  We provide a classification of type AI topological quantum systems in
dimension d=1,2,3,4 which is based on the equivariant homotopy properties of
"Real" vector bundles. This allows us to produce a fine classification able to
take care also of the non stable regime which is usually not accessible via
K-theoretic techniques. We prove the absence of non-trivial phases for one-band
AI free or periodic quantum particle systems in each spatial dimension by
inspecting the second equivariant cohomology group which classifies "Real" line
bundles. We also show that the classification of "Real" line bundles suffices
for the complete classification of AI topological quantum systems in dimension
d<4. In dimension d=4 the determination of different topological phases (for
free or periodic systems) is fixed by the second "Real Chern class which
provides an even labeling identifiable with the degree of a suitable map.
Finally, we provide explicit realizations of non trivial 4-dimensional free
models for each given topological degree.
