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
, 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 ref. [1].
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) spin networks and spin foams.
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