Resumen
In this memory, we will work with the Morse Inequalities, one of
the most important results of the Morse theory. Although our goal is to
continue the work that Robin Forman letf us when make the discrete version
of the Morse Theory, we will also discuss the classical theory. First of
all, we will start with homological algebra and simplicial complexes since
it will be very useful to understand the main results. We will continue
proving this inequalities following Milnor for the case of differentiable
Manifolds. Subsequently, We study analogous inequalities in the discrete
case as Forman gave us. In all this work Betti numbers and critical
simplices play an important role. Then, since we only had results for the
case of finite graphs, we look for other relationhips for the infinite
case by following the work of J.A. Vilches and others. Morse Inequalities
will be generalized so they are valid for infinite graphs and we will
present a new critical element for this type of graph: ray.