The Serre spectral sequence of a multiplicative fibration
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- by Yves Félix, Stephen Halperin and Jean-Claude Thomas PDF
- Trans. Amer. Math. Soc. 353 (2001), 3803-3831 Request permission
Abstract:
In a fibration $\Omega F \overset {\Omega j}{\rightarrow } \Omega X \overset {\Omega \pi }{\rightarrow }\Omega B$ we show that finiteness conditions on $F$ force the homology Serre spectral sequence with $\mathbb {F}_p$-coefficients to collapse at some finite term. This in particular implies that as graded vector spaces, $H_*(\Omega X)$ is “almost” isomorphic to $H_*(\Omega B)\otimes H_*(\Omega F)$. One consequence is the conclusion that $X$ is elliptic if and only if $B$ and $F$ are.References
- David J. Anick, Hopf algebras up to homotopy, J. Amer. Math. Soc. 2 (1989), no. 3, 417–453. MR 991015, DOI 10.1090/S0894-0347-1989-0991015-7
- Shôrô Araki, Steenrod reduced powers in the spectral sequences associated with a fibering. I, II, Mem. Fac. Sci. Kyūsyū Univ. A 11 (1957), 15–64, 81–97. MR 105681
- P. Hebroni, Sur les inverses des éléments dérivables dans un anneau abstrait, C. R. Acad. Sci. Paris 209 (1939), 285–287 (French). MR 14
- Armand Borel, Topics in the homology theory of fibre bundles, Lecture Notes in Mathematics, No. 36, Springer-Verlag, Berlin-New York, 1967. Lectures given at the University of Chicago, 1954; Notes by Edward Halpern. MR 0221507
- Sam Perlis, Maximal orders in rational cyclic algebras of composite degree, Trans. Amer. Math. Soc. 46 (1939), 82–96. MR 15, DOI 10.1090/S0002-9947-1939-0000015-X
- William Browder, On differential Hopf algebras, Trans. Amer. Math. Soc. 107 (1963), 153–176. MR 145530, DOI 10.1090/S0002-9947-1963-0145530-7
- Yves Félix and Stephen Halperin, Rational LS category and its applications, Trans. Amer. Math. Soc. 273 (1982), no. 1, 1–38. MR 664027, DOI 10.1090/S0002-9947-1982-0664027-0
- Yves Félix, Stephen Halperin, Jean-Michel Lemaire, and Jean-Claude Thomas, Mod $p$ loop space homology, Invent. Math. 95 (1989), no. 2, 247–262. MR 974903, DOI 10.1007/BF01393897
- Yves Félix, Stephen Halperin, and Jean-Claude Thomas, Hopf algebras of polynomial growth, J. Algebra 125 (1989), no. 2, 408–417. MR 1018954, DOI 10.1016/0021-8693(89)90173-7
- Yves Félix, Stephen Halperin, and Jean-Claude Thomas, Elliptic Hopf algebras, J. London Math. Soc. (2) 43 (1991), no. 3, 545–555. MR 1113392, DOI 10.1112/jlms/s2-43.3.545
- Yves Félix, Stephen Halperin, and Jean-Claude Thomas, Elliptic spaces, Bull. Amer. Math. Soc. (N.S.) 25 (1991), no. 1, 69–73. MR 1085825, DOI 10.1090/S0273-0979-1991-16027-1
- Yves Félix, Stephen Halperin, and Jean-Claude Thomas, Adams’ cobar equivalence, Trans. Amer. Math. Soc. 329 (1992), no. 2, 531–549. MR 1036001, DOI 10.1090/S0002-9947-1992-1036001-2
- Yves Félix, Stephen Halperin, and Jean-Claude Thomas, Torsion in loop space homology, J. Reine Angew. Math. 432 (1992), 77–92. MR 1184760, DOI 10.1016/S0040-9383(03)00048-X
- Y. Félix, S. Halperin, and J.-C. Thomas, The category of a map and the grade of a module, Israel J. Math. 78 (1992), no. 2-3, 177–196. MR 1194965, DOI 10.1007/BF02808056
- Yves Félix, Stephen Halperin, and Jean-Claude Thomas, Elliptic spaces. II, Enseign. Math. (2) 39 (1993), no. 1-2, 25–32. MR 1225255
- Yves Félix and Jean-Claude Thomas, Module d’holonomie d’une fibration, Bull. Soc. Math. France 113 (1985), no. 3, 255–258 (French, with English summary). MR 834038
- T. Ganea, A generalization of the homology and homotopy suspension, Comment. Math. Helv. 39 (1965), 295–322. MR 179791, DOI 10.1007/BF02566956
- Paul Goerss, Jean Lannes, and Fabien Morel, Vecteurs de Witt non commutatifs et représentabilité de l’homologie modulo $p$, Invent. Math. 108 (1992), no. 1, 163–227 (French). MR 1156389, DOI 10.1007/BF02100603
- Stephen Halperin, Rational fibrations, minimal models, and fibrings of homogeneous spaces, Trans. Amer. Math. Soc. 244 (1978), 199–224. MR 515558, DOI 10.1090/S0002-9947-1978-0515558-4
- Stephen Halperin, Universal enveloping algebras and loop space homology, J. Pure Appl. Algebra 83 (1992), no. 3, 237–282. MR 1194839, DOI 10.1016/0022-4049(92)90046-I
- Jean Lannes and Lionel Schwartz, À propos de conjectures de Serre et Sullivan, Invent. Math. 83 (1986), no. 3, 593–603 (French, with English summary). MR 827370, DOI 10.1007/BF01394425
- James P. Lin, Torsion in $H$-spaces. I, Ann. of Math. (2) 103 (1976), no. 3, 457–487. MR 425961, DOI 10.2307/1970948
- John McCleary, User’s guide to spectral sequences, Mathematics Lecture Series, vol. 12, Publish or Perish, Inc., Wilmington, DE, 1985. MR 820463
- John McCleary, Homotopy theory and closed geodesics, Homotopy theory and related topics (Kinosaki, 1988) Lecture Notes in Math., vol. 1418, Springer, Berlin, 1990, pp. 86–94. MR 1048178, DOI 10.1007/BFb0083695
- C. A. McGibbon and C. W. Wilkerson, Loop spaces of finite complexes at large primes, Proc. Amer. Math. Soc. 96 (1986), no. 4, 698–702. MR 826505, DOI 10.1090/S0002-9939-1986-0826505-X
- Saunders MacLane, Steinitz field towers for modular fields, Trans. Amer. Math. Soc. 46 (1939), 23–45. MR 17, DOI 10.1090/S0002-9947-1939-0000017-3
- John W. Milnor and John C. Moore, On the structure of Hopf algebras, Ann. of Math. (2) 81 (1965), 211–264. MR 174052, DOI 10.2307/1970615
- Mamoru Mimura and Hirosi Toda, Topology of Lie groups. I, II, Translations of Mathematical Monographs, vol. 91, American Mathematical Society, Providence, RI, 1991. Translated from the 1978 Japanese edition by the authors. MR 1122592, DOI 10.1090/mmono/091
- John C. Moore and Larry Smith, Hopf algebras and multiplicative fibrations. I, Amer. J. Math. 90 (1968), 752–780. MR 234455, DOI 10.2307/2373482
- John C. Moore and Larry Smith, Hopf algebras and multiplicative fibrations. II, Amer. J. Math. 90 (1968), 1113–1150. MR 238323, DOI 10.2307/2373293
- C. J. Everett Jr., Annihilator ideals and representation iteration for abstract rings, Duke Math. J. 5 (1939), 623–627. MR 13
- Sam Perlis, Maximal orders in rational cyclic algebras of composite degree, Trans. Amer. Math. Soc. 46 (1939), 82–96. MR 15, DOI 10.1090/S0002-9947-1939-0000015-X
- William M. Singer, Extension theory for connected Hopf algebras, J. Algebra 21 (1972), 1–16. MR 320056, DOI 10.1016/0021-8693(72)90031-2
- Larry Smith, The cohomology of stable two stage Postnikov systems, Illinois J. Math. 11 (1967), 310–329. MR 208597
- Saunders MacLane and O. F. G. Schilling, Infinite number fields with Noether ideal theories, Amer. J. Math. 61 (1939), 771–782. MR 19, DOI 10.2307/2371335
- George W. Whitehead, Elements of homotopy theory, Graduate Texts in Mathematics, vol. 61, Springer-Verlag, New York-Berlin, 1978. MR 516508
Additional Information
- Yves Félix
- Affiliation: Institut de Mathématiques, Université de Louvain-la-Neuve, B-1348 Louvain-la- Neuve, Belgium
- Stephen Halperin
- Affiliation: College of Computer, Mathematical and Physical Sciences, University of Maryland, College Park, Maryland 20742-3281
- Jean-Claude Thomas
- Affiliation: Faculté des Sciences, Université d’Angers, 49045 bd Lavoisier, Angers, France
- Received by editor(s): January 30, 1998
- Published electronically: April 24, 2001
- Additional Notes: Research of the second author was partially supported by an NSERC operating grant. Research of the first and third authors was partially supported by UMR-6093 au CNRS
This work was also partially supported by a NATO travel grant held by all three authors. - © Copyright 2001 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 353 (2001), 3803-3831
- MSC (2000): Primary 57T25, 55R20, 57T05
- DOI: https://doi.org/10.1090/S0002-9947-01-02801-X
- MathSciNet review: 1837260