Insight in the Rumin Cohomology and Orientability Properties of the Heisenberg Group

Abstract

The purpose of this study is to analyse two related topics: the Rumin cohomology and the H-orientability in the Heisenberg group H^n. In the first three chapters we carefully describe the Rumin cohomology with particular emphasis at the second order differential operator D, giving examples in the cases n=1 and n=2. We also show the commutation between all Rumin differential operators and the pullback by a contact map and, more generally, describe pushforward and pullback explicitly in different situations. Differential forms can be used to define the notion of orientability; indeed in the fourth chapter we define the H-orientability for H-regular surfaces and we prove that H-orientability implies standard orientability, while the opposite is not always true. Finally we show that, up to one point, a Möbius strip in H^1 is a H-regular surface and we use this fact to prove that there exist H-regular non-H-orientable surfaces, at least in the case n=1. This opens the possibility for an analysis of Heisenberg currents mod 2.