# 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 ${\displaystyle CW}$ , ${\displaystyle n}$ -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 ${\displaystyle Z}$  be a ${\displaystyle CW}$  complex that has the (three--dimensional) Quantum Spin Foam' (QSF) as a subspace. Furthermore, let ${\displaystyle f:Z\rightarrow QSS}$  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 ${\displaystyle n}$ -connected ${\displaystyle CW}$  model (Z,QSF) for the pair (QSS,QSF) such that}:

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

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