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Donghua Jiang
Un 3-polyGEM de cohomologie modulo 2 nilpotente
(A 3-polyGEM of nilpotent modulo 2 cohomology)
Annales de l'institut Fourier, 54 no. 4 (2004), p. 1053-1072, doi: 10.5802/aif.2043
Article PDF | Reviews MR 2111021 | Zbl 1065.55002
Class. Math.: 55N99, 55S45, 57T35, 55R20, 55T20
Keywords: polyGEM, Milgram spaces, Eilenberg-Moore spectral sequences
We give a counter-example of the following conjecture: if the reduced mod 2 cohomology of
any 1-connected polyGEM is of finite type and is not trivial, then it contains at least
one element of infinite height, i.e., non nilpotent.
[1] E.H. Brown & F.P. Peterson, “Whitehead products and cohomology operations”, Quart. J. Math. Oxford Ser. (2) 15 (1964), p. 116-120 MR 161341 | Zbl 0124.38401
[2] F. Cohen, “Communication privée”, , 2003
[3] E. Dror & Farjoun, Cellular spaces, null spaces and homotopy localization, Lecture Notes in Mathematics 1622, Springer-Verlag, Berlin, 1996 MR 1392221
[4] Y. Félix, S. Halperin, J.-M. Lemaire & J.-C. Thomas, “Mod $p$ loop space homology”, Invent. Math 95 (1989) no. 2, p. 247-262 MR 974903 | Zbl 0667.55007
[5] J. Grodal, The transcendence degree of the mod $p$ cohomology of finite Postnikov systems, 1996, p. 111-130 Zbl 0905.55012
[6] L. Kristensen, “On secondary cohomology operations”, Math. Scand 12 (1963), p. 57-82 MR 159333 | Zbl 0118.18303
[7] J. Lannes & et L. Schwartz, “À propos de conjectures de Serre et Sullivan”, Invent. Math 83 (1986) no. 3, p. 593-603 MR 827370 | Zbl 0563.55011
[8] J. Lannes & et L. Schwartz, “Sur les groupes d'homotopie des espaces dont la cohomologie modulo $2$ est nilpotente”, Israel J. Math 66 (1989) no. 1-3, p. 260-273 MR 1017166 | Zbl 0681.55012
[9] C.A. McGibbon & J.A. Neisendorfer, “On the homotopy groups of a finite-dimensional space”, Comment. Math. Helv. 59 (1984) no. 2, p. 253-257 MR 749108 | Zbl 0538.55010
[10] J. Milgram, “The structure over the Steenrod algebra of some 2-stage Postnikov systems”, Quart. J. Math. Oxford Ser. (2) 20 (1969), p. 161-169 MR 248811 | Zbl 0177.51501
[11] J.W. Milnor & J.C. Moore, “On the structure of Hopf algebras”, Annals of Mathematics (2) 81 (1965), p. 211-264 MR 174052 | Zbl 0163.28202
[12] J.-P. Serre, “Cohomologie modulo $2$ des complexes d'Eilenberg-MacLane”, Comment. Math. Helv 27 (1953), p. 198-232 MR 60234 | Zbl 0052.19501
[13] L. Smith, “The cohomology of stable two stage Postnikov systems”, Illinois J. Math 11 (1967), p. 310-329 Article | MR 208597 | Zbl 0171.21803
[14] N.E. Steenrod, Cohomology operations, Annals of Mathematics Studies, Princeton University Press, 1962 Zbl 0102.38104
Donghua Jiang
Un 3-polyGEM de cohomologie modulo 2 nilpotente
(A 3-polyGEM of nilpotent modulo 2 cohomology)
Annales de l'institut Fourier, 54 no. 4 (2004), p. 1053-1072, doi: 10.5802/aif.2043
Article PDF | Reviews MR 2111021 | Zbl 1065.55002
Class. Math.: 55N99, 55S45, 57T35, 55R20, 55T20
Keywords: polyGEM, Milgram spaces, Eilenberg-Moore spectral sequences
Résumé - Abstract
Bibliography
[2] F. Cohen, “Communication privée”, , 2003
[3] E. Dror & Farjoun, Cellular spaces, null spaces and homotopy localization, Lecture Notes in Mathematics 1622, Springer-Verlag, Berlin, 1996 MR 1392221
[4] Y. Félix, S. Halperin, J.-M. Lemaire & J.-C. Thomas, “Mod $p$ loop space homology”, Invent. Math 95 (1989) no. 2, p. 247-262 MR 974903 | Zbl 0667.55007
[5] J. Grodal, The transcendence degree of the mod $p$ cohomology of finite Postnikov systems, 1996, p. 111-130 Zbl 0905.55012
[6] L. Kristensen, “On secondary cohomology operations”, Math. Scand 12 (1963), p. 57-82 MR 159333 | Zbl 0118.18303
[7] J. Lannes & et L. Schwartz, “À propos de conjectures de Serre et Sullivan”, Invent. Math 83 (1986) no. 3, p. 593-603 MR 827370 | Zbl 0563.55011
[8] J. Lannes & et L. Schwartz, “Sur les groupes d'homotopie des espaces dont la cohomologie modulo $2$ est nilpotente”, Israel J. Math 66 (1989) no. 1-3, p. 260-273 MR 1017166 | Zbl 0681.55012
[9] C.A. McGibbon & J.A. Neisendorfer, “On the homotopy groups of a finite-dimensional space”, Comment. Math. Helv. 59 (1984) no. 2, p. 253-257 MR 749108 | Zbl 0538.55010
[10] J. Milgram, “The structure over the Steenrod algebra of some 2-stage Postnikov systems”, Quart. J. Math. Oxford Ser. (2) 20 (1969), p. 161-169 MR 248811 | Zbl 0177.51501
[11] J.W. Milnor & J.C. Moore, “On the structure of Hopf algebras”, Annals of Mathematics (2) 81 (1965), p. 211-264 MR 174052 | Zbl 0163.28202
[12] J.-P. Serre, “Cohomologie modulo $2$ des complexes d'Eilenberg-MacLane”, Comment. Math. Helv 27 (1953), p. 198-232 MR 60234 | Zbl 0052.19501
[13] L. Smith, “The cohomology of stable two stage Postnikov systems”, Illinois J. Math 11 (1967), p. 310-329 Article | MR 208597 | Zbl 0171.21803
[14] N.E. Steenrod, Cohomology operations, Annals of Mathematics Studies, Princeton University Press, 1962 Zbl 0102.38104
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