On the Gluing of germs of complex analytic spaces, Betti numbers and their structure

In this paper we introduce new classes of gluing of complex analytic spaces germs, called weakly large, large and strongly large. We give a description of their Poincaré series and, as applications, we give numerical criteria to determine when these classes of gluing of germs of complex analytic spaces are smooth, singular, complete intersections and Gorenstein in terms of their Betti numbers. In particular, we show that the gluing of the same germ of complex analytic space along of any subspace is always a singular germ.