  Topology in momentum space is the main characteristics of the ground states
of a system at zero temperature, the quantum vacua. The gaplessness of fermions
in bulk, on the surface or inside the vortex core is protected by topology.
Irrespective of the deformation of the parameters of the microscopic theory,
the energy spectrum of these fermions remains strictly gapless. This solves the
main hierarchy problem in particle physics. The quantum vacuum of Standard
Model is one of the representatives of topological matter alongside with
topological superfluids and superconductors, topological insulators and
semi-metals, etc. There is a number of of topological invariants in momentum
space of different dimensions. They determine universality classes of the
topological matter and the type of the effective theory which emerges at low
energy, give rise to emergent symmetries, including the effective Lorentz
invariance, and emergent gauge and gravitational fields. The topological
invariants in extended momentum and coordinate space determine the bulk-surface
and bulk-vortex correspondence, connecting the topology in bulk with the real
space. The momentum space topology gives some lessons for quantum gravity. In
effective gravity emerging at low energy, the collective variables are the
tetrad field and spin connections, while the metric is the composite object of
tetrad field. This suggests that the Einstein-Cartan-Sciama-Kibble theory with
torsion field is more relevant. There are also several scenarios of Lorentz
invariance violation governed by topology, including splitting of Fermi point
and development of the Dirac points with quadratic and cubic spectrum. The
latter leads to the natural emergence of the Horava-Lifshitz gravity.
