5. Minimality and arbitrary dimension

Reader card. This chapter performs the short dynamical calculation that kills every pairwise integer curvature and then assembles the block-supported theorem for any number of transverse generators. The coefficient group still has rank one; this is not a higher-rank absorption theorem.

We now prove that the integer transfer-function curvatures vanish for any finite number of transverse generators. The PV/Barlak comparison in Technical Engine A turns that calculation into vanishing of the operator \(K_1\)-defects.

5.1 Pairwise curvature is height-invariant

For each \(i\), choose \(h_i\in C(\Sigma,\mathbb Z)\) satisfying

\[ (\alpha_i-1)g=(\alpha_h-1)h_i. \]

For every pair define

\[ \kappa_{ij} =(\alpha_j-1)h_i-(\alpha_i-1)h_j. \]

Exactly as in Technical Engine A,

\[ \begin{aligned} (\alpha_h-1)\kappa_{ij} &=(\alpha_j-1)(\alpha_h-1)h_i -(\alpha_i-1)(\alpha_h-1)h_j\\ &=(\alpha_j-1)(\alpha_i-1)g -(\alpha_i-1)(\alpha_j-1)g\\ &=0. \end{aligned} \]

Thus

\[ \kappa_{ij}\in C(\Sigma,\mathbb Z)^{T_h}. \]

5.2 Minimality makes curvature constant

If an integer-valued continuous function is \(T_h\)-invariant and \(T_h\) is minimal, it is constant. Therefore

\[ \kappa_{ij}=m_{ij}\,1_\Sigma \]

for some integer \(m_{ij}\).

5.3 The constant has integral zero

The compact convex set of \(T_h\)-invariant probability measures is nonempty and is preserved by every \(T_i\). Since \(G=\mathbb Z^q\) is amenable, there is a probability measure \(\mu\) invariant under \(T_h\) and all of \(G\).

Integrating gives

\[ \begin{aligned} m_{ij} &=\int_\Sigma\kappa_{ij}\,d\mu\\ &=\int_\Sigma(\alpha_j-1)h_i\,d\mu -\int_\Sigma(\alpha_i-1)h_j\,d\mu\\ &=0. \end{aligned} \]

Hence

\[ \kappa_{ij}=0 \qquad\text{for all }i,j. \tag{5.1} \]

Minimality does identify the Barlak target with the actual \(K_1(B_h)\cong\mathbb Z\). The comparison theorem (A.4) and (5.1) therefore imply

\[ [\Omega_{ij}]=0\in K_1(C) \qquad\text{for all }i,j. \tag{5.2} \]

5.4 There are no higher LHS differentials

The class lies in

\[ E_2^{0,1} =H^0\!\left(G,H^1(H,M)\right). \]

The only possible outgoing differential is

\[ d_2:E_2^{0,1}\to E_2^{2,0}. \]

For \(r\geq3\),

\[ d_r:E_r^{0,1}\to E_r^{r,2-r} \]

has negative second degree, so its target is zero. Thus (5.1) removes the complete cohomological obstruction in the height-one row.

5.5 Completion of the block-supported proof

The remaining argument has four finite steps.

flowchart LR
    flow["Bounded clopen flows h_i"]
    curvature["κ_ij = 0"]
    gps["GPS: commuting symbols c_i"]
    block["Block support: ν = 1"]
    equivariant["Strict PE G-module"]

    flow --> curvature -->|"PV/Barlak comparison"| gps --> block --> equivariant

Chapter 1 constructs all \(W_i\) in one finite corner.
The PV/Barlak comparison theorem and (5.1) kill all actual \(K_1\)-curvatures.
The corner-groupoid, normalizer, orientation, and index lemmas in Chapter 3 allow one GPS retraction to make all symbols commute simultaneously.
Block support turns symbol equality into operator equality.

This proves the arbitrary-dimensional block-supported height-one theorem.

5.6 Why the result covers non-top heights

Nothing in Chapters 1–5 depends on a Rieffel height. They construct the coefficient equivariant module once. It may then be paired with any class in the transverse noncommutative torus that has a Rieffel–Heisenberg model.

If the transverse rank is \(q\), an elementary Rieffel module of height \(r\) uses

\[ M_T=\mathbb R^r\times\mathbb Z^{q-2r}, \qquad 0\leq2r\leq q. \]

The discrete factor is nontrivial below top height. The absorption proof does not alter it and imposes no top-dimensional condition.

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