Technical Engine A: the PV/Barlak curvature comparison
Reader card. This chapter proves the load-bearing bridge between the concrete square holonomy and the integer transfer-function curl. Its input is the construction of Chapter 1; its output is the typed identity between the height PV boundary of the holonomy and the curl. Readers seeking the construction before the certification should first read Square holonomy and integer curvature.
The equation numbers retain the prefix \(2\) because this is the technical proof of Step 2 in the constructive argument.
The individual transports in (1.4) need not commute. For two transverse generators, define
\[ \Omega =W_1\alpha_1(W_2) \bigl(W_2\alpha_2(W_1)\bigr)^{\ast} \in U(C), \qquad C=PM_n(B_h)P. \]
This is the holonomy around one transverse square.
flowchart TD
P00["P"]
P10["α_1(P)"]
P01["α_2(P)"]
P11["α_1α_2(P)"]
P11 -->|"α_1(W_2)"| P10
P10 -->|"W_1"| P00
P11 -->|"α_2(W_1)"| P01
P01 -->|"W_2"| P00
The two paths have the same source and range. Their ratio is \(\Omega\).
A.1 The discrete curvature
Recall the transfer equations
\[ (\alpha_i-1)g=(\alpha_h-1)h_i. \]
Set
\[ \kappa_{12} =(\alpha_2-1)h_1-(\alpha_1-1)h_2. \tag{A.1} \]
Commutativity of the three automorphisms gives
\[ \begin{aligned} (\alpha_h-1)\kappa_{12} &=(\alpha_2-1)(\alpha_h-1)h_1 -(\alpha_1-1)(\alpha_h-1)h_2\\ &=(\alpha_2-1)(\alpha_1-1)g -(\alpha_1-1)(\alpha_2-1)g\\ &=0. \end{aligned} \]
Thus
\[ \kappa_{12}\in M^{\alpha_h}. \]
Changing \(h_i\) changes \(\kappa_{12}\) by the appropriate transverse coboundary, so its cohomology class depends only on \([f]\).
A.2 The Pimsner–Voiculescu target
Write \(M=C(\Sigma,\mathbb Z)\). Because \(K_1(C(\Sigma))=0\), the height Pimsner–Voiculescu sequence gives canonical, transverse-equivariant identifications
\[ K_0(B_h)\cong M_{\alpha_h}, \qquad \partial_h:K_1(B_h)\xrightarrow{\cong}M^{\alpha_h}. \tag{A.2} \]
For the \(\mathbb Z^2\)-action generated by \(\alpha_i,\alpha_j\), Barlak's target is
\[ E_2^{2,-1} =H^2\!\left(\mathbb Z^2,K_1(B_h)\right) \cong K_1(B_h)_{\langle\alpha_i,\alpha_j\rangle}. \tag{A.3} \]
The following comparison for the block-supported action is the main result of this chapter:
\[ \partial_h[\Omega_{ij}] =\varepsilon\, [\kappa_{ij}] \quad\text{in}\quad \left(M^{\alpha_h}\right)_{\langle\alpha_i,\alpha_j\rangle}, \qquad \varepsilon\in\{1,-1\}. \tag{A.4} \]
The sign \(\varepsilon\) is fixed once the covariance convention, the orientation \(i\wedge j\), and Barlak's Bott convention are fixed. Only vanishing is used below.
When \(T_h\) is minimal, \(M^{\alpha_h}=\mathbb Z1_\Sigma\). Every transverse automorphism acts trivially on this copy of \(\mathbb Z\); equivalently, \(\alpha_i(U_h)=\lambda_iU_h\) is homotopic to \(U_h\). Hence the quotient in (A.3) is canonically \(K_1(B_h)\cong\mathbb Z\), and (A.4) is an equality in the actual \(K_1\)-group, not merely in a quotient.
A.3 The PV/Barlak comparison
There are two comparisons to make: the corner defect must be Barlak's compressed unitary, and Barlak's abstract \(d_2\) must become the integer curl (A.1) under (A.2). Section A.3.1 proves the operator comparison. Section A.3.2 constructs the filtered equivariant PV comparison and computes its exact-couple staircase.
A.3.1 The compressed unitary is the corner holonomy
For a fixed pair \(i,j\), put
\[ q=\begin{pmatrix}P&0\\0&0\end{pmatrix}. \]
The partial isometry \(W_i\) has initial projection \(\alpha_i(P)\) and final projection \(P\). Therefore
\[ v_i= \begin{pmatrix} W_i^\ast&1-\alpha_i(P)\\ 1-P&W_i \end{pmatrix} \in M_{2n}(B_h) \tag{A.5} \]
is a unitary and satisfies
\[ \alpha_i(q)=v_iqv_i^\ast. \]
Apply Barlak's Theorem 4.4 with \(v=v_i\) and \(w=v_j\). Its generalized Bott unitary is
\[ \kappa\!\left( q,\, v_j^\ast\alpha_j(v_i)^\ast\alpha_i(v_j)v_i \right). \tag{A.6} \]
Compressing the second entry of (A.6) by \(q\) gives
\[ W_j\alpha_j(W_i)\alpha_i(W_j^\ast)W_i^\ast =\Omega_{ij}^\ast. \tag{A.7} \]
The generalized Bott notation in (A.6) means the class of this corner unitary, extended by \(1-q\). Since \(P\) is full, corner Morita equivalence sends it to \([\Omega_{ij}^\ast]\). This proves the operator part needed for (A.4), including the only possible global sign. It does not prove the transfer-curl comparison.
A.3.2 The equivariant PV comparison with the integer curl
Theorem 4.4 of Barlak identifies his \(d_2\) with the generalized Bott unitary in (A.6), but a separate naturality argument is needed to identify that differential with the transfer-function curl. We give that argument using Barlak's Baum--Connes skeletal model before passing to his mapping-torus model. This makes the height PV triangle a cofiber sequence of filtered objects by applying one exact functor; no cone of \(1-\alpha_h\) is formed in the category of \(C^*\)-algebras.
Put \(A_0=C(\Sigma)\) and \(G=G_{ij}=\langle\alpha_i,\alpha_j\rangle\cong\mathbb Z^2\). In the block-supported case, \(G\) acts on \(B_h=A_0\rtimes_{\alpha_h}\mathbb Z\) by its action on the coefficients and fixes the height unitary. The height Toeplitz extension
\[ 0\longrightarrow A_0\otimes\mathcal K \longrightarrow\mathcal T_{\alpha_h} \longrightarrow B_h\longrightarrow0 \]
is therefore \(G\)-equivariant: \(G\) acts trivially on the Toeplitz shift and by \(\alpha_i,\alpha_j\) on \(A_0\). Its standard completely positive Toeplitz splitting is \(G\)-equivariant as well.
We record why the two equivalences used here are equivariant. For the correspondence \({}_{\alpha_h}A_0\), its Fock module \(\mathcal F_{\alpha_h}=\bigoplus_{n\geq0}({}_{\alpha_h}A_0)^{\otimes n}\) carries \[ g(\delta_n\otimes a)=\delta_n\otimes\alpha_g(a), \] because every \(\alpha_g\) commutes with \(\alpha_h\). The coefficient representation is covariant for this action, and both the Fock shift and the degree projections commute with it. The standard Toeplitz Kasparov inverse to the coefficient inclusion is built from this representation, the Fock shift, and the degree projections; the usual rotation/shift homotopy proving that the two Kasparov products are identities uses the same operators. It therefore commutes with the displayed \(G\)-action at every parameter value. This is the ordinary Toeplitz \(KK\)-equivalence proof, now regarded as a proof in \(KK^G\), and shows \[ A_0\longrightarrow\mathcal T_{\alpha_h} \quad\text{is a }KK^G\text{-equivalence}. \] Similarly, \(G\) acts trivially on \(\mathcal K\), so the rank-one corner and the standard \(G\)-equivariant stabilization bimodule show that \(A_0\to A_0\otimes\mathcal K\) is a \(KK^G\)-equivalence. Under these two equivalences, the ideal inclusion is the difference of the identity and the correspondence class \([\alpha_h]\), as in the standard equivariant Toeplitz proof of the PV sequence. Hence the extension gives the equivariant PV triangle
\[ \mathrm{kk}^G(A_0) \xrightarrow{\ u\ } \mathrm{kk}^G(A_0) \longrightarrow \mathrm{kk}^G(B_h) \longrightarrow \Sigma\mathrm{kk}^G(A_0), \tag{A.8} \]
where \(u=1-[\alpha_h]\), up to the standard rotation/sign convention. Below we use \(u_*=\alpha_h-1\) on \(K_0(A_0)\); replacing \(u\) by its negative changes only the global sign \(\varepsilon\) in (A.4).
We use the stable equivariant enhancement of Bunke--Engel--Land. Their Theorems 1.3--1.4 make \(\mathrm{KK}^G_{\mathrm{sep}}\) a stable \(\infty\)-category and send equivariantly semisplit extensions to cofiber sequences. Their Proposition 1.7 makes tensor product bi-exact, and Theorem 1.22 makes crossed product an exact functor on the enhanced categories.
Let
\[ \varnothing=Y_{-1}\subset Y_0\subset Y_1\subset Y_2=\mathbb R^2 \]
be the \(G\)-invariant cubical skeletal filtration used by Barlak. For a \(G\)-object \(X\) of equivariant \(KK\), define
\[ \mathscr C_p(X) =\Bigl(C_0(Y_p)\otimes X\Bigr)\rtimes G \quad\text{in }\mathrm{KK}_{\mathrm{sep}}. \]
Here tensor product carries the diagonal action. Proposition 1.7 and Theorem 1.22 cited above show that every \(\mathscr C_p\) is exact. The reduced and maximal choices agree because \(G\cong\mathbb Z^2\) is amenable. The inclusions \(Y_{p-1}\subset Y_p\) induce restriction \(\ast\)-homomorphisms \(C_0(Y_p)\to C_0(Y_{p-1})\), natural in \(X\). The \(G\)-action is proper on \(Y_p\), and \(Y_{p-1}\) is second countable, so the restriction extensions are equivariantly semisplit by Bunke--Engel--Land, Proposition 1.12(1). Thus the collection
\[ \mathscr C_\bullet: \mathrm{KK}^G_{\mathrm{sep}}\longrightarrow \operatorname{Fun}(\{2\to1\to0\to-1\}, \mathrm{KK}_{\mathrm{sep}}) \]
is one exact filtered functor. Applying it to (A.8) gives, for \(p=0,1,2\), the commutative diagram
\[ \begin{array}{ccccccc} \mathscr C_p(A_0)&\xrightarrow{u_p}&\mathscr C_p(A_0) &\longrightarrow&\mathscr C_p(B_h)&\longrightarrow& \Sigma\mathscr C_p(A_0)\ \big\downarrow&&\big\downarrow&&\big\downarrow&&\big\downarrow\ \mathscr C_{p-1}(A_0)&\xrightarrow{u_{p-1}}& \mathscr C_{p-1}(A_0)&\longrightarrow& \mathscr C_{p-1}(B_h)&\longrightarrow& \Sigma\mathscr C_{p-1}(A_0), \end{array} \tag{A.9} \]
whose rows are cofiber sequences. Equivalently, in the stable functor category there is a canonical filtered equivalence
\[ \Phi_\bullet: \operatorname{cofib}\!\left( u_\bullet:\mathscr C_\bullet(A_0)\to \mathscr C_\bullet(A_0) \right) \xrightarrow{\ \simeq\ } \mathscr C_\bullet(B_h). \tag{A.10} \]
This is the required coherent comparison. It is not a choice of unrelated triangulated-category cones: (A.10) is obtained by applying a single exact \(\infty\)-functor to the actual equivariant Toeplitz triangle.
For completeness, let
\[ \mathscr J_p(X)=\operatorname{fib}\!\left( \mathscr C_p(X)\to\mathscr C_{p-1}(X)\right). \]
The stable \(3\times3\) lemma applied to (A.9) gives the following diagram in which every displayed row and column is a cofiber sequence:
\[ \begin{array}{ccccc} \mathscr J_p(A_0)&\longrightarrow&\mathscr C_p(A_0)& \longrightarrow&\mathscr C_{p-1}(A_0)\ {\scriptstyle u}\big\downarrow&& {\scriptstyle u_p}\big\downarrow&& {\scriptstyle u_{p-1}}\big\downarrow\ \mathscr J_p(A_0)&\longrightarrow&\mathscr C_p(A_0)& \longrightarrow&\mathscr C_{p-1}(A_0)\ \big\downarrow&&\big\downarrow&&\big\downarrow\ \mathscr J_p(B_h)&\longrightarrow&\mathscr C_p(B_h)& \longrightarrow&\mathscr C_{p-1}(B_h). \end{array} \tag{A.11} \]
To justify the cell identification and its signs, note that \(Y_p\setminus Y_{p-1}\) is the disjoint union of free \(G\)-orbits of open \(p\)-cells, one orbit for each \(\mu\in T(p,2)\). The restriction extension identifies the fibre with the crossed product of the corresponding open-cell ideal. On each orbit, Green imprimitivity gives
\[ \left(C_0(G\times(0,1)^p)\otimes X\right)\rtimes G \ \simeq\ C_0((0,1)^p)\otimes X \ \simeq\ \Sigma^pX . \]
Taking the finite direct sum over the cell orbits gives
\[ \mathscr J_p(X)\simeq \bigoplus_{\mu\in T(p,2)}\Sigma^p X. \tag{A.12} \]
This equivalence is natural in \(X\): restriction, crossed-product descent, and the cell imprimitivity bimodule above all tensor with \(\operatorname{id}_X\). Fix the coordinate orientations \(e_i,e_j\) and choose the two-cell incidence convention for which
\[ \partial_1(e_k)=(\alpha_k-1)v,\qquad \partial_2(e_i\wedge e_j) =(\alpha_j-1)e_i-(\alpha_i-1)e_j . \]
The endpoint identifications of the \(e_k\)-cell give the first formula; traversing the four oriented sides of the square gives the second. Commutativity of \(\alpha_i\) and \(\alpha_j\) makes \(\partial_1\partial_2=0\). Applying the coefficient action therefore gives Barlak's Koszul maps
\[ \begin{array}{ccccc} M&\xrightarrow{\ d_G^0\ }&M\oplus M& \xrightarrow{\ d_G^1\ }&M,\\[2mm] g&\longmapsto& \bigl((\alpha_i-1)g,(\alpha_j-1)g\bigr),&&\\[-1mm] && (h_i,h_j)&\longmapsto& (\alpha_j-1)h_i-(\alpha_i-1)h_j . \end{array} \tag{A.13} \]
Equivalently, these are the degree-one and degree-two attaching maps in Barlak's Corollary 2.5. Reversing the chosen two-cell orientation changes the final formula by one global sign.
Combining the cell identifications (A.12) with the \(3\times3\) diagram (A.11) gives, through transverse degree two, the following diagram on the relevant homotopy groups:
\[ \begin{array}{ccccc} C^0(G;M)_1&\xrightarrow{\ d_G\ }& C^1(G;M)_1&\xrightarrow{\ d_G\ }& C^2(G;M)_1\\ {\scriptstyle\alpha_h-1}\big\downarrow&& {\scriptstyle\alpha_h-1}\big\downarrow&& {\scriptstyle\alpha_h-1}\big\downarrow\\ C^0(G;M)_0&\xrightarrow{\ d_G\ }& C^1(G;M)_0&\xrightarrow{\ d_G\ }& C^2(G;M)_0 . \end{array} \tag{A.13a} \]
The subscripts are height homological degrees. Diagram (A.13a) is not an extra chain model imposed on the filtered object: it is the homotopy-group diagram of the cell-fibre cofiber sequences in (A.11). This observation is what makes its staircase the exact-couple staircase.
Barlak's \(C_p\) in his Baum--Connes cofiltration is the concrete \(C^*\)-algebra representing \(\mathscr C_p\). Proposition 1.4 of his paper constructs compatible imprimitivity bimodules and, after Brown stabilization, a commutative isomorphism of the entire cofiltrations \(C_\bullet\otimes\mathcal K\cong F_\bullet\otimes\mathcal K\). Consequently (A.10)--(A.11) induce an isomorphism of exact couples, including the maps \(i,j,k\), with the mapping-torus exact couple used in his Proposition 4.2 and Theorem 4.4. Thus it remains only to calculate the \(d_2\) of the filtered cofiber.
The height PV sequence and \(K_1(A_0)=0\) give the exact sequence of \(G\)-modules
\[ 0\longrightarrow K_1(B_h) \xrightarrow{\ \partial_h\ }M \xrightarrow{\ \alpha_h-1\ }M \xrightarrow{\ \iota_h\ }K_0(B_h) \longrightarrow0. \tag{A.14} \]
The maps are \(G\)-equivariant because they come from the equivariant triangle (A.8). In particular, (A.14), not merely its kernel and cokernel, is the two-extension carried by the filtered comparison.
Filtered-cofiber staircase lemma. Under the edge identifications in (A.14), the differential \[ d_2^{0,0}:H^0(G,K_0(B_h))\longrightarrow H^2(G,K_1(B_h)) \] of the exact couple of \(\mathscr C_\bullet(B_h)\) is, up to the fixed global orientation sign, the two-extension connecting map of (A.14). Explicitly, if \(x\) is represented by \(g\in M\) and \[ (\alpha_i-1)g=(\alpha_h-1)h_i,\qquad (\alpha_j-1)g=(\alpha_h-1)h_j, \tag{A.15} \] then \[ d_2^{0,0}(x)= \left[(\alpha_j-1)h_i-(\alpha_i-1)h_j\right]. \tag{A.16} \]
Proof. Apply the spectrum-valued functor \(\operatorname{map}_{\mathrm{KK}}(\mathbb C,-)\) to (A.11). Through transverse degree two, its homotopy-group diagram consists of two copies of the Koszul complex (A.13), with vertical map \(\alpha_h-1\), and the homotopy groups of their cofiber. This assertion uses only the cell fibres (A.12); no vanishing is asserted for the odd \(K\)-groups of the intermediate stages \(\mathscr C_1(A_0)\).
We spell out the exact-couple chase. Lift \(x\in K_0(B_h)^G\) first to \(g\in M\). Its first cellular boundary is \(d_G^0g\). Equations (A.15) say precisely that this boundary is the vertical boundary of \(h=(h_i,h_j)\). In the first two columns of the \(3\times3\) diagram (A.11), exactness therefore produces a class
\[ \bar x_{\mathrm{cof}} \in K_0\!\left( \operatorname{cofib}(u_1:\mathscr C_1(A_0)\to \mathscr C_1(A_0))\right) \]
which maps to \(x\) at stage zero. Concretely, it is the cofiber lift represented by \((g;h_i,h_j)\): its first total boundary is
\[ \bigl((\alpha_i-1)g-(\alpha_h-1)h_i, (\alpha_j-1)g-(\alpha_h-1)h_j\bigr)=0. \]
This equality is a vanishing statement in the homotopy group of the relevant \(K\)-theory mapping spectrum. Exactness of the \(3\times3\) diagram supplies the displayed cofiber class; no literal homotopy between the original projections is being asserted or chosen. The filtered equivalence (A.10) sends \(\bar x_{\mathrm{cof}}\) to a class in \(K_0(\mathscr C_1(B_h))\) mapping to \(x\). It is therefore exactly an admissible first-stage lift in Barlak's definition (Appendix A, Remark A.2), rather than merely an element with the same eventual image.
Apply the next oriented attaching map. The \(g\)-part has already been cancelled by (A.15), so the upper boundary is
\[ d_G^1h=(\alpha_j-1)h_i-(\alpha_i-1)h_j. \]
Moreover,
\[ (\alpha_h-1)d_G^1h=d_G^1(\alpha_h-1)h =d_G^1d_G^0g=0, \]
so exactness of the left column of (A.11) identifies this element with a class in \(K_1(B_h)=\ker(\alpha_h-1)\). By the definition \(d_2=ji^{-1}k\), this is the second-page differential.
For clarity, the choice indeterminacy is entirely the algebraic indeterminacy of the two-extension. Put \(u=\alpha_h-1\) and \(N=\ker u\). If another lift of \(x\) is \(g'=g+ua\), then
\[ d_Gg'=u(h+d_Ga). \]
Every compatible choice \(h'\) with \(uh'=d_Gg'\) consequently has
\[ h'=h+d_Ga+n,\qquad n\in C^1(G;N). \]
It follows that
\[ d_Gh'=d_Gh+d_Gn, \]
so the resulting element of \(H^2(G;N)\) changes only by a degree-two Koszul coboundary. This simultaneously handles the choices of \(g\), of \(h\), and of the exact-couple lift, without invoking projection homotopies. This proves (A.16). \(\square\)
The argument is coefficient-functorial and proves the promised universal form:
Universal PV/Barlak comparison lemma. Let \(A\) be a separable \(\mathbb Z^2\)-\(C^*\)-algebra with generators \(\alpha_i,\alpha_j\), let \(\theta\in\operatorname{Aut}(A)\) commute with that action, and put \(B=A\rtimes_\theta\mathbb Z\). Assume that the induced \(\mathbb Z^2\)-action on \(B\) fixes the implementing unitary and that \(K_1(A)=0\). Set \(M=K_0(A)\). For \(x\in K_0(B)^{\mathbb Z^2}\), choose \(g,h_i,h_j\in M\) with \[ \iota_\theta(g)=x,\qquad (\alpha_i-1)g=(\theta-1)h_i,\qquad (\alpha_j-1)g=(\theta-1)h_j. \] Then Barlak's differential satisfies \[ \partial_\theta\!\left(d_2^{0,0}(x)\right) =\varepsilon\left[ (\alpha_j-1)h_i-(\alpha_i-1)h_j \right] \in H^2\!\left(\mathbb Z^2,\ker(\theta-1)\right). \] The sign \(\varepsilon\) depends only on the PV and cellular orientation conventions.
Indeed, replace \(A_0,\alpha_h,B_h\) by \(A,\theta,B\) in (A.8)--(A.16). The exact skeletal functor, the cell-fibre \(3\times3\) diagrams, and the staircase proof are unchanged; the only coefficient hypothesis used in the two-extension is \(K_1(A)=0\).
At the group level the height direction is homological. Put the two copies of \(M\) in homological degrees one and zero:
\[ C_1=M\xrightarrow{\alpha_h-1}C_0=M, \qquad H_0(C_\bullet)=M_{\alpha_h}, \qquad H_1(C_\bullet)=M^{\alpha_h}. \tag{A.17} \]
Thus the mixed page is written
\[ H^a\!\left(G_{ij},H_b(\langle\alpha_h\rangle,M)\right), \]
not with height group cohomology in the same degree labels. Equivalently, Poincare duality for \(\mathbb Z\) rewrites the input as height cohomological degree one and the target as degree zero: \(H^1(\mathbb Z,M)=M_{\alpha_h}\) and \(H^0(\mathbb Z,M)=M^{\alpha_h}\). We use homological labels because they match the two-term PV complex \(C_1\to C_0\) directly. The edge maps in degrees zero and one are respectively the height PV identifications \(\iota_h^{-1}\) and \(\partial_h\) from (A.2). The filtered exact-couple isomorphism (A.10)--(A.11) gives the commutative comparison
\[ \begin{array}{ccc} H^0\!\left(G_{ij},K_0(B_h)\right) &\xrightarrow{\ d_{2,\mathrm{Barlak}}^{0,0}\ }& H^2\!\left(G_{ij},K_1(B_h)\right)\\ {\scriptstyle\iota_h^{-1}}\quad\big\downarrow&& {\scriptstyle\partial_h}\quad\big\downarrow\\ H^0\!\left(G_{ij},M_{\alpha_h}\right) &\xrightarrow{\ \varepsilon d_{2,\mathrm{mix}}\ }& H^2\!\left(G_{ij},M^{\alpha_h}\right). \end{array} \tag{A.18} \]
The lower arrow is the transgression from \(H^0(G_{ij},H_0)\) to \(H^2(G_{ij},H_1)\). The sign \(\varepsilon\) is fixed by the PV boundary and square orientations. Under \(\iota_h^{-1}\), the projection \(q\) in Section A.3.1 is represented by \(g+L\), where \(L\) is the trivial stabilization from Chapter 1. Constants have zero transverse boundary and do not change \(\kappa_{ij}\). Thus (A.16) sends \([q]\) to \([\kappa_{ij}]\). Combining (A.18) with the operator calculation (A.6)--(A.7) proves (A.4). The comparison is therefore a theorem, not an additional hypothesis.
A.4 More than two transverse generators
For \(G=\mathbb Z^q\), define for every pair
\[ \kappa_{ij} =(\alpha_j-1)h_i-(\alpha_i-1)h_j. \tag{A.19} \]
Barlak's operator-theoretic \(d_2\)-class has one component for each oriented coordinate two-plane. Its primary corner curvatures vanish exactly when
\[ [\Omega_{ij}]=0\in K_1(C) \qquad\text{for all }i,j. \]
The integer curls (A.19) have the same coordinate indexing. Applying the proved comparison (A.4) to every coordinate two-plane identifies the two families.
This is necessary for exact commuting transports. It is not yet sufficient: a \(K_1\)-trivial normalizer defect can still be nontrivial as a homeomorphism or as a diagonal phase. The following two chapters remove those layers in order.
A.5 The full-field primary curvature
The exact-couple comparison above uses block support because the transverse automorphisms had to fix the height unitary. The conclusion that the full-field and block-supported defects have the same primary \(K_1\)-class, however, follows by a direct homotopy.
For \(0\leq t\leq1\), define automorphisms of \(B_h\) by
\[ \beta_i^t(a)=\alpha_i(a)\quad(a\in C(\Sigma)), \qquad \beta_i^t(U_h)=e^{2\pi it\Theta_{ih}}U_h. \tag{A.20} \]
Thus \(\beta_i^0\) is the block-supported action and \(\beta_i^1\) is the action induced by the full field. The clopen projection \(P\) and the finite PE transports \(W_i\) constructed in Chapter 1 do not depend on \(t\), and \(\beta_i^t(P)=\alpha_i(P)\) for every \(t\). Hence the source and range identities for \(W_i\) hold for every \(t\), and
\[ \Omega_{ij}(t) =W_i\beta_i^t(W_j) \bigl(W_j\beta_j^t(W_i)\bigr)^{\ast} \in U(PM_n(B_h)P). \tag{A.21} \]
Each \(W_i\) is a finite sum of clopen-weighted powers of \(U_h\). On a term \(aU_h^m\), the only new factor in \(\beta_i^t\) is \(e^{2\pi it m\Theta_{ih}}\). Therefore (A.21) is a norm-continuous path in the fixed corner, and
\[ [\Omega_{ij}(1)]=[\Omega_{ij}(0)] \quad\text{in }K_1(PM_n(B_h)P). \tag{A.22} \]
This proves the full-field homotopy lemma. By the block-supported PV/Barlak comparison, minimality of the selected height then forces the primary \(K_1\)-curvatures for the full field to vanish as well. What the path does not remove is the locally constant diagonal phase left after the GPS symbol correction; Chapter 4 treats that phase and retains its finite-stage coboundary hypothesis.