Rational homotopy groups and Koszul algebras
[Groupes d'homotopie rationnels et algèbres de Koszul]
Comptes Rendus. Mathématique, Tome 335 (2002) no. 1, pp. 53-58.

Soient X et Y deux CW-espaces de type fini (X connexe, Y simplement connexe), tels que l'anneau de cohomologie H * (Y,) soit un k-recalibrage de H * (X,). Si H * (X,) est une algèbre de Koszul, alors l'algèbre de Lie graduée π * (ΩY) est le k-recalibrage de gr * (π 1 X). Si Y est un espace formel, alors l'implication réciproque est vraie aussi, et l'espace Y est coformel. De plus, si X est formel, avec algèbre de cohomologie de Koszul, on trouve des isomorphismes de groupes filtrés entre le complété de Malcev de π1X, le complété de [ΩS 2k+1 ,ΩY], et le groupe de Milnor–Moore d'applications de cogèbres entre H * (ΩS 2k+1 ,) et H * (ΩY,).

Let X and Y be finite-type CW-spaces (X connected, Y simply connected), such that the ring H * (Y,) is a k-rescaling of H * (X,). If H * (X,) is a Koszul algebra, then the graded Lie algebra π * (ΩY) is the k-rescaling of gr * (π 1 X). If Y is a formal space, then the converse holds, and Y is coformal. Furthermore, if X is formal, with Koszul cohomology algebra, there exist filtered group isomorphisms between the Malcev completion of π1X, the completion of [ΩS 2k+1 ,ΩY], and the Milnor–Moore group of coalgebra maps from H * (ΩS 2k+1 ,) to H * (ΩY,).

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/S1631-073X(02)02420-2
Papadima, Stefan 1 ; Suciu, Alexander I. 2

1 Institute of Mathematics of the Romanian Academy, PO Box 1-764, RO-70700 Bucharest, Romania
2 Department of Mathematics, Northeastern University, Boston, MA 02115, USA
@article{CRMATH_2002__335_1_53_0,
     author = {Papadima, Stefan and Suciu, Alexander I.},
     title = {Rational homotopy groups and {Koszul} algebras},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {53--58},
     publisher = {Elsevier},
     volume = {335},
     number = {1},
     year = {2002},
     doi = {10.1016/S1631-073X(02)02420-2},
     language = {en},
     url = {http://archive.numdam.org/articles/10.1016/S1631-073X(02)02420-2/}
}
TY  - JOUR
AU  - Papadima, Stefan
AU  - Suciu, Alexander I.
TI  - Rational homotopy groups and Koszul algebras
JO  - Comptes Rendus. Mathématique
PY  - 2002
SP  - 53
EP  - 58
VL  - 335
IS  - 1
PB  - Elsevier
UR  - http://archive.numdam.org/articles/10.1016/S1631-073X(02)02420-2/
DO  - 10.1016/S1631-073X(02)02420-2
LA  - en
ID  - CRMATH_2002__335_1_53_0
ER  - 
%0 Journal Article
%A Papadima, Stefan
%A Suciu, Alexander I.
%T Rational homotopy groups and Koszul algebras
%J Comptes Rendus. Mathématique
%D 2002
%P 53-58
%V 335
%N 1
%I Elsevier
%U http://archive.numdam.org/articles/10.1016/S1631-073X(02)02420-2/
%R 10.1016/S1631-073X(02)02420-2
%G en
%F CRMATH_2002__335_1_53_0
Papadima, Stefan; Suciu, Alexander I. Rational homotopy groups and Koszul algebras. Comptes Rendus. Mathématique, Tome 335 (2002) no. 1, pp. 53-58. doi : 10.1016/S1631-073X(02)02420-2. http://archive.numdam.org/articles/10.1016/S1631-073X(02)02420-2/

[1] Baues, H.J. Commutator Calculus and Groups of Homotopy Classes, London Math. Soc. Lecture Note Series, 50, Cambridge University Press, Cambridge, 1981

[2] D.C. Cohen, F.R. Cohen, M. Xicoténcatl, Lie algebras associated to fiber-type arrangements, Preprint, 2000, available at arXiv:math.AT/0005091

[3] Cohen, F.R.; Gitler, S. Loop spaces of configuration spaces, braid-like groups, and knots, Cohomological Methods in Homotopy Theory, Progress in Math., 196, Birkhäuser, Basel, 2001, pp. 59-78

[4] F.R. Cohen, T. Sato, On groups of homotopy groups, loop spaces, and braid-like groups, Preprint, 2001

[5] Koschorke, U.; Rolfsen, D. Higher dimensional link operations and stable homotopy, Pacific J. Math., Volume 139 (1989), pp. 87-106

[6] Markl, M.; Papadima, S. Homotopy Lie algebras and fundamental groups via deformation theory, Ann. Inst. Fourier, Volume 42 (1992), pp. 905-935

[7] Milnor, J.W.; Moore, J.C. On the structure of Hopf algebras, Ann. of Math., Volume 81 (1965), pp. 211-264

[8] Papadima, S. Campbell–Hausdorff invariants of links, Proc. London Math. Soc., Volume 75 (1997), pp. 641-670

[9] S. Papadima, A.I. Suciu, Homotopy Lie algebras, lower central series, and the Koszul property, Preprint, 2001, available at arXiv:math.AT/0110303

[10] Papadima, S.; Yuzvinsky, S. On rational K[π,1] spaces and Koszul algebras, J. Pure Appl. Algebra, Volume 144 (1999), pp. 157-167

[11] Quillen, D. Rational homotopy theory, Ann. of Math., Volume 90 (1969), pp. 205-295

[12] Shelton, B.; Yuzvinsky, S. Koszul algebras from graphs and hyperplane arrangements, J. London Math. Soc., Volume 56 (1997), pp. 477-490

[13] Shiga, H.; Yagita, N. Graded algebras having a unique rational homotopy type, Proc. Amer. Math. Soc., Volume 85 (1982), pp. 623-632

[14] Sullivan, D. Infinitesimal computations in topology, Inst. Hautes Études Sci. Publ. Math., Volume 47 (1977), pp. 269-331

Cité par Sources :