  This essay summarizes the efforts required to build a program of a unified,
low-dimension topology that allows characterizing all these flat space-times.
Since spatiotemporal manifolds are topological spaces equipped with metrics,
their properties are characterized by Clifford algebras in hypercomplex rings
associative with unity, so that Galileo's transformations are induced by a dual
number; the Lorentz transformations, by a perplexed number and the Euclid
transformations, by a complex number. This fact led us to establish an internal
automorphism in the ring of hybrid numbers that acts as a map of the manifolds
and induces the space-time metric based on the quality (characteristic) of the
associated hypercomplex unit. From this automorphism, we built hybrid
trigonometric functions, which we call Poincar\'e functions, which allowed us
to deduce general properties of space-time, hyperbolic, parabolic and
elliptical geometries and the groups SO (3), SO (4) and SO (1, 3). This
approach allows us to highlight the global properties of space-time, suggests
methods for geodynamic models and allows us to interpret anti-matter as matter
in a Euclidean space-time where the nature of time is imaginary.
