# PlanetPhysics/Operator Algebra and Complex Representation Theorems 2

### CW-complex representation theorems in quantum operator algebra and quantum algebraic topology

\htmladdnormallink{QAT {http://planetphysics.us/encyclopedia/QuantumOperatorAlgebra5.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 ref. [1]. 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 ${\displaystyle i=n}$.