(Co)bordism groups in PDEs.

We introduce a geometric theory of PDEs, by obtaining existence theorems of smooth and singular solutions. In this framework, following our previous results on (co)bordisms in PDEs, we give characterizations of quantum and integral (co)bordism groups and relate them to the formal integrability of PDEs. An explicitly proof that the usual Thom-Pontrjagin construction in (co)bordism theory can be generalized also to the case of singular integral (co)bordism in the category of differential equations is given. In fact, we prove the existence of a spectrum that characterizes the singular integral (co)bordism groups in PDEs. Moreover, a general method that associates in a natural way Hopf algebras (\textit{full $ p$-Hopf algebras, $ 0\le p\le n-1$}), to any PDE $ E_k\subset J^k_n(W)$, just introduced in \cite{PRA28, PRA52}, is further studied. Applications to particular important classes of PDEs are considered. In particular, we carefully consider the Navier-Stokes equation $(N)$ and explicitly calculate their quantum and integral bordism groups. An existence theorem of solutions of $(NS)$ with change of sectional topology is obtained. Relations between integral bordism groups and causal integral manifolds, causal tunnel effects, and the full $ p$-Hopf algebras, $ 0\le p\le 3$, for the Navier-Stokes equation are determined.