Beyond a rank-one height direction

Research direction. This chapter explains the obstruction tower and the missing higher-rank normalizer theorem. It is a roadmap, not an input to the height-one theorem. In particular, pairwise square curvature need not exhaust the coherence data when the coefficient rank exceeds one.

The theorem is arbitrary-dimensional in the number of transverse directions, but it uses one distinguished height generator. This distinction is structural.

What changes for \(H=\mathbb Z^s\)

Let \(H\cong\mathbb Z^s\) be the coefficient subgroup and \(G\cong\mathbb Z^q\) the transverse subgroup. For \(M=C(\Sigma,\mathbb Z)\), the Lyndon--Hochschild--Serre page becomes

\[ E_2^{a,b}=H^a\!\left(G,H^b(H,M)\right), \qquad 0\leq b\leq s. \]

This chapter uses cohomological height degree. Technical Engine A instead labels the height PV complex homologically. For the duality group \(H\cong\mathbb Z^s\), with its standard orientation,

\[ H^b(H,M)\cong H_{s-b}(H,M). \]

Thus in rank one \(H^1(\mathbb Z,M)=H_0(\mathbb Z,M)=M_{\mathbb Z}\), the coinvariants. The two chapters use Poincare-dual labels for the same height term; no grading reversal is intended.

When \(s=1\), the class used in this book lies in \(E_2^{0,1}\). Its only possible outgoing differential is \(d_2\). That is why a pairwise square curvature is the complete integral obstruction.

For \(s>1\), coefficient classes can have height degree \(b>1\), and then

\[ d_r:E_r^{0,b}\longrightarrow E_r^{r,b-r+1} \]

can be nonzero for several values of \(r\). Rectangular commutators no longer exhaust the coherence data.

flowchart TD
    R1["rank-one height: b = 1"]
    D2["only d_2 can leave E_r^(0,1)"]
    SQ["pairwise square curvature"]
    GPS["GPS retraction for one minimal homeomorphism"]
    RS["higher-rank height: b > 1 possible"]
    DR["d_2, d_3, ..., d_(b+1)"]
    HC["higher coherence data"]
    NG["no direct GPS replacement known"]
    R1 --> D2 --> SQ --> GPS
    RS --> DR --> HC --> NG

The second missing ingredient

The Giordano--Putnam--Skau normalizer splitting is tied to the ordered orbit structure of a single minimal Cantor homeomorphism. It converts an index-zero full-group element into the AF kernel and retracts the normalizer onto a cyclic centralizer.

A higher-rank coefficient groupoid has no canonical analogue of that cyclic retraction. A complete extension therefore needs both:

1. a cohomological analysis of all higher differentials; and
2. a finite PE normalizer theorem that turns their vanishing into coherent compact-open bisections.

The rank-one theorem should be viewed as the base case in which these two problems collapse to \(d_2\) plus GPS.