# PlanetPhysics/CW Complex Representation Theorems

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

\htmladdnormallink{QAT {http://planetphysics.us/encyclopedia/QAT.html} theorems for quantum state spaces of spin networks and quantum spin foams based on $CW$ , $n$ -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 $Z$  be a $CW$  complex that has the (three--dimensional) Quantum Spin Foam' (QSF) as a subspace. Furthermore, let $f:Z\rightarrow QSS$  be a map so that $\displaystyle f \mid QSF = 1_{QSF''$ , with QSS being an arbitrary, local quantum state space (which is not necessarily finite). There exists an $n$ -connected $CW$  model (Z,QSF) for the pair (QSS,QSF) such that}:

$f_{*}:\pi _{i}(Z)\rightarrow \pi _{i}(QST)$ ,

is an isomorphism for $i>n$  and it is a monomorphism for $i=n$ . The $n$ -connected $CW$  model is unique up to homotopy equivalence. (The $CW$  complex, $Z$ , 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 $(QSS,QSF)$  of topological spaces defined as in Lemma 1 , with QSF nonempty, there exist $n$ -connected $CW$  models $f:(Z,QSF)\rightarrow (QSS,QSF)$  for all $n\geq 0$ . Such models can be then selected to have the property that the $CW$  complex $Z$  is obtained from QSF by attaching cells of dimension $n>2$ , and therefore $(Z,QSF)$  is $n$ -connected. Following Lemma 01 one also has that the map: $f_{*}:\pi _{i}(Z)\rightarrow \pi _{i}(QSS)$  which is an isomorphism for $i>n$ , and it is a monomorphism for $i=n$ .