  Topological Lifshitz transitions involve many types of topological structures
in momentum and frequency-momentum spaces: Fermi surfaces, Dirac lines, Dirac
and Weyl points, etc. Each of these structures has their own topological
invariant ($N_1$, $N_2$, $N_3$, $\tilde N_3$, etc.), which supports the
stability of a given topological structure. The topology of the shape of Fermi
surfaces and Dirac lines, as well as the interconnection of the objects of
different dimensions, lead to numerous classes of Lifshitz transitions. The
consequences of Lifshitz transitions are important in different areas of
physics. The singularities emerging at the transition may enhance the
transition temperature to superconductivity; the Lifshitz transition can be in
the origin of the small masses of elementary particles in our Universe; the
black hole horizon serves as the surface of Lifshitz transition between the
vacua with type-I and type-II Weyl points; etc.
