Smooth maps, pullback path spaces, connections, and torsions

Chen, Kuo Tsai

doi:10.1090/S0002-9947-1986-0854088-1

Smooth maps, pullback path spaces, connections, and torsions

Author: Kuo Tsai Chen
Journal: Trans. Amer. Math. Soc. 297 (1986), 617-627
MSC: Primary 58A12; Secondary 55N10, 55T20, 58A40
DOI: https://doi.org/10.1090/S0002-9947-1986-0854088-1
MathSciNet review: 854088
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Abstract: By generalizing the local version of the usual differential geometric notion of connections and that of torsions, a model for the pullback path space of a smooth map is constructed from the induced map of the de Rham complexes. The pullback path space serves not only as a homotopy fiber but also as a device reflecting differentiable properties of the smooth map. Applications are discussed.

[1] David J. Anick, A model of Adams-Hilton type for fiber squares, Illinois J. Math. 29 (1985), no. 3, 463–502. MR 786733
[2] Kuo Tsai Chen, Connections, holonomy and path space homology, Differential geometry (Proc. Sympos. Pure Math., Vol. XXVII, Part 1, Stanford Univ., Stanford, Calif., 1973) Amer. Math. Soc., Providence, R. I., 1975, pp. 39–52. MR 0440540
[3] Kuo Tsai Chen, Reduced bar constructions on de Rham complexes, Algebra, topology, and category theory (a collection of papers in honor of Samuel Eilenberg), Academic Press, New York, 1976, pp. 19–32. MR 0413151
[4] Kuo-Tsai Chen, Extension of 𝐶^{∞} function algebra by integrals and Malcev completion of 𝜋₁, Advances in Math. 23 (1977), no. 2, 181–210. MR 0458461, https://doi.org/10.1016/0001-8708(77)90120-7
[5] Kuo Tsai Chen, Iterated path integrals, Bull. Amer. Math. Soc. 83 (1977), no. 5, 831–879. MR 0454968, https://doi.org/10.1090/S0002-9904-1977-14320-6
[6] Kuo-Tsai Chen, Pullback de Rham cohomology of the free path fibration, Trans. Amer. Math. Soc. 242 (1978), 307–318. MR 0478190, https://doi.org/10.1090/S0002-9947-1978-0478190-7
[7] Pierre Deligne, Phillip Griffiths, John Morgan, and Dennis Sullivan, Real homotopy theory of Kähler manifolds, Invent. Math. 29 (1975), no. 3, 245–274. MR 0382702, https://doi.org/10.1007/BF01389853
[8] Pierre-Paul Grivel, Formes différentielles et suites spectrales, Ann. Inst. Fourier (Grenoble) 29 (1979), no. 3, ix, 17–37 (French, with English summary). MR 552958
[9] V. K. A. M. Gugenheim, On a modified Eilenberg-Moore theorem, Geometric applications of homotopy theory (Proc. Conf., Evanston, Ill., 1977), II, Lecture Notes in Math., vol. 658, Springer, Berlin, 1978, pp. 177–190. MR 513574
[10] V. K. A. M. Gugenheim and J. Peter May, On the theory and applications of differential torsion products, American Mathematical Society, Providence, R.I., 1974. Memoirs of the American Mathematical Society, No. 142. MR 0394720
[11] Richard M. Hain, Twisting cochains and duality between minimal algebras and minimal Lie algebras, Trans. Amer. Math. Soc. 277 (1983), no. 1, 397–411. MR 690059, https://doi.org/10.1090/S0002-9947-1983-0690059-3
[12] Stephen Halperin, Rational fibrations, minimal models, and fibrings of homogeneous spaces, Trans. Amer. Math. Soc. 244 (1978), 199–224. MR 0515558, https://doi.org/10.1090/S0002-9947-1978-0515558-4
[13] Stephen Halperin and James Stasheff, Obstructions to homotopy equivalences, Adv. in Math. 32 (1979), no. 3, 233–279. MR 539532, https://doi.org/10.1016/0001-8708(79)90043-4
[14] Jean-Marie Lemaire, Modèles minimaux pour les algèbres de chaînes, Publ. Dép. Math. (Lyon) 13 (1976), no. 3, 13–26 (French). MR 0461500
[15] Daniel Quillen, Rational homotopy theory, Ann. of Math. (2) 90 (1969), 205–295. MR 0258031, https://doi.org/10.2307/1970725
[16] Larry Smith, Lectures on the Eilenberg-Moore spectral sequence, Lecture Notes in Mathematics, Vol. 134, Springer-Verlag, Berlin-New York, 1970. MR 0275435
[17] Dennis Sullivan, Infinitesimal computations in topology, Inst. Hautes Études Sci. Publ. Math. 47 (1977), 269–331 (1978). MR 0646078
[18] Daniel Tanré, Homotopie rationnelle: modèles de Chen, Quillen, Sullivan, Lecture Notes in Mathematics, vol. 1025, Springer-Verlag, Berlin, 1983 (French). MR 764769
[19] Wen Tsün Wu, Theory of 𝐼*-functor in algebraic topology. Effective calculation and axiomatization of 𝐼*-functor on complexes, Sci. Sinica 19 (1976), no. 5, 647–664. MR 0645387