PROOF OF THE 4-DIMENSIONAL SMOOTH POINCARE' CONJECTURE IN PDE's ALGEBRAIC TOPOLOGY 

The 4-dimensional smooth Poincaré conjecture has been proved in the following work:

[1] A. Prástaro, 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. Prástaro, Exotic heat PDE's. Commun. Math. Anal. 10(1)(2011), 64-81. arXiv: 1006.4483[math.GT]. 

[3] A. Prástaro, 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. Prástaro 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 Poincaré conjecture is true.