# PlanetPhysics/CW Complex Representation Theorems

### CW-complex representation theorems in quantum algebraic topologyEdit

*\htmladdnormallink{QAT* {http://planetphysics.us/encyclopedia/QAT.html} theorems for quantum state spaces of spin networks and quantum spin foams based on , -connected models and fundamental theorems.}

Let us consider first a lemma in order to facilitate the proof of the following theorem concerning spin networks and quantum spin foams.

**Lemma**
Let be a complex that has the (three--dimensional) Quantum Spin `Foam' (QSF) as a subspace. Furthermore, let be a map so that **Failed to parse (syntax error): {\displaystyle f \mid QSF = 1_{QSF'' }**
, with QSS being an arbitrary, local quantum state space (which is not necessarily finite). There exists an -connected model (Z,QSF) for the pair (QSS,QSF) such that}:

,

is an isomorphism for and it is a monomorphism for . The -connected model is unique up to homotopy equivalence. (The complex, , considered here is a homotopic `hybrid' between QSF and QSS).

**Theorem 2.** (Baianu, Brown and Glazebrook, 2007: In Section 9 of a recent NAQAT preprint).
For every pair of topological spaces defined as in **Lemma 1** ,
with QSF nonempty, there exist -connected models
for all . Such models can be then selected to have the property that the complex
is obtained from QSF by attaching cells of dimension , and therefore is -connected.
Following **Lemma 01** one also has that the map:
which is an isomorphism for , and it is a
monomorphism for .

*Note* See also the definitions of (quantum) *\htmladdnormallink{spin networks and spin foams* {http://planetphysics.us/encyclopedia/SpinNetworksAndSpinFoams.html}.}