PROOF OF THE 4 DIMENSIONAL SMOOTH POINCARE CONJECTURE IN PDE s ALGEBRAIC TOPOLOGY The 4 dimensional smooth Poincar conjecture has been proved in the following work 1 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 2014 471532 ISBN 978 1 4939 1124 0 print 978 1 4939 1124 0 eBook DOI 10 1007 978 1 4939 1124 0 arXiv 1101 0283 math AT For complementary information see also the following works 2 A Pr staro Exotic heat PDE s Commun Math Anal 10 1 2011 64 81 arXiv 1006 4483 math GT 3 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 2012 369419 ISBN 978 3 642 28820 3 Print 978 3 28821 0 Online DOI 10 1007 978 3 642 28821 0 arXiv 1009 1176 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 1 and 2 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 1 2 3 4 5 6 one has also that any smooth n dimensional homotopy sphere M is diffeomorphic to S n In particular the case n 4 is related to the proof that the smooth Poincar conjecture is true