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$ , $ 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 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.



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