Differential cohomotopy versus differential cohomology for M-theory and differential lifts of Postnikov towers

Abstract

We compare the description of the M-theory form fields via cohomotopy versus that via integral cohomology. The conditions for lifting the latter to the former are identified using obstruction theory in the form of Postnikov towers, where torsion plays a central role. A subset of these conditions are shown to correspond compatibly to existing consistency conditions, while the rest are new and point to further consistency requirements for M-theory. Bringing in the geometry leads to a differential refinement of the Postnikov tower, which should be of independent interest. This provides another confirmation that cohomotopy is the proper generalized cohomology theory to describe these fields.