Iterated path integrals

Bulletin of the American Mathematical Society

Volumn 83, Issue 5, 1977, Pages 831-879

  Chen, Kuo Tsai ^a  

^a UNIVERSITY OF ILLINOIS AT URBANA CHAMPAIGN   (United States)

Author keywords

Bar and cobar constructions; Differentiable spaces; Differential forms; Eilenberg Moore spectral sequence; Fibration; Fundamental groups; Loop space homplogy; Path space calculus; Twisting cochain

Indexed keywords

EID: 84966202575     PISSN: 02730979     EISSN: None     Source Type: Journal    
DOI: 10.1090/S0002-9904-1977-14320-6     Document Type: Article

References (75)

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.
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.
3. L. V. Ahlfors and L. Sario, Riemann surfaces, Princeton Univ. Press, Princeton, N. J., 1960. MR 22 #5729.
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.
5. A. Asada, Algebraic cohomology of loop spaces, J. Fac. Sci. Shinshu Univ. 4 (1969), 1-23. MR 41 #9241.
6. A. Asada, Representations of loop spaces and fibre bundles, J. Fac. Sci. Shinshu Univ. 4 (1969), 39-56. MR 41 #9256.
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.
8. R. Bott, The space of loops on a Lie group, Michigan Math. J. 5 (1958), 35-61. MR 21 #1589.
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.
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.
11. E. H. Brown, Jr., Twisted tensor products. I, Ann. of Math. (2) 69 (1959), 223-246. MR 21 #4423.
12. H. Cartan, Sur les groupes d'Eilenberg-MacLane 1(II, ń). I. Methode des contructions, Proc. Nat. Acad. Sci. U. S. A. 40 (1954), 467-471. MR 16, 390.
13. K.-T. Chen, Iterated integrals and exponential homomorphisms, Proc. London Math. Soc. (3) 4 (1954), 502-512. MR 17, 394.
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.
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.
16. K.-T. Chen, Formal differential equations, Ann. of Math. (2) 73 (1961), 110-133. MR 27 #371.
17. K.-T. Chen, Iterated path integrals and generalized paths, Bull. Amer. Math. Soc. 73 (1967), 935-938. MR 36 #806.
18. K.-T. Chen, Algebraic paths, J. Algebra 10 (1968), 8-36. MR 37 #5204.
19. K.-T. Chen, Covering-space-like algebras, J. Algebra 13 (1969), 308-326. MR 41 #8403.
20. K.-T. Chen, An algebraic dualization of fundamental groups, Bull. Amer. Math. Soc. 75 (1969), 1020-1024. MR 41 #5455.
21. K.-T. Chen, Algebras of iterated path integrals and fundamental groups, Trans. Amer. Math. Soc. 156 (1971), 359-379. MR 43 # 1069.
22. K.-T. Chen, Differential forms and homotopy groups, J. Differential Geometry 6 (1971), 231-246.
23. K.-T. Chen, Free subalgebras of loop space homology and Massey products, Topology 11 (1972), 237-243. MR 45 #4411.
24. K.-T. Chen, Iterated integrals of differential forms and loop space homology, Ann. of Math. (2) 97 (1973), 217-246.
25. K.-T. Chen, Fundamental groups, nilmanifolds and iterated integrals, Bull. Amer. Math. Soc. 79 (1973), 1033-1035. MR 47 #7676.
26. K.-T. Chen, Iterated integrals, fundamental groups and covering spaces, Trans. Amer. Math. Soc. 206 (1975), 83-98.
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.
28. K.-T. Chen, Solvability on manifolds by quadratures permitting only integrals, Bull. Amer. Math. Soc. 80 (1974), 1210-1212. MR 50 #5845.
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.
30. 1, 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.
32. W. G. Dwyer, Strong convergence of the Eilenberg-Moore spectral sequence, Topology 13 (1974), 255-265.
33. S. Eilenberg, Singular homology in differentiable manifolds, Ann. of Math. (2) 48 (1947), 670-681. MR 9, 52.
34. S. Eilenberg and S. MacLane, 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.
36. S. Eilenberg and J. C. Moore, Limits and spectral sequences, Topology 1 (1962), 1-23. MR 26 #6229.
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.
39. V. K. A. M. Gugenheim, On the chain-complex of a fibration, Illinois J. Math. 16 (1972), 398-414. MR 46 #891.
40. P. J. Hilton and S. Wylie, Homology theory, Cambridge Univ. Press, New York, 1960. MR 22 #5963.
41. S.-T. Hu, Homotopy theory, Academic Press, New York and London, 1959. MR 21 #5186.
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.
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.
45. E. R. Kolchin, Differential algebra and algebraic groups, Academic Press, New York, 1973.
46. D. Kraines, Massey higher products, Trans. Amer. Math. Soc. 124 (1966), 431-449. MR 34 #2010.
47. D. Kraines, Primitive chains and H(ŪX), Topology 8 (1969), 31-38. MR 38 #6581.
48. S. MacLane, Homology, Academic Press, New York; Springer-Verlag, Berlin, 1963. MR 28 #122.
49. W. Magnus, Beziehungen zwischen Gruppen und Idealen in einem speziellen Ring, Math. Ann. III (1935), 259-280.
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.
51. J. P. May, Matric Massey products, J. Algebra 12 (1969), 533-568. MR 39 #289.
52. J. W. Milnor, Isotopy of links, Algebraic Geometry and Topology, Princeton Univ. Press, Princeton, N. J., 1957, pp. 280-306. MR 19, 1070.
53. J. W. Milnor, Morse theory, Ann. of Math. Studies, no. 51, Princeton Univ. Press, Princeton, N. J., 1963. MR 29 #634.
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.
55. A. L. Oniščik, On cohomologies of spaces of paths, Mat. Sb. 44 (86) (1958), 3-52; English transi., Amer. Math. Soc. Transl.(2) 25 (1963), 283-334. MR 20 #6101.
56. A. N. Paršin, A generalization of Jacobian variety, Izv. Akad. Nauk SSSR Ser Mat. 30 (1966), 175-182; English transi., Amer. Math. Soc. Transi. (2) 84 (1969), 187-196. MR 33 #4956.
57. R. Ree, Lie elements and an algebra associated with shuffles, Ann. of Math. (2) 68 (1958), 210-220. MR 20 #6447.
58. R. Ree, Generalized Lie elements, Canad. J. Math. 12 (1960), 493-502. MR 22 #5698.
59. J.-P. Serre, Homologie singuliére des espaces fibres. Applications, Ann. of Math. (2) 54 (1951), 425-505. MR 13, 574.
60. L. Smith, Homological algebra and the Eilenberg-Moore spectral sequence, Trans. Amer. Math. Soc. 129 (1967), 58-93. MR 35 #7337.
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.
62. J. W. Smith, The de Rham theorem for general spaces, Tôhoku Math. J. (2) 18 (1966), 115-137. MR 34 #2027.
63. J. Stallings, Homology and central series of groups, J. Algebra 2 (1965), 170-181. MR 31 #232.
64. U. Stammbach, Homology in group theory, Springer-Verlag, Berlin, 1973.
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.
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.
67. M. E. Sweedler, Hopf algebras, Benjamin, New York, 1969. MR 40 #5705.
68. A. Weil, Sur les théorèmes de de Rham, Comment. Math. Helv. 26 (1952), 119-145. MR 14, 307.
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.
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.
71. H. Whitney, Geometric integration theory, Princeton Univ. Press, Princeton, N. J., 1957. MR 19, 309.
72. W. T. Wu, Theory of I*-functors in algebraic topology, Sci. Sinica 18 (1975), 464-482.
73. W. G. Dwyer, Exotic convergence of the Eilenberg-Moore spectral sequence, Illinois J. Math. 19(1975), 607-617.
74. V. K. A. M. Gugenheim, On the multiplicative structure of the de Rham theory, J. Differential Geometry 11 (1976), 309-314.
75. V. K. A. M. Gugenheim, On Chen's iterated integrals (to appear).