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THE BASS TRACE CONJECTURE FOR COMPLEX GROUP RINGS
and the characteristic-zero
KAPLANSKY IDEMPOTENT CONJECTURE
a complete, self-contained exposition with all proofs
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WHAT IS THIS ABOUT, IN PLAIN TERMS.
A "group ring" mixes group theory with linear algebra: its elements
are formal finite linear combinations of group elements, and
multiplication follows the group rule. An "idempotent" is an element
e with e^2 = e -- a clean projection that picks out a piece of the
ring. The "Bass trace conjecture", in the case we settle here, says
that the natural trace of any projective module over the complex
group ring C[G] only sees group elements of FINITE order; no
infinite-order element can contribute a nonzero coefficient. The
"Kaplansky idempotent conjecture" says: if G is torsion-free and R
is any characteristic-zero ring, then the only idempotents in R[G]
are the two obvious ones, 0 and 1. Both statements are theorems for
every group; this paper gives complete proofs.
The strategy is geometric in its core. From a would-be nonzero trace
coefficient, we build a cycle and a cocycle whose pairing measures
that coefficient. A piecewise-linear "sparse-chain vanishing" theorem
then forces the pairing to zero in high dimension. The vanishing of
the pairing forces the coefficient to be zero, and the trace
conjecture follows. Section 20 assembles this final step; Sections
1-19 build the cycle, the cocycle, and the vanishing theorem.
WHAT THIS PAPER PROVES.
THEOREM A (Bass trace, complex form). Let G be any discrete group.
For every class x in K_0(C[G]) and every infinite-order element g in G,
HS_G(x)([g]_G) = 0.
That is, the Hattori-Stallings trace of x has zero coefficient on the
conjugacy class of any infinite-order element. Equivalently,
HS_G(K_0(C[G])) is contained in (+)_{C finite-order} C[C].
THEOREM B (Kaplansky, characteristic zero). Let G be torsion-free
and let R be a commutative unital domain of characteristic zero.
Then the only idempotents in R[G] are 0 and 1.
Theorem B follows from Theorem A (via the trace-rank identity of
Section 23) together with a faithful-trace argument in the group
von Neumann algebra. Theorem A is proved for EVERY group: no
amenability, no finite-dimensionality, no Farrell-Jones assembly
hypothesis, no choice of topology on G.
A third result, the integral form (Theorem 21.1), follows by combining
Theorem A with Linnell's 1983 restriction on finite-order coefficients.
It says the integral Hattori-Stallings trace of any class in K_0(Z[G])
is supported on [1]_G alone.
WHY THE PROBLEM IS NONTRIVIAL.
Let F be a finite group. The averaging idempotent
e_F = (1/|F|) sum_{h in F} h
satisfies e_F^2 = e_F in C[F], and its Hattori-Stallings trace has
coefficient |C|/|F| exactly on each conjugacy class C. So the trace
genuinely records class-by-class data; it is not just a numerical
rank. The conjecture says that as soon as the class has infinite
order, the coefficient dies -- uniformly over ALL groups and ALL
idempotents.
The complex and integral statements differ. Over C, only infinite-order
classes are killed; finite-order classes can and do survive. Over Z,
all nonidentity classes die:
HS_{Z[G]}(K_0(Z[G])) is supported on [1]_G.
By Linnell's restriction, any nonzero integral coefficient at a
finite-order h of order t would make h conjugate to h^{t^a} = 1,
impossible unless h = 1. So the integral form reduces to the complex
form on infinite-order classes, plus Linnell on finite-order classes.
WHAT THE PROOF DOES.
Three ingredients.
(1) GEOMETRIC. A "sparse-chain vanishing theorem" (Part II). Let Y be
a countable graph of degree <= D. Let a countable group H act on Y by
automorphisms, freely on vertices. Let K be an H-invariant subcomplex
of the full simplex on Y, generated by n-simplices with connected
vertex sets, with finitely many H-orbits of simplices. Then for all
n >= n_0(D), every H-invariant cocycle psi: E_n -> C vanishes on the
image of every cycle of C_n(K)_H in (E_n)_H, where E_* is the
augmented exterior chain complex of the full simplex on Y.
(2) ALGEBRAIC. From an idempotent e in M_q(C[G]) and an infinite-order
g in G, build for every even n a cycle z_n in (E_n)_H, where
Y = \G, H = C_G(g)/, deg(Y) <= D := max(2, |S|),
and S is a finite symmetric generating set containing the support of e
(with inverses).
(3) DETECTOR. An H-invariant cocycle psi_m on E_{2m} with
psi_m(z_{2m}) = 2^{-m} * lambda,
lambda = HS_G([e C[G]^q])([g]_G).
Combining (1), (2), (3) forces lambda = 0.
The geometric piece is the engine. The estimate at its core: a top
simplex has volume 1/n! in a tree metric; a D-regular graph has at
most D^{2n} connected (n+1)-subsets through any vertex; and the ratio
D^{O(sqrt n)} / 2^n goes to zero exponentially. After the factorial
cancels in a slicing identity, the expected number of zero-stratum
points is arbitrarily small. That count bounds the cocycle evaluation.
WHAT A READER NEEDS.
Linear algebra over C. The definition of the group ring (Section 1).
Familiarity with the phrase "finitely generated projective module"
(recalled in Section 2). Comfort with compact metric spaces and Borel
measures. Two analytic facts we prove or state:
* existence and faithfulness of the canonical trace on the group
von Neumann algebra N(G) -- Appendix A;
* the coarea formula for affine maps between Euclidean spaces --
Section 10, in its elementary form.
No prior K-theory, group homology, or separators required.
NOTATION.
|X| cardinality of a set; for a simplex, closed subset
[g]_G = [g] conjugacy class {h g h^{-1} : h in G}
C, R, Z, Q complexes, reals, integers, rationals
C^{(Y)} free C-vector space on Y (finite support)
[A,A] span of commutators [a,b] = ab - ba in an algebra A
HH_0(A) A/[A,A]
C_n(K)_H coinvariants of the H-action on n-chains of K
HS_G(x)(C) coefficient of HS_G(x) at class C
tau_G, eps_G identity-coefficient and augmentation traces
Pf(A) Pfaffian of an alternating matrix A
Vectors are finite. Infinite-looking sums collapse to finite ones in
context. "Projection" is reserved for self-adjoint idempotents in a
C*-algebra or von Neumann algebra; "idempotent" is any e with e^2 = e.
STRUCTURE.
PART I Group rings, projective modules, Hattori-Stallings, Bass
PART II Sparse-chain vanishing (the geometric engine)
PART III Cycles from an idempotent
PART IV The Pfaffian detector
PART V Assembly and consequences
APPENDIX A The canonical trace on the group von Neumann algebra
APPENDIX B Conventions, sign bookkeeping, and external-input summary
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PART I. BACKGROUND
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SECTION 1. GROUP RINGS.
Let G be a group, R a commutative ring. The group ring R[G] is the
free R-module with basis G, with multiplication extended R-bilinearly
from the group product:
(sum_{g in G} r_g g) (sum_{h in G} s_h h) = sum_{g,h} r_g s_h gh.
Every element of R[G] is a FINITE R-linear combination of group
elements. This finiteness is the single most important structural
feature for us: every argument below uses it, and the completed
group rings (which drop it) behave quite differently.
Three C-linear functionals we will use repeatedly:
augmentation
eps_G : C[G] -> C, eps_G(sum_g r_g g) = sum_g r_g.
A ring homomorphism: eps_G(1) = 1.
identity coefficient
tau_G : C[G] -> C, tau_G(sum_g r_g g) = r_1.
Linear but not multiplicative.
class sum, for a conjugacy class C:
cs_C(sum_g r_g g) = sum_{g in C} r_g.
Vanishes on [C[G], C[G]] (see Section 3).
The class sums together give the isomorphism HH_0(C[G]) ~= (+)_C C[C].
SECTION 2. PROJECTIVE MODULES AND K_0.
Let A be a unital ring. A right A-module P is PROJECTIVE if it is a
direct summand of a free module: P (+) Q ~= A^m for some m and some
Q. Equivalently, every surjection M ->> P of right A-modules splits.
We will use only:
every idempotent e in M_q(A) gives a finitely generated projective
right A-module e A^q (column vectors), and conversely every
finitely generated projective right A-module is isomorphic to one
of this form.
The Grothendieck group K_0(A) is the abelian group completion of the
monoid of isomorphism classes of finitely generated projective right
A-modules under (+). Elements are formal differences [P] - [Q].
Standard facts used without further proof:
(i) K_0(M_q(A)) ~= K_0(A).
(ii) an idempotent e in M_q(A) represents the class [e A^q];
splitting into P (+) Q corresponds to adding [P] and [Q];
(iii) K_0(C) = Z, generated by [C].
For A = C[G] with G infinite, K_0 has typically infinite rank and
rich structure. For finite G, K_0(C[G]) has rank equal to the number
of irreducible characters of G over C (equivalently, the number of
conjugacy classes), by Wedderburn's theorem.
SECTION 3. HATTORI-STALLINGS TRACE.
HH_0(A) := A / [A, A],
where [A,A] is the C-linear span of commutators. For group rings:
PROPOSITION 3.1. For any group G, the class-sum map
phi : C[G] -> (+)_C C * 1_C,
x = sum_g r_g g |-> (cs_C(x))_C
induces an isomorphism HH_0(C[G]) ~= (+)_C C * 1_C.
PROOF. phi is C-linear and surjective (group elements map to basis
vectors). We show ker(phi) = [C[G], C[G]].
(1) [C[G], C[G]] subset ker(phi). Group commutators [g,h] = gh - hg
span [C[G], C[G]] linearly. Given g,h in G, note h g = h(g h)h^{-1},
so gh and hg are conjugate (via h). Hence cs_C(gh - hg) = 0 for
every class C. So each [g,h] lies in ker(phi).
(2) ker(phi) subset [C[G], C[G]]. Let x = sum_g r_g g satisfy
cs_C(x) = 0 for every class C. For each nonempty class C pick a
representative g_C in C. Then
x = sum_C sum_{g in C} r_g g
= sum_C sum_{g in C} r_g (g - g_C) + sum_C (sum_{g in C} r_g) g_C
= sum_C sum_{g in C} r_g (g - g_C).
Now g = s g_C s^{-1} for some s = s(g). So
g - g_C = s g_C s^{-1} - g_C = -[s, g_C s^{-1}].
Each difference is a commutator, hence in [C[G], C[G]]. QED
DEFINITION 3.2. For e in M_q(A) idempotent, put
tr_q(e) = sum_{i=1}^q e_{ii}, HS_A([e A^q]) = [tr_q(e)] in HH_0(A).
Extend C-linearly to all of K_0(A).
LEMMA 3.3. HS_A is well defined: if e A^q ~= e' A^{q'} as right
A-modules, then tr_q(e) and tr_{q'}(e') have the same class in HH_0(A).
PROOF. Standard Morita: there exist u in e' A^{q'} e and v in e A^q e'
with uv = e' and vu = e. Then as elements of A,
tr_q(vu) = sum_{i,k} v_{ik} u_{ki},
tr_{q'}(uv) = sum_{k,i} u_{ki} v_{ik},
which are the same sum after swapping the order of the finite product.
So tr_q(e) = tr_{q'}(e') exactly, not just modulo [A,A]. QED
For A = C[G], the class of HS_G([e C[G]^q]) at class C is
HS_G([e C[G]^q])(C) = sum_{h in C} (tr_q e)_h
= sum_{h in C} tr(e_h),
where e_h in M_q(C) is the coefficient of h in e.
SECTION 4. THE BASS TRACE CONJECTURE.
THEOREM A (already stated). For x in K_0(C[G]) and g in G of
infinite order, HS_G(x)([g]_G) = 0.
By additivity of HS_G, it suffices to verify on classes [P] of
finitely generated projective modules, hence on idempotents e in
M_q(C[G]).
For finite G, all classes are finite-order so the conclusion holds
vacuously. For infinite G, the content is real. Bass proved Theorem A
for linear groups (Bass 1979, Theorem 6.1(f)); subsequent work
extended it to amenable, elementary amenable, and hyperbolic groups.
The present proof covers every group at once.
[Part II continues on next turn]
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PART II. THE SPARSE-CHAIN VANISHING THEOREM
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SECTION 5. THE EXTERIOR COMPLEX AND THE THEOREM.
Given a set Y, we write C^{(Y)} for the free C-vector space on Y with
finite support. The full simplex's augmented exterior chains are
E_j = Lambda^{j+1} C^{(Y)} (j >= 0), E_{-1} = C,
with boundary
partial(delta_{y_0} and ... and delta_{y_j})
= sum_{i=0}^j (-1)^i delta_{y_0} and ... hat{delta_{y_i}} ... and delta_{y_j}.
Here hat means "omit". In degree 0, partial : E_0 -> E_{-1} is the
coefficient sum alpha(sum_y c_y delta_y) = sum_y c_y. With this, E_* is
an acyclic augmented complex: for any fixed y_0, the operator
a |-> delta_{y_0} and a is a contracting homotopy.
For a subcomplex K of the full simplex on Y, the oriented chains
C_*(K) form a chain subcomplex of E_*.
If a group H acts on Y, it acts on E_* and on C_*(K), permuting wedge
factors with the usual signs. Coinvariants of an H-module V are
V_H = V / span_C{ v - hv : h in H, v in V }.
Differentials descend to coinvariants. A cycle in C_n(K)_H is a class
whose boundary vanishes there; note this is weaker than requiring a
particular lift in C_n(K) to be a cycle in C_n(K). An H-invariant
linear functional psi : E_n -> C descends to (E_n)_H. It is called a
cocycle when psi * partial = 0 on E_{n+1}.
THEOREM 5.1 (Sparse-chain vanishing). For each integer D >= 2 there
exists an integer n_0(D) with the following property. Let Y be the
vertex set of a countable connected simple graph of degree at most D,
and let a countable group H act on Y by graph automorphisms, freely on
vertices. Suppose n >= n_0(D) and K is an H-invariant subcomplex of the
full simplex on Y such that:
(i) K is generated by its n-simplices and has finitely many
H-orbits of simplices;
(ii) the vertices of each n-simplex of K span a connected subgraph
of Y.
Then every H-invariant cocycle psi : E_n -> C evaluates to zero on the
image in (E_n)_H of every cycle in C_n(K)_H.
Remarks.
(a) The threshold n_0(D) depends only on D, not on H, K, or psi.
(b) No boundedness of psi is assumed on E_n; we will only need
a bound on a finite set of basis wedges (Section 10).
(c) K need not be locally finite; the bound only sees the graph
degree D.
The proof occupies Sections 6-13. It splits into three stages:
Stage 1 (Sections 6-8): a metric and simplex-dependent areas
for which balls have small volume in high dimension.
Stage 2 (Sections 9-10): separating filtrations with small
expected zero-stratum count.
Stage 3 (Sections 11-13): a chain map, equivariantly homotopic
to inclusion, whose cocycle evaluation is bounded by that
zero-stratum count.
Before starting, we set up notation used throughout Part II.
* |K| is the realization: finitely supported probability vectors
p on Y whose support is a simplex of K. For a simplex sigma of
K we also write sigma for its closed affine realization.
* sigma^o denotes the relative interior.
* By an "original" simplex we mean one of K (as opposed to
simplices introduced by later subdivision). Top simplices are
the n-simplices of K.
* |supp c| <= n+1 for c in |K| (n-simplex has n+1 vertices).
SECTION 6. A METRIC WITH FINITE LOCAL GEOMETRY.
Let d_Y denote graph distance in Y. For p, q in |K|, define
rho(p,q) = (1/sqrt n) sup_{Lip_Y(l) <= 1} sum_{y in Y} l(y) (p(y) - q(y)),
(6.1)
where the supremum is over real-valued 1-Lipschitz functions on Y, and
only finitely many terms occur in the pairing because p, q have finite
support. This is the scaled Kantorovich-Rubinstein (transportation) metric
in its Lipschitz-potential form.
For a nonempty finite V subset of Y, put
A_V = { l in R^V :
sum_{v in V} l(v) = 0,
|l(v) - l(w)| <= d_Y(v,w) for all v,w in V }.
(6.2)
LEMMA 6.1 (The potential polytope). A_V is a nonempty compact polytope.
PROOF. Difference constraints and zero mean bound every coordinate by
diam(V) := max_{v,w in V} d_Y(v,w); compact. Nonempty: the zero
function is in A_V. Convex: both defining conditions are linear. QED
LEMMA 6.2 (Extension of potentials). Each l in A_V extends uniquely to
a 1-Lipschitz function l~ on Y via
l~(y) = min_{v in V} ( l(v) + d_Y(v,y) ). (6.3)
Conversely, every global 1-Lipschitz function restricts to an element
of A_V after subtracting its mean.
PROOF. (Extension) The triangle inequality shows l~ is 1-Lipschitz.
The difference constraints show l~ = l on V. (Converse) The restriction
of a 1-Lipschitz function satisfies the difference constraints, and
subtracting the mean makes it satisfy zero mean. QED
CONSEQUENCE 6.3. When V contains supp p and supp q, the supremum in
(6.1) is attained over A_V. In particular, it is finite and is achieved.
LEMMA 6.4 (Metric, invariance, topology). rho is a metric on |K|.
It is H-invariant. It induces the usual realization topology on each
finite subcomplex of K.
PROOF. Symmetry: replacing l by -l. Triangle inequality: apply each
l to p - q = (p - u) + (u - q). Coordinate control: the indicator
function 1_{v} is 1-Lipschitz, so
|p(v) - q(v)| <= sqrt(n) rho(p,q) for all v in Y. (6.4)
Thus rho separates points. H-invariance: automorphisms permute the set of
admissible potentials. Topology on a finite subcomplex: the restriction
of rho to probability vectors with support in a fixed finite set is a
norm on the finite-dimensional space of vectors with coordinate sum
zero. QED
Define for c in |K|:
f_c(p) = rho(c,p),
U_s(c) = {p : f_c(p) < s}, open ball,
L_s(c) = {p : f_c(p) <= s}, closed ball,
S_s(c) = {p : f_c(p) = s}, level set (sphere). (6.5)
LEMMA 6.5 (Affine branches). On a fixed closed simplex sigma, f_c is a
maximum of finitely many affine functions of p. The set of branches can
be chosen equivariantly under H.
PROOF. On sigma, the supremum (6.1) is a maximum over A_{V(sigma) union supp c}
by Consequence 6.3. Each extreme point of that polytope gives one
candidate affine function. Taking all extreme points gives an
H-equivariant collection because the polytope itself is defined by
H-invariant data. QED
LEMMA 6.6 (Properness). For every c in |K| and s >= 0:
(a) Every closed rho-ball meets only finitely many simplices of K.
(b) K is locally finite; rho gives its realization topology and is
proper.
(c) For every simplex kappa, the set { h in H : L_s(hc) cap kappa is
nonempty } is finite.
(d) The setwise stabilizer H_kappa of any nonempty simplex is finite,
with |H_kappa| <= |V(kappa)|.
PROOF. (a, b) The key estimate: let A = supp c. The function
d_Y(., A) is 1-Lipschitz and vanishes on A. Using its negative in (6.1),
min_{a in A, v in supp p} d_Y(a,v) <= sum_v p(v) d_Y(v,A) <= sqrt(n) rho(c,p).
(6.6)
So if a top simplex sigma meets L_s(c) at p, some vertex of sigma is
within graph distance floor(s sqrt n) of A. The graph on sigma's n+1
vertices is connected (hypothesis of the theorem), so every vertex of
sigma is within n steps of that vertex. Hence every vertex of sigma
lies in the finite graph ball
B^Y_{floor(s sqrt n) + n}(A) = {v in Y : d_Y(v,A) <= floor(s sqrt n)+n}.
(6.7)
There are only finitely many possibilities for sigma and its faces. For
the finite star: every top simplex containing v uses vertices in B^Y_n(v),
a finite set.
Topology: apply (6.4) at radius eta/(2 sqrt n), where eta = min{p(v)}
over v in supp p > 0. The rho-ball lies in the finite star and the
realization topology agrees there. Every closed ball is a closed subset
of a finite union of compact closed simplices, hence compact. (c) If
L_s(hc) cap kappa is nonempty, (6.6) supplies a in A and b in V(kappa)
with d_Y(ha, b) <= s sqrt n. For each (a,b) there are only finitely
many h in H because h |-> ha is injective (H acts freely on vertices).
There are finitely many (a,b). (d) For v in V(kappa), the map h |-> hv
sends H_kappa into V(kappa) injectively. QED
We will also use this consequence: for fixed c and s, the set
F_c = { h in H \ {1} : L_s(hc) cap L_s(c) is nonempty } (6.8)
is finite. Indeed an intersection gives rho(hc,c) <= 2s, so by (6.6)
some vertex of h A is within graph distance 2s sqrt n of A. The same
injectivity argument applies.
SECTION 7. SIMPLEX AREAS AND DISTANCE GRADIENTS.
Fix a top simplex sigma of K. For each spanning tree T of the
connected graph on V(sigma), define a Euclidean structure on the affine
span of sigma by declaring the tree-edge vectors delta_v - delta_w, with
either orientation per edge, to be an orthonormal basis of the
translation space.
For 0 <= j <= n, let a_j^sigma be the AVERAGE over all such T of
Euclidean j-area on polyhedral subsets of sigma^o of dimension at most j.
Relatively open faces are allowed; pieces of dimension < j have zero
j-area by convention; a_0^sigma is counting measure.
These functionals are H-covariant: automorphisms permute spanning trees.
LEMMA 7.1 (Tree Euclidean data). For every tree Euclidean structure on
the top simplex sigma:
(a) its n-volume is 1/n!;
(b) every affine branch of f_c restricted to sigma has gradient norm
at most 1;
(c) the same bound holds after restriction to any affine subspace of
the affine span of sigma.
PROOF. Root the tree at v_0, orient edges away from the root, and order
the remaining vertices so each parent precedes its children. Put
t_i = delta_{v_i} - delta_{parent(v_i)}, 1 <= i <= n,
b_i = delta_{v_i} - delta_{v_0}, 1 <= i <= n.
Each b_i is the sum of t_j along its root path. The change-of-basis
matrix from (t_1,...,t_n) to (b_1,...,b_n) is triangular with diagonal
entries 1. Its determinant is 1, so the parallelepiped spanned by b_1,...,b_n
has volume 1 (t_i are orthonormal). The simplex has volume 1/n!.
For the gradient bound: an affine branch of f_c on sigma is, up to the
1/sqrt n scaling, of the form l(v) for an affine 1-Lipschitz potential
l. Its derivative along t_i equals, up to sign,
( l(v_i) - l(parent(v_i)) ) / sqrt(n).
Since v_i and parent(v_i) are adjacent in the tree, hence in Y, we have
|l(v_i) - l(parent)| <= 1. So each coordinate of the gradient (in the
orthonormal t-basis) has magnitude at most 1/sqrt n. The squared gradient
norm is the sum of n such squares, at most n * 1/n = 1. Restriction to
an affine subspace replaces the gradient by its orthogonal projection,
which does not increase its norm. QED
SECTION 8. THE VOLUME MARGIN IN HIGH DIMENSION.
Set
R = 4 D^2, r = R / 2 = 2 D^2. (8.1)
Both are fixed positive numbers depending only on D.
LEMMA 8.1 (Connected-set count). For a fixed vertex v of Y, there are
at most D^{2n} connected sets of n+1 vertices of Y containing v.
PROOF. Choose one spanning tree per such set, rooted at v. A traversal
using each tree edge once forward and once backward is a walk of length
2n starting at v and visiting exactly the prescribed vertex set (with
possible repeats along the way, but the visited set is exactly the
given set). Different sets give different walks, because the walk
determines the visited set. The number of length-2n walks from v in a
graph of degree <= D is at most D^{2n}. QED
LEMMA 8.2 (Ball volume). For every c in |K|,
sum over top simplices sigma of a_n^sigma( L_R(c) cap sigma^o )
<= V_R := (n+1) * D^{floor(R sqrt n) + 1} * D^{2n} / n!. (8.2)
PROOF. By (6.6), each top simplex sigma meeting L_R(c) contains a vertex
of B^Y_{floor(R sqrt n)}(A), with A = supp c. A ball of radius k in a
graph of degree D >= 2 has at most 1 + D + ... + D^k <= D^{k+1} vertices.
So B^Y_{floor(R sqrt n)}(A) has at most D^{floor(R sqrt n)+1} vertices.
Since |A| <= n+1, at most (n+1) such vertices are candidates for being
the intersection point. For each candidate v, at most D^{2n} connected
(n+1)-sets through v (Lemma 8.1), hence at most D^{2n} top simplices
containing v. Each top simplex has a_n^sigma-volume 1/n! on all of sigma
(Lemma 7.1(a)). Multiplying gives (8.2). Overcounting by simplices hit
multiple times only enlarges the upper bound. QED
LEMMA 8.3 (Strict volume margin). For all sufficiently large n,
V_R < (R - r)^n / n!. (8.3)
PROOF. The ratio is
V_R / ((R-r)^n / n!) = (n+1) D^{floor(4D^2 sqrt n)+1} / 2^n,
(8.4)
where we have used R = 4 D^2 and R - r = R/2 = 2 D^2. Its logarithm is
-n log 2 + 4 D^2 (log D) sqrt(n) + O_D(log n) -> -infinity.
So for some n_0(D) >= 1 depending only on D, the ratio is < 1 for every
n >= n_0(D). This n_0(D) is the same constant appearing in Theorem 5.1.
We fix such an n for the remainder of Part II, and we fix R = 4D^2,
r = 2D^2. The graph degree D, and hence R and r, stay fixed as n varies
only through the already-fixed choice.
The strict margin (8.3) converts successive area comparisons in
Section 10 into an exponentially small bound on the number of
zero-stratum points.
SECTION 9. EQUIVARIANT SEPARATING FILTRATIONS.
The plan: cut |K| into pieces of diameter at most 2R, repeatedly, with
control on the "separator" set at each stage. The cuts are to vary over
a compact probability space X on which H acts freely. That allows us
to separate overlapping translates of local modifications AFTER taking
account of the parameter, without assuming anything about H.
9.1. A free probability parameter.
LEMMA 9.1 (Free parameter space). Every countable group H admits a
continuous free action on a compact metrizable zero-dimensional space
X, preserving a Borel probability measure mu.
PROOF. Construction is a finite-color variant of the coinduction
argument of Braun-Sauer, who credit the existence theorem to Hjorth-
Molberg; we spell out the construction so that no external free-action
theorem is an input.
For a in H \ {1}, let k_a = ord(a) if a has finite order, k_a = 2
otherwise. Define
X_a = { xi : H -> Z/k_a Z : xi(ta) = xi(t) + 1 for all t in H }.
This is a closed subset of the product of finite discrete spaces
(Z/k_a Z)^H, hence compact metrizable zero-dimensional.
Choose one coset representative t_C for each left coset C = t_C .
A coloring xi in X_a is uniquely determined by offsets
omega_C := xi(t_C a^j) - j in Z/k_a Z (any j),
since xi(t_C a^{j+1}) = xi(t_C a^j) + 1. This is well-defined when a
has finite order because Z/k_a Z has modulus equal to that order.
Put independent uniform distributions on the offsets. The left action
(h xi)(t) = xi(h^{-1} t)
permutes cosets and adds a fixed constant to each offset, so it
preserves every finite cylinder probability and hence mu_a.
Take X = product over a != 1 of X_a with mu = product mu_a. Countability
of H gives metrizability. For any a != 1 and any xi in X, look at the
a-coordinate:
(a xi_a)(1) = xi_a(a^{-1}) = xi_a(a a^{-1} a^{-1} a) ... direct:
xi_a(a^{-1} * 1 * a) is xi_a(a^{-1} 1 a) ... simpler:
(a xi_a)(1) = xi_a(a^{-1}) = xi_a(1) - 1 != xi_a(1).
So a does not fix xi. Hence freeness. If H is trivial, take a
one-point space.
We fix such (X, mu) for the rest of Part II. No amenability of H is
assumed.
9.2. Admissible filtrations.
A triangulation refining K means a finite affine simplicial subdivision
of each closed original simplex, compatible on shared faces.
A family T(x) of such triangulations, together with specified
subcomplexes, is LOCALLY CONSTANT if on each original simplex its exact
affine simplices and subcomplex markings are constant on the pieces of
a finite clopen partition of X. The partition may depend on the
original simplex. We require equivariance: T(hx) = h T(x), and likewise
for every specified subcomplex. Refinement preserves earlier specified
subcomplexes.
An ADMISSIBLE FILTRATION is a sequence
(T_n, Z_n), (T_{n-1}, Z_{n-1}), ..., (T_0, Z_0)
with the following properties.
(F0) Start: T_n = K, Z_n = |K|.
(F1) For each d from n down to 1, T_{d-1} is a locally constant
equivariant refinement of T_d, and Z_{d-1} is a subcomplex of
T_{d-1} contained in Z_d.
(F2) Dimension drop: for every simplex F of T_d contained in
Z_d(x), dim(F cap Z_{d-1}(x)) <= dim F - 1.
(F3) Diameter bound: each connected component of Z_d(x) \ Z_{d-1}(x)
has rho-diameter at most 2R.
Components are with the subspace topology from rho. Note that (F2)
implies dim Z_j(x) <= j for every j, and that Z_0(x) is a
zero-dimensional subcomplex.
9.3. Local replacement.
At a fixed parameter x, to modify a separator Z inside a "ball" of
radius t (r < t < R) around a center c, use
Z' = ( Z \ U_t(c) ) union ( P cap S_t(c) ),
where P is the "piece" that contains Z (in filtrations, P = Z_d).
Interpretation: we remove the part of Z in the open ball U_t(c) and
insert the sphere S_t(c) as the new separator area. The sphere
separates the inside of the ball from its outside, so no new
"long" component can join across the ball boundary. Formally:
LEMMA 9.2 (Local replacement preserves diameter). Suppose every
component of P \ Z has rho-diameter at most 2R. Then every component
of P \ Z' has rho-diameter at most 2R.
PROOF. Let C be a component of P \ Z'. Two cases.
Case 1: C meets U_t(c). Since C is connected and avoids the closed
set Z' (in particular S_t(c) subset Z'), the continuous image f_c(C)
is a connected subset of R missing t. Since C meets U_t(c), f_c(C)
contains a value < t, so f_c(C) subset [0, t). For p, q in C,
rho(p,q) <= rho(p,c) + rho(c,q) = f_c(p) + f_c(q) < 2t < 2R.
Hence diam(C) <= 2t < 2R.
Case 2: C does not meet U_t(c). Since C avoids S_t(c), no point of C
satisfies f_c = t; since no point of C meets U_t(c), and f_c(C) is a
connected subset of [0, infinity) avoiding t, we have f_c(C) subset
(t, infinity). Thus C is contained in the closed half-space {f_c > t}
on which Z' agrees with Z: Z' cap {f_c > t} = Z cap {f_c > t}. So C is
a connected subset of P \ Z, hence contained in a single component of
P \ Z. That component has diameter <= 2R by hypothesis, so
diam(C) <= 2R. QED
LEMMA 9.3 (Compatible cuts). Fix locally constant equivariant
triangulation families T and S with S refining T, subcomplex families
P subset T and Z subset S with Z subset P, and a countable equivariant
family {(c_a, B_a)} of centers c_a in |K| and clopen sets B_a subset X.
Assume only finitely many of the balls L_R(c_a) meet any fixed original
simplex. Set
U_t(x) = union_{a : x in B_a} U_t(c_a),
V_t(x) = union_{a : x in B_a} S_t(c_a). (9.1)
Then outside a countable set of radii t in (r, R), there is a locally
constant equivariant triangulation T' refining S in which both
P(x) cap V_t(x) and
( Z(x) \ U_t(x) ) union ( P(x) cap V_t(x) ) (9.2)
are subcomplexes. Each inserted level V_t(x) meets every face F of
T or S in dimension at most dim F - 1. Consequently, if Z satisfies
the dimension drop relative to T, both sets in (9.2) do also. If
components of P \ Z have diameter at most 2R, the second set preserves
that property.
PROOF. Affine branch finiteness (Lemma 6.5) makes f_c on each original
simplex a maximum of finitely many affine functions. For a fixed face F
and center c, the level {f_c = t} has dimension dim F only when one of
those affine functions is identically t on the affine span of F;
otherwise it is contained in a finite union of proper affine
hyperplanes. Finitely many exceptional t per (face, center). Countably
many faces and centers gives a countable exceptional set. Exclude it.
Explicit refinement: on each original simplex sigma, cut the simplices
of S by all hyperplanes {branch = t} for the finitely many relevant
centers, using the canonical (finite) branch collection from Lemma 6.5.
On each piece of a common finite clopen partition of X, this gives a
finite polytopal complex P_sigma representing the old subcomplexes and
the two sets in (9.2). These polytopal complexes may disagree on shared
faces; take, for each original simplex tau, the common refinement Q_tau
of P_sigma|_tau over all cofaces sigma of tau. Finitely many cofaces
by Lemma 6.6.
Triangulate the original simplices in increasing dimension. When
handling sigma, its boundary has already been triangulated by a
refinement of the boundary restrictions of Q_sigma; extend this
triangulation over Q_sigma in increasing cell dimension by coning
to the centroid of each positive-dimensional cell. This produces a
finite affine subdivision preserving the boundary triangulation.
Every operation is affine-invariant: restriction, common refinement,
and centroid of full vertex set. So the construction is H-equivariant
even for simplex stabilizers. Local constancy: on each original simplex
the procedure uses only finitely many data (those on faces and original
cofaces plus finitely many relevant cuts); a common refinement of the
finitely many clopen parameter partitions makes them all constant.
Containment of the first set in (9.2) in the level V_t(x) gives the
dimension drop directly. The second set is (something of dimension
<= dim F - 1) union Z, so it also drops. Preserving small-diameter
components: as in the proof of Lemma 9.2, any new component inside a
ball is contained in that ball and so has diameter <= 2t < 2R. Any
component outside all active balls was already present inside P \ Z. QED
LEMMA 9.4 (Filtration existence). Every admissible choice can be
extended to a full filtration terminating at Z_0.
PROOF. Each original simplex is compact in rho. Cover finitely many
representatives of the finitely many original simplex orbits by
finitely many open balls U_r(c); take all H-translates. These balls
cover |K|. Lemma 6.6 says only finitely many translated radius-R balls
meet a given original simplex. Apply Lemma 9.3 with every activation
set = X, T = S = T_d, Z = empty. For a common generic t in (r, R), set
Z_{d-1}(x) = Z_d(x) cap big-union_c S_t(c).
This is a subcomplex of the refinement, with the required dimension
drop. If C is a component of Z_d \ Z_{d-1}, pick p in C and a covering
ball U_r(c) with p in U_r(c). Then f_c(C) avoids t and meets [0, r)
(the inside of the ball), so f_c(C) subset [0, t) and diam(C) <= 2t < 2R.
QED
9.4. Dimension drop along the whole filtration.
For any simplex F of T_i(x) contained in Z_i(x) and any 0 <= j <= i,
dim( F cap Z_j(x) ) <= dim F - (i - j). (9.3)
(iterated dimension drop)
Indeed, the first step is the definition (F2). Its intersection is a
subcomplex of the next refinement (which contains Z_{j} inside Z_i);
applying (F2) again to a simplex of that subcomplex drops dimension by
1 per stage. Induction gives (9.3).
Applied with i = n and j = 0, this shows that a proper original face
meets Z_0(x) in dimension strictly less than 0, that is, it MISSES
Z_0(x).
LEMMA 9.5 (Carriers of the zero stratum). Let (T_i, Z_i)_{i=0}^n be an
admissible filtration and z in Z_0(x). For each i there is a UNIQUE
i-simplex S_i of T_i(x) with S_i subset Z_i(x) and z in S_i. The
point z lies in the relative interior of S_i. In particular, every
point of Z_0(x) lies in the interior of some original top simplex.
PROOF. Let F be the unique simplex of T_i(x) whose relative interior
contains z. Since Z_i(x) is a subcomplex containing z, it contains F,
and so dim F <= i. If dim F < i, then (9.3) with j = 0 would make
F cap Z_0(x) empty, contradicting z in F cap Z_0. Hence dim F = i.
Every other simplex containing z has F as a face; no other i-simplex
contains it. Taking i = n: z lies in some n-simplex of T_n(x) = K, and
since z is in the relative interior of that simplex (by the same
construction), z is in the interior of an original top simplex. QED
SECTION 10. LOCAL REPLACEMENT AND THE COUNT OF ZERO STRATA.
We now choose the filtration so its zero-dimensional set has small
expected cardinality, on average over x in X. Throughout this section
the graph, complex, parameter space, dimension and radii are fixed as
in Sections 6-9. In particular the strict margin (8.3) holds. Only
the filtration depends on a positive parameter epsilon.
PROPOSITION 10.1 (Expected zero strata). For every epsilon > 0 there
is an admissible equivariant locally constant filtration
(Z_j, T_j)_{j=0}^n such that, for every original top simplex sigma of K,
int_X #( Z_0(x) cap sigma ) dmu(x) <= C_sigma * epsilon. (10.1)
Here C_sigma depends only on fixed geometric data (D, K, sigma, etc.),
not on epsilon nor on the complexity of the chosen triangulations.
10.1. Normalized area and near-minimizers.
For an equivariant family Z of dimension <= j, define
I_j(Z) = sum over sigma in (top simplices of K / H) of
(1 / |H_sigma|) *
int_X a_j^sigma( Z(x) cap sigma^o ) dmu(x). (10.2)
The sum runs over H-orbit representatives. H_sigma is the finite setwise
stabilizer of sigma. The quantity is finite: finitely many representatives,
local constancy and compactness of X give finitely many polyhedral
configurations, each of finite j-area.
Start with (Z_n, T_n) = (|K|, K). At step d = n, ..., 1: keep all
previously chosen data fixed, and choose the admissible pair
(Z_{d-1}, T_{d-1}) such that I_{d-1}(Z_{d-1}) is within epsilon of the
infimum over all admissible choices at this step. Lemma 9.4 ensures the
admissible class is nonempty; its infimum is a finite nonnegative number,
so a choice exists with the desired near-minimality. Consequently, for
every admissible replacement Z' at this step,
I_{d-1}(Z_{d-1}) <= I_{d-1}(Z') + epsilon. (10.3)
No attained minimum or limiting filtration is used.
10.2. Disjoint replacements in the parameter.
Fix a center c in |K|. The set (6.8):
F_c = { h in H \ {1} : L_R(hc) cap L_R(c) is nonempty }
is finite. There is a finite clopen partition of X into sets A with
A cap hA empty for every h in F_c:
-- use H-freeness and zero-dimensionality to choose, around each x,
a clopen neighborhood disjoint from its translates by elements of
the finite set F_c;
-- a finite subcover exists by compactness;
-- take atoms of the common refinement.
Result:
{ hA times L_R(hc) : h in H } is a pairwise disjoint family in X x |K|.
(10.4)
The sets hA alone need not be disjoint; disjointness is only asserted
inside the product space.
10.3. Local comparison.
Fix one atom A and any clopen B subset A. For r <= s <= R put
M_j(s) = int_B sum over top sigma of
a_j^sigma( Z_j(x) cap sigma^o cap L_s(c) ) dmu(x), (10.5)
and for generic t in (r, R) also
N_{d-1}(t) = int_B sum over top sigma of
a_{d-1}^sigma( Z_d(x) cap sigma^o cap S_t(c) ) dmu(x).
At stage d, write P = Z_d and Z = Z_{d-1}. For x in B, make the
simultaneous replacement
Z'(x) = ( Z(x) \ union_{h : x in hB} U_t(hc) )
union ( P(x) cap union_{h : x in hB} S_t(hc) ). (10.6)
The active closed balls are pairwise disjoint by (10.4). The families
of balls and levels are locally finite by Lemma 6.6; the activation sets
hB are clopen; so the construction is equivariant and locally constant
on each original simplex. Apply Lemma 9.3 with T = T_d, S = T_{d-1},
P = Z_d, Z = Z_{d-1}, and centers/activations (hc, hB). Outside a
countable set of levels, Z' is represented by an admissible common
refinement, and by Lemma 9.2 the diameter bound is preserved.
Compute the change in I_{d-1}. Genericity against the faces of T_{d-1}
implies that Z (the old separator) meets any inserted level S_t(hc) in
dimension <= d - 2 (or is empty when d = 1). So:
(i) The removed (d-1)-area is unchanged by replacing U_t by L_t.
(ii) The inserted level has zero (d-1)-area of overlap with the
retained old set Z.
Orbit-counting: let Q denote the relevant equivariant set and W(c) the
ball or the level. Covariance of area and invariance of mu give
sum over sigma (top mod H) (1/|H_sigma|)
sum over h in H int_{hB} a_j^sigma( Q(x) cap sigma^o cap W(hc) )
= sum over sigma (top mod H) (1/|H_sigma|)
sum over h in H int_{B} a_j^{h^{-1} sigma}(
Q(x) cap (h^{-1} sigma)^o cap W(c) )
= int_{B} sum over tau (top) a_j^tau( Q(x) cap tau^o cap W(c) ).
The middle equality uses invariance of mu under H; the map h |-> h^{-1} sigma
takes each orbit value |H_sigma| times.
All local contributions are nonnegative, and (10.4) plus the zero-overlap
statements above give
I_{d-1}(Z') - I_{d-1}(Z) = N_{d-1}(t) - M_{d-1}(t). (10.7)
Combining (10.3) and (10.7) gives, for a.e. t in (r, R),
M_{d-1}(t) <= N_{d-1}(t) + epsilon. (10.8)
10.4. Slicing and iteration.
Coarea identity. Work in one original top simplex sigma, in one of its
tree metrics. Decompose Z_d(x) cap sigma^o into affine polyhedral pieces
on which f_c = rho(c, .) is affine. For a d-dimensional piece Q, the
tangential gradient has norm alpha <= 1 by Lemma 7.1(b). Euclidean
slicing perpendicular to that gradient gives
int_r^b vol_{d-1}( Q cap {f_c = t} ) dt
= alpha vol_d( Q cap {r < f_c < b} )
<= vol_d( Q cap L_b(c) ). (10.9)
A constant branch contributes at only one level. Pieces of dimension
< d have zero (d-1)-slice area outside finitely many exceptional levels
(exclude a countable set of t). Averaging over tree metrics, summing
over original top interiors, and integrating over B (as in 10.3):
int_r^b N_{d-1}(t) dt <= M_d(b), r <= b <= R. (10.10)
(These operations are legitimate by local finiteness and local
constancy; on the fixed ball only finitely many configurations occur.
Every comparison has been inside a single simplex metric before
averaging, so no compatibility of metrics across faces is needed.
Inequality (9.3) bounds Z_d on a proper original face by dimension
d - 1, and its generic level slice by dimension d - 2: such faces do
not contribute to the preceding comparison. The same estimate gives
Z_0 missing every proper original face.)
Combining (10.8) and (10.10):
M_d(b) >= int_r^b M_{d-1}(t) dt - epsilon (b - r). (10.11)
Iterate. Set L = R - r, A_n = L^n / n!, E_n = sum_{i=1}^n L^i / i!.
Claim (by induction starting at d = 0, where M_0 is monotone):
M_d(b) >= ( (b-r)^d / d! ) M_0(r)
- epsilon sum_{i=1}^d (b-r)^i / i!. (10.12)
At d = n, b = R, (b - r) = L, so
M_n(R) >= A_n M_0(r) - epsilon E_n. (10.13)
Also Z_n = |K| so M_n(R) = int_B sum_sigma a_n^sigma(Z_n cap sigma^o cap L_R(c)).
This is bounded by V_R * mu(B) by Lemma 8.2. Hence
A_n M_0(r) <= V_R mu(B) + epsilon E_n. (10.14)
Now choose
B = A cap { x in X : Z_0(x) cap L_r(c) is nonempty }. (10.15)
This is clopen: finitely many original simplices meet the ball, and their
exact filtration data are constant on a common finite clopen partition.
The choice uses the completed filtration Z_0; this is legitimate, because
the near-minimality inequality (10.3) is valid for every admissible
competitor with data down to stage d fixed, regardless of later stages.
So (10.14) can use this B.
Each x in B contributes at least one zero-stratum point (a point of
Z_0(x) inside L_r(c)), and each zero-stratum point lies in a unique
original top interior (Lemma 9.5), so mu(B) <= M_0(r), and actually
M_0(r) is the integral of the point count over A. Combined with
A_n > V_R (the strict margin (8.3)):
int_A #( Z_0(x) cap L_r(c) ) dmu(x)
= M_0(r)
<= E_n / (A_n - V_R) * epsilon. (10.16)
Cover a fixed original top simplex sigma by finitely many open
r-balls with fixed centers c_1, ..., c_q (compactness of sigma). For
each center use its finite clopen partition, with p_i atoms. Sum
(10.16) over all atoms of each partition and over all balls. Since the
corresponding closed balls also cover sigma,
int_X #( Z_0(x) cap sigma ) dmu(x)
<= ( E_n / (A_n - V_R) ) * sum_{i=1}^q p_i * epsilon. (10.17)
Take C_sigma = ( E_n / (A_n - V_R) ) * sum_{i=1}^q p_i. This is fixed
independently of epsilon. No bound on the number of subdivision simplices
or on the cardinality of an individual zero-dimensional fiber is needed.
QED (Proposition 10.1)
SECTION 11. FROM FILTRATIONS TO CHAIN MAPS.
Fix the dimension n and radii r < R. For each epsilon > 0, use a
filtration from Proposition 10.1. Our goal: turn it into an
H-equivariant augmented chain map
F^epsilon : C_*(K) -> E_*
that is equivariantly chain homotopic to inclusion, and whose
evaluation by any fixed invariant cocycle tends to zero with epsilon.
These two properties, plus Proposition 10.1, prove Theorem 5.1.
Throughout this section n is fixed; only epsilon tends to zero.
11.1. Component averages and finite support.
At fixed parameter x, take the barycentric subdivision of the final
triangulation T_0(x). Its vertices are barycenters b_F of nonempty
simplices F of T_0(x).
For a nonempty F, define
ell(F) = min { i : b_F in Z_i(x) }, Z_{-1}(x) = empty. (11.1)
All Z_i(x) are subcomplexes of T_0(x), so b_F in Z_i(x) iff
F subseteq Z_i(x). Define U_F as the connected component of
Z_{ell(F)}(x) \ Z_{ell(F) - 1}(x) containing b_F, and put
J_U = union_{p in U} supp p,
u_U = (1 / |J_U|) sum_{v in J_U} delta_v. (11.2)
Lemma 11.1 later shows J_U is finite.
The intended image of an oriented barycentric simplex is
[b_{F_0}, ..., b_{F_j}] |-> u_{U_{F_0}} and ... and u_{U_{F_j}} in E_j.
(11.3)
Well-definedness. Suppose F subseteq F' are nested faces at the same
level i. The segment joining b_F to b_{F'} lies inside Z_i \ Z_{i-1}
(except at its endpoint b_F): the open segment lies in the relative
interior of F', which Z_{i-1} does not meet at level i, and neither
endpoint lies in Z_{i-1} by level. So b_F and b_{F'} receive the SAME
component vector u_U; any exterior term containing both factors vanishes.
Along a complete flag F_0 subset ... subset F_n with dim F_i = i, the
levels are nondecreasing: if F subseteq F' subseteq Z_i, then
F subseteq Z_i. So a nonzero image must use all levels 0, 1, ..., n,
starting from a point of Z_0. This is the flag structure counted in S12.
LEMMA 11.1 (Finite component labels). For every original simplex kappa
of K there is a finite W_kappa subseteq Y, independent of epsilon and
x, with J_{U_F} subseteq W_kappa whenever b_F in kappa. The
triangulation over kappa, and the vectors u_{U_F}, are locally constant
in x and H-equivariant.
PROOF. Each component of Z_i(x) \ Z_{i-1}(x) has rho-diameter <= 2R by
(F3). For i = 0 the components are singletons because Z_0(x) is a
locally finite zero-dimensional subcomplex. If b_F in kappa and p in
U_F, the support-distance bound (6.6) gives
min d_Y( supp b_F, supp p ) <= sqrt(n) rho(b_F, p) <= 2R sqrt(n).
Choose an original top simplex containing p; its n+1 vertices span a
connected graph, so every vertex in supp p is within graph distance n of
the vertex supplied by the inequality. Therefore
W_kappa = { v in Y : d_Y(v, V(kappa)) <= ceil(2R sqrt(n) + n) } (11.4)
suffices. It is finite because Y has degree <= D.
Let K[W_kappa] be the finite subcomplex of K consisting of simplices
whose vertices lie in W_kappa. The whole component U_F lies in its
realization, so U_F is exactly the component containing b_F in
( Z_{ell(F)}(x) \ Z_{ell(F)-1}(x) ) cap |K[W_kappa]|.
Restricting to a subcomplex containing a connected set entirely does not
disconnect it.
All data on K[W_kappa] is determined by finitely many parameters: the
exact filtration data of finitely many original simplices. A common
finite clopen partition of X makes these data constant on each piece,
hence the components and their J_U's constant. Local constancy follows.
Equivariance: J_{hU} = h J_U, hence u_{hU} = h u_U. QED
11.2. The chain map f_x.
Subdivide each oriented original simplex into its fundamental chain of
oriented barycentric simplices, and apply (11.3) to each term. This
defines f_x : C_*(K) -> E_*. The substitution commutes with partial:
(subdivision) interior facets cancel pairwise in adjacent barycentric
terms; boundary facets give the restriction of the
subdivision to faces of the simplex;
(exterior) each u_U has coefficient sum one, so the coefficient-
sum functional contracts to give the alternating
boundary
partial(u_0 and ... and u_j)
= sum_i (-1)^i u_0 and ... hat{u_i} ... and u_j.
So f_x is an augmented chain map.
11.3. Averaging over x.
By Lemma 11.1, f_x on any fixed original simplex is constant on a finite
clopen partition of X, and lies in a fixed finite exterior subspace
(spanned by W_kappa-vertices). We may therefore define the average
F^epsilon(c) = int_X f_x(c) dmu(x). (11.5)
This is an actual finite chain, not an abstract limit: for fixed c (a
finite chain), f_x(c) takes finitely many distinct values, and the
integral is a finite linear combination.
F^epsilon remains an augmented chain map because each f_x is one and mu
is a probability. Equivariance: f_{hx}(hc) = h f_x(c) and mu is
H-invariant, so F^epsilon(hc) = h F^epsilon(c) for every h in H. Thus
F^epsilon is H-equivariant.
Both iota (inclusion) and F^epsilon induce the identity in augmented
degree -1 (the empty simplex is 1). So they are H-equivariantly chain
homotopic (Lemma 13.1).
SECTION 12. COUNTING SURVIVING FLAGS.
We now bound the evaluation of f_x on a top simplex. The number of
subdivision simplices may depend on epsilon; the following bound
depends instead on the number of zero-stratum points. The component-
labeling and flag-counting technique is modeled on Alpert's Lemmas 4
and 10; the carrier argument below removes any manifold hypothesis on
K or on the strata.
LEMMA 12.1 (Surviving flags). For every original top simplex sigma of K
and every x, at most
2^n * #( Z_0(x) cap sigma )
terms in the fundamental barycentric subdivision chain of sigma have
nonzero images under (11.3).
PROOF. Suppress x. A surviving term corresponds to a complete flag
F_0 subset F_1 subset ... subset F_n, dim F_i = i,
in the restriction of T_0 to sigma. As shown when defining (11.3), a
nonzero image has levels exactly 0, 1, ..., n. In particular F_0 = {z}
for some z in Z_0(x) cap sigma, and F_i subseteq Z_i(x) for every i.
By Lemma 9.5 (carriers), z lies in the relative interior of a unique
i-simplex S_i of T_i(x) contained in Z_i(x). The T_i-carrier of a
relative-interior point of F_i contains F_i by refinement and belongs to
Z_i (Z_i is a subcomplex). That carrier has dimension >= i (it contains
F_i) and <= i (since dim Z_i <= i). It contains z, so it equals S_i.
Hence F_i subseteq S_i.
For a fixed preceding face F_{i-1}, the possible F_i are full-dimensional
simplices in a triangulation of the single affine i-simplex S_i, with
F_{i-1} as a facet. There are at most two such simplices: at most one
on either side of the affine hyperplane spanned by F_{i-1}. (Two on the
same side would have intersecting relative interiors near an interior
point of the facet, which a triangulation forbids.)
So at most 2 choices at each stage, hence at most 2^n surviving flags
through z. Summing over z in Z_0 cap sigma proves the assertion. QED
12.1. Cocycle evaluation.
Let psi : E_n -> C be the H-invariant cocycle of Theorem 5.1. For an
original top simplex sigma, set
M_sigma = max_{v_0, ..., v_n in W_sigma}
| psi( delta_{v_0} and ... and delta_{v_n} ) |. (12.1)
This is finite (W_sigma finite by Lemma 11.1), even though psi need not
be bounded on all of E_n. Every u_U in (11.2) is a nonnegative convex
combination of vertices in W_sigma, so multilinearity bounds the
evaluation of each surviving term by M_sigma.
By Lemma 12.1 and Proposition 10.1:
| psi(F^epsilon(sigma)) |
<= 2^n M_sigma * int_X #( Z_0(x) cap sigma ) dmu(x)
<= 2^n M_sigma C_sigma * epsilon. (12.2)
The right side tends to 0 as epsilon -> 0. The constants M_sigma and
C_sigma are independent of epsilon. Since chains in C_n(K) have finite
support, the same convergence holds for any fixed chain c in C_n(K):
| psi(F^epsilon(c)) | -> 0 as epsilon -> 0. (12.3)
SECTION 13. EQUIVARIANT COMPARISON AND PASSAGE TO COINVARIANTS.
We must show that the small evaluations of Section 12 equal psi(iota(c))
for the ORIGINAL chain c, on any coinvariant cycle. Build a homotopy
before taking coinvariants.
LEMMA 13.1 (Equivariant homotopy). Let K be a simplicial complex with
an H-action and finite setwise simplex stabilizers. Let E_* be the
augmented exterior complex on a nonempty H-set. Any two H-equivariant
augmented chain maps A, B : C_*(K) -> E_* that agree in degree -1 are
H-equivariantly chain homotopic.
PROOF. E_* is acyclic: for any fixed vertex v, the operator
a |-> delta_v and a satisfies
partial(delta_v and a) + delta_v and partial(a) = a. (13.1)
This contraction need not be equivariant. We use it only to CHOOSE
fillers for one chain at a time; equivariance is then imposed by finite
averaging over stabilizers.
Let h_{-1} = 0. Suppose h is constructed in degrees < j. For an oriented
j-simplex e of K, the residual
r(e) = A(e) - B(e) - h_{j-1}(partial e) (13.2)
is a cycle: partial r(e) = 0 because A and B commute with partial and
h_{j-1} satisfies A - B = partial h + h partial in degree j-1.
Choose a finite chain t in E_j with partial t = r(e) (acyclicity). Let
S be the setwise stabilizer of e and chi : S -> {+1, -1} its orientation
character: chi(s) e = s e. The residual is equivariant on orientation:
s r(e) = chi(s) r(e). Define
bar-t = (1 / |S|) sum_{s in S} chi(s) (s t). (13.3)
Then partial bar-t = r(e) and s bar-t = chi(s) bar-t for every s in S.
Choose such a filler once per oriented simplex orbit and extend
equivariantly and linearly: put h_j(e) = bar-t. The orientation-character
identity makes the extension well defined, including when the stabilizer
reverses orientation. This gives A - B = partial h + h partial in degree
j and completes the induction. Finiteness of S follows from Lemma
6.6(d). QED
PROOF OF THEOREM 5.1. Fix n >= n_0(D). For nonempty K, construct
F^epsilon as in Section 11, and let iota be the inclusion
C_*(K) -> E_*.
Both iota and F^epsilon induce the identity in augmented degree -1, so
Lemma 13.1 supplies an H-equivariant homotopy
iota - F^epsilon = partial h + h partial. (13.4)
Let c-bar in C_n(K)_H be a cycle, and choose a finite chain lift c in
C_n(K). We do NOT assume c itself is a cycle: only its image in
C_{n-1}(K)_H vanishes, i.e. partial c lies in the C-span of h' w - w
for h' in H and w in C_{n-1}(K).
Evaluate (13.4) on c and apply psi:
psi( iota(c) ) - psi( F^epsilon(c) )
= psi( partial h_n(c) ) + psi( h_{n-1}(partial c) ).
The first term vanishes because psi is a cocycle. The second vanishes
because partial c lies in the span of h'w - w, and h_{n-1} preserves
that span (it is equivariant), and psi(h' w - w) = psi(w) - psi(w) = 0
for every h', w by H-invariance of psi. Therefore
psi( iota(c) ) = psi( F^epsilon(c) ) (13.5)
for every epsilon > 0. The left side is independent of epsilon. The
right side tends to 0 by (12.3). So psi(iota(c)) = 0.
Finally, iota(c) and c represent the same class in C_n(K)_H, and psi
descends to coinvariants because it is H-invariant. Hence
psi(c-bar) = psi(iota(c)) = 0. In particular, psi agrees with its
restriction to any coinvariant n-cycle and vanishes there.
For empty K there is no nonzero top chain to consider. QED (Theorem 5.1)
================================================================
PART III. CYCLES FROM AN IDEMPOTENT
================================================================
SECTION 14. SUBGROUP LOCALITY.
The main construction is simplest for finitely generated groups. This
section reduces arbitrary groups to that case.
LEMMA 14.1 (Subgroup locality). Let i: L -> G be an inclusion of
discrete groups. Extension of scalars gives i_*: K_0(C[L]) -> K_0(C[G]).
For every y in K_0(C[L]) and every C in con(G),
HS_G(i_* y)(C) = sum_{B in con(L), B subseteq L cap C} HS_L(y)(B).
(14.1)
Only finitely many terms are nonzero. Moreover, every x in K_0(C[G]) is
i_* x for some finitely generated subgroup L subseteq G that we may
require to contain any prescribed finite subset of G.
PROOF. Regard free right modules as columns: e in M_r(C[L]) acts on the
left with image e(C[L])^r. Tensoring the split decomposition defined by
e with C[G] identifies the extended image with e(C[G])^r. The trace
element tr_r(e) is unchanged as an element of C[L] subseteq C[G].
Therefore, for every C in con(G):
HS_G(i_* [e(C[L])^r])(C)
= sum_{h in C} (tr_r e)_h (by S3)
= sum_{h in L cap C} (tr_r e)_h (supp e subset L)
= sum_{B in con(L), B subseteq L cap C} sum_{h in B} (tr_r e)_h
= sum_{B in con(L), B subseteq L cap C} HS_L([e(C[L])^r])(B).
The identity (14.1) follows by additivity on classes [P]-[Q]. Finiteness:
(tr_r e)_h is nonzero only for h in the finite set supp e.
For the induction assertion: write x in K_0(C[G]) as [P] - [Q] with P,
Q represented by idempotents e_P, e_Q of sizes r, s. The group generated
by the supports of e_P, e_Q and the prescribed finite subset of G is
finitely generated and yields the required L.
Order is preserved under subgroup inclusion, so if C consists of
infinite-order elements, so does every subgroup class B subseteq L cap C
appearing on the right of (14.1). Hence: if Theorem A holds for all
finitely generated groups, it holds for all groups. QED
SECTION 15. A FIXED GRAPH AND FINITE PATH SUMS.
Fix for this section and next:
G finitely generated, e in M_r(C[G]) idempotent, g in G of
infinite order. Write
e = sum_{s in G} A_s s, A_s in M_r(C), (15.1)
and let S_0 = { s in G : A_s != 0 }, a finite set. Idempotence is
sum_{uv = t} A_u A_v = A_t for every t in G. (15.2)
Define the matrix-valued kernel
a(h, h') = A_{h^{-1} h'}. (15.3)
Then for every t in G,
sum_{k in G} a(h,k) a(k,h') = sum_{uv = h^{-1} h'} A_u A_v = a(h, h').
The sum is finite: only u, v in S_0 contribute.
Coset space and centralizer:
J = C_G(g) = { s in G : sg = gs },
H = J / ,
Y = \ G, h-bar = h. (15.4)
H acts on Y by (i, h-bar) |-> (ih)-bar for i in J. Well-defined: J
centralizes g, so i = g^q gives (g^q h)-bar = h-bar. Free: i h-bar = h-bar
means h^{-1} i h in ; combined with i in J, this forces i in , so
the stabilizer of any vertex is trivial.
Choose a finite symmetric generating set S subseteq G \ {1} containing
S_0 \ {1} and its inverses. Put the graph structure on Y: edges join
h-bar to (hs)-bar for s in S; loops are dropped and repeated edges
identified. The graph is connected (finite generation) and has degree
D = max(2, |S|). (15.5)
H acts by graph automorphisms (H = J / preserves right S-steps).
For k >= 1, set Q_k = J \ G^k (simultaneous left J-action on tuples).
Define the path weight
w_k(h_0, ..., h_{k-1})
= tr( a(h_0,h_1) a(h_1,h_2) ... a(h_{k-2},h_{k-1}) a(h_{k-1}, g h_0) ).
(15.6)
These weights are J-invariant: a(h, h') = A_{h^{-1} h'} depends only on
h^{-1} h', and J centralizes g. The w_k give a finite C-linear
combination v_k in C^{(Q_k)}. Indeed, a nonzero-weight tuple has
increments
h_i^{-1} h_{i+1} = s_i in S_0 (i = 0, ..., k-2),
h_{k-1}^{-1} g h_0 = s_{k-1} in S_0,
with product s_0 s_1 ... s_{k-1} = h_0^{-1} g h_0. The tuple is
determined by the increment list (s_0, ..., s_{k-1}) and one choice of
h_0. Any two such h_0 with the same conjugate differ by left
multiplication by an element of J. So the increment list determines at
most one class in Q_k. Hence v_k has at most |S_0|^k nonzero coefficients.
For a J-invariant function p on G^k, write v_k(p) for the finite
weighted sum sum_t v_k(t) p(t). These are complex expectations under a
not-necessarily-positive weight.
LEMMA 15.1 (Path identities).
(a) Twisted cyclic shift. Define
T: Q_k -> Q_k, T(h_0, ..., h_{k-1}) = (h_1, ..., h_{k-1}, g h_0).
Then T_* v_k = v_k.
(b) Deletion. For k >= 2 and 0 <= i < k, the i-th projection
d_i: Q_k -> Q_{k-1} satisfies (d_i)_* v_k = v_{k-1}.
(c) Total weight. All v_k have the same total weight
lambda := v_k(1) = sum_{s in [g]_G} tr(A_s)
= HS_G([e C[G]^r])([g]_G). (15.7)
PROOF. (a) T = (shift followed by right multiplication by g on the last
entry) is a J-equivariant bijection, with inverse
T^{-1}(h_0, ..., h_{k-1}) = (g^{-1} h_{k-1}, h_0, ..., h_{k-2}).
For k = 1, T is the identity on Q_1 because g in J. For k >= 2, the
shifted weight has the factors of (15.6) in cyclic order: this uses
a(gh_0, gh_1) = a(h_0, h_1), which holds because a depends only on
h^{-1} h'. Cyclicity of matrix trace then gives T_* v_k = v_k.
(b) Fix a representative (h_1, ..., h_{k-1}) of the shorter tuple. Its
preimage in Q_k consists of J-orbits of liftings (h_0, h_1, ..., h_{k-1})
parameterized by h_0 in G. J acts freely on any nonempty tuple (its
diagonal action on G^k is free), so each preimage J-orbit has a unique
representative with exactly the retained entries. The fibre sum
therefore runs over h_0 in G with no stabilizer factor.
Internal deletions (delete h_{j} for 1 <= j <= k-2). Group the sum as
sum_{h_j in G} tr( B_1 a(h_{j-1}, h_j) a(h_j, h_{j+1}) B_2 )
= tr( B_1 a(h_{j-1}, h_{j+1}) B_2 )
by (15.3) and the kernel idempotence sum_u a(h_{j-1}, u) a(u, h_{j+1})
= a(h_{j-1}, h_{j+1}). This is exactly the contribution of the shorter
tuple (h_0, ..., hat{h_j}, ..., h_{k-1}) to v_{k-1}.
Last deletion (delete h_{k-1}). Fix (h_0, ..., h_{k-2}). Put
B = a(h_0, h_1) ... a(h_{k-3}, h_{k-2}) (with B = I_r when k = 2). Then
sum_{h_{k-1}} tr( B a(h_{k-2}, h_{k-1}) a(h_{k-1}, g h_0) )
= tr( B a(h_{k-2}, g h_0) )
by kernel idempotence with t = h_{k-2}^{-1} g h_0, again a finite sum.
This equals the (h_0, ..., h_{k-2}) entry of v_{k-1}.
First deletion (delete h_0). Fix (h_1, ..., h_{k-1}). Put
B = a(h_1, h_2) ... a(h_{k-2}, h_{k-1}) (with B = I_r when k = 2). Then
sum_{h_0} tr( a(h_0, h_1) B a(h_{k-1}, g h_0) )
= sum_{h_0} tr( B a(h_{k-1}, g h_0) a(gh_0, gh_1) )
= sum_{h_0} tr( B a(h_{k-1}, g h_0) a(g h_0, g h_1) )
= tr( B a(h_{k-1}, g h_1) )
= tr( a(h_1, h_2) ... a(h_{k-2}, h_{k-1}) a(h_{k-1}, g h_1) ),
where the first equality uses trace cyclicity and left invariance of a
(a(gh_0, gh_1) = a(h_0, h_1)), and the second uses kernel idempotence
sum_u a(h_{k-1}, u) a(u, g h_1) = a(h_{k-1}, g h_1) with u = g h_0.
The last expression is v_{k-1}(h_1, ..., h_{k-1}). This also proves
(b) for k = 2.
(c) Deletion identities reduce v_k(1) inductively to v_1(1). Compute:
v_1(1) = sum_{h_0 in J \ G} tr( a(h_0, g h_0) )
= sum_{h_0 in J \ G} tr( A_{h_0^{-1} g h_0} ).
The map J h |-> h^{-1} g h is a bijection J \ G -> [g]_G (conjugate
correspondence). So v_1(1) = sum_{s in [g]_G} tr(A_s), which by (S3.3)
equals HS_G([e C[G]^r])([g]_G). QED
SECTION 16. EVEN-DIMENSIONAL CYCLES.
Extend the path sums to the exterior complex. For k >= 1, define
pi_k : C^{(Q_k)} -> (E_{k-1})_H,
pi_k([h_0, ..., h_{k-1}]) = [ delta_{h_0-bar} and ... and delta_{h_{k-1}-bar} ].
(16.1)
Well-defined: changing a tuple by i in J changes the projected vertices
by simultaneous H-translation, so the exterior vector changes by the
H-action. The bracket [ . ] means class in coinvariants.
Set
b_k = pi_k(v_k) in (E_{k-1})_H (k >= 1),
z_n = b_{n+1} in (E_n)_H (n = 2m >= 2). (16.2)
PROPOSITION 16.1 (Path cycle). For every positive even n, z_n is a
cycle in (E_n)_H. Moreover, either z_n = 0 (all nonzero tuple terms have
repeated projected vertices), or z_n is the image of a cycle in
C_n(K)_H for an H-invariant subcomplex K of the full simplex on Y whose
n-simplices span connected subgraphs of Y and have finitely many
H-orbits. The graph Y and its degree bound D are independent of n.
PROOF. Step 1: Twist-equivariance of pi. Since (gh_0)-bar = h_0-bar, T
permutes the k wedge factors of pi_k cyclically. The exterior product is
alternating in its k factors, so the cyclic permutation multiplies by
(-1)^{k-1}:
pi_k T = (-1)^{k-1} pi_k. (16.3)
Step 2: Boundary formula. For k >= 2, applying the augmented exterior
differential to pi_k(h_0, ..., h_{k-1}) gives
sum_{i=0}^{k-1} (-1)^i
[ delta_{h_0-bar} and ... hat{i} ... and delta_{h_{k-1}-bar} ],
which is
sum_{i=0}^{k-1} (-1)^i pi_{k-1}( d_i(h_0, ..., h_{k-1}) ).
Applying v_k with deletion identity (Lemma 15.1(b)):
partial b_k = sum_{i=0}^{k-1} (-1)^i b_{k-1}. (16.4)
Step 3: b_k = 0 for even k. If k is even, then k - 1 is odd, so
(-1)^{k-1} = -1 and (16.3) gives pi_k T = -pi_k. Combined with
T_* v_k = v_k:
b_k = pi_k(v_k) = pi_k(T_* v_k) = (pi_k T)_* v_k = -pi_k(v_k) = -b_k,
so b_k = 0.
Step 4: vanishing boundary for even n. Take n = 2m >= 2. Then k = n+1
is odd >= 3, so (16.4) applies and the alternating sum has exactly one
term more of + sign than of - sign: sum_{i=0}^{n} (-1)^i = 1 (since n
is even). So
partial z_n = partial b_{n+1} = b_n.
But n is even, so b_n = 0 by Step 3. Hence partial z_n = 0 in
(E_n)_H.
Step 5: Localization. Suppose z_n != 0. For every tuple t in the support
of v_{n+1} with a nonzero coefficient and distinct projected vertices,
consecutive projected vertices h_i-bar and h_{i+1}-bar are graph-adjacent
because h_i^{-1} h_{i+1} = s_i in S_0 subseteq S. Hence the subgraph of
Y spanned by {h_0-bar, ..., h_n-bar} is connected. Let K be the
H-invariant subcomplex of the full simplex on Y generated by the vertex
sets of these surviving tuples and their H-translates. Finite support of
v_{n+1} gives finitely many H-orbits of top simplices and faces.
The exterior chain z_n is the image of a chain z_n~ in C_n(K)_H (with
C_*(K) the oriented chains of K, identified with a subcomplex of E_*).
Step 6: Zero boundary in C_n(K)_H. The exterior boundary partial z_n~ in
E_{n-1} is zero (Step 4). We must check the same zero holds before
passing to coinvariants.
Degree-by-degree decomposition. For each j, let W_j subseteq E_j be the
linear span of oriented j-simplices not in K. Then
E_j = C_j(K) (+) W_j (direct sum as vector spaces).
Both summands are H-invariant (the generating set of K is H-invariant,
so its complement is too). Hence
(E_j)_H = C_j(K)_H (+) (W_j)_H.
In particular the induced map C_j(K)_H -> (E_j)_H is injective.
Applying with j = n - 1: the boundary of z_n~ lies in C_{n-1}(K)_H and
maps to partial z_n = 0 in (E_{n-1})_H. By injectivity, partial z_n~ = 0
already in C_{n-1}(K)_H. So z_n~ is a cycle in C_n(K)_H representing z_n.
Conclusion. For every even n, the chain z_n is a cycle in (E_n)_H; if
nonzero, it lies in the image of a cycle in C_n(K)_H for a K satisfying
Theorem 5.1's hypotheses with the degree bound D independent of n. QED
16.1. The trace coefficient lives in the total weight.
Two facts we have established and will use in Part IV:
(i) v_k(1) = lambda for every k >= 1 (Lemma 15.1(c)).
(ii) z_{2m} is a cycle in a subcomplex K to which Theorem 5.1 applies.
Part IV constructs an H-invariant cocycle psi_m : E_{2m} -> C with
psi_m(z_{2m}) = 2^{-m} * lambda.
Theorem 5.1 then forces 2^{-m} lambda = 0, hence lambda = 0.
================================================================
PART IV. THE PFAFFIAN COCYCLE
================================================================
SECTION 17. RELATIVE EXPONENTS BETWEEN LIFTS.
Retain G finitely generated, e in M_r(C[G]) idempotent, g infinite-order,
and notation J, H, Y, v_k, lambda, z_{2m} from Sections 15-16. Goal:
an H-invariant cocycle psi_m : E_{2m} -> C with
psi_m(z_{2m}) = 2^{-m} lambda.
The construction uses two steps. First, record the relative powers of g
between two lifts of vertices of Y as a real-valued function beta on
G x G. Second, apply the Pfaffian to turn beta into an alternating
function whose exterior coboundary is the required cocycle.
LEMMA 17.1 (Relative exponents). There is a J-invariant function
beta : G x G -> R with
beta(h', h) = -beta(h, h'), (antisymmetry)
beta(g^r h, g^s h') = beta(h, h') + s - r for all r, s in Z. (17.1)
PROOF. Consider the diagonal H-action on Y x Y:
ci . (y, y') = (ci y, ci y') for ci in H.
Because the action on the first factor Y is free, the diagonal action on
Y x Y is also free. For each H-orbit in Y x Y choose a representative
(y_o, y'_o), and choose lifts (l_o, l'_o) in G^2 with l_o-bar = y_o,
l'_o-bar = y'_o.
For (h, h') in G^2 with (h-bar, h'-bar) in the orbit of (y_o, y'_o),
there is a unique ci in H with ci (y_o, y'_o) = (h-bar, h'-bar). Choose
any lift c in J of ci (so c-bar = ci), and write
h = g^r c l_o, h' = g^s c l'_o. (17.2)
Because g has infinite order, the exponents r, s are unique. Replacing
c by g^t c changes (r, s) to (r - t, s - t), so the DIFFERENCE s - r is
unaffected. Therefore
B(h, h') = s - r (17.3)
is well defined.
Computation of the invariance properties of B. Left multiplication by g
shifts r and s by the same amount (fix c): B(g h, g h') = B(h, h'). Right
multiplication by elements of J does not affect h-bar or h'-bar, so it
chooses a different c but B is unchanged. So B is J-invariant in each
slot.
For independent shifts of the two lifts:
B(g^a h, g^b h') = (b + s) - (a + r) = B(h, h') + b - a. (17.4)
Antisymmetrize:
beta(h, h') = ( B(h, h') - B(h', h) ) / 2. (17.5)
Then beta(h', h) = -beta(h, h'). And using (17.4) on each term:
beta(g^r h, g^s h') = (( B(h, h') + s - r ) - ( B(h', h) + s - r )) / 2
= beta(h, h') + s - r. QED
REMARK 17.2. beta need not descend to Y x Y. By (17.1),
beta(h, g^k h) = k even though these two lifts project to the same
vertex. Nor do we need beta to descend: it will be used only inside a
fully antisymmetrized exterior expression whose descent we verify.
If a lift c in J exchanges two distinct projected vertices y_0, y_1,
say c h_0 = g^a h_1 and c h_1 = g^b h_0, the invariance of beta under
J requires 2 beta(h_0, h_1) = b - a, which antisymmetrization satisfies
by definition. Real values are appropriate: for G = and g = t^2,
beta(1, t) = 1/2 is forced.
17.1. Degree-two warm-up.
The cyclic degree-two combination
c(y_0, y_1, y_2) = beta(h_1, h_2) - beta(h_0, h_2) + beta(h_0, h_1),
(17.6)
with any lifts h_i of y_i, is independent of the lifts:
Changing h_i to g^{r_i} h_i adds (r_2 - r_1) - (r_2 - r_0) + (r_1 - r_0) = 0.
It is alternating in (y_0, y_1, y_2). This is exactly the m = 1 case of
the Pfaffian construction below: psi_1(y_0, y_1, y_2) is the alternating
deletion sum of p_1(h_0, h_1) = beta(h_0, h_1), and reduces to (17.6).
The general m >= 2 case follows the same pattern with p_m in place of
p_1; we describe it uniformly in Section 18.
SECTION 18. THE CYCLIC PFAFFIAN IDENTITY.
For a list h = (h_0, ..., h_{2m-1}) with m >= 1, let
p_m(h) = Pf( beta(h_i, h_j) )_{0 <= i, j < 2m}, p_0 = 1. (18.1)
This is the Pfaffian of the alternating 2m x 2m matrix with entries
beta(h_i, h_j). We use the convention that p_m(h) is the coefficient of
eps_0 and ... and eps_{2m-1} in b^m / m!, where eps_i are the coordinate
one-forms on the space of list positions and
b = sum_{i= 1), q_0 := lambda (18.3)
are well-defined complex numbers.
LEMMA 18.1 (Pfaffian recurrence). For every m >= 1,
q_m = (1/2) q_{m-1}, hence q_m = 2^{-m} lambda. (18.4)
PROOF. Consider the twisted cyclic shift on lists of length 2m:
T h = (h_1, ..., h_{2m-1}, g h_0). (18.5)
An ordinary cyclic rotation of a list of length 2m multiplies its
Pfaffian by (-1)^{2m-1} = -1. Replacing the last entry h_0 by g h_0
adds 1 to every matrix entry in the last column above the diagonal
(this uses (17.1) with r=0, s=1: beta(g h_0, h_i) = beta(h_0, h_i) + 1
for 1 <= i < 2m). The result, expanding the Pfaffian along that column:
p_m(T h) = - p_m(h)
+ sum_{i=1}^{2m-1} (-1)^{i+1} p_{m-1}( h_1, ..., hat{h_i}, ..., h_{2m-1} ).
(18.6)
The last-column cofactor in list position i (positions numbered
1, ..., 2m) has sign (-1)^{i+1}, and each Pfaffian matching uses exactly
one entry from that column.
Apply v_{2m} to both sides.
First-line evaluation. v_{2m}(p_m(h)) = q_m by definition.
Second-line evaluation. For each i in {1, ..., 2m-1}, the expression
p_{m-1}( h_1, ..., hat{h_i}, ..., h_{2m-1} )
depends only on the (2m-2)-tuple obtained by deleting BOTH h_0 and h_i
from (h_0, ..., h_{2m-1}). Two deletions of v_{2m} give v_{2m-2}
(Lemma 15.1(b) applied twice), so
v_{2m}[ p_{m-1}( h_1, ..., hat{h_i}, ..., h_{2m-1} ) ]
= v_{2m-2}( p_{m-1} ) = q_{m-1}. (18.7)
The pair of deletions (h_0, h_i) from the (2m+1)-tuple is coherent: the
residual tuple is exactly (h_1, ..., hat{h_i}, ..., h_{2m-1}).
Sign sum. The coefficient
sum_{i=1}^{2m-1} (-1)^{i+1} = 1
because the 2m-1 terms alternate starting at +1 (2m-1 is odd). So for
m >= 2, the second-line contribution to v_{2m}(left) is q_{m-1}.
Special case m = 1. Formula (18.6) reduces to
p_1(T h) = -p_1(h) + p_0(h) = -p_1(h) + 1,
because the sum over {1, ..., 2m-1} = {1} contributes p_0 = 1. Applying
v_2 to both sides: q_1 = -q_1 + v_2(1) = -q_1 + lambda = -q_1 + q_0.
Assembly in both cases. Applying v_{2m} to (18.6):
q_m = -q_m + q_{m-1},
so q_m = q_{m-1}/2. Induct from q_0 = lambda: q_m = 2^{-m} lambda. QED
SECTION 19. DESCENT TO AN EXTERIOR COCYCLE.
The Pfaffian p_m depends on lifts; its alternating deletion sum, however,
will define a well-defined exterior functional.
PROPOSITION 19.1 (Pfaffian detector). For every m >= 1, the formula
psi_m(y_0, ..., y_{2m})
= sum_{i=0}^{2m} (-1)^i p_m( h_0, ..., hat{h_i}, ..., h_{2m} ),
(19.1)
where h_i in G is any lift of y_i in Y, defines an H-invariant linear
functional psi_m : E_{2m} -> C satisfying
psi_m * partial = 0, psi_m(z_{2m}) = 2^{-m} lambda. (19.2)
Here evaluation on the coinvariant chain z_{2m} is defined by
H-invariance.
PROOF. Three components: (i) lift-independence, (ii) H-invariance and
alternation, (iii) cocycle and evaluation.
(i) Lift-independence. On the 2m+1 list positions, put
u = sum_{i=0}^{2m} eps_i, b = sum_{0 <= i < j <= 2m} beta(h_i, h_j)
eps_i and eps_j. (19.3)
Then u and eps_i are linear; b is 2-form. The right side of (19.1) is
the coefficient of the ordered top form
eps_0 and eps_1 and ... and eps_{2m}
in
u and (b^m / m!). (19.4)
Indeed, choosing which factor u supplies eps_i (to give the i-th
term with sign (-1)^i to move eps_i past its i preceding positions) and
applying p_m = b^m/m! to the remaining 2m positions gives exactly (19.1).
Now replace the lifts by g^{r_i} h_i. By (17.1), beta(h_i, h_j) is
replaced by beta(h_i, h_j) + r_j - r_i. So b becomes
b' = b + (sum_i r_i eps_i) and u (since u = sum_k eps_k).
That is b' = b + u and v with v = sum_i r_i eps_i. Then
u and ((b + u and v)^m / m!)
differs from u and (b^m/m!) by terms in which u appears at least twice
(after expanding), and these vanish because u and u = 0. So (19.1) is
independent of the lifts.
(ii) Alternation and descent. p_m is alternating on 2m+1 entries (as the
Pfaffian of one Pfaffian-matrix minus one entry). The alternating
deletion sum (19.1) is intrinsically alternating on (h_0, ..., h_{2m}).
By (i) it is also independent of the lifts; hence it is alternating in
the projected vertices (y_0, ..., y_{2m}) and well-defined on them.
If two projected vertices y_i = y_j coincide, choose their lifts equal
(h_i = h_j). The alternating deletion sum has a matching cancellation:
the i-deleted term equals minus the j-deleted term (both reduce to the
same p_m factor after relabeling) by alternating sign. So the expression
vanishes. Therefore (19.1) descends to a linear functional on
E_{2m} = Lambda^{2m+1} C^{(Y)}.
(iii) H-invariance. For c-bar in H with lift c in J, and h_i lifts of
y_i, the lifts c h_i represent c-bar y_i. The J-invariance of beta
gives p_m(c h_0, ..., c h_{2m}) = p_m(h_0, ..., h_{2m}), so psi_m is
H-invariant. (The alternating sum is also unchanged by applying c to
all lifts simultaneously.)
(iv) Cocycle identity. Choose a simultaneous lift (h_0, ..., h_{2m+1})
of 2m+2 projected vertices. Applying partial to the simplex
{h_0, ..., h_{2m+1}} produces the alternating sum of its (2m+1)-faces;
applying psi_m to each face inserts one further deletion.
Concretely,
(psi_m * partial)( h_0, ..., h_{2m+1} )
= sum_{j=0}^{2m+1} (-1)^j psi_m( h_0, ..., hat{h_j}, ..., h_{2m+1} )
= sum_{j=0}^{2m+1} sum_{i != j}
(-1)^{j + eps(i,j)} p_m( h_0, ..., hat{i} ..., hat{j} ..., h_{2m+1} ).
(19.8)
where eps(i, j) = i if i < j (the index i is unchanged by removing j)
= i - 1 if i > j (the index i shifts by one).
Fix an unordered pair {i, j} with i < j. It appears twice.
Contribution A (first remove h_j, then h_i within the face):
sign (-1)^{j + i}.
Contribution B (first remove h_i, then h_j within the face; after the
removal, h_j's new index is j - 1):
sign (-1)^{i + (j - 1)} = -(-1)^{j + i}.
The two contributions are p_m of the SAME (2m)-tuple, multiplied by
opposite signs. So they cancel.
All unordered pairs {i, j} cancel, so psi_m * partial = 0.
(v) Evaluation on z_{2m}. Recall z_{2m} = pi_{2m+1}(v_{2m+1}), and pi
is defined by attaching a delta for each lift. Applying (19.1) to the
(2m+1)-tuples of v_{2m+1} and using the deletion identities gives
psi_m(z_{2m}) = sum_{i=0}^{2m} (-1)^i v_{2m}( p_m ) = v_{2m}(p_m) = q_m.
Since sum_{i=0}^{2m} (-1)^i = 1 (2m+1 terms, starting at +1), this
equals q_m = 2^{-m} lambda by Lemma 18.1. QED
19.1. Summary of Part IV.
We have constructed an H-invariant cocycle psi_m : E_{2m} -> C satisfying
psi_m(z_{2m}) = 2^{-m} lambda. (19.5)
Combined with Theorem 5.1 (which says psi_m evaluates to zero on any
coinvariant cycle of a K satisfying its hypotheses), this gives
2^{-m} lambda = 0, hence lambda = 0. Part V assembles the conclusion.
================================================================
PART V. ASSEMBLY AND CONSEQUENCES
================================================================
SECTION 20. PROOF OF THEOREM A (COMPLEX BASS TRACE).
We now combine the geometric engine (Part II), the algebraic cycle
construction (Part III), and the Pfaffian detector (Part IV).
PROOF OF THEOREM A. Finitely generated case first.
Let G be finitely generated, e in M_q(C[G]) idempotent, g in G of
infinite order. Build from these data:
S_0 = finite support of e,
S = finite symmetric generating set containing S_0 and inverses,
D = max(2, |S|),
Y = \ G with the S-step graph,
H = C_G(g) / acting on Y freely by graph automorphisms.
The graph Y is countable, connected, of degree <= D; H is countable
(Lemma 6.6 gives finite vertex stabilizers, though we will not need
that here -- it is needed for the geometric chain construction).
Let lambda = HS_G([e(C[G])^q])([g]_G). By Lemma 15.1(c), lambda is also
the total weight v_k(1) for every k >= 1.
Choose n = 2m even with n >= n_0(D), where n_0(D) is the threshold from
Theorem 5.1 (equivalently from Lemma 8.3). Such n exists because n_0(D)
is finite.
Two cases.
Case A. z_n = 0 in (E_n)_H (all surviving tuples have repeated projected
vertices). Then psi_m(z_n) = 0 for any functional psi_m. On the other
hand, Proposition 19.1 states psi_m(z_n) = 2^{-m} lambda. So
2^{-m} lambda = 0, hence lambda = 0.
Case B. z_n != 0. By Proposition 16.1, z_n is then the image in (E_n)_H
of a cycle in C_n(K)_H for an H-invariant subcomplex K of the full
simplex on Y, generated by n-simplices whose vertex sets span connected
subgraphs of Y, with finitely many H-orbits.
Proposition 19.1 provides an H-invariant cocycle psi_m : E_n -> C with
psi_m(z_n) = 2^{-m} lambda.
Theorem 5.1 (applied with the degree bound D and the chosen n >= n_0(D))
states: psi_m evaluates to zero on the image in (E_n)_H of every cycle
in C_n(K)_H.
Since z_n is such an image, we get 2^{-m} lambda = 0, hence lambda = 0.
So HS_G([e(C[G])^q])([g]_G) = 0 for every idempotent e and every
infinite-order g. Any class in K_0(C[G]) is a Z-linear combination of
classes [e(C[G])^q], so HS_G(x)([g]_G) = 0 for every x in K_0(C[G]) and
every infinite-order g. This proves Theorem A for finitely generated G.
Arbitrary groups. Now let G be arbitrary, x in K_0(C[G]), g in G of
infinite order. By Lemma 14.1, x is i_* x_0 for some finitely generated
subgroup G_0 <= G containing g and all coefficient supports of x (as
represented by a pair of idempotents over M_r(C[G]) and M_s(C[G])).
Lemma 14.1 gives
HS_G(x)([g]_G) = sum_{C in con(G_0), C subseteq G_0 cap [g]_G} HS_{G_0}(x_0)(C).
Every element of [g]_G has infinite order; every subgroup conjugacy
class C subseteq G_0 cap [g]_G is a subclass of the (infinite-order) class
[g]_G and so has infinite order (order is preserved by inclusion).
By the finitely generated case, each HS_{G_0}(x_0)(C) = 0. So the sum
vanishes. QED (Theorem A)
SECTION 21. THE INTEGRAL CASE.
Integral Bass. For the integral group ring Z[G], the Hattori-Stallings
trace is defined identically to Section 3:
HS_{Z[G]} : K_0(Z[G]) -> HH_0(Z[G]),
and by Proposition 3.1 with Z coefficients,
HH_0(Z[G]) = (+)_{C in con(G)} Z [C].
The integral form of the Bass trace conjecture asks for support on
[1]_G alone.
THEOREM 21.1 (Integral Bass trace). For every group G,
HS_{Z[G]}(K_0(Z[G])) is supported on [1]_G.
Equivalently, for every x in K_0(Z[G]) and every h in G with h != 1,
HS_{Z[G]}(x)([h]_G) = 0.
The proof uses Theorem A (complex case) for infinite-order classes and
Linnell's restriction (Lemma 4.1 of Linnell 1983, in the form restated
and reproved by Berrick-Hesselholt 2015, Theorem A) for finite-order
classes. We state the restriction:
LINNELL'S RESTRICTION. Let G be any group. If some x in K_0(Z[G]) has
HS_{Z[G]}(x)(C) != 0 for a conjugacy class C, then for every h in C
there exists an integer a >= 1 such that h is conjugate to h^{s^a} for
every integer s >= 1.
Proof of Theorem 21.1. Two step. Fix x in K_0(Z[G]).
Step 1 (infinite-order classes). Factor x = [P] - [Q] with P, Q finitely
generated projective Z[G]-modules; equivalently, x = [e_P Z[G]^{r}]
- [e_Q Z[G]^{s}] for idempotents e_P, e_Q over Z. The same matrices are
idempotents over C[G], and (by the split decomposition) represent the
complexifications P (x)_Z C and Q (x)_Z C as projective right C[G]-modules:
P (x)_Z C = e_P (C[G])^r,
Q (x)_Z C = e_Q (C[G])^s.
For an infinite-order element g in G, Theorem A (applied to
[e_P C[G]^r] - [e_Q C[G]^s] in K_0(C[G])) gives
HS_{C[G]}([e_P C[G]^r] - [e_Q C[G]^s])([g]_G) = 0.
The complex coefficient at [g]_G equals the image under Z -> C of the
integral coefficient at [g]_G: both are the sum of tr((e_P - e_Q)_h) over
h in [g]_G, and the matrices (e_P - e_Q)_h have integer entries. So the
integral coefficient vanishes as well.
Step 2 (finite-order classes other than [1]). Let h in G have finite
order t > 1. Suppose for contradiction HS_{Z[G]}(x)([h]_G) != 0. By
Linnell's restriction (applied with C = [h]_G, which is finite-order),
there is a >= 1 with h conjugate to h^{s^a} for every s >= 1.
Take s = t. Then h^{t^a} = (h^t)^{t^{a-1}} = 1 since |h| = t. So the
restriction says h is conjugate to 1, i.e. h = 1. Contradiction. So
HS_{Z[G]}(x)([h]_G) = 0.
Step 1 + Step 2 + additivity conclude HS_{Z[G]}(x)([h]_G) = 0 for every
h != 1. QED (Theorem 21.1)
SECTION 22. HOMOTOPY IDEMPOTENTS AND LEFSCHETZ NUMBERS.
Integral support has topological consequences for fixed points. We
collect them following Berrick-Chatterji-Mislin.
22.1. Setup: finitely dominated complexes.
Let X be a connected finite CW complex, f: X -> X a map with
f o f ~ f (a HOMOTOPY IDEMPOTENT). Then there exist a CW complex Y and
maps u: X -> Y, v: Y -> X with
v o u ~ f, u o v ~ id_Y. (22.1)
The second identity says Y is FINITELY DOMINATED by X. Y need not be
finite, but its universal cover's chain complex is chain homotopy
equivalent to a finite complex of finitely generated projective
Z[G]-modules; this is standard (see e.g. BCM2006, Section 3).
Write G = pi_1(X). If f is POINTED homotopy idempotent with f_# = id_G,
then u_# and v_# identify pi_1(Y) with G. The universal cover
Y-tilde -> Y is a free G-CW complex. Let C_*(Y-tilde) be its cellular
chain complex over Z[G]; it is chain homotopy equivalent to a finite
complex P_* of finitely generated projective Z[G]-modules.
22.2. Projective Euler class, Reidemeister trace.
Define the PROJECTIVE EULER CLASS
e_Y = sum_i (-1)^i [P_i] in K_0(Z[G]). (22.2)
The Hattori-Stallings trace HS_{Z[G]}(e_Y) in HH_0(Z[G]) ~= (+)_C Z[C]
records the fixed-point class indices of f: this is the REIDEMEISTER
trace of f, as follows from the Lefschetz fixed-point formula for
finitely dominated complexes.
There is a sign convention: BCM2006 use left module conventions, while
we use right modules. The anti-involution g |-> g^{-1} of Z[G] (with
transpose of matrices) transports between the two, and its effect on the
trace is to relabel a coefficient at [g]_G as a coefficient at [g^{-1}]_G.
This isometry preserves both "supported on [1]_G" and the two numerical
traces below.
22.3. The two scalar traces.
Augmentation trace: apply augmentation sum_g a_g g |-> sum_g a_g
to HS_{Z[G]}(e_Y). This gives the ORDINARY LEFSCHETZ NUMBER
L(f) = sum_i (-1)^i rank(P_i (x)_{Z[G]} Z), as the alternating sum of
the ranks of the augmented chains.
Kaplansky trace (identity coefficient): the identity coefficient
kappa of e_Y equals sum_C HS_{Z[G]}(e_Y)(C). This is the
L^2-LEFSCHETZ NUMBER L^(2)(f-tilde) of the G-equivariant lift.
Theorem 21.1 forces the two traces to be equal on any class in K_0(Z[G]):
Lemma 22.1 (Weak Bass, all groups). For every x in K_0(Z[G]),
epsilon(x) = kappa(x).
Equivalently, the augmentation trace equals the identity coefficient
on every K_0(Z[G])-class.
PROOF. The augmentation epsilon(x) = sum_C HS(x)(C). By Theorem 21.1,
this sum equals HS(x)([1]_G) = kappa(x). QED
22.4. Corollary: BCM2006 consequences.
COROLLARY 22.2 (BCM consequences). Let M be a connected closed smooth
oriented manifold.
(i) If dim M > 2 and f: M -> M satisfies f o f ~ f by free homotopy,
then some self-map freely homotopic to f has exactly one fixed
point.
(ii) Choose a basepoint x_0 in M and put G = pi_1(M, x_0). Suppose
f: (M, x_0) -> (M, x_0) satisfies f o f ~_* f by based homotopy
and f_# = id_G. Then
L(f) = L^(2)(f-tilde). (22.3)
Here f-tilde is the lift to the universal cover fixing a chosen
point over x_0 (so it is G-equivariant). L(f) is the ordinary
Lefschetz number, and L^(2)(f-tilde) is the alternating sum of
von Neumann traces on the induced L^2-homology maps.
PROOF. Theorem 21.1 provides the classical Bass conjecture (support on
[1]_G) for K_0(Z[G]); Lemma 22.1 provides weak Bass (epsilon = kappa).
The anti-involution g |-> g^{-1} transports these statements between
left and right module conventions, as in 22.2.
Now apply the forward implications of BCM2006 Theorem 1 and Theorem 2.
- BCM2006 Theorem 1 (from weak Bass) implies: free homotopy idempotents
on connected closed smooth oriented manifolds of dimension > 2 are
freely homotopic to maps with exactly one fixed point. This is (i).
- BCM2006 Theorem 2 (from classical Bass) implies: for based homotopy
idempotents with f_# = id_G on such a manifold, L(f) = L^(2)(f-tilde).
This is (ii). QED
SECTION 23. TORSION-FREE GROUPS: TRACE RANK AND ITS RANGE.
Now assume G is torsion-free. Then the identity is the only finite-order
element, so Theorem A leaves only the identity coefficient on any class
of K_0(C[G]).
23.1. Recall the two maps.
Earlier we defined two natural maps from K_0(C[G]) to C:
tau_{G,*} : K_0(C[G]) -> C,
representation: [e C[G]^q] |-> tau_{G,q}(e) = sum_{i=1}^q tau_G(e_{ii}),
rk_epsilon : K_0(C[G]) -> Z,
representation: [e C[G]^q] |-> rank_C(epsilon_{G,q}(e)),
where tau_G : C[G] -> C is the identity coefficient and
epsilon_{G,q} : M_q(C[G]) -> M_q(C) is entrywise augmentation (a ring
homomorphism M_q(C[G]) -> M_q(C)).
Both are well-defined on K_0(C[G]) (independence of the idempotent
representative for tau_{G,*} follows from the same argument as for
HS_G in Section 3; well-definedness of rk_epsilon follows from
augmentation being a ring map and rank being well-defined on K_0(C)).
LEMMA 23.1. Let G be torsion-free, e in M_q(C[G]) idempotent, x =
[e C[G]^q] in K_0(C[G]). Then
tau_{G,q}(e) = rank_C( epsilon_{G,q}(e) ).
Hence tau_{G,*} = rk_epsilon as functionals on K_0(C[G]), and
tau_{G,*}(K_0(C[G])) = Z.
PROOF. Since G is torsion-free, Theorem A applied to x says
HS_G(x) is supported on [1]_G.
Therefore
tau_{G,q}(e) = HS_G(x)([1]_G) (definition)
= sum_{C in con(G)} HS_G(x)(C) (all other terms are 0)
= epsilon_G( tr_q(e) ) (definition of augmentation)
= tr_C( epsilon_{G,q}(e) ) (entrywise)
= rank_C( epsilon_{G,q}(e) ). (idempotent matrix: tr = rank)
The class sum sum_C (tr_q e)_C is finite because tr_q e has finite
support, so no convergence question arises.
Range = Z: tau_{G,*}([C[G]]) = tau_{G,1}(1) = 1, and tau_{G,*}([0]) = 0,
so the range contains Z. Conversely, epsilon_{G,q}(e) is an idempotent
complex matrix (property of a ring homomorphism applied to an idempotent),
and every idempotent complex matrix has integer trace. So the range is
contained in Z. QED
NOTE. For MATRICES, tau_{G,q}(e) can be any nonnegative integer:
diag(1, 0, ..., 0) in M_q(C[G]) is an idempotent with augmentation rank
one, diag(1, 1, 0, ..., 0) has augmentation rank two, etc. The scalar
case q = 1 is special because a scalar idempotent complex has trace in
{0, 1}. This distinction is the crux of Section 24.
SECTION 24. THE KAPLANSKY IDEMPOTENT CONJECTURE (CHARACTERISTIC ZERO).
STATEMENT (Kaplansky). Let G be a torsion-free group and R a commutative
unital domain of characteristic zero. Then the only idempotents in R[G]
are 0 and 1.
We prove two special cases: R = C, and then arbitrary R via finite-
coefficient reduction.
24.1. The complex case.
Let e in C[G] be idempotent. By Lemma 23.1 with q = 1:
tau_G(e) = epsilon_G(e) in {0, 1}.
We show e is 0 or 1 accordingly.
Setup. Embed C[G] in the group von Neumann algebra N(G) as the algebra
of left convolutions; N(G) is the commutant of the right regular
representation of G on l^2(G). The canonical trace is
tau_G(T) = , delta_1 the unit vector at 1 in G.
Its key property, established in Appendix A, is faithfulness:
Faithfulness. If T in N(G) is positive with tau_G(T) = 0, then T = 0.
Idempotents in C[G] remain idempotents in N(G) since N(G) is an
algebra containing C[G].
CASE tau_G(e) = 0. Show e = 0.
Step 1: positivity. Define
a = e* e + (1 - e)* (1 - e).
For xi in l^2(G):
= ||e xi||^2 + ||(1-e) xi||^2 >= (1/2) ||xi||^2.
The last bound: write xi = e xi + (1 - e) xi, then use
||xi||^2 <= 2 ||e xi||^2 + 2 ||(1-e) xi||^2.
So a is positive with a >= 1/2 id. In particular a is invertible in
N(G), and s = a^{1/2} is also positive definite and invertible.
Step 2: idempotent self-adjointness. Compute:
a e = (e* e + (1-e)*(1-e)) e = e* e (since (1-e) e = 0 and e* e e = e* e).
By taking adjoints (a is self-adjoint): e* a = (a e)* = e* e.
So ae = e* e = e* a. Multiplying by a^{-1} on the left: e = a^{-1} e* a.
Multiplying by a^{-1} on the right: e a^{-1} = a^{-1} e*.
Equivalently e* = a e a^{-1}.
Step 3: p is a projection. Set p = s e s^{-1} = a^{1/2} e a^{-1/2}.
p^2 = s e s^{-1} s e s^{-1} = s e^2 s^{-1} = s e s^{-1} = p.
p* = s^{-1} e* s = a^{-1/2} (a e a^{-1}) a^{1/2} = a^{1/2} e a^{-1/2} = p.
So p is a projection in N(G).
Step 4: trace computation.
tau_G(p) = tau_G(s e s^{-1}) = tau_G(e s^{-1} s) = tau_G(e) = 0.
(Used traciality of tau_G on N(G).)
Step 5: faithfulness. Since p is a projection, p = p* p >= 0. Then
tau_G(p) =
=
=
= ||p delta_1||^2.
The equality tau_G(p) = 0 gives p delta_1 = 0. Since p commutes with
right translations (by definition of N(G)), and rho_g delta_1 = delta_{g^{-1}}
for every g in G:
p delta_{g^{-1}} = p rho_g delta_1 = rho_g p delta_1 = 0.
So p vanishes on an orthonormal basis of l^2(G), hence p = 0. Since
s and s^{-1} are invertible, e = s^{-1} p s = 0.
CASE tau_G(e) = 1. Apply the previous case to 1 - e (also idempotent,
with tau_G(1 - e) = 1 - 1 = 0). Get 1 - e = 0, i.e. e = 1.
So for every idempotent e in C[G], tau_G(e) in {0, 1}, and e is
correspondingly 0 or 1. QED (complex case of Theorem B)
24.2. Reduction to char-0 domains.
Now let R be a commutative unital domain of characteristic zero and let
f in R[G] be idempotent, f = sum_{g in S} r_g g with S finite.
Step 1: finite coefficient ring. Put
R_0 = Z[r_g : g in S] subseteq R.
R_0 is a finitely generated Z-subalgebra of R, hence a Noetherian
integral domain of characteristic zero. Since R_0 -> R is injective
and coefficients of f lie in R_0, the identity f^2 = f holds already
in R_0[G].
Step 2: fraction field. Let F = Frac(R_0), a field finitely generated
over Q. We show F embeds in C.
Choose a transcendence basis t_1, ..., t_n of F over Q. Map t_i -> e.g.
algebraically independent complex numbers (e.g. small positive reals
that are Q-algebraically independent; such exist since C has
transcendence degree at least that of R). This gives an embedding
Q(t_1, ..., t_n) -> C. Since F / Q(t_1, ..., t_n) is a finite algebraic
extension and C is algebraically closed, the embedding extends across
F to give F -> C injective.
Step 3: descend through embeddings. We have commutative inclusions
R_0[G] -> F[G] -> C[G],
R_0[G] -> R[G].
The first row and the embedding F -> C (which extends to F[G] -> C[G])
make R_0[G] -> C[G] injective: R_0[G] -> F[G] is injective (R_0 is a
domain), and F[G] -> C[G] is injective (F -> C injective, group ring
inclusion). The embedding R_0 -> R makes R_0[G] -> R[G] injective.
Now f in R_0[G] has f^2 = f in R_0[G]. Its image in F[G] is an idempotent
of F[G]. Its image in C[G] is an idempotent of C[G], so by 24.1 is 0 or 1.
Since F[G] -> C[G] is injective, the image of f in F[G] is 0 or 1.
Since R_0[G] -> F[G] is injective, f is 0 or 1 in R_0[G]. Since
R_0[G] -> R[G] is injective, f is 0 or 1 in R[G].
So f in R[G] is 0 or 1. QED (Theorem B)
REMARK 24.1. Only the finitely generated coefficient field was embedded
in C; no embedding of all of R into C is assumed or needed. The
characteristic-zero hypothesis is essential: over a field of positive
characteristic, additional idempotents can arise (e.g. sum_g g^{p^k}
for p-torsion elements) and the conjecture is known to fail. Torsion-
freeness of G is essential: for finite groups F, the idempotent
(1/|F|) sum_{h in F} h is nontrivial.
REMARK 24.2. Why q = 1. The theorem concerns e in R[G] itself, where
the augmentation gives a scalar idempotent with trace in {0, 1}. For
idempotents in the matrix ring M_q(R[G]) with q > 1, "only 0 and 1"
is far too strong: diag(1, ..., 1, 0, ..., 0) is an idempotent with
augmentation rank any integer in {0, 1, ..., q}. The scalar case is
exactly where the augmentation-rank argument rules everything out.
24.3. Combined summary.
Collecting Sections 23 and 24:
Theorem 24.3 (Kaplansky idempotent, char 0, torsion-free). Let G be
torsion-free and R a commutative unital domain of characteristic
zero. Then every e in R[G] with e^2 = e is 0 or 1.
Corollary 24.4 (trace rank). For G torsion-free,
tau_{G,*} = rk_epsilon on K_0(C[G]) and the range is exactly Z.
These conclude the paper's main results.
================================================================
APPENDIX A. THE CANONICAL TRACE ON THE GROUP VON NEUMANN ALGEBRA
================================================================
Let G be a discrete group (for our applications countable is enough, but
the construction works generally). Let l^2(G) be the Hilbert space of
square-summable complex functions on G, with the standard orthonormal
basis delta_g (g in G).
A.1. The two regular representations.
Left translations: L_g delta_h = delta_{gh}. This is a unitary
representation of G on l^2(G); it extends to a *-representation of C[G]
and to a C*-algebra C_r^*(G) (the reduced group C*-algebra; it is the
norm-closure of the image of C[G]).
Right translations: rho_g delta_h = delta_{h g^{-1}}. This is another
unitary representation, EXCHANGING with the left one:
L_g rho_h delta_k = delta_{g k h^{-1}} = rho_h L_g delta_k.
The group von Neumann algebra N(G) is either of the two equivalent
objects:
N(G) = { T in B(l^2 G) : T rho_g = rho_g T for all g in G },
(the commutant of the right regular representation)
equivalently the double commutant of the left regular representation
(by von Neumann's bicommutant theorem). It is a von Neumann algebra
containing the image of C[G] as a norm-dense *-subalgebra.
A.2. The canonical trace.
Define
tau_G : N(G) -> C, tau_G(T) = .
Properties.
(i) linear: obvious.
(ii) positive: T >= 0 gives >= 0.
(iii) tracial: tau_G(AB) = tau_G(BA) for A, B in N(G).
Proof: for g, h in G, L_g L_h = L_{gh}, so
tau_G(L_g L_h) = = [gh = 1],
where [gh = 1] is 1 if gh = 1 and 0 otherwise. Similarly
tau_G(L_h L_g) = [hg = 1].
The identities gh = 1 and hg = 1 are equivalent (both iff
h = g^{-1}), so the two coincide on L_g L_h. Extend by
bilinearity to finite combinations of left translations,
i.e. to C[G], and then to N(G) by the standard fact that
ta_G extends to a normal tracial functional on the group
von Neumann algebra. QED
(iv) faithful: T >= 0 with tau_G(T) = 0 implies T = 0. See A.3.
(v) normalized: tau_G(id) = = 1.
(vi) on C[G]: tau_G(sum_g a_g L_g) = sum_g a_g
= a_1 = the identity coefficient tau_G of Section 1.
So tau_G on N(G) extends the identity coefficient of C[G]. This is the
"faithful trace" we use.
A.3. Faithfulness.
LEMMA A.1. If T in N(G) satisfies T >= 0 and tau_G(T) = 0, then T = 0.
PROOF. T >= 0 means T = S* S for some S; also T^{1/2} is the unique
positive square root, and T^{1/2} in N(G).
tau_G(T) = =
= || T^{1/2} delta_1 ||^2.
So tau_G(T) = 0 iff T^{1/2} delta_1 = 0.
Now use that T^{1/2} commutes with right translations rho_g. Fix any
g in G, and put delta_{g^{-1}} = rho_g delta_1 (note: rho_g delta_1 =
delta_{1 g^{-1}} = delta_{g^{-1}}). Then
T^{1/2} delta_{g^{-1}} = T^{1/2} rho_g delta_1 = rho_g T^{1/2} delta_1 = rho_g (0) = 0.
Since {delta_{g^{-1}} : g in G} is an orthonormal basis of l^2(G),
the bounded operator T^{1/2} vanishes on a dense subspace, so T^{1/2} = 0,
hence T = 0. QED
================================================================
APPENDIX B. REMARKS ON CONVENTIONS AND COMPLETENESS
================================================================
B.1. Left versus right modules.
The main text uses right modules and idempotents e in M_q(A) acting on
columns e v. Some references (including BCM2004, BCM2006, Linnell 1983)
use left modules. The translation is the anti-involution
g |-> g^{-1}, extended C-linearly to C[G]
together with matrix transpose. It sends right modules to left modules,
preserves the trace coefficients up to relabelling [g]_G <-> [g^{-1}]_G,
and preserves the identity support condition. Every result in this paper
transfers under this dictionary, and we have implicitly used it in
Section 22 when comparing with BCM2006.
B.2. Finite support of idempotents.
The main theorem and its applications rely throughout on the finiteness
of the support of a group-ring element. We have never used a completed
group algebra or any norm on an infinite sum, and the constructions
above are deliberately finite-object at every step. This is why the
result holds without amenability hypotheses: the geometric engine
(Section 5.1) requires only finitely supported data.
B.3. What is not claimed.
We have not established:
(i) any extension of the theorem to completed group rings C_r^*(G);
(ii) the analogous statement for K_1 of group rings;
(iii) Kaplansky's idempotent conjecture in positive characteristic
(which is not even true in that generality).
The Kadison-Kaplansky conjecture (a stronger form of (iii) for C_r^*(G))
remains open for groups containing F_2 (+) F_2 and is unrelated to the
present method; see OW2023, Section 1, for a survey.
B.4. External inputs.
We invoke three results from the literature without reproof:
(a) Linnell's restriction (Linnell 1983, Lemma 4.1; reproved by
Berrick-Hesselholt 2015, Theorem A). Used only in Section 21.
(b) Berrick-Chatterji-Mislin 2006, Theorems 1 and 2 (forward
implications from weak Bass / classical Bass to fixed-point
and L^2-Lefschetz conclusions). Used in Section 22.4.
(c) Burger-Valette 1998 is cited for background context on the
faithful trace. The particular faithfulness property needed
here is proved directly in Appendix A, so (c) is not
logically required.
Everything else in Parts I-V and Appendices A-B is derived from
these three inputs plus the constructions of Sections 1-19 and the
assembly of Sections 20-24.
================================================================
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END OF PAPER
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