Index

O-surgery, 250 I-jet, 283, 285

A-cusp-curves, 549 (a, b)-realization, 336 abelian representations, 85 action functional, 456 adapted pair, 350 admissible P, 235 algebre de Lie, 116 almost complex structure compati­

ble with w, 535 almost self-dual, 365, 375, 402

- configuration, 357, 362, 365, 367,369,393,411,419

ampleness, 585 anti-self dual, 591

- connection, 203, 437, 438, 441 anti-self-duality, 436 application moment, 109, 111, 113,

115, 117-119 Arnold conjecture, 124,283,297,

445, 446, 483, 556, 575 asymptotically fiat, 4, 43

- manifold, 5, 6, 32, 44, 49, 50, 53, 54

Atiyah-Jones conjecture, 305 axioms, 525

- of symplectic capacities, 525

Bennequin invariants, 99 bi-invariant, 529

- metric, 529 bicomplex, 632, 633 bicomplexes, 628 Bishop family, 338 blowing down, 439 Bogomolny equation, 3, 6, 43, 355,

360 Borromean rings, 253, 439 Bott-Shulman construction, 634 braid group, 100, 101, 103

Brieskorn sphere, 77, 253 bubbling, 304, 307 bubbling off, 587

c-homeomorhism, 188 c-regular, 186, 189 c-symplectomorphism, 192 CEwcapacity, 190, 191 Calabi-Yau manifold, 507, 515, 645 Caley transform, 466 calibrated, 337 canonical line bundle, 530 Cartan involution, 562, 563 Casson invariant, 99, 627 Cazimir functions, 455 center map, 382 Cerf diagram, 287 chain homotopy, 231 chain isomorphism, 96 characteristic foliation, 337, 338,

547, 554 charge, 7,8,46,637,640,647,676 Chern class, 203 Chern-Simons

- function, 78, 89, 96, 177,203 - functional, 107, 123, 165, 196,

210, 218, 268, 301, 537, 622 - gauge theory, 637, 638, 648,

665,667 - invariant, 203, 627 - theory, 647, 649, 654, 657,

661,668,669,676 Chern-Weil

- integral, 215, 231 - theory, 226, 232

circle bundle, 588, 589, 595, 598 Clark duality, 330 classifying space, 299, 301 Clifford module, 17, 18 closed characteristic, 530, 531, 552 coadjoint orbits, 100, 103 coadjoint representation, 100, 102

680

cobordism, 197-202, 224, 437, 442 compact Hermitian symmetric

space, 557, 560, 561, 566 compact smooth category, 303-305,

314 compact smooth framed category,

312,313 complex-like orientation, 337 configuration, 374, 375 conformally invariant, 640, 642 Conley-Zehnder index, 484, 501,

502, 505, 507, 508, 510, 515 connected sum decomposition, 88 connection, 3, 8-10, 14,35,43,44,

47-50, 53-55, 57, 100, 103, 196, 301, 355, 374, 409, 410, 412, 416, 421, 591, 627, 638, 639, 645, 647-649, 653, 665, 670, 671

connection deformed-flat, 215 contact

- form, 284 - isotopy, 286, 289, 294 - structure, 99 - transformation, 285, 288, 291

convex integration, 585 cotangent bundle, 100-103 coupling, 637, 640, 642, 646, 648,

649, 656, 657, 663, 664, 674-676 courbe hyperelliptique, 109, 112,

113 cup product, 136, 148, 160, 269,

279,610 curvature, 3, 4, 35, 43, 48, 54-57,

593, 649, 671 - 2-form,79

D-cohomology, 603, 604 de Rham-bar bicomplex, 628 deformation complex, 361 deformed instanton equations, 219,

224 deformed product, 279 degree of a gauge transformation,

237 Dehn surgery, 85, 242

Index

Dehn twist, 102 determinant line bundle, 82 Dirac monopole, 360, 370 Dirac string, 370 discriminant, 1I5-l18, 121 displace, 190 displacement energy, 529, 579 distance function, 186 Donaldson

- invariant, 107,435--438, 587, 588-591, 597, 600, 602, 604, 605

- polynomial, 435, 438 - Donaldson polynomial invari-

ants, 124 double complex, 628, 631

effective homology class, 445, 452 Eilenberg-Maclane space, 271 Einstein-Kahler, 560 elliptic, 333

- complex, 84 - orbit, 328

energy, 7, 593, 595, 651, 656, 657, 674

equation de Lax, III equivariant

- cohomology, 152, 153, 155, 161, 468, 473, 475

- differential forms, 631 - moment map, 563 - Morse complex, 156 - Morse theory, 447, 453 - signature, 467

Euler characteristic, 576, 577, 579-581

exact - diffeomorphism, 556 - isotopy, 555, 556 - Lagrangian submanifold, 556 - symplectic diffeomorphisms,

556 - triangle, 195, 212, 244, 247

excision, 245 - property, 201

Index

- theorem, 206

fatgraphs, 652, 659 fiber product, 599, 600 fibered product, 591, 592 fiberwise cone, 470 first Chern class, 445 flag, 100, 101, 103, 305 flat connection, 12, 13, 79, 80, 85,

205,206, 212, 216, 220, 221, 225, 226, 228, 231, 234, 237, 357, 627

Floer - class, 207 - co-complex, 262 - cohomology, 314, 319, 484,

500, 559, 587-590, 592, 596, 598, 600, 603, 604

- cohomology groups, 503, 508, 514

- complex, 262, 620 - differential, 204 - exact triangle, 435, 437, 438,

441, 442 - functor, 239 - grading, 229 - group, 268 - homology, 99, 125, 132, 146,

163, 164, 195,216,218,240, 268, 297, 298, 301, 318, 435-437, 441, 447, 535, 593, 597, 599,601

- homology groups, 197, 200-202

- homotopy, 299 - homotopy type, 297, 317-319,

322 - pairing, 262 - theory, 638, 647, 654, 655

Floer-Morse function, 298, 299, 301 flot geodesique, 109-111, 117, 118 flow category, 299, 300, 302, 304-

308,314-316, 318 flux, 550 foliation, 582

framed knot, 242 framed surgery, 95

681

framing, 304, 312, 313, 316 Fredholm, 18,22,24,43,46,60,

361, 363, 385, 386, 388, 428, 592, 606 - index, 6, 487 - operator, 216, 300, 301

front, 458, 466 Fukaya

- complex, 274 - construction, 261 - -Floer complex, 262

G-connections, 107 G-equivariant cohomology, 153,

162 gauge, 370, 372, 374, 375, 379, 412

- equivalence classes, 627 - group, 10, 20, 24, 43, 44, 48,

53, 65, 78, 610, 639, 647, 667 - theory, 302, 305, 587, 638,

639,658 - transformation, 379

generalized Laurent series, 498 generating family, 284, 292, 293,

462, 466, 467 - of the legendrian submanifold,

285 generating function, 292, 293, 460,

463, 464 geodesic flows, 103 geodesics, 102, 659 geodesique, 109, 111, 114, 119-121 geometric triangle, 208, 222 ghost number, 641-645, 653, 654,

656,661,666-669,671 ghosts, 637, 640, 642-644, 672 Gluck's surgery, 336 gluing, 198,261,267,416,612

- map, 131 - theorems, 172 - trajectories, 166

grading, 318 - function, 196

682

- of Floer homology, 301, 302 greek generating function, 289, 291 Gromov width, 526 Grothendieck splitting, 560

h-principle, 575, 582, 584 Hamiltonian

- diffeomorphism, 445, 453 - field, 576 - flow, 579 - isotopy, 455, 464, 555, 556 - symplectomorphisms, 550 - vector field, 105

Hamiltonians of knots, 89 handle, 435, 439, 440, 442 handle slide, 249 Hausdorff distance, 186 Hausdorff metric, 188 Hegaard splitting, 90 Hermitian symmetric space, 559 Higgs field, 4, 5, 8, 12, 15,23,31,

33, 46, 65, 71 Hilbert's zeroes theorem, 454 Hofer-Zehnder capacity, 527 holomorphic

- bisectional curvature, 560 - filling, 337 - foliation, 548

holonomic, 284 holonomy, 79, 547, 550-553 homogeneity, 455 homogeneous

- diffeomorphisms, 455 - functions, 455 - hamiltonian isotopy, 460 - vector fields, 455

homological 3-sphere, 99 homology 3-sphere, 77, 124, 195,

435, 437, 439, 440, 589, 598, 627

homology sphere, 86, 243, 257, 438, 587, 588

Hopf link, 254 hypersurface of co-Lipschitz type,

532

Index

hypersurface of contact type, 531

index bundle, 592, 596 index theory, 331 inner-regular, 189-191 instanton, I, 107, 197,216,217,

231,308,637,654,655,657-659,669, 670 - connection, 203 - equation, 214, 215 - Floer cohomology, 622 - homology, 77 -homology groups, 82, 195 - homology of a knot, 85 - -invariants, 77 - moduli space, 204, 263, 276

intersection form, 437-439 irreducible connections, 80 irreducible flat connections, 203,

357 isotopic forms, 546 isotopy, 286 isotopy compactly supported, 286,

290 iteration formula, 331

J-convex, 337, 338 J-flat,337 J -holomorphic, 549

- curves, 516 - disc ll, 338 - sphere, 486-488, 492, 495,

497,514,516,520 J -pseudoconvex boundary, 539 join, 469

K -theory, 302, 314, 322, 592, 596 K3 surface, 435, 438, 439, 442, 515 KAM theory, 327, 329 Kahler, 445

- form, 548 - manifold, 305, 560, 562

Kahler-Einstein manifolds, 562 Killing vector field, 561, 562 Kirby calculus, 195, 246, 248 knot complement, 85, 86

Index

Koszul complex, 468, 475 Kiinneth spectral sequence, 469

Lagrange - fibration, 100 - submanifold, 99

Lagrangian - foliation, 339 - submanifold, 105, 106, 555,

556, 575, 576 - subspace, 465, 576 - torus, 190 - embedding, 335

least action principle, 455 Legendre

- submanifold, 99 - transform, 330

Legendrian, 285 - submanifolds, 284

Leray spectral sequences, 473 Lie

- algebra, 100, 102 - field, 285 - group, 3,47, 61, 99, 103, 605

local intersection index, 339

magnetic charge, 4 manifold with comers, 594 Maslov index, 537 mass, 4, 12, 32, 44, 46

- of the singularity, 519 Massey product, 271 Milnor fiber, 99, 102,439 Milnor fibrations, 103 minimal Chern number, 484 minimal toric manifold, 452 mirror symmetry, 638, 647 moduli space, 1,3,5-8, 13,27,44,

196, 197,216,227,229,231, 233, 259, 308, 357, 437, 438, 441, 640, 642, 652, 659-661, 665, 666, 670, 675 - of gradient lines, 126 - of instantons, 257

momentum map, 480

683

monodromy, 79, 100-103, 553 monopole, 1, 3, 5, 8, 37, 356, 360,

361, 372 - moduli space, 356 - moment, 7

monotone, 483, 485 - Lagrangian submanifold, 558 - symplectic manifold, 304, 446

Morse - complex, 124, 134, 136, 137,

139, 146, 158, 269, 609 - inequalities, 124, 127, 537 - theory, 125, 297, 298, 308,588

Morse-Bott - complex, 124, 139, 146, 150,

151, 159 - function, 124, 146, 147, 150-

152, 155, 159, 273 - theory, 137, 155

Morse-Novikov theory of multi­valued functionals, 447

Morse-Smale - flow, 79, 505 - gradient flows, 81 - vector field, 609

natural connection, 547 Newton lattices, 454 non-Lagrangian submanifold, 576 normal submanifold, 576, 578 normalized embedding, 347 Novikov homology, 486, 507

open string, 638, 652, 653, 658, 659,661,666, 670-672, 674-676

orbites coadjointes, 109, 110, 118 outer-regular, 189, 191

'if-compatible, 545, 546, 549, 550 Palais-Smale condition, 609 parallel transport, 80 partial differential relation, 575 phase function, 284

- quadratic, 290 Poincare

- dual, 442, 591

684

- metric, 548 - sphere, 77

Poisson - manifolds, 634 - structure, 102 - variety, 455

polarization, 298, 300, 301, 305, 319, 320

polarized manifold, 299, 300, 301, 304,319

polydisc, 336 polynomial invariant, 280 polynomial symplectomorphism, 99 Pontrjagin class, 79 positively semitransversal, 340 pro-spectrum, 299, 300, 313, 314,

317,318,320-323 probleme de Neumann, 109, III product connection, 83 projectively flat connection, 198,

201 pseudo-holomorphic, 298 pseudo-holomorphic curves, 105-

107,556 pseudo-isotopic forms, 546 pseudo-isotopy, 549 pseudo gradient vector field, 609

quadratic, 284 quadrique, 109, 111, 112, 114,

117,119-121 quantization, 593 quasi-functions, 284 quasi-periodic, 329 quasi-periodic behaviour, 327

rational surface, 545 ray space, 469-471 real form, 561 reduced equivariant cohomology,

471 reduced moduli space, 259 reduced semi-infinite cohomology,

477 reducible flat connection, 221

Index

regular chains, 187 regular energy surface, 530, 531 relative

- Donaldson invariant, 107, 438, 441, 587, 590, 601, 602

- Floer groups, 538 - second Chern class, 196 - Tk -equivariant cohomology,

470 representation space, 627 restricted contact type, 189 restricted gauge group, 236 Ricci curvature, 560 Riemann-Roch theorem, 487 rigidity theorem, 185 ruled surfaces, 545, 554

50(3),435,438,441,442,605 - connections, 436

SU(2), 152, 163,302,589 SU2 , 3, 43, 47,58,65,66 Sard-Smale theorem, 491 secondary Pontrjagin class, 79 Segal conjecture, 299, 321 self-dual, 6, 15, 17, 22, 32, 33, 35,

43,650 - configuration, 360, 417 - connections, 357 - equation, 377 - monopole, 7, 44

self-duality, 645 self-indexing: strictly, 138 self-indexing: weakly, 138 self-intersection, 589, 605 semi-infinite cohomology, 476 semi-infinite dimensional cycles,

298 semi-infinite equivariant cohomol-

ogy, 476 Serre spectral sequence, 271, 475 sheaf,474 short foliation, 340 Siegel upper half plane, 548 signature, 99 simple J -holomorphic map, 489

Index

simple J -holomorphic sphere, 487, 488

simple map, 487 simplicial decomposition, 595 singular, 519 space of self-dual connections, 81 space-time, 637, 638, 649, 652,

653, 669, 670, 672, 674 - curvature, 648

spectral sequence, 205, 468, 598 stable category, 313, 316, 317, 319-

322 string field, 654 string theory, 637, 638, 642-644,

647, 653, 654, 656, 657, 659, 661-663, 665, 669, 671, 672, 676

surgery - cobordism, 87, 244 - on K, 199 - triangle, 87, 88

symplectic - capacity, 185, 188,525 - cylinder, 525 - diffeomorphism, 105, 106, 540,

541 - Floer homology, 619 - homology, 535,540 - neighbourhood theorem, 553 - reduction, 627 - ruled surface, 547 - shape, 336, 351

symplectic ally regular, 189, 190 symplectic ally unknotted, 335 systemes integrables, 109, III

tamed,337 temporal gauge, 81 topological

- category, 303, 305, 309 - field theory, 588, 637, 639,

654,657 - gravity, 640--642, 657, 665 - sigma model, 637, 638, 640,

641, 643, 644, 654, 660, 665

685

tores de Liouville, 109, 118, 120, 121

toric - manifold, 445, 448, 450, 455,

462, 479 - symplectic manifold, 449, 451 - variety, 450, 451

torus bundles, 240 torus-equivariant cohomology, 468 totally real submanifold, 577 transversal vector field, 576

universal G bundle, 605 unknotted, 335 unknottedness of Lagrangian sur­

faces, 335

vacuum configuration, 11-13, 18 vertex operator, 642, 644, 656, 662,

663 vertex operators, 642, 644, 662

Wahl-Neumann theorem, 99 wavefront, 287, 288 weakly monotone, 484, 485, 489 weight homomorphism, 497 Weinstein Conjecture, 531 Weizenbock formulas, 18, 35 Weyl group, 100 world-sheet, 637, 638, 641, 644,

648, 649, 652, 654, 656, 659, 669, 674, 675

Yang-Mills functional, 123 Yang-Mills theory, 638 Yang-Mills-Higgs

- action, 5,46 - theory, 3, 4, 43, 44, 53 - configurations, 359 - equations, 359, 360 - functional, 355, 357, 360, 409

,Zj8-grading, 435, 587 'zj8'z-graded, 598 ,Zj8'z-grading, 302

Progress in Mathematics

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Geometries in Interaction Special issue in honor of Mikhael Gromov Reprint aus GAFA, Vol. 5 (1995), No.2

Y. Eliashberg f R. Schoen, School of Stanford, CA, USAf V. Milman f L. Polterovich, School of Mathematical Sciences, Tel Aviv University, Israel (Eds)

1995. 444 pages. Hardcover ISBN 3-7643-5260-4

In the last decades of the xx century tremendous progress has been achieved in geometry. The discovery of deep interrelations between geometry and other fields, including algebra, analysis and topology, has pushed it into the mainstream of modern mathematics. This Special Issue, Geometries in Interaction, in honour of Mikhail Gromov contains 14 papers (originally published in Geometric And Functional Analysis vol. 5.2) which give a wide panorama of recent fundamental developments in modern geometry and its related subjects.

The contributors to this volume are 1. Bourgain, 1. Cheeger, J. Cogdell, A. Connes, Y. Eliashberg, H. Hofer, F. Lalonde, W. Luo, G. Margulis, D. McDuff H. Moscovici, G. MoSlow, S. Novikov, G. Perelman, 1. Piatetski-Shapiro, G. Pisier, X. Rong, Z. Rudnick, D. Salamon, P. Sarnak, R. Schoen, M. Shubin, 1(. Wysocki, and E. Zehnder.

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1994. 342 pages. Hardcover ISBN 3-7643-5066-0

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