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

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.

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.

@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},
     pages = {905--935},
     publisher = {Institut Fourier},
     address = {Grenoble},
     volume = {42},
     number = {4},
     year = {1992},
     doi = {10.5802/aif.1315},
     mrnumber = {93j:55017},
     zbl = {0760.55010},
     language = {en},
     url = {http://www.numdam.org/articles/10.5802/aif.1315/}
}
TY  - JOUR
AU  - Markl, Martin
AU  - Papadima, Stefan
TI  - Homotopy Lie algebras and fundamental groups via deformation theory
JO  - Annales de l'Institut Fourier
PY  - 1992
SP  - 905
EP  - 935
VL  - 42
IS  - 4
PB  - Institut Fourier
PP  - Grenoble
UR  - http://www.numdam.org/articles/10.5802/aif.1315/
DO  - 10.5802/aif.1315
LA  - en
ID  - AIF_1992__42_4_905_0
ER  - 
%0 Journal Article
%A Markl, Martin
%A Papadima, Stefan
%T Homotopy Lie algebras and fundamental groups via deformation theory
%J Annales de l'Institut Fourier
%D 1992
%P 905-935
%V 42
%N 4
%I Institut Fourier
%C Grenoble
%U http://www.numdam.org/articles/10.5802/aif.1315/
%R 10.5802/aif.1315
%G en
%F AIF_1992__42_4_905_0
Markl, Martin; Papadima, Stefan. Homotopy Lie algebras and fundamental groups via deformation theory. Annales de l'Institut Fourier, Tome 42 (1992) no. 4, pp. 905-935. doi : 10.5802/aif.1315. http://www.numdam.org/articles/10.5802/aif.1315/

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

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

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

[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

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

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

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

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

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

[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 | Zbl

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

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

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

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

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

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

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

[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 | Zbl

[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 | Zbl

[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

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

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

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

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

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

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

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

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

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

[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 | Zbl

Cité par Sources :