Iterated path integrals

Kuo-Tsai Chen

Bulletin of the American Mathematical Society

Bull. Amer. Math. Soc.
Volume 83, Number 5 (1977), 831-879.

Iterated path integrals

Kuo-Tsai Chen

Full-text: Open access

PDF File (4376 KB)

Article information

Source
Bull. Amer. Math. Soc., Volume 83, Number 5 (1977), 831-879.

Dates
First available in Project Euclid: 4 July 2007

Permanent link to this document
https://projecteuclid.org/euclid.bams/1183539443

Mathematical Reviews number (MathSciNet)
MR0454968

Zentralblatt MATH identifier
0389.58001

Subjects
Primary: 58A99: None of the above, but in this section 55D35
Secondary: 53B15: Other connections 53C65: Integral geometry [See also 52A22, 60D05]; differential forms, currents, etc. [See mainly 58Axx] 55H20 49F05

Citation

Chen, Kuo-Tsai. Iterated path integrals. Bull. Amer. Math. Soc. 83 (1977), no. 5, 831--879. https://projecteuclid.org/euclid.bams/1183539443

References

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.
Zentralblatt MATH: 0086.37502
Mathematical Reviews (MathSciNet): MR90045
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.
Zentralblatt MATH: 0071.16403
Mathematical Reviews (MathSciNet): MR77929
Digital Object Identifier: doi:10.1007/BF02564350
3. L. V. Ahlfors and L. Sario, Riemann surfaces, Princeton Univ. Press, Princeton, N. J., 1960. MR 22 #5729.
Zentralblatt MATH: 0196.33801
Mathematical Reviews (MathSciNet): MR114911
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.
Zentralblatt MATH: 0222.58002
Mathematical Reviews (MathSciNet): MR279832
5. A. Asada, Algebraic cohomology of loop spaces, J. Fac. Sci. Shinshu Univ. 4 (1969), 1-23. MR 41 #9241.
Zentralblatt MATH: 0322.55013
Mathematical Reviews (MathSciNet): MR264650
6. A. Asada, Representations of loop spaces and fibre bundles, J. Fac. Sci. Shinshu Univ. 4 (1969), 39-56. MR 41 #9256.
Zentralblatt MATH: 0322.55026
Mathematical Reviews (MathSciNet): MR264665
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.
Zentralblatt MATH: 0112.25904
Mathematical Reviews (MathSciNet): MR144949
Digital Object Identifier: doi:10.1090/S0002-9947-1963-0144949-8
8. R. Bott, The space of loops on a Lie group, Michigan Math. J. 5 (1958), 35-61. MR 21 #1589.
Zentralblatt MATH: 0096.17701
Mathematical Reviews (MathSciNet): MR102803
Digital Object Identifier: doi:10.1307/mmj/1028998010
Project Euclid: euclid.mmj/1028998010
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.
Zentralblatt MATH: 0052.19301
Mathematical Reviews (MathSciNet): MR60233
Digital Object Identifier: doi:10.1007/BF02564566
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.
Zentralblatt MATH: 0101.39702
Mathematical Reviews (MathSciNet): MR105694
Digital Object Identifier: doi:10.2307/2372843
11. E. H. Brown, Jr., Twisted tensor products. I, Ann. of Math. (2) 69 (1959), 223-246. MR 21 #4423.
Zentralblatt MATH: 0199.58201
Mathematical Reviews (MathSciNet): MR105687
Digital Object Identifier: doi:10.2307/1970101
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.
Zentralblatt MATH: 0055.41801
Mathematical Reviews (MathSciNet): MR65160
Digital Object Identifier: doi:10.1073/pnas.40.6.467
13. K. -T. Chen, Iterated integrals and exponential homomorphisms, Proc. London Math. Soc. (3) 4 (1954), 502-512. MR 17, 394.
Zentralblatt MATH: 0058.25603
Mathematical Reviews (MathSciNet): MR73174
Digital Object Identifier: doi:10.1112/plms/s3-4.1.502
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.
Zentralblatt MATH: 0077.25301
Mathematical Reviews (MathSciNet): MR85251
Digital Object Identifier: doi:10.2307/1969671
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.
Zentralblatt MATH: 0097.25803
Mathematical Reviews (MathSciNet): MR106258
16. K. -T. Chen, Formal differential equations, Ann. of Math. (2) 73 (1961), 110-133. MR 27 #371.
Zentralblatt MATH: 0098.05702
Mathematical Reviews (MathSciNet): MR150370
Digital Object Identifier: doi:10.2307/1970284
17. K. -T. Chen, Iterated path integrals and generalized paths, Bull. Amer. Math. Soc. 73 (1967), 935-938. MR 36 #806.
Zentralblatt MATH: 0179.19601
Mathematical Reviews (MathSciNet): MR217717
Digital Object Identifier: doi:10.1090/S0002-9904-1967-11858-5
Project Euclid: euclid.bams/1183529114
18. K. -T. Chen, Algebraic paths, J. Algebra 10 (1968), 8-36. MR 37 #5204.
Zentralblatt MATH: 0204.33803
Mathematical Reviews (MathSciNet): MR229630
Digital Object Identifier: doi:10.1016/0021-8693(68)90102-6
19. K. -T. Chen, Covering-space-like algebras, J. Algebra 13 (1969), 308-326. MR 41 #8403.
Zentralblatt MATH: 0181.32501
Mathematical Reviews (MathSciNet): MR263803
Digital Object Identifier: doi:10.1016/0021-8693(69)90077-5
20. K. -T. Chen, An algebraic dualization of fundamental groups, Bull. Amer. Math. Soc. 75 (1969), 1020-1024. MR 41 #5455.
Zentralblatt MATH: 0182.57101
Mathematical Reviews (MathSciNet): MR260834
Digital Object Identifier: doi:10.1090/S0002-9904-1969-12345-1
Project Euclid: euclid.bams/1183530830
21. K. -T. Chen, Algebras of iterated path integrals and fundamental groups, Trans. Amer. Math. Soc. 156 (1971), 359-379. MR 43 # 1069.
Zentralblatt MATH: 0217.47705
Mathematical Reviews (MathSciNet): MR275312
Digital Object Identifier: doi:10.1090/S0002-9947-1971-0275312-1
22. K. -T. Chen, Differential forms and homotopy groups, J. Differential Geometry 6 (1971), 231-246.
Zentralblatt MATH: 0229.58002
Mathematical Reviews (MathSciNet): MR380858
Project Euclid: euclid.jdg/1214430406
23. K. -T. Chen, Free subalgebras of loop space homology and Massey products, Topology 11 (1972), 237-243. MR 45 #4411.
Zentralblatt MATH: 0238.55006
Mathematical Reviews (MathSciNet): MR295345
Digital Object Identifier: doi:10.1016/0040-9383(72)90010-9
24. K. -T. Chen, Iterated integrals of differential forms and loop space homology, Ann. of Math. (2) 97 (1973), 217-246.
Zentralblatt MATH: 0227.58003
Mathematical Reviews (MathSciNet): MR380859
Digital Object Identifier: doi:10.2307/1970846
25. K. -T. Chen, Fundamental groups, nilmanifolds and iterated integrals, Bull. Amer. Math. Soc. 79 (1973), 1033-1035. MR 47 #7676.
Zentralblatt MATH: 0271.58001
Mathematical Reviews (MathSciNet): MR319130
Digital Object Identifier: doi:10.1090/S0002-9904-1973-13318-X
Project Euclid: euclid.bams/1183534997
26. K. -T. Chen, Iterated integrals, fundamental groups and covering spaces, Trans. Amer. Math. Soc. 206 (1975), 83-98.
Zentralblatt MATH: 0301.58006
Mathematical Reviews (MathSciNet): MR377960
Digital Object Identifier: doi:10.1090/S0002-9947-1975-0377960-0
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.
Zentralblatt MATH: 0312.53010
Mathematical Reviews (MathSciNet): MR440540
28. K. -T. Chen, Solvability on manifolds by quadratures permitting only integrals, Bull. Amer. Math. Soc. 80 (1974), 1210-1212. MR 50 #5845.
Zentralblatt MATH: 0295.58006
Mathematical Reviews (MathSciNet): MR353361
Digital Object Identifier: doi:10.1090/S0002-9904-1974-13682-7
Project Euclid: euclid.bams/1183536030
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.
Zentralblatt MATH: 0341.57034
Mathematical Reviews (MathSciNet): MR413151
30. K. -T. Chen, Extension of C^∞ function algebra by integrals and Malcev completion of π₁, Advances in Math. 23 (1977), 181-210.
Zentralblatt MATH: 0345.58003
Mathematical Reviews (MathSciNet): MR458461
Digital Object Identifier: doi:10.1016/0001-8708(77)90120-7
31. P. Deligne, P. Griffiths, J. Morgan and D. Sullivan, Real homotopy theory of Kähler manifolds, Invent. Math. 29 (1975), 245-274.
Zentralblatt MATH: 0312.55011
Mathematical Reviews (MathSciNet): MR382702
Digital Object Identifier: doi:10.1007/BF01389853
32. W. G. Dwyer, Strong convergence of the Eilenberg-Moore spectral sequence, Topology 13 (1974), 255-265.
Zentralblatt MATH: 0303.55012
Mathematical Reviews (MathSciNet): MR394663
Digital Object Identifier: doi:10.1016/0040-9383(74)90018-4
33. S. Eilenberg, Singular homology in differentiable manifolds, Ann. of Math. (2) 48 (1947), 670-681. MR 9, 52.
Zentralblatt MATH: 0029.41904
Mathematical Reviews (MathSciNet): MR21314
Digital Object Identifier: doi:10.2307/1969134
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.
Zentralblatt MATH: 0050.39304
Mathematical Reviews (MathSciNet): MR56295
Digital Object Identifier: doi:10.2307/1969820
35. S. Eilenberg and J. C. Moore, Homology and fibrations. I, Comment. Math. Helv. 40 (1966), 199-236. MR 34 # 3579.
Zentralblatt MATH: 0148.43203
Mathematical Reviews (MathSciNet): MR203730
Digital Object Identifier: doi:10.1007/BF02564371
36. S. Eilenberg and J. C. Moore, Limits and spectral sequences, Topology l (1962), 1-23. MR 26 #6229.
Zentralblatt MATH: 0104.39603
Mathematical Reviews (MathSciNet): MR148723
Digital Object Identifier: doi:10.1016/0040-9383(62)90093-9
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.
Zentralblatt MATH: 0148.17104
Mathematical Reviews (MathSciNet): MR206950
39. V. K. A. M. Gugenheim, On the chain-complex of a fibration, Illinois J. Math. 16 (1972), 398-414. MR 46 #891.
Zentralblatt MATH: 0238.55015
Mathematical Reviews (MathSciNet): MR301736
Project Euclid: euclid.ijm/1256065766
40. P. J. Hilton and S. Wylie, Homology theory, Cambridge Univ. Press, New York, 1960. MR 22 #5963.
Zentralblatt MATH: 0091.36306
Mathematical Reviews (MathSciNet): MR115161
41. S. -T. Hu, Homotopy theory, Academic Press, New York and London, 1959. MR 21 #5186.
Zentralblatt MATH: 0088.38803
Mathematical Reviews (MathSciNet): MR106454
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.
Zentralblatt MATH: 0124.14902
Mathematical Reviews (MathSciNet): MR141728
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.
Zentralblatt MATH: 0119.37502
Mathematical Reviews (MathSciNet): MR152974
45. E. R. Kolchin, Differential algebra and algebraic groups, Academic Press, New York, 1973.
Zentralblatt MATH: 0264.12102
Mathematical Reviews (MathSciNet): MR568864
46. D. Kraines, Massey higher products, Trans. Amer. Math. Soc. 124 (1966), 431-449. MR 34 #2010.
Zentralblatt MATH: 0146.19201
Mathematical Reviews (MathSciNet): MR202136
Digital Object Identifier: doi:10.1090/S0002-9947-1966-0202136-1
47. D. Kraines, Primitive chains and H_*(ΩX), Topology 8 (1969), 31-38. MR 38 #6581.
Zentralblatt MATH: 0176.52404
Mathematical Reviews (MathSciNet): MR238305
Digital Object Identifier: doi:10.1016/0040-9383(69)90028-7
48. S. Mac Lane, Homology, Academic Press, New York; Springer-Verlag, Berlin, 1963. MR 28 #122.
Zentralblatt MATH: 0328.18009
Mathematical Reviews (MathSciNet): MR1344215
49. W. Magnus, Beziehungen zwischen Gruppen und Idealen in einem speziellen Ring, Math. Ann. 111 (1935), 259-280.
Mathematical Reviews (MathSciNet): MR1512992
Digital Object Identifier: doi: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.
Zentralblatt MATH: 0212.55904
Mathematical Reviews (MathSciNet): MR254832
51. J. P. May, Matric Massey products, J. Algebra 12 (1969), 533-568. MR 39 #289.
Zentralblatt MATH: 0192.34302
Mathematical Reviews (MathSciNet): MR238929
Digital Object Identifier: doi:10.1016/0021-8693(69)90027-1
52. J. W. Milnor, Isotopy of links, Algebraic Geometry and Topology, Princeton Univ. Press, Princeton, N. J., 1957, pp. 280-306. MR 19, 1070.
Zentralblatt MATH: 0080.16901
Mathematical Reviews (MathSciNet): MR92150
53. J. W. Milnor, Morse theory, Ann. of Math. Studies, no. 51, Princeton Univ. Press, Princeton, N. J., 1963. MR 29 #634.
Zentralblatt MATH: 0108.10401
Mathematical Reviews (MathSciNet): MR163331
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.
Zentralblatt MATH: 0163.28202
Mathematical Reviews (MathSciNet): MR174052
Digital Object Identifier: doi:10.2307/1970615
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.
Zentralblatt MATH: 0127.38903
Mathematical Reviews (MathSciNet): MR99662
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.
Zentralblatt MATH: 0189.21501
Mathematical Reviews (MathSciNet): MR196770
57. R. Ree, Lie elements and an algebra associated with shuffles, Ann. of Math. (2) 68 (1958), 210-220. MR 20 #6447.
Zentralblatt MATH: 0083.25401
Mathematical Reviews (MathSciNet): MR100011
Digital Object Identifier: doi:10.2307/1970243
58. R. Ree, Generalized Lie elements, Canad. J. Math. 12 (1960), 493-502. MR 22 #5698.
Zentralblatt MATH: 0093.25503
Mathematical Reviews (MathSciNet): MR114880
Digital Object Identifier: doi:10.4153/CJM-1960-044-x
59. J.-P. Serre, Homologie singulière des espaces fibrés. Applications, Ann. of Math. (2) 54 (1951), 425-505. MR 13, 574.
Zentralblatt MATH: 0045.26003
Mathematical Reviews (MathSciNet): MR45386
Digital Object Identifier: doi:10.2307/1969485
60. L. Smith, Homological algebra and the Eilenberg-Moore spectral sequence, Trans. Amer. Math. Soc. 129 (1967), 58-93. MR 35 #7337.
Zentralblatt MATH: 0177.51402
Mathematical Reviews (MathSciNet): MR216504
Digital Object Identifier: doi:10.1090/S0002-9947-1967-0216504-6
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.
Zentralblatt MATH: 0197.19702
Mathematical Reviews (MathSciNet): MR324702
62. J. W. Smith, The de Rham theorem for general spaces, Tôhoku Math. J. (2) 18 (1966), 115-137. MR 34 #2027.
Zentralblatt MATH: 0146.19402
Mathematical Reviews (MathSciNet): MR202154
Digital Object Identifier: doi:10.2748/tmj/1178243443
Project Euclid: euclid.tmj/1178243443
63. J. Stallings, Homology and central series of groups, J. Algebra 2 (1965), 170-181. MR 31 #232.
Zentralblatt MATH: 0135.05201
Mathematical Reviews (MathSciNet): MR175956
Digital Object Identifier: doi:10.1016/0021-8693(65)90017-7
64. U. Stammbach, Homology in group theory, Springer-Verlag, Berlin, 1973.
Zentralblatt MATH: 0272.20049
Mathematical Reviews (MathSciNet): MR382477
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.
Zentralblatt MATH: 0234.55020
Mathematical Reviews (MathSciNet): MR321079
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.
Zentralblatt MATH: 0319.58005
Mathematical Reviews (MathSciNet): MR370611
67. M. E. Sweedler, Hopf algebras, Benjamin, New York, 1969. MR 40 #5705.
Zentralblatt MATH: 0194.32901
Mathematical Reviews (MathSciNet): MR252485
68. A. Weil, Sur les théorèmes de de Rham, Comment. Math. Helv. 26 (1952), 119-145. MR 14, 307.
Zentralblatt MATH: 0047.16702
Mathematical Reviews (MathSciNet): MR50280
Digital Object Identifier: doi:10.1007/BF02564296
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.
Zentralblatt MATH: 0141.19202
Mathematical Reviews (MathSciNet): MR111056
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.
Zentralblatt MATH: 0030.07902
Mathematical Reviews (MathSciNet): MR20255
Digital Object Identifier: doi:10.1073/pnas.33.5.117
71. H. Whitney, Geometric integration theory, Princeton Univ. Press, Princeton, N. J., 1957. MR 19, 309.
Zentralblatt MATH: 0083.28204
Mathematical Reviews (MathSciNet): MR87148
72. W. T. Wu, Theory of I^*- functors in algebraic topology, Sci. Sinica 18 (1975), 464-482.
Zentralblatt MATH: 0336.55019
Mathematical Reviews (MathSciNet): MR645386
73. W. G. Dwyer, Exotic convergence of the Eilenberg-Moore spectral sequence, Illinois J. Math. 19 (1975), 607-617.
Zentralblatt MATH: 0328.55014
Mathematical Reviews (MathSciNet): MR383409
Project Euclid: euclid.ijm/1256050669
74. V. K. A. M. Gugenheim, On the multiplicative structure of the de Rham theory, J. Differential Geometry 11 (1976), 309-314.
Zentralblatt MATH: 0344.55004
Mathematical Reviews (MathSciNet): MR418083
Project Euclid: euclid.jdg/1214433428
75. V. K. A. M. Gugenheim, On Chen's iterated integrals (to appear).
Zentralblatt MATH: 0391.58004

Project Euclid

mathematics and statistics online

Bulletin of the American Mathematical Society

Iterated path integrals

More by Kuo-Tsai Chen

Article information

Citation

References

American Mathematical Society

More like this