(Co)bordism groups in PDEs

We introduce a geometric theory of PDEs, by obtaining existence theorems of smooth and singular solutions. Within this framework, following our previo us results on (co)bordisms in PDEs, we give characterizations of quantum an d integral (co)bordism groups and relate them to the formal integrability o f PDEs. An explicit proof that the usual Thom-Pontryagin construction in (c o)bordism theory can be generalized also to a singular integral (co)bordism on the category of differential equations is given. In fact, we prove the existence of a spectrum that characterizes the singular integral (co)bordis m groups in PDEs. Moreover, a general method that associates, in a natural way, Hopf algebras (full p-Hopf algebras, 0 less than or equal to p less th an or equal to n - 1), to any PDE, recently introduced, is further studied. Applications to particular important classes of PDEs are considered. In pa rticular, we carefully consider the Navier-Stokes equation (NS) and explici tly calculate their quantum and integral bordism groups. An existence theor em of solutions of (NS) with a change in sectional topology is obtained. Re lations between integral bordism groups and causal integral manifolds, caus al tunnel effects, and the full p-Hopf algebras, 0 less than or equal to p less than or equal to 3, for the Navier-Stokes equation are determined.