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This paper aims to show that there exists a triangulation of the Heisenberg group $\mathbb{H}^n$ into singular simplexes with regularity properties on both the low-dimensional and high-dimensional layers. For low dimensions, we request our simplexes to be horizontal while, for high dimensions, we define a notion of straight simplexes using exponential and logarithmic maps and we require our simplexes to have high-dimensional straight layers. A triangulation with such simplexes is first constructed on a general polyhedral structure and then extended to the whole Heisenberg group. In this paper we also provide some explicit examples of grid and triangulations., 22 pages

This paper aims to define and study currents and slices of currents in the Heisenberg group $\mathbb{H}^n$. Currents, depending on their integration properties and on those of their boundaries, can be classified into subspaces and, assuming their support to be compact, we can work with currents of finite mass, define the notion of slices of Heisenberg currents and show some important properties for them. While some such properties are similarly true in Riemannian settings, others carry deep consequences because they do not include the slices of the middle dimension $n$, which opens new challenges and scenarios for the possibility of developing a compactness theorem. Furthermore, this suggests that the study of currents on the first Heisenberg group $\mathbb{H}^1$ diverges from the other cases, because that is the only situation in which the dimension of the slice of a hypersurface, $2n-1$, coincides with the middle dimension $n$, which triggers a change in the associated differential operator in the Rumin complex., 28 pages

This paper aims to define and study a notion of orientability in the Heisenberg sense ($\mathbb{H}$-orientability) for the Heisenberg group $\mathbb{H}^n$. In particular, we define such notion for $\mathbb{H}$-regular $1$-codimensional surfaces. Analysing the behaviour of a M\"obius Strip in $\mathbb{H}^1$, we find a $1$-codimensional $\mathbb{H}$-regular, but not Euclidean-orientable, subsurface. Lastly we show that, for regular enough surfaces, $\mathbb{H}$-orientability implies Euclidean-orientability. As a consequence, we conclude that non-$\mathbb{H}$-orientable $\mathbb{H}$-regular surfaces exist in $\mathbb{H}^1$., Comment: 19 pages

The purpose of this study is to analyse two related topics: the Rumin cohomology and the $\mathbb{H}$-orientability in the Heisenberg group $\mathbb{H}^n$. 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 $\mathbb{H}$-orientability for $\mathbb{H}$-regular surfaces and we prove that $\mathbb{H}$-orientability implies standard orientability, while the opposite is not always true. Finally we show that, up to one point, a M\"obius strip in $\mathbb{H}^1$ is a $\mathbb{H}$-regular surface and we use this fact to prove that there exist $\mathbb{H}$-regular non-$\mathbb{H}$-orientable surfaces, at least in the case $n=1$. This opens the possibility for an analysis of Heisenberg currents mod $2$., Comment: 111 pages, 1 figure