Summary
We first give a new proof of a conjecture of J.-P. Serre on the homotopy of finite complexes, which was recently proved by C. McGibbon and J. Neisendorfer. The emphasis is on a property of the mod. 2 homology of certain spaces: their “quasi-boundedness” as right modules over the Steenrod algebra. This property is preserved when one goes from a simply connected space to its loop space and also when one takes a covering of anH-space. Then we show that this notion of quasi-boundedness simplifies H. Miller's proof of D. Sullivan's conjecture on the contractibility of the space of pointed maps from the classifying space of the groupe ℤ/2 into a finite complex.
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Références
G. Carlsson: G.B. Segal's Burnside ring conjecture for (ℤ/2)k. Topology22, 83–103 (1983)
S. Eilenberg, J. Moore: Homology and fibrations. Comment. Math. Helv.40, 199–236 (1966)
J. Lannes, S. Zarati: Sur lesU-injectifs (A paraître aux Annales de l'E.N.S.)
C. McGibbon, J. Neisendorfer: On the homotopy groups of a finite dimensional space. Comment. Math. Helv.59, 253–257 (1984)
H. Miller: The Sullivan conjecture on maps from classifying spaces. Ann. Math.120, p. 39–87 (1984)
H. Miller: Correction to “the Sullivan conjecture on maps from classifying spaces”. Ann. Math.121, 605–609 (1985)
H. Miller: Massey-Peterson towers and maps from classifying spaces. Algebraic Topology Aarhus 1982. Lect. Notes Math.1051, pp. 401–417
J. Milnor, J. Moore: On the structure of Hopf algebras. Ann. Math. (2)81, 211–264 (1965)
J. Moore, L. Smith: Hopf algebras and multiplicative fibrations II. Am. J. Math.90, 1113–1150 (1968)
D. Rector: Steenrod operations in the Eilenberg-Moore spectral sequence. Comment. Math. Helv.45, 540–552 (1970)
J.-P. Serre: Cohomologie modulo 2 des complexes d'Eilenberg-MacLane Comment. Math. Helv.27, 198–232 (1953)
L. Smith: On Künneth theorem. I. The Eilenberg-Moore spectral sequence. Math. Z.116, 94–140 (1970)
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Lannes, J., Schwartz, L. A propos de conjectures de Serre et Sullivan. Invent Math 83, 593–603 (1986). https://doi.org/10.1007/BF01394425
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DOI: https://doi.org/10.1007/BF01394425