Chapter 6 Arc geometry and algebra: foliations, moduli spaces, string topology and field theory Ralph M. Kaufmann Contents Introduction .................................... 256 1 The spaces .................................. 260 1.1 The basic idea ............................. 260 1.2 Windowed surfaces with partial measured foliations ......... 260 1.3 The spaces of weighted arcs ...................... 261 1.4 Different pictures for arcs ....................... 261 1.5 Quasi-filling families, arc graphs and dual ribbon graphs ....... 264 1.6 Foliation picture ............................ 265 2 The gluing and the operad structures ..................... 266 2.1 Standard gluing for foliations ..................... 266 2.2 Cyclic operad structure: the scaling approach of [31] ......... 269 2.3 Chains and homology ......................... 271 2.4 Singular homology and singular chains ................ 271 2.5 Open/cellular chains .......................... 272 2.6 Modular structure: the approach of [32] ................ 273 2.7 S 1 action ................................ 274 2.8 Twist gluing .............................. 276 2.9 Variations on the gluings ........................ 276 3 Framed little discs and the Gerstenhaber and BV structures ......... 277 3.1 Short overview ............................. 277 3.2 (Framed) little discs and (spineless) cacti ............... 280 3.3 Cellular structure ............................ 282 3.4 The BV operator ............................ 284 3.5 The associator ............................. 287 4 Moduli space, the Sullivan-PROP and (framed) little discs ......... 289 4.1 Moduli spaces ............................. 289 4.2 Operad structure on moduli spaces ................... 290 4.3 The Sullivan quasi-PROP ....................... 291 5 Stops, stabilization and the Arc spectrum .................. 294 5.1 Stops: adding a unit .......................... 294