3. Correcting the transports: the GPS normalizer retraction
Reader card. Input: finite-PE bisection transports whose pairwise corner \(K_1\)-defects vanish. Output: finite-PE gauge corrections making all underlying bisection symbols commute simultaneously. This chapter turns a stable index statement into exact geometric compatibility; it does not remove the separate magnetic diagonal phase.
In plain language, the argument puts the matrix colors into one ordered height orbit, separates each transverse transport into an index-zero local rearrangement and a successor-commuting part, and absorbs the local rearrangement by a PE gauge.
This chapter is the geometric heart of the proof. It upgrades \(K_1\)-triviality to exact commutation of the underlying finite PE bisections.
3.1 The corner as a one-dimensional orbit groupoid
Write \(P=\operatorname{diag}(p_1,\ldots,p_n)\). The full diagonal projection determines the clopen unit space
\[ Y=\{(x,a):p_a(x)=1\} \subset\Sigma\times\{1,\ldots,n\}. \tag{3.1} \]
Fullness here is concrete: Chapter 1 arranged that at least one \(p_a(x)\) is \(1\) for every \(x\). For fixed \(x\), order
\[ \Lambda_x =\{(m,a)\in\mathbb Z\times\{1,\ldots,n\}: p_a(T_h^m x)=1\} \]
lexicographically, first by \(m\) and then by the colour \(a\). Define \(\psi(x,a)\) to be the next coloured point in this ordered set. Since every height level has at least one colour, the height displacement of one successor step is either \(0\) or \(1\). It is locally constant because the \(p_a\) are clopen. Hence
\[ \psi:Y\to Y \tag{3.2} \]
is a homeomorphism. Its orbit through \((x,a)\) is exactly the set of all coloured points above the \(T_h\)-orbit of \(x\), so minimality of \(T_h\) implies minimality of \(\psi\).
The map which sends a power of the successor to the corresponding height arrow and colour change is an isomorphism
\[ Y\rtimes_\psi\mathbb Z \cong \bigl((\Sigma\rtimes_{T_h}\mathbb Z) \times\mathcal R_n\bigr)|_Y, \tag{3.3} \]
where \(\mathcal R_n\) is the full equivalence relation on the colour set. Thus the coloured full corner is exactly a Cantor minimal transformation groupoid, not merely Morita equivalent to one.
flowchart LR
y0["(x, colour 1)"]
y1["(T_h^a x, colour 3)"]
y2["(T_h^b x, colour 2)"]
y3["(T_h^c x, colour 1)"]
y0 -->|"ψ"| y1 -->|"ψ"| y2 -->|"ψ"| y3
3.2 The topological full group and its index
Let
\[ \Gamma=\tau[\psi] \]
be the topological full group. Every \(\gamma\in\Gamma\) has a locally constant orbit cocycle \(n_\gamma:Y\to\mathbb Z\) such that
\[ \gamma(y)=\psi^{n_\gamma(y)}(y). \]
There is a canonical index homomorphism
\[ I:\Gamma\to\mathbb Z, \qquad \Gamma_0=\ker I. \]
For the corresponding coefficient-one normalizer unitary \(u_\gamma\), the \(K_1\)-class is exactly this index under
\[ K_1(C(Y)\rtimes_\psi\mathbb Z)\cong\mathbb Z. \]
3.3 Transverse transports normalize the full group
Let \(Y_i\) be the unit space of \(\alpha_i(P)\). The automorphism \(\alpha_i\) gives an orientation-preserving groupoid isomorphism from the reduction over \(Y\) to the reduction over \(Y_i\). The coefficient-free symbol of \(W_i\) is a compact-open height bisection from \(Y_i\) to \(Y\). Their composite is therefore a homeomorphism
\[ S_i=\operatorname{symb}(W_i\alpha_i)|_Y. \]
It induces an automorphism of the groupoid in (3.3). Conjugating a compact-open full bisection by this automorphism again gives a compact-open full bisection. Equivalently, if \(\gamma(y)=\psi^{n_\gamma(y)}y\) with locally constant \(n_\gamma\), then the orbit cocycle of \(S_i\gamma S_i^{-1}\) is a finite composition of the locally constant cocycles of \(S_i,S_i^{-1}\), and \(\gamma\). Consequently
\[ S_i\in N(\Gamma), \]
the normalizer of \(\Gamma\) in \(\operatorname{Homeo}(Y)\).
We determine the orientation without identifying two different integer cocycles. For \(x\in\Sigma\), let \(\Lambda_x\) be the coloured set of units over the original height orbit of \(x\). It is ordered first by the original height coordinate \(m\in\mathbb Z\) and then by the finite colour. A level contributes at most the fixed finite number of colours. Because the corner is full and clopen and the height system is minimal, the contributing levels are syndetic: finitely many height translates cover the ambient unit space, so gaps between consecutive contributing levels are uniformly bounded. It follows that the two ends of a \(\psi\)-orbit are exactly the ends
\[ m\longrightarrow+\infty, \qquad m\longrightarrow-\infty \tag{3.4} \]
of the original height coordinate.
The map induced by \(\alpha_i\) preserves these two ends because \(\alpha_iT_h=T_h\alpha_i\). The compact-open bisection supplied by \(W_i\) has only finitely many branches. Hence there is an \(R_i<\infty\) such that each branch changes the original height coordinate by an integer of absolute value at most \(R_i\). It also preserves both ends. Their composite \(S_i\) therefore satisfies
\[ S_i(+\infty)=+\infty, \qquad S_i(-\infty)=-\infty. \tag{3.5} \]
This argument concerns the original height coordinate only. It does not assert that the successor exponent \(n_\gamma\), or the cocycle of the coloured system, equals that height coordinate.
3.4 The Giordano–Putnam–Skau decomposition
Define
\[ C^\varepsilon(\psi) ={} \{c\in\operatorname{Homeo}(Y): c\psi c^{-1}=\psi^{\pm1}\}. \]
Giordano–Putnam–Skau prove that multiplication gives
\[ N(\Gamma) \cong \Gamma_0\rtimes C^\varepsilon(\psi). \]
Consequently there is a homomorphic retraction
\[ r:N(\Gamma)\to C^\varepsilon(\psi), \qquad \ker r=\Gamma_0. \]
Write the GPS factorization first as
\[ S_i=\eta_i c_i, \qquad \eta_i\in\Gamma_0, \qquad c_i\psi c_i^{-1}=\psi^{\varepsilon_i}, \quad \varepsilon_i\in\{1,-1\}. \]
Every element of \(\Gamma\) has a bounded, locally constant successor exponent on compact \(Y\), so \(\eta_i\) preserves the two ends. If \(\varepsilon_i=-1\), then \(c_i\psi^ky=\psi^{-k}c_i y\), and \(c_i\) exchanges them. This contradicts (3.5), because both \(S_i\) and \(\eta_i\) preserve the ends. Thus \(\varepsilon_i=1\), and the factorization is
\[ S_i=\eta_i c_i, \qquad \eta_i\in\Gamma_0, \qquad c_i\psi=\psi c_i. \tag{3.6} \]
3.5 Apply the retraction to the square defect
The symbol of the square defect is the commutator
\[ \operatorname{symb}(\Omega_{ij})=[S_i,S_j] \]
up to the selected commutator convention. Applying \(r\) gives
\[ [c_i,c_j] =r\!\left(\operatorname{symb}(\Omega_{ij})\right). \tag{3.7} \]
Here is the required \(K_1\)-to-index comparison, with all maps typed. Let
\[ \partial_\psi: K_1(C(Y)\rtimes_\psi\mathbb Z) \longrightarrow K_0(C(Y)) \tag{3.8} \]
be the height PV boundary. Minimality gives \(K_0(C(Y))^\psi=\mathbb Z[1_Y]\), so exactness and the usual crossed-product convention give
\[ \partial_\psi[u_\gamma] =\varepsilon_\psi I(\gamma)[1_Y], \qquad \varepsilon_\psi\in\{1,-1\}. \tag{3.9} \]
Indeed, GPS Section 5 constructs the Fredholm homomorphism \(K_1(C(Y)\rtimes_\psi\mathbb Z)\to\mathbb Z\), proves that it sends \([u_\gamma]\) to \(I(\gamma)\), and sends the implementing unitary \([u_\psi]\) to \(1\). Since the height PV boundary sends \([u_\psi]\) to \(\varepsilon_\psi[1_Y]\), the two homomorphisms agree up to this single sign. Equation (3.9), rather than a boundary map into \(K_1\), is the precise comparison.
Let
\[ \Phi:C(Y)\rtimes_\psi\mathbb Z \xrightarrow{\cong}C=PM_n(B_h)P \tag{3.10} \]
be the algebra isomorphism induced by the groupoid isomorphism (3.3). A finite PE normalizer \(w\in C\) with symbol \(\gamma\) satisfies
\[ w=\Phi(d\,u_\gamma) \]
for a locally constant diagonal unitary \(d\in C(Y)\). On a finite clopen partition choose logarithms of the finitely many values of \(d\); hence \([d]=0\) and
\[ [w]=\Phi_{\ast}[u_\gamma]. \tag{3.11} \]
This avoids any assertion that a preselected numerical PV boundary is literally preserved by corner Morita equivalence. What is needed is only vanishing: \(\Phi_*\) is an isomorphism, and the inclusion of the full corner \(j:C\hookrightarrow M_n(B_h)\) is a \(K_1\)-isomorphism. The complete comparison actually used is therefore
\[ \begin{array}{ccccc} K_1(C(Y)\rtimes_\psi\mathbb Z) &\xrightarrow[\cong]{\ \Phi_{\ast}\ }& K_1(C) &\xrightarrow[\cong]{\ j_{\ast}\ }& K_1(M_n(B_h))\\ [u_\gamma]&\longmapsto&[w]&\longmapsto&j_{\ast}[w]. \end{array} \tag{3.11a} \]
Therefore (3.9)--(3.11) give
\[ [w]=0\text{ in }K_1(C) \quad\Longleftrightarrow\quad I(\gamma)=0. \tag{3.12} \]
Applied to the square defect, this says
\[ I\!\left(\operatorname{symb}(\Omega_{ij})\right)=0 \quad\Longleftrightarrow\quad [\Omega_{ij}]=0\text{ in }K_1(C). \tag{3.13} \]
Assume \([\Omega_{ij}]=0\) in the actual corner \(K_1\)-group. This is an input to the exact two-stage lemma. In the minimal-height theorem it follows from the integer-curl calculation and the PV/Barlak comparison (A.4); minimality removes the coinvariant quotient. Under this actual \(K_1\)-vanishing hypothesis,
\[ \operatorname{symb}(\Omega_{ij})\in\Gamma_0 =\ker r. \]
Equation (3.7) therefore yields
\[ [c_i,c_j]=1 \qquad\text{for every }i,j. \tag{3.14} \]
3.6 Lift the correction back to PE operators
Lift \(\eta_i^{-1}\) by its coefficient-one compact-open bisection unitary \(z_i\), and set
\[ W_i'=z_iW_i. \]
Since \(\eta_i\) has a finite clopen orbit partition, \(z_i\) is a finite PE height transport. The corrected semilinear operator \(W_i'\alpha_i\) has symbol \(c_i\).
The same retraction \(r\) is applied to every generator. Thus (3.14) holds simultaneously for all \(q\) directions; there is no iterative pairwise correction that could spoil an earlier pair.
flowchart LR
S["S_i in N(Γ)"]
factor["S_i = η_i c_i"]
eta["η_i in Γ_0<br/>finite PE gauge"]
c["c_i commutes with ψ"]
S --> factor
factor --> eta
factor --> c
eta -->|"multiply by η_i^−1"| c
3.7 What has—and has not—been proved
We have proved exact commutation of the symbols \(c_i\). If all operators are coefficient-one bisections and the transverse action preserves coefficient-one bisections, this already gives exact operator commutation. That is the block-supported case.
If \(\alpha_i(U_h)\) carries a magnetic phase, two operators with the same bisection can differ by a diagonal unitary. The next chapter identifies that residual defect.