Dictionary for tiling theorists
This chapter translates the main objects between tiling dynamics, groupoids, operator algebras, and the proof. It is not a replacement for the Glossary; it is a map between mathematical dialects.
Reader card. Read across a row whenever an operator-algebraic term obscures a geometric operation. The last column says why that object is present in the proof.
| Tiling or dynamical language | Operator-algebraic language | Role in the proof |
|---|---|---|
| Finite patch condition | Clopen cylinder \(C\subseteq\Sigma\) | Local coefficient data |
| Integer combination of patches | \(f\in C(\Sigma,\mathbb Z)\) | Signed pattern weight |
| Patch frequency | \(\mu(1_C)\) | Trace of a clopen projection |
| Translation orbit | Transformation groupoid | Arrows recording allowed translations |
| Covariant observable | Crossed-product element | Combines pattern coefficients and translations |
| Move mass along a height orbit | Add \((T_h-1)a\) | Does not change the height coinvariant class |
| Bounded local matching | Finite-PE compact-open bisection | Explicit module transport |
| Finite coloring or extra labels | Matrix stabilization | Supplies room for matchings without multiplying the graded class |
| Two routes around a square | \(W_i\alpha_i(W_j)\) and \(W_j\alpha_j(W_i)\) | Tests whether transports form a group action |
| Holonomy around the square | \(\Omega_{ij}\) | Operator obstruction to strict equivariance |
| Curl of bounded flows | \(\kappa_{ij}\) | Integer form of the same primary obstruction |
| Local orbit rearrangement | Topological-full-group element | Gauge used to correct transports |
| Index of an orbit rearrangement | Corner \(K_1\)-class | Detects whether GPS can absorb its non-AF part |
| Magnetic phase accumulated by translations | Twisted multiplier or Busby--Smith cocycle | Produces Pfaffian terms and the residual phase obstruction |
| Magnetic translation plane | Rieffel--Heisenberg module | Supplies the noncommutative-torus factor |
| Allowed integrated density of states | Trace of a \(K_0\)-class | Gap label |
Four distinctions to keep visible
A function versus its coinvariant class
A function \(f\) is concrete patch data. Its class \([f]\) forgets bounded height redistributions. Transverse invariance is imposed on \([f]\), not necessarily on the chosen representative \(f\).
Equality in \(K\)-theory versus an explicit isomorphism
The equality
\[ [P]=[\alpha_i(P)] \]
says that two projective modules have the same stable class. The theorem needs an explicit, finite-PE transport between chosen representatives. The clopen-flow construction supplies that extra locality.
Vanishing index versus exact commutation
The equation
\[ [\Omega_{ij}]=0\in K_1 \]
does not literally say \(\Omega_{ij}=1\). The GPS step turns zero index into finite-PE gauge corrections whose geometric symbols commute. In a magnetic field one must still check the remaining diagonal phase.
A gap-label group versus open spectral gaps
The trace range of \(K_0\) constrains the values that gaps may carry across covariant Hamiltonians. It does not say that every class is realized by an open gap of one fixed Hamiltonian. That is a further spectral problem.
A minimal amount of \(K\)-theory
Only three ideas are needed on a first reading.
- A projection represents a stable vector bundle or finite projective module.
- \(K_0\) permits formal differences of projections, which is why signed pattern data can be used.
- A unitary represents a \(K_1\)-class. Here that class measures the index of square holonomy.
The PV sequence computes the \(K\)-theory of crossing by one copy of \(\mathbb Z\). The technical curvature chapter uses it to translate the unitary class \([\Omega_{ij}]\) into the integer curl \(\kappa_{ij}\).
A minimal amount of \(KK\)-theory
For this proof, it is enough initially to regard \(KK\)-theory as a category in which:
- homomorphisms and geometric correspondences define generalized maps;
- homotopy-equivalent constructions become equal;
- exact sequences become exact triangles; and
- compatible constructions can be composed by the Kasparov product.
The book uses this language because \(1-\alpha_h\) is an additive \(KK\)-morphism rather than a \(*\)-homomorphism from which one could form an ordinary algebraic mapping cone. The enhanced equivariant framework provides the coherent cofibers needed to compare the PV filtration with Barlak's differential. Readers may use the comparison formula as a named theorem until reaching Part III.
A minimal amount of Morita equivalence
A full corner \(PAP\) and the algebra \(A\) carry the same stable module theory. Passing to a colored full corner lets the proof replace a matrix projection by the orbit groupoid of a single minimal successor map. This is not a claim that the corner and the original algebra are literally equal; it is a controlled equivalence preserving the relevant \(K\)-theory and trace data.