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FAILURE OF RATIONAL INJECTIVITY FOR MAXIMAL COARSE ASSEMBLY
A complete, friendly, self-contained proof tour
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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.
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THE THEOREM AND ITS CONSEQUENCE (formal)
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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.
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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.