Remote Access Bulletin of the American Mathematical Society

Bulletin of the American Mathematical Society

ISSN 1088-9485(online) ISSN 0273-0979(print)

Journals Home Search My Subscriptions Subscribe
Previous issue | This issue | Most recent issue | All issues (1891–Present) | Next issue | Previous article | Articles in press | Recently published articles | Next article
Request Permissions
 
 

 

Iterated path integrals


Author: Kuo-Tsai Chen
Journal: Bull. Amer. Math. Soc. 83 (1977), 831-879
MSC (1970): Primary 58A99, 55D35; Secondary 53B15, 53C65, 55H20, 49F05
DOI: https://doi.org/10.1090/S0002-9904-1977-14320-6
MathSciNet review: 0454968
Full-text PDF

References | Similar Articles | Additional Information

References [Enhancements On Off] (What's this?)

  • 1. J. F. Adams, On the cobar construction, Colloque de topologie algébrique (Louvain, 1956), George Thone, Liège; Masson, Paris, 1957, pp. 81-87. MR 19, 759. MR 90045
  • 2. J. F. Adams and P. J. Hilton, On the chain algebra of a loop space, Comment. Math. Helv. 30 (1956), 305-330. MR 17, 1119. MR 77929
  • 3. L. V. Ahlfors and L. Sario, Riemann surfaces, Princeton Univ. Press, Princeton, N. J., 1960. MR 22 #5729. MR 114911
  • 4. S. I. Al'ber, The topology of functional manifolds and the calculus of variation in the large, Uspehi Mat. Nauk 25 (154) (1970), no. 4, 57-122 = Russian Math. Surveys 25 (1970), no. 4, 51-117. MR 43 #5553. MR 279832
  • 5. A. Asada, Algebraic cohomology of loop spaces, J. Fac. Sci. Shinshu Univ. 4 (1969), 1-23. MR 41 #9241. MR 264650
  • 6. A. Asada, Representations of loop spaces and fibre bundles, J. Fac. Sci. Shinshu Univ. 4 (1969), 39-56. MR 41 #9256. MR 264665
  • 7. G. Baumslag, On the residual finiteness of generalized free products of nilpotent groups, Trans. Amer. Math. Soc. 106 (1963), 193-209. MR 26 #2489. MR 144949
  • 8. R. Bott, The space of loops on a Lie group, Michigan Math. J. 5 (1958), 35-61. MR 21 #1589. MR 102803
  • 9. R. Bott and H. Samelson, On the Pontryagin product in spaces of paths, Comment. Math. Helv. 27 (1953), 320-337 (1954). MR 15, 643. MR 60233
  • 10. R. Bott and H. Samelson, Applications of the theory of Morse to symmetric spaces, Amer. J. Math. 80 (1958), 964-1029. MR 21 #4430. MR 105694
  • 11. E. H. Brown, Jr., Twisted tensor products. I, Ann. of Math. (2) 69 (1959), 223-246. MR 21 #4423. MR 105687
  • 12. H. Cartan, Sur les groupes d'Eilenberg-Mac Lane H (II, n). I. Méthode des contructions, Proc. Nat. Acad. Sci. U. S. A. 40 (1954), 467-471. MR 16, 390. MR 65160
  • 13. K. -T. Chen, Iterated integrals and exponential homomorphisms, Proc. London Math. Soc. (3) 4 (1954), 502-512. MR 17, 394. MR 73174
  • 14. K. -T. Chen, Integration of paths, geometric invariants and a generalized Baker-Hausdorff formula, Ann. of Math. (2) 65 (1957), 163-178. MR 19, 12. MR 85251
  • 15. K. -T. Chen, Integration of paths--a faithful representation of paths by non-commutative formal power series, Trans. Amer. Math. Soc. 89 (1958), 395-407. MR 21 #4992. MR 106258
  • 16. K. -T. Chen, Formal differential equations, Ann. of Math. (2) 73 (1961), 110-133. MR 27 #371. MR 150370
  • 17. K. -T. Chen, Iterated path integrals and generalized paths, Bull. Amer. Math. Soc. 73 (1967), 935-938. MR 36 #806. MR 217717
  • 18. K. -T. Chen, Algebraic paths, J. Algebra 10 (1968), 8-36. MR 37 #5204. MR 229630
  • 19. K. -T. Chen, Covering-space-like algebras, J. Algebra 13 (1969), 308-326. MR 41 #8403. MR 263803
  • 20. K. -T. Chen, An algebraic dualization of fundamental groups, Bull. Amer. Math. Soc. 75 (1969), 1020-1024. MR 41 #5455. MR 260834
  • 21. K. -T. Chen, Algebras of iterated path integrals and fundamental groups, Trans. Amer. Math. Soc. 156 (1971), 359-379. MR 43 # 1069. MR 275312
  • 22. K. -T. Chen, Differential forms and homotopy groups, J. Differential Geometry 6 (1971), 231-246. MR 380858
  • 23. K. -T. Chen, Free subalgebras of loop space homology and Massey products, Topology 11 (1972), 237-243. MR 45 #4411. MR 295345
  • 24. K. -T. Chen, Iterated integrals of differential forms and loop space homology, Ann. of Math. (2) 97 (1973), 217-246. MR 380859
  • 25. K. -T. Chen, Fundamental groups, nilmanifolds and iterated integrals, Bull. Amer. Math. Soc. 79 (1973), 1033-1035. MR 47 #7676. MR 319130
  • 26. K. -T. Chen, Iterated integrals, fundamental groups and covering spaces, Trans. Amer. Math. Soc. 206 (1975), 83-98. MR 377960
  • 27. K. -T. Chen, Connections, holonomy and path space homology, Differential Geometry (Proc. Sympos. Pure Math., vol 27), Part 1, Amer. Math. Soc., Providence, R. I., 1975, pp. 39-52. MR 440540
  • 28. K. -T. Chen, Solvability on manifolds by quadratures permitting only integrals, Bull. Amer. Math. Soc. 80 (1974), 1210-1212. MR 50 #5845. MR 353361
  • 29. K. -T. Chen, Algebra, Topology and Category Theory, A Collection of Papers in Honor of Samuel Eilenberg, Academic Press, New York 1976, 19-32. MR 413151
  • 30. K. -T. Chen, Extension of C1, Advances in Math. 23 (1977), 181-210.
  • 31. P. Deligne, P. Griffiths, J. Morgan and D. Sullivan, Real homotopy theory of Kähler manifolds, Invent. Math. 29 (1975), 245-274. MR 382702
  • 32. W. G. Dwyer, Strong convergence of the Eilenberg-Moore spectral sequence, Topology 13 (1974), 255-265. MR 394663
  • 33. S. Eilenberg, Singular homology in differentiable manifolds, Ann. of Math. (2) 48 (1947), 670-681. MR 9, 52. MR 21314
  • 34. S. Eilenberg and S. Mac Lane, On the groups of H(π, n). I, Ann. of Math. (2) 58 (1953), 55-106. MR 15, 54.
  • 35. S. Eilenberg and J. C. Moore, Homology and fibrations. I, Comment. Math. Helv. 40 (1966), 199-236. MR 34 # 3579. MR 203730
  • 36. S. Eilenberg and J. C. Moore, Limits and spectral sequences, Topology l (1962), 1-23. MR 26 #6229. MR 148723
  • 37. E. Friedlander, P. A. Griffiths and J. Morgan, Homotopy theory and differential forms, Seminario di Geometria, Firenz (1972), mimeographed.
  • 38. T. Ganea, On the loop spaces of projective spaces, J. Math. Mech. 16 (1967), 853-858. MR 34 #6766. MR 206950
  • 39. V. K. A. M. Gugenheim, On the chain-complex of a fibration, Illinois J. Math. 16 (1972), 398-414. MR 46 #891. MR 301736
  • 40. P. J. Hilton and S. Wylie, Homology theory, Cambridge Univ. Press, New York, 1960. MR 22 #5963. MR 115161
  • 41. S. -T. Hu, Homotopy theory, Academic Press, New York and London, 1959. MR 21 #5186. MR 106454
  • 42. H. H. Johnson, A generalization of K.-T. Chen's invariants for paths under transformation groups, Trans. Amer. Math. Soc. 105 (1962), 453-461. MR 25 #5125. MR 141728
  • 43. W. Klingenberg, Mimeographed notes on closed geodesics (to appear in book form).
  • 44. S. Kobayashi and K. Nomizu, Foundations of differential geometry. Vol. I, Interscience, New York, 1963. MR 27 #2945. MR 152974
  • 45. E. R. Kolchin, Differential algebra and algebraic groups, Academic Press, New York, 1973. MR 568864
  • 46. D. Kraines, Massey higher products, Trans. Amer. Math. Soc. 124 (1966), 431-449. MR 34 #2010. MR 202136
  • 47. D. Kraines, Primitive chains and H, Topology 8 (1969), 31-38. MR 38 #6581. MR 238305
  • 48. Saunders Mac Lane, Homology, Classics in Mathematics, Springer-Verlag, Berlin, 1995. Reprint of the 1975 edition. MR 1344215
  • 49. Wilhelm Magnus, Beziehungen zwischen Gruppen und Idealen in einem speziellen Ring, Math. Ann. 111 (1935), no. 1, 259–280 (German). MR 1512992, https://doi.org/10.1007/BF01472217
  • 50. W. S. Massey, Higher order linking numbers, Conf. on Algebraic Topology (Univ. of Illinois at Chicago Circle, 1968), Univ. of Illinois at Chicago Circle, Chicago, 1969, pp. 174-205. MR 40 #8039. MR 254832
  • 51. J. P. May, Matric Massey products, J. Algebra 12 (1969), 533-568. MR 39 #289. MR 238929
  • 52. J. W. Milnor, Isotopy of links, Algebraic Geometry and Topology, Princeton Univ. Press, Princeton, N. J., 1957, pp. 280-306. MR 19, 1070. MR 92150
  • 53. J. W. Milnor, Morse theory, Ann. of Math. Studies, no. 51, Princeton Univ. Press, Princeton, N. J., 1963. MR 29 #634. MR 163331
  • 54. J. W. Milnor and J. C. Moore, On the structure of Hopf algebras, Ann. of Math. (2) 81 (1965), 211-264. MR 30 #4259. MR 174052
  • 55. A. L. Oniščik, On cohomologies of spaces of paths, Mat. Sb. 44 (86) (1958), 3-52; English transl., Amer. Math. Soc. Transl. (2) 25 (1963), 283-334. MR 20 #6101. MR 99662
  • 56. A. N. Parsin, A generalization of Jacobian variety, Izv. Akad. Nauk SSSR Ser Mat. 30 (1966), 175-182; English transl., Amer. Math. Soc. Transl. (2) 84 (1969), 187-196. MR 33 #4956. MR 196770
  • 57. R. Ree, Lie elements and an algebra associated with shuffles, Ann. of Math. (2) 68 (1958), 210-220. MR 20 #6447. MR 100011
  • 58. R. Ree, Generalized Lie elements, Canad. J. Math. 12 (1960), 493-502. MR 22 #5698. MR 114880
  • 59. J.-P. Serre, Homologie singulière des espaces fibrés. Applications, Ann. of Math. (2) 54 (1951), 425-505. MR 13, 574. MR 45386
  • 60. L. Smith, Homological algebra and the Eilenberg-Moore spectral sequence, Trans. Amer. Math. Soc. 129 (1967), 58-93. MR 35 #7337. MR 216504
  • 61. L. Smith, Lectures on the Eilenberg-Moore spectral sequence, Lecture Notes in Math., vol. 134, Springer-Verlag, Berlin and New York, 1970. MR 43 # 1191. MR 324702
  • 62. J. W. Smith, The de Rham theorem for general spaces, Tôhoku Math. J. (2) 18 (1966), 115-137. MR 34 #2027. MR 202154
  • 63. J. Stallings, Homology and central series of groups, J. Algebra 2 (1965), 170-181. MR 31 #232. MR 175956
  • 64. U. Stammbach, Homology in group theory, Springer-Verlag, Berlin, 1973. MR 382477
  • 65. J. D. Stasheff, H-spaces and classifying spaces-Foundations and recent developments, Algebraic Topology (Proc. Sympos. Pure Math., vol. 22), Amer. Math. Soc., Providence, R. I., 1971, pp. 247-272. MR 47 #9612. MR 321079
  • 66. D. Sullivan, Differential forms and the topology of manifolds, Manifolds--Tokyo 1973 (Proc. Internat. Conf. on Manifolds, Tokyo, 1973), Math. Soc. Japan, Univ. of Tokyo Press, Tokyo, 1975, pp. 37-49. MR 50 # 11236. MR 370611
  • 67. M. E. Sweedler, Hopf algebras, Benjamin, New York, 1969. MR 40 #5705. MR 252485
  • 68. A. Weil, Sur les théorèmes de de Rham, Comment. Math. Helv. 26 (1952), 119-145. MR 14, 307. MR 50280
  • 69. A. Weil, Introduction à l'étude des variétés kählériennes (Publ. Inst. Math. Univ. Nancago, VI), Actualités Sci. Ind. No. 1267, Hermann, Paris, 1958. MR 22 # 1921. MR 111056
  • 70. J. H. C. Whitehead, An expression of Hopf's invariant as an integral, Proc. Nat. Acad. Sci. U.S.A. 33 (1947), 117-123. MR 8, 525. MR 20255
  • 71. H. Whitney, Geometric integration theory, Princeton Univ. Press, Princeton, N. J., 1957. MR 19, 309. MR 87148
  • 72. W. T. Wu, Theory of I, Sci. Sinica 18 (1975), 464-482. MR 645386
  • 73. W. G. Dwyer, Exotic convergence of the Eilenberg-Moore spectral sequence, Illinois J. Math. 19 (1975), 607-617. MR 383409
  • 74. V. K. A. M. Gugenheim, On the multiplicative structure of the de Rham theory, J. Differential Geometry 11 (1976), 309-314. MR 418083
  • 75. V. K. A. M. Gugenheim, On Chen's iterated integrals (to appear).

Similar Articles

Retrieve articles in Bulletin of the American Mathematical Society with MSC (1970): 58A99, 55D35, 53B15, 53C65, 55H20, 49F05

Retrieve articles in all journals with MSC (1970): 58A99, 55D35, 53B15, 53C65, 55H20, 49F05

Additional Information

DOI: https://doi.org/10.1090/S0002-9904-1977-14320-6