' has support inside a finite metric neighborhood of F, and its entries are compact. Density of the finite-support algebras inside K_Z closes the argument. QED. 4.3. Multipliers ----------------- We need bounded finite-propagation operators with NON-compact entries; the canonical case is the unitary implementing a group action. Lemma (finite-propagation multipliers). Every bounded finite- propagation operator D on H_Z acts by left and right multiplication on C[Z], and these actions extend to a multiplier of A_Z. This assignment is a faithful *-homomorphism, and ||D||_op <= ||D||_M(A_Z) <= c_Z(R) ||D||_op, if prop(D) <= R. For an operator supported on a single matching, the multiplier norm equals the concrete norm. Proof. Product of D with T in C[Z] has finite propagation and compact entries, so is again in C[Z]. Decompose D into (T^{(a)}) matchings as in Lemma 4.1 (now D need NOT be locally compact, only bounded finite propagation). For each matching operator D_a with entries of norm <= 1: * (D_a)^* D_a is diagonal positive contraction. * W = (1 - (D_a)^* D_a)^{1/2} is a bounded diagonal operator (square roots of 0 <= x <= 1). * For any T in C[Z], the identity T^* T - (D_a T)^* (D_a T) = (W T)^* (W T) holds on the algebraic level, and both sides lie in C[Z] (W is bounded, so WT is bounded finite propagation and locally compact). * Apply any *-representation pi of C[Z]. The left side is ||pi(T)||^2 - ||pi(D_a T)||^2, and the right side is ||pi(W T)||^2 >= 0. So ||pi(D_a T)|| <= ||pi(T)||. * Therefore ||D_a T||_max <= ||T||_max. Applying the same bound to D_a^* and taking adjoints gives the corresponding right-multiplication bound. The algebraic compatibility identities (associativity, adjunction) pass to the completion. Summing over the matchings gives ||D T||_max <= (2 N_R - 1) ||T||_max. By symmetry, ||D||_M(A_Z) <= c_Z(R) ||D||_op. The assignment (D -> multiplication by D) is faithful: the concrete representation of A_Z on H_Z is nondegenerate (it contains K(H_Z)), so it extends to the multiplier algebra, and it sends the multiplier of D back to the concrete operator D. So the multiplier assignment is faithful. Particular case: a single matching. Then the bound is 1, not c_Z(R), so ||D||_M(A_Z) = ||D||_op. QED. In particular, any operator identity involving bounded finite- propagation operators remains valid as an identity of multipliers of A_Z. What is NOT claimed: that the concrete representation of the entire maximal Roe algebra is faithful (that fails for expanding graphs). 4.4. Coarse functoriality -------------------------- Let h : Z -> W be a coarse map: preimages of bounded sets are bounded, and for every R there is S with d_Z(x,y) <= R implying d_W(hx,hy) <= S. Choose an "exact covering isometry" V_h : H_Z -> H_W: an isometry sending the summand over z into the summand over h(z), using orthogonal ranges when several vertices share an image. Possible because the vertex fibers h^{-1}(w) are finite and H_0 is infinite dimensional. Proposition (coarse functoriality). Conjugation by V_h defines a *-homomorphism h_A = Ad(V_h) : A_Z -> A_W. The induced map on K-theory is independent of the choice of V_h. These induced maps respect composition, and a covering isometry over identity (i.e., h = id_Z) induces the identity on K_*(A_Z). Proof. For T in C[Z], coarse control bounds prop(V_h T V_h^*). Each of its vertex entries is a finite sum of compact operators, because h^{-1}(w) is finite for each w. So conjugation is a *-homomorphism C[Z] -> C[W]. The universal property of the maximal completion extends it canonically to a *-homomorphism A_Z -> A_W. Independence of V_h. Let V_0, V_1 be two covering isometries. Define, for 0 <= lambda <= pi/2, V_lambda : H_Z -> H_W oplus H_W, V_lambda(xi) = (cos lambda V_0 xi, sin lambda V_1 xi). Each V_lambda is an isometry. Conjugation by V_lambda lands in M_2(A_W). Its four matrix entries have compact vertex entries and one common propagation bound for each fixed T in C[Z]. Moreover, ||V_lambda T V_lambda^* - V_mu T V_mu^*||_op <= 2 ||T||_op |lambda - mu|. Lemma 4.1 (fixed-propagation norm control) applied to the four matrix entries promotes this estimate to MAXIMAL-norm continuity on C[Z]. Contractivity and density extend it to A_Z. The two endpoints of lambda (0 and pi/2) are Ad(V_0) and Ad(V_1) in the two standard matrix corners, which induce the SAME stabilized K-theory map. QED. Notation. For the specific maps h = f, i_+, i_- from section 2.10, write f_A : A_Y -> A_B, i_{+,A}, i_{-,A} : A_B -> A_X for the induced *-homomorphisms. Any statements about their K-theory maps are by Proposition 4.4 independent of the covering choices. 4.5. The canonical odd index formula ------------------------------------- Let r > 0. Regard a point of P_r(Z) as a finitely supported probability measure on Z whose support has diameter at most r. Equip P_r(Z) with the 1-Wasserstein metric. This is a proper metric that induces the usual topology. There is a natural grouping of the ample module over P_r(Z) into "vertex blocks": choose a countable dense E subset P_r(Z) containing the vertices, and for each dense point choose one support vertex. This partitions E into blocks H_v (all dense points whose assigned vertex is v), and each block is an infinite-dimensional separable Hilbert space, so isomorphic to H_0. Under this regrouping, the algebraic Roe algebra of P_r(Z) is EXACTLY C[Z] (propagations differ by at most 2r, local compactness translates), and the identification extends to the maximal completions by universality. Now the odd K-homology cycle formula. On P_r(Z), a K-homology class in K_1 is represented (up to stabilization and by adding degenerate cycles) by a bounded self-adjoint operator F satisfying: [F, a] is compact for every a in C_0(P_r(Z)) (pseudolocality); a(F^2 - 1) is compact for every a in C_0(P_r(Z)) (perturbation). The cycle can be taken to have finite propagation: let phi_v be square roots of the barycentric coordinate functions; F' = sum over v of phi_v F phi_v is a finite-propagation cycle (propagation <= 2r) equivalent to F. After regrouping into vertex blocks, F is a self-adjoint multiplier of A_Z (Lemma 4.3), with F^2 - 1 in C[Z] (Lemma 4.2). Set p = (1+F)/2. Then p in M(A_Z), p^2 - p in C[Z] subset A_Z, u_F = exp(2 pi i p) in 1 + A_Z, and mu_Z^max([F]) = [u_F] in K_1(A_Z). This is the "boundary formula" of the assembly map for odd K-homology. It agrees with the Paschke duality computation of Oyono-Oyono-Yu 2009 (Corollary 2.18) and Schick-Seyedhosseini 2021 (Proposition 2.22, Propositions 2.27-2.28, Corollary 2.29). We take it as the DEFINITION of mu_Z^max for what follows. Passing to the direct limit over r -> infinity gives the assembly map on KX_1(Z). 4.6. Naturality of the end classes ----------------------------------- Proposition (naturality). For beta in KX_1(B) and alpha = (i_+)_* beta - (i_-)_* beta in KX_1(X): mu_X^max(alpha) = (i_{+,A})_* mu_B^max(beta) - (i_{-,A})_* mu_B^max(beta). (*) Proof. Both classes come from scale-3 cycles, where i_{+,3} and i_{-,3} are proper simplicial inclusions of P_3(B) into P_3(X). Choose the dense-set modules so that the target dense set contains the source dense set, and support-vertex choices agree on the images. The resulting module isometry V is an exact covering isometry for the end maps (after regrouping). For F representing the source odd class (finite propagation, self-adjoint, pseudolocal, F^2 - 1 locally compact), its push-forward to the target is F_tilde = V F V^* - (1 - V V^*). The second term is a degenerate cycle (it lives on the complement of V V^* and is trivially pseudolocal and equal to -1 there). Properness of i_+ (and i_-) ensures that pullbacks of C_0(P_3(X)) functions lie in C_0(P_3(B)), so F_tilde is pseudolocal and F_tilde^2 - 1 is locally compact. Coarse control preserves finite propagation. Writing p = (1+F)/2 we have p_tilde = V p V^*. The exponential formula is compatible with conjugation in the maximal completion: the function h(z) = (exp(2 pi i z) - 1) / (z^2 - z) extends to an entire function (its numerator vanishes to first order at z = 0 and z = 1). Approximate h uniformly by polynomials on a disk containing the spectra of p and p_tilde. Since exp(2 pi i p) - 1 = (p^2 - p) h(p), and p^2 - p lies in C[B], every polynomial identity holds in the algebraic Roe algebra and passes to maximal norm by continuity. So exp(2 pi i p_tilde) - 1 = i_{+,A} ( exp(2 pi i p) - 1 ). Taking K-theory classes, this is the statement that the two assembly maps commute with i_+ (and analogously i_-). Apply to beta, subtract the two, and use the definition of the Rips direct-limit assembly map. Independence of the covering choices follows from Proposition 4.4. QED. ================================================================================ PART 5. A TRANSFER USING KAZHDAN'S PROPERTY (T) ================================================================================ We need to show that (f_A)_* : K_1(A_Y) -> K_1(A_B) is SURJECTIVE, so that we can lift the analytic class to a class on Y. The map f : Y -> B forgets the fiber. Its associated natural map (f_A)_* on K-theory does not normally have a splitting, because the fibers Y_u grow without bound in size. Kazhdan's property (T) of Gamma = SL_3(Z) supplies a canonical projection onto "fiber constants" that makes the transfer well-behaved. 5.1. The fiber-constant isometry V ----------------------------------- For each u in B, define xi_u = |F_u|^{-1/2} * (sum over y in F_u of delta_y) in l^2(F_u), the normalised indicator of the fiber F_u = f^{-1}(u). These vectors define an isometry V : H_B -> H_Y, V (delta_u tensor h) = xi_u tensor h. V has no finite propagation bound (the fibers have unbounded diameter), so it is NOT a covering isometry. The transfer must be built differently. 5.2. Kazhdan's property (T) ----------------------------- Gamma = SL_3(Z) has Kazhdan's property (T). We use only the following concrete consequence, which follows from any of the standard characterisations of (T): for the fixed finite symmetric generating set S of Gamma (section 2.3), there is kappa > 0 such that every unitary representation pi of Gamma, every unit vector v in ker(pi(S - 1))^perp (i.e., v orthogonal to the invariant vectors), satisfies sup over s in S of ||pi(s) v - v|| >= kappa. Equivalently, for the "Laplacian" Delta = 1 - (1/|S|) sum over s in S of s in C^*_max(Gamma), the spectrum of Delta in any representation satisfies spec(Delta) subset of {0} union [epsilon, 2], with epsilon = kappa^2 / (2|S|) > 0. In particular, the projection onto invariant vectors in every representation is the value of the continuous functional calculus at 0 on Delta. So the projection "constants in each fiber" belongs to M(A_Y): specifically, define P_n = (1 - Delta/2)^n in M(A_Y), converging in maximal norm to a projection P in M(A_Y) whose concrete image is V V^*. Then ||P_n - P||_M(A_Y) -> 0. (K) Each P_n is finite-propagation and preserves every fiber (in the sense that P_n maps l^2(F_u) tensor H_0 into itself). Its concrete image converges to V V^*, the concrete constants-per-fiber projection. 5.3. Transporting matrix entries --------------------------------- For each pair u, v in B and cell entry T_{uv} : l^2(F_v) -> l^2(F_u), we need a "transport" operator that moves T_{uv} from one fiber to another coherently. Two cases. * Same component B_j. Choose a shortest edge path from v to u in B_j (and the reverse path from u to v). Horizontal transport along this path is a permutation O_{uv} : l^2(F_v) -> l^2(F_u), O_{uu} = identity. * Different components. Use a rank-one isometry O_{uv} = |xi_u> 2(R + R'), these two paths have the SAME total seam crossing numbers. Indeed, their lifted coordinate displacements differ by a multiple of l_j and have difference of absolute value at most 2(R + R') in each coordinate. That multiple is zero. Since U_j and V_j commute in Q_{2j}, horizontal transport depends only on these seam totals. So O_{uv} O_{vw} = O_{uw} for every summand under consideration. The defect vanishes on all sufficiently large components. The omitted components and any cross- component entries lie in a finite union of finite graphs, so D has finite vertex support. PD = 0 in A_Y. On the concrete Hilbert space H_Y, Equations (K) above give L(T) V = V T and V^* L(T) = T V^* (the concrete image of P is V V^*, and O_{uv} xi_v = xi_u). So the CONCRETE image of PD is zero. To obtain PD = 0 in the MAXIMAL completion: choose a finite union E of components of Y containing the vertex support of D. Every P_n preserves each fiber, so P_n D has vertex support in E. Lemma 4.2 identifies its maximal norm with its concrete operator norm (finite vertex support). On H_Y, P_n D -> 0 in operator norm (P_n -> VV^* and V V^* D = 0 concretely). Hence P_n D -> 0 in MAXIMAL norm too. On the other hand, by (K), P_n D -> P D in A_Y. Therefore P D = 0 in A_Y. QED. 5.4. The transfer and its splitting with f_A ---------------------------------------------- Define the transfer Phi : A_B -> A_Y, Phi(T) = P L(T) for T in C[B], extension to A_B by universality. Proposition (transfer). Phi is a *-homomorphism, and (f_A Phi)_* = id on K_*(A_B). In particular (f_A)_* : K_1(A_Y) -> K_1(A_B) is SURJECTIVE. Proof that Phi is a *-homomorphism. P is a projection commuting with every L(T) (lemma 5.3 statement). For T, T' in C[B]: Phi(T) Phi(T') = P L(T) P L(T') = P L(T) L(T') = P L(T T') (because P D = 0 and P^2 = P) = Phi(T T'). The lift L preserves adjoints, so Phi is a *-homomorphism on C[B]. The maximal-norm universality extends it to a *-homomorphism A_B -> A_Y. Computing f_A Phi. Let J : H_Y -> H_B be a covering isometry for f (it sends the summand at y in F_u isometrically into the summand at u, with orthogonal ranges for distinct y in the same fiber). For T in C[B], continuity in maximal norm gives f_A Phi(T) = limit over n -> infinity of J P_n L(T) J^* in A_B. All operators in this limit have propagation at most prop(T) on B, because P_n preserves each fiber (so propagation past the base projection is invisible). In CONCRETE operator norm they converge to J V V^* L(T) J^* = J V T V^* J^*. The isometry J V : H_B -> H_B lies over the identity of B, so J V T V^* J^* belongs to C[B] and has the same propagation bound. The difference from each approximant also has this fixed propagation bound. Lemma 4.1 (fixed-propagation norm control) promotes the concrete convergence to MAXIMAL-norm convergence. Therefore f_A Phi(T) = Ad(J V)(T). By density this identity holds on A_B. Finally, Proposition 4.4 says that a covering isometry over the identity induces the identity on K-theory. Apply it to J V: the induced map on K_* is the identity. So (f_A Phi)_* = id on K_*(A_B). In particular, for any x in K_1(A_B), the class Phi_*(x) in K_1(A_Y) satisfies (f_A)_*(Phi_*(x)) = x. This means (f_A)_* : K_1(A_Y) -> K_1(A_B) is surjective. QED. ================================================================================ PART 6. SLIDING BETWEEN THE TWO ENDS ================================================================================ We have an analytic lift to Y (part 5), and naturality says mu_X^max(alpha) = i_+ mu_B(beta) - i_- mu_B(beta). To see it vanish, we need the two endpoint maps on Y -> X to induce the SAME map on K_1. The two endpoint maps i_+ f, i_- f : Y -> X are separated by a distance growing with j. They are not uniformly close. Instead, we place a copy of an operator at every height and rotate the whole stack by one height. The common interior copies cancel on K-theory. 6.1. Successive slices ----------------------- For -3j <= t <= 3j, let s_{j,t} : Y_j -> X_j, s_{j,t}(u, q) = (u, q_reduced, t), where q_reduced is the reduction of q in Q_{2j} to Q_{k_j(t)}. Each s_{j,t} is 1-Lipschitz (both Cayley edges and horizontal matchings go to edges or vertices at heights t). Consecutive slices are close: d_X(s_{j,t}(y), s_{j,t+1}(y)) = 1 for all y in Y_j. (slice) Choose exact covering isometries D_{j,t} : H_{Y_j} -> H_{X_j} over these vertex maps. They exist because each vertex summand is a copy of H_0. For the two endpoints (-3j and +3j) choose D_{j,-3j} and D_{j,+3j} compatible with the covering isometries defining i_{-,A} f_A and i_{+,A} f_A. For fixed j, the ranges of the D_{j,t} for distinct t are orthogonal (they lie at distinct heights). 6.2. Two stacks modulo compacts -------------------------------- Let q : A_X -> A_X / K_X be the quotient (K_X = compact operators on H_X, identified as K_Z in Lemma 4.2). For T in C[Y], write T_j = chi_{Y_j} T chi_{Y_j}. The difference T - direct sum over j of T_j has finite vertex support and is compact (Lemma 4.2). So T_j behaves like T modulo compacts. From a single slice, build two stacks S_0(T) = direct sum over j of sum over t = -3j to +3j - 1 of D_{j,t} T_j D_{j,t}^* S_1(T) = direct sum over j of sum over t = -3j to +3j - 1 of D_{j,t+1} T_j D_{j,t+1}^*. The number of copies grows with j, but ranges are orthogonal, so ||S_a(T)||_op <= ||T||_op and ||S_a(T)||_max <= ||T||_max. For prop(T) <= R, the 1-Lipschitz condition gives prop(S_a(T)) <= R. Every vertex entry is compact (finite sum of isometric images of compact entries of T_j, since Y_j is finite). So S_a(T) in C[X]. Multiplicativity modulo compacts. The maps S_a are not multiplicative before the quotient. But for T, T' in C[Y], the compression defect (T T')_j - T_j T'_j = chi_{Y_j} T (1 - chi_{Y_j}) T' chi_{Y_j} is zero except for finitely many j (because T, T' have bounded propagation). Each nonzero term is compact on H_{Y_j}, since it is a matrix of compact operators over a finite vertex set. Copying into the finitely many heights keeps it compact. So overline{S_a} = q S_a : C[Y] -> A_X / K_X is a *-homomorphism. By maximal universality, it extends to a *-homomorphism with domain A_Y. 6.3. The rotation between the stacks ------------------------------------- We show overline{S_0} and overline{S_1} induce the SAME map on K-theory. Doubling the target module: W_{lambda, j, t} : H_{Y_j} -> H_{X_j} oplus H_{X_j}, W_{lambda, j, t} = (cos lambda D_{j,t}, sin lambda D_{j,t+1}), for 0 <= lambda <= pi/2 and -3j <= t < 3j. Each is an isometry. For distinct pairs (j, t), ranges are orthogonal (different components, or different heights in each of the two factor copies). So they assemble to a single isometry W_lambda. Define F_lambda(T) = W_lambda (direct sum over (j,t) of T_j) W_lambda^*. Its diagonal blocks are cos^2 lambda S_0(T) and sin^2 lambda S_1(T). The off-diagonal blocks are sums of D_{j,t} T_j D_{j,t+1}^* and counterparts. If prop(T) <= R, (slice) gives propagation at most R + 1 for these. Their vertex entries are again finite sums of compact operators. So F_lambda(T) in M_2(C[X]), prop(F_lambda(T)) <= R + 1. Same compact-defect calculation shows that the images of F_lambda in M_2(A_X / K_X) are *-homomorphisms on C[Y], extending to A_Y. Continuity. Orthogonality gives ||W_lambda - W_mu|| = ((cos lambda - cos mu)^2 + (sin lambda - sin mu)^2)^{1/2} <= |lambda - mu|. So in concrete operator norm, ||F_lambda(T) - F_mu(T)||_op <= 2 ||T||_op |lambda - mu|. The difference has propagation <= R + 1, uniformly in lambda. Lemma 4.1 (fixed-propagation norm control) promotes this to maximal-norm bound ||F_lambda(T) - F_mu(T)||_{M_2(A_X)} <= C_R ||T||_op |lambda - mu|, for a constant C_R independent of j, lambda, mu. Point-norm continuity on the dense algebra C[Y], then extension by contractivity, gives point-norm continuity on A_Y. At lambda = 0, F_0(T) is overline{S_0}(T) in the first matrix corner. At lambda = pi/2, F_{pi/2}(T) is overline{S_1}(T) in the second matrix corner. The corner identifications on K-theory agree. So (overline{S_0})_* = (overline{S_1})_*. (equal) 6.4. Cancellation of the interior ---------------------------------- Define the interior stack I(T) = direct sum over j of sum over t = -3j + 1 to 3j - 1 of D_{j,t} T_j D_{j,t}^* . Its image overline{I} = q I is a *-homomorphism on A_Y by the same compact-defect argument. The endpoints. Define E_- = i_{-,A} f_A, E_+ = i_{+,A} f_A. Consider a single component (j fixed, no cross-component). In S_0, the t = -3j term is exactly E_-(T_j), and in S_1, the t = 3j - 1 term is exactly E_+(T_j). All other terms of both stacks lie in the strictly interior heights -3j + 1, ..., 3j - 1. Cross-component pieces are compact (Lemma 5.3 defect argument with chi_{Y_j} in place of F_u). So modulo K_X, overline{S_0}(T) = overline{I}(T) + q E_-(T), overline{S_1}(T) = overline{I}(T) + q E_+(T). The two summands in each expression have ORTHOGONAL RANGES: one is supported on interior heights and the other on an endpoint. Orthogonal sums on K_1. If phi and psi are orthogonal *-homomorphisms, and u is a unitary in the domain algebra with u - 1 in the ideal, then 1 + (phi + psi)(u - 1) = (1 + phi(u - 1)) (1 + psi(u - 1)), and multiplication of unitaries adds their K_1 classes. So K_1(phi + psi) = K_1(phi) + K_1(psi) whenever phi, psi have orthogonal ranges. Applied to (equal): q_* (E_-)_* + overline{I}_* = q_* (E_+)_* + overline{I}_*. Cancel overline{I}_* on the two sides: q_* (E_-)_* = q_* (E_+)_* in K_1(A_X/K_X). Injection of q_*. Look at the cyclic six-term exact sequence of the short exact sequence 0 -> K_X -> A_X -> A_X / K_X -> 0, in particular the segment K_1(K_X) -> K_1(A_X) -> K_1(A_X / K_X) -> K_0(K_X) -> K_0(A_X). Since K_1(K_X) = 0 (Lemma 4.2: K_X = K(H_X) has trivial K_1), the map q_* : K_1(A_X) -> K_1(A_X / K_X) is injective. So the equality q_*(E_-)_* = q_*(E_+)_* in K_1(A_X / K_X) lifts to equality (E_-)_* = (E_+)_* in K_1(A_X). We have proven (i_{-,A} f_A)_* = (i_{+,A} f_A)_* on K_1(A_Y). (agent) ================================================================================ PART 7. PUTTING EVERYTHING TOGETHER ================================================================================ We now assemble the theorem. Theorem (main). There exist finite connected graphs X_j with a common vertex-degree bound, a coarse disjoint union X, and an infinite-order class alpha in KX_1(X) with mu_X^max(alpha) = 0. Hence mu_X^max tensor Q is not injective. Proof. The graphs X_j (finite, connected, uniformly bounded degree) and their coarse disjoint union X were constructed in Part 2. The class alpha in KX_1(X) was defined in Part 3, and Proposition 3.10 showed it has infinite order. By Proposition 5.4 (transfer), the map (f_A)_* : K_1(A_Y) -> K_1(A_B) is surjective. So there exists b in K_1(A_Y) with (f_A)_*(b) = mu_B^max(beta) in K_1(A_B). By the naturality of the maximal coarse index (Proposition 4.6) and the equality of endpoint maps (Part 6): mu_X^max(alpha) = (i_{+,A})_* mu_B^max(beta) - (i_{-,A})_* mu_B^max(beta) = (i_{+,A})_* (f_A)_*(b) - (i_{-,A})_* (f_A)_*(b) = (i_{+,A} f_A)_*(b) - (i_{-,A} f_A)_*(b) = 0. Here we used naturality of assembly to express mu_X^max(alpha) = i_+ mu_B(beta) - i_- mu_B(beta), then substituted mu_B(beta) = (f_A)_*(b), then recognized (i_+ f_A)_* = (i_- f_A)_* at the K_1 level from (agent). Since alpha has infinite order, its image in KX_1(X) tensor Q is nonzero. So mu_X^max tensor Q is not injective. QED. Concluding remarks on what is proven and what is not. * This is a counterexample to RATIONAL injectivity of maximal coarse assembly. It says nothing about injectivity with integer (integral) coefficients; torsion obstructions can survive rationalization. * The space X is a coarse disjoint union of finite graphs of uniformly bounded vertex degree. Such spaces have bounded geometry, and the paper's class alpha belongs to the actual Rips colimit for this space. * The failure is not a surjectivity failure. It is a genuine injectivity failure: the analytic image is zero, but the topological class is not. * Every step except the "graph of bounded degree" input is completely general. The specific graph construction (tori + SL_3(Z) congruence quotients + long collar) is what realises the "topological class survives but analytic class vanishes" pattern. * Compare with the companion paper "A counterexample to the coarse Novikov conjecture" (OpenAI 2026), which proves the analogous statement for REDUCED coarse assembly. The theorem here strengthens that to the MAXIMAL setting. The step from reduced to maximal is genuinely new and is the reason for the length of Parts 4-6. * The main open question after this paper is what additional hypotheses on a bounded-geometry graph union GUARANTEE rational injectivity of maximal assembly. Prior positive results (Yu on finite asymptotic dimension, Yu on uniform Hilbert embedding, Chen-Wang-Yu on fibred coarse embedding, Guo-Wang-Zhang on A-by-FCE structures) all require some such hypothesis. The present counterexample shows that bounded geometry alone is not enough.