CW-complex representation theorems in QAT

CW-complex representation theorems in quantum algebraic topology

QAT theorems for quantum state spaces of spin networks and quantum spin foams based on $CW$

-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 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 $i=n$ .

Note See also the definitions of (quantum) spin networks and spin foams.

"CW-complex representation theorems in QAT" is owned by bci1.
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