  We discuss the momentum-space topology of 3+1 and 2+1 strongly correlated
fermionic systems. For the 3+1 systems the important universality class is
determined by the topologically stable Fermi points in momentum space. In the
extreme limit of low energy the condensed matter system of this universality
class acquires all the symmetries, which we know today in high energy physics:
Lorentz invariance, gauge invariance, general covariance, etc. The chiral
fermions as well as gauge bosons and gravity field arise as fermionic and
bosonic collective modes of the system. This introduces the conceptual
similarity between such condensed matter and the quantum vacuum of Standard
Model, which also contains the Fermi point. For the 2+1 fermionic systems the
momentum space topology gives rise to the quantization of different physical
parameters, such as Hall conducivity and spin-Hall conductivity, and to quantum
phase transitions with the abrupt change of the momentum-space topology. The
combined p-space and r-space topology determines in particular the topology of
the energy spectrum of fermion zero modes living within the topological
defects; fermionic charges of the topologically nontrivial extended objects;
axial anomaly; etc.
