Homotopy Lie algebras and fundamental groups via deformation theory
Annales de l'Institut Fourier, Volume 42 (1992) no. 4, p. 905-935

We formulate first results of our larger project based on first fixing some easily accessible invariants of topological spaces (typically the cup product structure in low dimensions) and then studying the variations of more complex invariants such as π * ΩS (the homotopy Lie algebra) or gr * π 1 S (the graded Lie algebra associated to the lower central series of the fundamental group). We prove basic rigidity results and give also an application in low-dimensional topology.

Nous donnons les premiers résultats de notre plus vaste projet en fixant d’abord quelques invariants facilement accessibles des espaces topologiques (par exemple le cup-produit en basses dimensions) et en étudiant alors la variation d’invariants plus complexes tels que π*ΩS (l’algèbre de Lie homotopique) ou bien gr * π 1 S (l’algèbre de Lie graduée associée aux séries centrales du groupe fondamental). Nous donnons des résultats fondamentaux de rigidité, ainsi qu’une application à la topologie en basses dimensions.

@article{AIF_1992__42_4_905_0,
     author = {Markl, Martin and Papadima, Stefan},
     title = {Homotopy Lie algebras and fundamental groups via deformation theory},
     journal = {Annales de l'Institut Fourier},
     publisher = {Imprimerie Louis-Jean},
     address = {Gap},
     volume = {42},
     number = {4},
     year = {1992},
     pages = {905-935},
     doi = {10.5802/aif.1315},
     mrnumber = {93j:55017},
     zbl = {0760.55010},
     language = {en},
     url = {https://aif.centre-mersenne.org/item/AIF_1992__42_4_905_0}
}
Markl, Martin; Papadima, Stefan. Homotopy Lie algebras and fundamental groups via deformation theory. Annales de l'Institut Fourier, Volume 42 (1992) no. 4, pp. 905-935. doi : 10.5802/aif.1315. aif.centre-mersenne.org/item/AIF_1992__42_4_905_0/

[1] D. Anick, Non-commutative graded algebras and their Hilbert series, J. of Algebra, (1)78 (1982), 120-140. | Zbl 0502.16002

[2] D. Anick, Connections between Yoneda and Pontrjagin algebras, Lect. Notes in Math. 1051, Springer-Verlag, 1984, pp. 331-350. | MR 87e:55025 | Zbl 0542.57035

[3] D. Anick, Inert sets and the Lie algebra associated to a group, Journ. of Algebra, 111-1 (1987), 154-165. | MR 89d:20031 | Zbl 0635.20015

[4] I.K. Babenko, On analytic properties of the Poincaré series of loop spaces, Matem. Zametki, 27 (1980), 751-765, in Russian ; English transl. in Math. Notes 27 (1980).

[5] B. Berceanu, Şt. Papadima, Cohomologically generic 2-complexes and 3-dimensional Poincaré complexes, preprint.

[6] K.-T. Chen, Iterated integral of differential forms and loop space homology, Ann. of Mathematics, (2)97 (1973), 217-246. | Zbl 0227.58003

[7] K.-T. Chen, Differential forms and homotopy groups, J. of Differential Geometry, 6 (1971), 231-246. | MR 52 #1755 | Zbl 0229.58002

[8] K.-T. Chen, Commutator calculs and link invariants, Proc. Amer. Math. Soc., 3 (1952), 44-55. | MR 13,721d | Zbl 0049.40402

[9] B. Cenkl, R. Porter, Malcev's completion of a group and differential forms, J. of Differential Geometry, 15 (1980), 531-542. | MR 82k:55013 | Zbl 0491.20033

[10] Y. Félix, Dénombrement des types de k-homotopie. Théorie de la déformation, Bull. Soc. Math. France, (3)108 (1980). | Numdam | MR 82i:55011 | Zbl 0452.55005

[11] Y. Félix, S. Halperin, Rational LS category and its applications, Trans. Amer. Math. Society, 273 (1982), 1-38. | MR 84h:55011 | Zbl 0508.55004

[12] Y. Félix, J.-C. Thomas, Sur la structure des espaces de LS catégorie deux, Illinois J. of Math., (4)30 (1986), 574-593. | MR 87k:55016 | Zbl 0585.55010

[13] P.A. Griffiths, J.W. Morgan, Rational homotopy theory and differential forms, Progress in Math. 16, Birkhäuser, 1981. | MR 82m:55014 | Zbl 0474.55001

[14] W. Greub, S. Halperin, R. Vanstone, Connections, curvature and cohomology, vol. III, Academic Press, 1976. | MR 53 #4110 | Zbl 0372.57001

[15] S. Halperin, J.-M. Lemaire, Suites inertes dans les algèbres de Lie graduées, Math. Scand., 61 (1987), 39-67. | MR 89e:55022 | Zbl 0655.55004

[16] P.J. Hilton, U. Stambach, A course in homological algebra, Graduate texts in Mathematics 4, Springer-Verlag, 1971. | MR 49 #10751 | Zbl 0238.18006

[17] S. Halperin, J.D. Stasheff, Obstructions to homotopy equivalences, Advances in Math., 32 (1979), 233-279. | MR 80j:55016 | Zbl 0408.55009

[18] S. Kojima, Nilpotent completions and Lie rings associated to link groups, Comment. Math. Helv., 58 (1983) 115-134. | MR 85b:57008 | Zbl 0528.57004

[19] T. Kohno, Série de Poincaré-Koszul associée aux groupes de tresses pures, Invent. Math., 82 (1985), 57-75. | MR 87c:32015a | Zbl 0574.55009

[20] J.P. Labute, The determination of the Lie algebra associated to the lower central series of a group, Trans. Amer. Math. Society, (1) 288 (1985), 51-57. | MR 86b:20049 | Zbl 0576.20022

[21] J.P. Labute, The Lie algebra associated to the lower central series of a link group and Murasugi's conjecture, Proc. Amer. Math. Soc., 109,4 (1990), 951-956. | MR 90k:20065 | Zbl 0696.20035

[22] C. Löfwall, On the subalgebra generated by the one-dimensional elements in the Yoneda Ext-algebra, Algebra, algebraic topology and their interactions, Proc. Stockholm 1983, Lect. Notes in Math. 1183, Springer-Verlag, 1986, pp. 291-338. | Zbl 0595.16020

[23] J.-M. Lemaire, F. Sigrist, Dénombrement des types d'homotopie rationnelle, C. R. Acad. Sci. Paris, 287 (1978), 109-112. | MR 80b:55009 | Zbl 0382.55005

[24] M. Markl, Şt. Papadima, Geometric decompositions, algebraic models and rigidity theorems, Journ. of Pure and Appl. Algebra, 71 (1991), 53-73. | MR 92f:55016 | Zbl 0728.55005

[25] S.B. Priddy, Koszul resolutions, Trans. Amer. Math. Society, 152 (1970), 39-60. | MR 42 #346 | Zbl 0261.18016

[26] Şt. Papadima, The rational homotopy of Thom spaces and the smoothing of homology classes, Comment. Math. Helv., 60 (1985), 601-614. | MR 87e:57030 | Zbl 0592.57025

[27] D. Quillen, Rational homotopy theory, Ann. of Math., 90 (1969), 205-295. | MR 41 #2678 | Zbl 0191.53702

[28] D. Sullivan, Infinitesimal computations in topology, Publ. Math. IHES, 47 (1977), 269-331. | Numdam | MR 58 #31119 | Zbl 0374.57002

[29] J.-P. Serre, Lie algebras and Lie groups, Benjamin, 1965. | MR 36 #1582 | Zbl 0132.27803

[30] J.D. Stasheff, Rational Poincaré duality spaces, Illinois J. of Math., 27 (1983), 104-109. | MR 85c:55012 | Zbl 0488.55010

[31] D. Tanré, Homotopie rationnelle. Modèles de Chen, Quillen, Sullivan, Lect. Notes in Mathem. 1025, Springer-Verlag, 1983. | Zbl 0539.55001

[32] D. Tanré, Cohomologie de Harrison et type d'homotopie rationnelle, Algebra, algebraic topology and their interactions, Proc. Stockholm 1983, Lect. Notes in Math. 1183, Springer-Verlag, 1986, pp. 361-370. | MR 87m:55015 | Zbl 0594.55013