With a single 3-simplex (by identifying its faces in couples) it is possible to make 39 compact delta-complexes that can be grouped in 8 classes of complexes having the same homology groups (see mypaper Table 1).
My question is the following:
is there any book and/or paper where this complexes are studied and classified?
If not, how can I tell which one is homeomorphic to which one? I know that this is not a simple problem to solve (for example it is a non computable problem for D>4 where D is the dimension of the space) but I guess there must be some simple algorithm that can be applied to such a simple case (one simplex).
The reason for the my question is that there is a huge number of compact 3-delta-complexes, that can be grouped in 292 homology classes (see same link as before Tables 1, 2 and 3), composed of no more then 3 simplexes but the 3-torus (which I thought to be one of the simplest 3D space and which, if I am not wrong, has homology groups H0=Z, H1=Z^3, H2=Z^3, H3=Z) is not among them.
If the criteria for defining a 3-space to be simple and basic is the number of simplexes to make it as a delta-complex, then there are at least 292 spaces (although probably not all prime spaces) simpler then it.
These 292 spaces, what are they???