Quantum and Integral (Co)Bordisms in Partial Differential Equations

Abstract

Characterizations of quantum bordisms and integral bordisms in PDEs by means of subgroups of usual bordism groups are given. More precisely, it is proved that integral bordism groups can be expressed as extensions of quantum bordism groups and these last are extensions of subgroups of usual bordism groups. Furthermore, a complete cohomological characterization of integral bordism and quantum bordism is given. Applications to particular important classes of PDEs are considered. Finally, we give a complete characterization of integral and quantum singular bordisms by means of some suitable characteristic numbers. Some examples of interesting PDEs which arise in physics are also considered where existence of solutions with change of sectional topology (tunnel effect) is proved. As an application, we relate integral bordism to the spectral term \( E_1^{0,n - 1} \) that represents the space of conservation laws for PDEs. This also gives a general method to associate in a natural way a Hopf algebra to any PDE.