UNIVERSITY OF HELSINKI DEPARTMENT OF MATHEMATICS AND STATISTICS Insight in the Rumin Cohomology and Orientability Properties of the Heisenberg Group GIOVANNI CANARECCI Licentiate Thesis November 2018 Insight in the Rumin Cohomology and Orientability Properties of the Heis- enberg Group Licentiate Thesis Abstract: The purpose of this study is to analyse two related topics: the Rumin cohomology n and the H-orientability in the Heisenberg group H . 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 pull- back by a contact map and, more generally, describe pushforward and pullback explicitly in different situations. Differential forms can be used to define the no- tion of orientability; indeed in the fourth chapter we define the H-orientability for H-regular surfaces and we prove that H-orientability implies standard orientabil- ity, while the opposite is not always true. Finally we show that, up to one point, a Mobius¨ strip in H1 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. Contact information Giovanni Canarecci email address: giovanni.canarecci@helsinki.fi Office room: B418 Department of Mathematics and Statistics P.O. Box 68 (Pietari Kalmin katu 5) FI-00014 University of Helsinki Finland 3 “Beato colui che sa pensare al futuro senza farsi prendere dal panico” “Lucky is he who can think about the future without panicking” 4 Contents Introduction 7 1 Preliminaries 11 1.1 Lie Groups and Left Translation . 11 n 1.2 Definition of H ........................... 13 n 1.2.1 Left Invariance and Horizontal Structure on H ...... 15 n 1.2.2 Distances on H ...................... 22 n 1.2.3 Dimensions and Integration on H ............. 23 2 Differential Forms and Rumin Cohomology 25 2.1 The Rumin Complex . 25 2.2 Cohomology of H1 ......................... 29 2.3 Cohomology of H2 ......................... 30 n 3 Pushforward and Pullback in H 33 3.1 Definitions and Properties . 33 n 3.2 Commutation of Pullback and Rumin Complex in H ....... 35 3.3 Derivative of Compositions, Pushforward and Pullback . 41 3.3.1 General Maps . 41 3.3.2 Contact Maps . 44 3.3.3 Contact diffeomorphisms . 50 3.3.4 Higher Order . 53 4 Orientability 59 n 4.1 H-regularity in H ......................... 60 4.2 The Mobius¨ Strip in H1 ....................... 62 4.3 Comparing Orientabilities . 65 n 4.3.1 H-Orientability in H .................... 65 4.3.2 Invariances . 70 4.3.3 Comparison . 72 5 6 CONTENTS 5 Appendices 79 A Proof of the Explicit Rumin Complex in H2 ............ 79 1 2 B General Rumin Differential Operator dc in H and H ....... 84 n C Dimension of the Rumin Complex in H .............. 90 D Explicit Computation of the Commutation between Pullback and Rumin Complex . 93 E Calculations for the Mobius¨ Strip . 99 Bibliography 109 Acknowledgments 111 Chapter 1 Appendix A, Chapter 2 B and C Appendix D Chapter 3 Appendix E Chapter 4 Introduction The purpose of this study is to analyse two related topics: the Rumin cohomology and the orientability of a surface in the most classic example of Sub-Riemannian n geometry, the Heisenberg group H . Our work begins with a quick definition of Lie groups, Carnot groups and left translation operators, moving then to define the Heisenberg group and its proper- ties. There are many references for an introduction on the Heisenberg group; here n we used, for example, parts of [4], [5], [8] and [10]. The Heisenberg Group H , 2n 1 n ¥ 1, is a p2n 1q-dimensional manifold denoted pR ;¦;dccq where the group operation ¦ is given by ¢ ¢ ¢ 1 1 1 1 1 1 1 1 x x px;y;tq ¦ px ;y ;t q : x x ;y y ;t t ¡ xJ ; 1 y 2n 2 y y R ¢ 1 1 n 1 0 In with x;y;x ;y P R , t;t P R and J . Additionaly, the Heisenberg ¡In 0 Group is a Carnot group of step 2 with algebra h h1 ` h2. The first layer h1 has a standard orthonormal basis of left invariant vector fields which are called horizontal: # B ¡ 1 B Xj x j 2 y j t; B 1 B Yj y j 2 x j t; j 1;:::;n: They hold the core property that rXj;Yjs Bt : T for each j, where T alone spans the second layer h2 and is called the vertical direction. By definition, the horizontal subbundle changes inclination at every point (see Figure 1), allowing movement from any point to any other point following only horizontal paths. The Carnot–Caratheodory´ distance dcc, then, measures the distance between any two points along curves whose tangent vectors are horizontal. The topological dimension of the Heisenberg group is 2n 1, while its Hausdorff dimension with respect to the Carnot-Caratheodory´ distance is 2n 2. Such di- mensional difference leads to the existence of a natural cohomology called Rumin cohomology and introduced by M. Rumin in 1994 (see [19]), whose behaviour is significantly different from the standard de Rham one. This is not the only effect 7 8 INTRODUCTION Figure 1: Horizontal subbundle in the first Heisenberg Group H1.1 of the dimensional difference: another is that there exist surfaces regular in the Heisenberg sense but fractal in the Euclidean sense (see [12]). With a dual argument, one can associate at vector fields Xj’s,Yj’s and T the cor- responding differential forms: dx j’s and dy j’s for Xj’s and Yj’s respectively, and 1 ¸n q : dt ¡ px dy ¡ y dx q 2 j j j j j1 for T. They also divide as horizontal and vertical in the same way as before. These differential forms compose the complexes that, in the Heisenberg group, are described by the Rumin cohomology (see [19] and 5.8 in [8]). Rumin forms are compactly supported on an open set U and their sets are denoted by Dk pUq, H where # k Dk pUq : W ; for k 1;:::;n H Ik Dk pUq : Jk; for k n 1;:::;2n 1; H with Wk the space of k-differential forms, Ik ta ^ q b ^ dq { a P Wk¡1; b P Wk¡2u and Jk ta P Wk { a ^ q 0; a ^ dq 0u. The Rumin cohomology is the cohomology of this complex: dQ dQ dQ dQ dQ dQ 0 Ñ Ñ D1 pUq Ñ ¤¤¤ Ñ Dn pUq ÑD Dn 1pUq Ñ Dn 2pUq Ñ ¤¤¤ Ñ D2n 1pUq Ñ 0 R H H H H H pr s q r s where d is the standard differential operator and, for k n, dQ a Ik : da Ik 1 ; while, for k ¥ n 1, dQ : d| : The second order differential operator D is defined Jk as ¡ ¡ © © ¡1 pr s n q ¡p q ^ D a I : d a L da | n 1 q h1 whose presence reflects the difference between the topological and Hausdorff di- n¡1 n 1 mensions of the space. In the definition above L : h1 Ñ h1, Lpwq : 1pictures shown with permission of the author Anton Lukyanenko. INTRODUCTION 9 dq ^ w, is a diffeomorphism among differential forms. In Chapter 2 we will carefully describe the cohomology and we show its complete behaviour in the cases n 1 and n 2. In particular we show how to compute the second order operator D explicitly. In the appendices to this chapter we follow the presentation in [10] and explain how it is possible to write the Rumin dif-¨ ferential operators as one operator d , reducing then the complex to Dk pUq;d c H c (Appendix B). We also discuss the dimensions of the spaces in the Rumin com- plex (Appendix C). The differential operators dQ and D look much more complicated than the standard operator d and one could wonder whether they also hold the property of commut- ing with the pullback by a mapping. We show in Chapter 3 that this is true for contact maps, a map whose pushforward sends horizontal vectors to horizontal n n vectors. Namely one has that for a contact map f : H Ñ H the following rela- tions hold: # ¦ ¦ f dQ dQ f for k n; f ¦D D f ¦ for k n: We also show the behaviour of pushfoward and pullback in several situations in this setting, for which a useful starting point is [14]. Differential forms can be used to define the notion of orientability, so it is natural to ask whether the Rumin forms provide a different kind of orientability respect to the standard definition. In Chapter 4 we show that this is indeed the case. First, we have to notice how in the Heisenberg group it is natural to give an ad hoc definition of regularity for surfaces, the H-regularity (see [7] and [8]) which, roughly speak- ing, locally requires the surface to be a level set of a function with non-vanishing horizontal gradient. The points such gradient is null are called characteristic (see, for instance, [1] and [15]) and must usually be avoided. For such surfaces we give a new definition of orientability (H-orientability) along with some properties. In particular we show that it behaves well with respect to the left-translations and 2 the anisotropic dilation drpx;y;tq prx;ry;r tq. Furthermore, we prove that H- orientability implies standard orientability, while the opposite is not always true.