Surgery and bordism groups in quantum partial differential equations. I: The quantum Poincaré conjecture

In this work, in two parts, we continue to develop the geometric theory of quantum PDE’s, introduced by us starting from 1996. (The second part is quoted in Prástaro [A. Prástaro, Surgery and bordism groups in quantum partial differential equations. II: Variational quantum PDE’s, Nonlinear Anal. TMA, in press ( 10.1016/j.na.2008.10.063)]) This theory has the purpose to build a rigorous mathematical theory of PDE’s in the category D S of noncommutative manifolds ( quantum (super)manifolds), necessary to encode physical phenomena at microscopic level (i.e., quantum level). Aim of the present paper is to report on some new issues in this direction, emphasizing an interplaying between surgery, integral bordism groups and conservations laws. In particular, a proof of the Poincaré conjecture, generalized to the category D S , is given by using our geometric theory of PDE’s just in such a category.