INTRODUCTION TO FOLIATIONS AND LIE GROUPOIDS

By U. MARAK

Prerequisites 1 1 Foliations 4 11 Definition andfirst examples 5 12 Alternative definitions of foliations 9 13 Constructions of foliations 14 2 Holonomy andstability 19 21 Holonomy 20 22 Riemannian foliations 25 23 Local Reeb stability 30 24 Orbifolds 34 25 Global Reeb stability in codimension 1 44 26 Thurston’s stability theorem 49 3 Two classical theorems 56 31 Haefliger’s theorem 57 311 Review of Morse functions 58 312 Morse functions into codimension 1 foliations 60 313 Proof of Haefliger’s theorem 62 32 Novikov’s theorem 65 321 Vanishing cycles 66 322 Existence of a compact leaf 71 323 Existence of a Reeb component 77 4 Molino’s theory 81 41 Transverse parallelizability 82 411 Homogeneous foliations 82 412 Transversely parallelizable foliations 86 42 Principal bundles 92 421 Connections on principal bundles 93 422 Transverse principal bundles 98 43 Lie foliations andMolino’ s theorem 101 431 Lie foliations 102 432 The Darboux cover 103 433 Molino’s structure theorem 108 5 Lie groupoids 110 51 Definition andfirst examples 111 52 The monodromy and holonomy groupoids 117 53 Some general constructions 121 54 Equivalence of Lie groupoids 127 55 Etale groupoids 134 56 Proper groupoids and orbifolds 140 57 Principal bundles over Lie groupoids 144 6 Lie algebroids 149 61 The Lie algebroidof a Lie groupoid 150 62 Definition ande xamples of Lie algebroids 153 63 Lie theory for Lie groupoids 157 64 Integrability andd evelopable foliations 160 References and further reading 166