proof of the dimensional smooth poincare' conjecture in pde's algebraic topology the dimensional smooth poincar conjecture has been proved in the following work a pr staro exotic pde's mathematics without boundaries surveys in interdisciplinary research eds p m pardalos and th m rassias springer heidelberg new york dordrecht london isbn print ebook doi arxiv math at for complementary information see also the following works a pr staro exotic heat pde's commun math anal arxiv math gt a pr staro exotic heat pde's ii essays in mathematics and its applications dedicated to stephen smale eds p m pardalos and th m rassias springer heidelberg new york dordrecht london isbn print online doi arxiv math at the method used is the algebraic topology of pde's formulated by a pr staro in a series of early published works on the geometric theory of pde's and the characterization of their global solutions by means of suitable bordism groups in particular in and it is proved the following theorem theorem integral h cobordism in ricci flow pde's the ricci flow equation for n dimensional riemannian manifolds admits that starting from a n dimensional sphere s n we can dynamically arrive into a finite time to any n dimensional homotopy sphere m when this is realized with a smooth solution i e solution with characteristic flow without singular points then s n is diffeomorphic to m the other homotopy spheres sigma n that are homeomorphic to s n only are reached by means of singular solutions in particular for n belonging equal to one has also that any smooth n dimensional homotopy sphere m is diffeomorphic to s n in particular the case n is related to the proof that the smooth poincar conjecture is true