PlanetPhysics/CW Complex Representation Theorems

CW-complex representation theorems in quantum algebraic topologyEdit

\htmladdnormallink{QAT {} 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 {}.}