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 $$CW$$-, $$n$$-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 $$f \mid QSF = 1_{QSF'' $$, 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 ref. . 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 $$i=n$$.

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