HEISENBERG COVER TORSION AND THE PRECISE FINITE-CW TRANSFER DEFECT
==============================================================
Written: 2026-10-02 12:10:09 UTC. Target AIM-GEOMETRIC_GROUP_THEORY-0134.
This is a proof-first auxiliary packet. It does NOT resolve the general
closed-aspherical-manifold question, and is not a publication/readiness
decision. Novelty is undetermined; all published inputs are credited below.

MAIN DIRECT RESULT
-------------------
For every finite-index subgroup L of the integral Heisenberg group H, let
Lambda be its projection to Z^2, d=[Z^2:Lambda], and
L intersect Z(H)=<c^r>, r>0. Then

  r divides d,   [H:L]=d*r,
  L_ab = Z^2 direct_sum Z/(d/r),
  |Tor H_1(N_L;Z)|=d/r=[H:L]/r^2 <= [H:L].

Here N_L is the corresponding closed Heisenberg nilmanifold cover. In
particular normalized logarithmic H_1 torsion tends to zero for ANY
sequence of such subgroups whose indices tend to infinity, not merely the
congruence tower or normal subgroups. This is an exact subgroup calculation,
not a transfer of the determinant counterexample to manifold homology.

1. COORDINATE CONVENTIONS AND GENERAL SUBGROUP PRESENTATION
----------------------------------------------------------
Write H as integer triples (x,y,z), identified with matrices
[[1,x,z],[0,1,y],[0,0,1]]. Multiplication and inverse are

  (x,y,z)(X,Y,Z)=(x+X,y+Y,z+Z+xY),
  (x,y,z)^-1=(-x,-y,-z+xy).

Set a=(1,0,0), b=(0,1,0), c=(0,0,1). The source's commutator convention is
[g,h]=g^-1 h^-1 g h. For these class-two coordinates,

  [g,h]=(0,0,xY-Xy),   [a,b]=c.

The center is exactly <c>: commuting with a forces y=0, and commuting
with b forces x=0. Projection pi(x,y,z)=(x,y) has kernel <c>.

For finite-index L, its projection Lambda has a lattice basis
(A,0),(B,C), with A,C>0 and 0<=B<A. Its index is d=AC. Choose lifts
u=(A,0,alpha), v=(B,C,beta) in L and w=c^r generating the central
intersection. Every element of L has a unique collected expression

  u^m v^n w^ell,   m,n,ell integers.

Existence follows by matching the projection with u^m v^n and correcting
the remaining central element by w^ell. For uniqueness, first project to
the independent lattice basis; m=n=0 follows, and then c^(r ell)=1 forces
ell=0. We have [u,v]=c^d, so r divides d. Therefore L has the exact
presentation

  <u,v,w | [u,w]=[v,w]=1, [u,v]=w^(d/r)>.

To justify that there are no omitted relations, the displayed presentation
collects every word to u^m v^n w^ell; its map to L is injective by the
uniqueness just proved. Abelianizing leaves precisely one relation
(d/r)*w=0, giving Z^2 direct_sum Z/(d/r). The factor Z/1 is trivial.

2. INDEX, NORMALITY AND SHARP UNIVERSAL TORSION BOUNDS
----------------------------------------------------
Let P=pi^-1(Lambda). Then [H:P]=d and P=L*<c>. Central cosets show
[P:L]=[<c>:L intersect <c>]=r, proving [H:L]=dr without any normality
assumption. With n=[H:L] and t=|Tor L_ab|, the exact identity is

  n=r^2*t,   1<=t<=n.

This last inequality is sharp: L=<a^d,b,c> has r=1, index d and torsion d.
Thus a universal bound t<=sqrt(n), for example, would be false. The
logarithm rather than the raw torsion order is what is sublinear:

  0<=log(t)/n<=log(n)/n<=2/sqrt(n) ->0.

The last elementary bound follows by integrating 1/x<=1/sqrt(x) on
[1,n]. Another useful bound is log(t)/n<=1/r^2, since log(t)<=t.

Conjugation by a and b changes only the central coordinate. In the chosen
basis, L is normal in H if and only if r divides A, B and C, equivalently
Lambda is contained in r Z^2. Necessity follows by conjugating u and v;
sufficiency follows because those central changes lie in <c^r>, which
preserves the generators. Thus normal L also satisfies r^2|d and n>=r^3.
For a nested residual tower, the central steps r tend to infinity: the
intersection of the subgroups rZ is trivial, and these nested positive
integer steps cannot remain bounded. Either bound above then supplies the
vanishing normalized logarithmic torsion. No matrix determinant is needed.

3. ACTUAL NILMANIFOLD HOMOLOGY AND THE CONGRUENCE TOWER
-----------------------------------------------------
Let H_R be the real Heisenberg Lie group with the same coordinate law and
N=H\H_R. Integer translations yield a compact quotient; a fundamental
domain can be reduced successively in x,y,z to a unit box. The lattice acts
freely and properly, and H_R is R^3 as a manifold. Hence N is a closed
oriented aspherical three-manifold. It admits a Riemannian metric. For any
finite-index L, N_L=L\H_R is its connected finite cover.

The presentation above computes its H_1. Integral Poincare duality gives
H_2=H^1=Hom(H_1,Z)=Z^2; H_0=H_3=Z. Therefore the complete homology is

  H_0=Z, H_1=Z^2 direct_sum Z/t, H_2=Z^2, H_3=Z.

The only nontrivial homological torsion is in degree one. The source uses
logarithmic integral torsion rho^Z=sum_j(-1)^j log|Tor H_j|, so here
rho^Z(N_L)=-log(t).

For q=3^i, the congruence subgroup is the kernel of reduction to H_3(Z/q):

  H_i={(x,y,z): q divides x,y,z}=<a^q,b^q,c^q>.

Indeed a^(qm)b^(qn)c^(q ell) has central coordinate q^2 mn+q ell,
which gives every triple divisible by q. The reduction is surjective with
q^3 elements, so [H:H_i]=q^3; this is a normal subgroup. Its projected
lattice has d=q^2 and central step r=q. Since

  [a^q,b^q]=c^(q^2)=(c^q)^q,

we obtain H_1(N_i)=Z^2 direct_sum Z/q. The tower is strictly decreasing
with index ratio 27. Every integer coordinate divisible by all 3^i must
be zero, proving intersection_i H_i={1}. Thus this is an actual tower
meeting the source's finite-index, normal, strict and residual conditions.
Its normalized middle-degree logarithmic torsion is

  log(q)/q^3 ->0.

The executable's printed torsion-order/index fractions 1/q^2 are NOT these
logarithmic quantities; they are distinct controls and should not be mixed.

4. CREDITED L2 INPUT; NO DEDUCTION FROM FINITE TORSION ALONE
----------------------------------------------------------
Wolfgang Lueck, Approximating L2-invariants and homology growth,
arXiv:1203.2827v3 (October2012), Corollary1.13, printed p5, applies to a
CLOSED ASPHERICAL manifold whose fundamental group contains a nontrivial
elementary amenable normal subgroup. It proves vanishing of the L2-Betti
numbers and L2-torsion, and of the normalized integral homology torsion
limits for residual finite-index normal systems. I read that statement
and its proof in the primary version; the source lane separately pins it.

Our N is closed and aspherical as above; <c> is a nontrivial infinite
cyclic, therefore elementary amenable, normal subgroup of H. The exact
source hypotheses match. Hence all b_j^(2)(tildeN)=0 and
rho^(2)(tildeN)=0. In dimension3, k=1, the AIM expression is minus the
normalized H_1 torsion limit, also zero by the direct calculation. This
is a known positive special case, NOT a counterexample to the target.

Do not infer rho^(2)=0 just from vanishing L2-Betti numbers or from the
finite logarithmic torsion limit. Those implications are false in general;
the credited elementary-amenable-normal-subgroup theorem is essential.

Primary URL: https://arxiv.org/pdf/1203.2827v3
Local retained source: ../sources/pdfs/luck-1203.2827.pdf
SHA256: bcc933d828bcdbc8ae969d68c716d760beb3d98d74dc9c0560bccfaacc52ef75
The publication is GAFA23(2)(2013),622-663. The local filename omits the
version suffix; the checked embedded document is the v3 source above.

5. EXACT q=3 REGULAR DETERMINANT AND A CONCRETE NONUNIT TEST
---------------------------------------------------------
Holger Kammeyer, A counterexample to the determinant approximation
conjecture, arXiv:2609.15567v1 (14September2026), Theorem2 and Section2,
uses f=1-2a+2b and this congruence tower. Its finite operators are
invertible and their normalized determinants are at most 5^(1/3), whereas
the infinite Fuglede-Kadison determinant is2. This published result is
credited, not claimed as ours. The finite calculation below reproduces
its smallest nontrivial level directly with standard-library integers.

Enumerate Q=H/H_1 by its 27 triples modulo3. On the column basis indexed
by g, define R_a e_g=e_(ga), and similarly for b,c. Let T=R_a-R_b and
M=identity-2T. Exact Bareiss elimination gives

  det(M)=1729=7*13*19.

Independent modular Gaussian eliminations at7,13,19,23,101,1009 match this
integer. M is identity modulo2, its determinant is odd, and its row/column
absolute-sum norms are at most5. The exact primitive-central control

  E=2*identity-R_c-R_c^2,  E^2=3E, trace(E)=54,
  T^3 E=0 but T^2 E!=0

shows the two primitive-central blocks have total dimension18 and are
genuinely nilpotent of degree3 for T. This reproduces Kammeyer's local
mechanism at q=3. The remaining dimension9 block is the abelian quotient.
Its determinant is also1729. An analytic independent calculation is

  product_(alpha^3=beta^3=1)(1-2alpha+2beta)
    =product_(alpha^3=1)[(1-2alpha)^3+8]
    =7*(-11-18omega)*(-11-18omega^2)
    =7*(121-198+324)=1729,

where omega is a primitive cubic root. The normalized determinant is
1729^(1/27); the integer inequality 1729<=5^9 checks its quoted bound
without a floating root. No extrapolation from q=3 proves the all-q
published theorem.

This determinant is also a decisive nonunit/surjectivity test over ZH.
If right multiplication by f were onto ZH, there would be u with uf=1.
Reducing to ZQ would give an integer one-sided inverse for M, forcing
det(M)=+-1. Since its determinant is1729, coker(R_f:ZH->ZH) is nonzero.
The finite coinvariant cokernel has order1729. Thus an infinite group-ring
module defect is certified by an exact finite calculation, not by assuming
that an injective infinite operator is invertible.

6. A GENUINE FINITE-CW TRANSFER, AND ITS PRECISE FAILED HYPOTHESES
--------------------------------------------------------------
The following is the parent-proposed repaired transfer, checked here at
the attaching-map and chain level. It is an adjacent finite-CW result,
not an extension to a closed aspherical manifold.

Start with the actual nilmanifold N, and Y2=N wedge S^2. Its fundamental
group remains H. The universal cover of Y2 consists of contractible tildeN
with one sphere at each lifted wedge point. Collapsing the contractible
subcomplex gives a wedge of spheres indexed by H, so Hurewicz identifies
pi_2(Y2) with the free ZH module generated by the new sphere. Attach a
3-cell along the class f in that module to form Y. This does not change
pi_1. A pinched/whiskered sphere representative realizes the coefficients
1,-2,2 on the sphere at1,a,b, with every whisker in the original one-
skeleton. Its cellular degree in the old N two-cells is zero. Consequently
there is an ACTUAL direct-sum chain decomposition

  C_*(tildeY;Z)=C_*(tildeN;Z) direct_sum
                 [ZH --R_f--> ZH, in degrees3->2].

The old N attaching maps have no component in the added sphere, so there
is no hidden mixed differential in either direction. The decomposition
persists after passing to every finite congruence cover. It gives

  H_1(Y_i)=H_1(N_i),
  H_2(Y_i)=Z^2 direct_sum coker(M_q),
  H_3(Y_i)=Z,
  rho^Z(Y_i)=log|det(M_q)|-log(q).

Here finite M_q is invertible over Q by the published/odd-determinant
argument, so its kernel is zero and its cokernel finite. At q=3 the added
torsion has order1729. Since1729 is squarefree, that finite abelian group
is cyclic; no full Smith decomposition is needed to know its order.

For the infinite complex use Kammeyer's proved injectivity and determinant
class calculation for R_f, not just the convention of deleting zero
singular values from a determinant. A square injective von Neumann
operator has zero-dimensional reduced cokernel by its polar decomposition.
Thus the two-term summand is L2-acyclic, and its L2-torsion is +log2
(the degree3 differential has sign (-1)^(3+1)). By the known nilmanifold
L2-acyclicity/torsion zero and direct-sum additivity,

  b_j^(2)(tildeY)=0 for all j,   rho^(2)(tildeY)=log2.

Kammeyer's ALL-q finite determinant bound then gives

  limsup_i rho^Z(Y_i)/q^3 <= (log5)/3 < log2=rho^(2)(tildeY).

This supplies the failure of a modified integral-torsion approximation
statement for this finite connected three-dimensional CW complex. The
gap is at least log2-(log5)/3. The checker verifies only q=3, not the
all-q input or the infinite Fuglede-Kadison calculation, which are borrowed
from the primary paper. Whether this consequence was stated previously
is not settled by this calculation; no first/novel result claim is made.

WHY THIS CANNOT REFUTE THE ACTUAL AIM TARGET: tildeY is not contractible.
Its ordinary H_2 is coker(R_f), already proved nonzero by the1729 test.
Hence Y is not aspherical. It is also not a closed manifold. Replacing the
original manifold by this space violates TWO explicit source hypotheses.
The latter failure has its own exact certificate: if Y were a closed
three-manifold, its connected27-sheeted cover Y_1 would be one. Its H_3=Z
forces orientability, so integral Poincare duality would make H_2 torsion-
free. The actual H_2 has a subgroup of order1729, a contradiction.
Keeping only dimension3, the correct fundamental group and L2-acyclicity
does not repair those violations. For the original nilmanifold itself,
the source's middle-degree equality remains the known zero=zero case.

Primary Kammeyer URL: https://arxiv.org/html/2609.15567v1
This exact group-ring example and its infinite/all-q bounds are published
prior art. The standalone subgroup and attachment proofs here are auxiliary
derivations with undetermined novelty, not a full-source closure.

7. REPRODUCIBILITY AND NEXT STEP
-------------------------------
check_heisenberg.py uses only the Python standard library, reads/writes no
files and uses explicit exceptions rather than assert. The actual isolated
run passed45,905 checks: 1,359 collected-subgroup fixtures, 19 independent
finite quotient subgroup/index controls, the exact27x27 determinant and
modular/block controls, and eight congruence levels. The saved receipt is
heisenberg-exact-run-001.json. Finite fixtures do not replace the universal
presentation proof, the all-level residual-intersection proof or the
credited analytic L2 theorems.

Replay:
  python3 -I -B check_heisenberg.py
  python3 -I -B -O check_heisenberg.py

Safe remaining target step: any attempted manifold/aspherical realization
must explicitly eliminate the ordinary universal-cover homology defect
without canceling the determinant torsion contribution. No such bridge is
provided here. Preserve this failed-transfer obstruction rather than count
the finite-CW result as an answer to AIM-GEOMETRIC_GROUP_THEORY-0134.
