Constructing product fibrations by means of a generalization of a theorem of Ganea
HTML articles powered by AMS MathViewer
- by Paul Selick PDF
- Trans. Amer. Math. Soc. 348 (1996), 3573-3589 Request permission
Abstract:
A theorem of Ganea shows that for the principal homotopy fibration $\Omega B\to F\to E$ induced from a fibration $F\to E\to B$, there is a product decomposition $\Omega (E/F)\approx \Omega B\times \Omega (F*\Omega B)$. We will determine the conditions for a fibration $X\to Y\to Z$ to yield a product decomposition $\Omega (Z/Y)\approx X\times \Omega (X*Y)$ and generalize it to pushouts. Using this approach we recover some decompositions originally proved by very computational methods. The results are then applied to produce, after localization at an odd prime $p$, homotopy decompositions for $\Omega {J_{k}\left (S^{2n}\right )}$ for some $k$ which include the cases $k=p^{t}$. The factors of $\Omega {J_{p^{t}}\left (S^{2n}\right )}$ consist of the homotopy fibre of the attaching map $S^{2np^{t}-1}\to {J_{p^{t}-1}\left (S^{2n}\right )}$ for ${J_{p^{t}}\left (S^{2n}\right )}$ and combinations of spaces occurring in the Snaith stable decomposition of $\Omega ^{2} S^{2n+1}$.References
- David Anick, Differential algebras in topology, Research Notes in Mathematics, vol. 3, A K Peters, Ltd., Wellesley, MA, 1993. MR 1213682
- Frederick R. Cohen, The unstable decomposition of $\Omega ^{2}\Sigma ^{2}X$ and its applications, Math. Z. 182 (1983), no. 4, 553–568. MR 701370, DOI 10.1007/BF01215483
- Frederick R. Cohen, Thomas J. Lada, and J. Peter May, The homology of iterated loop spaces, Lecture Notes in Mathematics, Vol. 533, Springer-Verlag, Berlin-New York, 1976. MR 0436146, DOI 10.1007/BFb0080464
- F. R. Cohen, J. C. Moore, and J. A. Neisendorfer, Torsion in homotopy groups, Ann. of Math. (2) 109 (1979), no. 1, 121–168. MR 519355, DOI 10.2307/1971269
- F. R. Cohen, J. C. Moore, and J. A. Neisendorfer, The double suspension and exponents of the homotopy groups of spheres, Ann. of Math. (2) 110 (1979), no. 3, 549–565. MR 554384, DOI 10.2307/1971238
- F. R. Cohen, J. C. Moore, and J. A. Neisendorfer, Exponents in homotopy theory, Algebraic topology and algebraic $K$-theory (Princeton, N.J., 1983) Ann. of Math. Stud., vol. 113, Princeton Univ. Press, Princeton, NJ, 1987, pp. 3–34. MR 921471
- Albrecht Dold and Richard Lashof, Principal quasi-fibrations and fibre homotopy equivalence of bundles, Illinois J. Math. 3 (1959), 285–305. MR 101521
- Albrecht Dold and René Thom, Quasifaserungen und unendliche symmetrische Produkte, Ann. of Math. (2) 67 (1958), 239–281 (German). MR 97062, DOI 10.2307/1970005
- T. Ganea, A generalization of the homology and homotopy suspension, Comment. Math. Helv. 39 (1965), 295–322. MR 179791, DOI 10.1007/BF02566956
- Tadasi Nakayama, On Frobeniusean algebras. I, Ann. of Math. (2) 40 (1939), 611–633. MR 16, DOI 10.2307/1968946
- 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
- J. Peter May, Classifying spaces and fibrations, Mem. Amer. Math. Soc. 1 (1975), no. 1, 155, xiii+98. MR 370579, DOI 10.1090/memo/0155
- 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 Milnor, On spaces having the homotopy type of a $\textrm {CW}$-complex, Trans. Amer. Math. Soc. 90 (1959), 272–280. MR 100267, DOI 10.1090/S0002-9947-1959-0100267-4
- Tsuyoshi Fujiwara (ed.), Proceedings of the 3rd Symposium on Semigroups, Osaka University, Department of Mathematics, Osaka, 1980. Held at the Inter-University Seminar House of Kansai, Kobe, August 4–6, 1979. MR 571690
- Joseph A. Neisendorfer, The exponent of a Moore space, Algebraic topology and algebraic $K$-theory (Princeton, N.J., 1983) Ann. of Math. Stud., vol. 113, Princeton Univ. Press, Princeton, NJ, 1987, pp. 35–71. MR 921472
- Joseph A. Neisendorfer, $3$-primary exponents, Math. Proc. Cambridge Philos. Soc. 90 (1981), no. 1, 63–83. MR 611286, DOI 10.1017/S0305004100058539
- J. A. Neisendorfer and P. S. Selick, Some examples of spaces with or without exponents, Current trends in algebraic topology, Part 1 (London, Ont., 1981) CMS Conf. Proc., vol. 2, Amer. Math. Soc., Providence, R.I., 1982, pp. 343–357. MR 686124
- Paul Selick, Odd primary torsion in $\pi _{k}(S^{3})$, Topology 17 (1978), no. 4, 407–412. MR 516219, DOI 10.1016/0040-9383(78)90007-1
- Paul Selick, A spectral sequence concerning the double suspension, Invent. Math. 64 (1981), no. 1, 15–24. MR 621768, DOI 10.1007/BF01393932
- Paul Selick, Moore conjectures, Algebraic topology—rational homotopy (Louvain-la-Neuve, 1986) Lecture Notes in Math., vol. 1318, Springer, Berlin, 1988, pp. 219–227. MR 952582, DOI 10.1007/BFb0077805
- P. Selick, Homology and some homotopy decompositions of the James filtration on spheres, Trans. Amer. Math. Soc. 348 (1996), 3549–3572.
- V. P. Snaith, A stable decomposition of $\Omega ^{n}S^{n}X$, J. London Math. Soc. (2) 7 (1974), 577–583. MR 339155, DOI 10.1112/jlms/s2-7.4.577
- James D. Stasheff, $H$-spaces and classifying spaces: foundations and recent developments, Algebraic topology (Proc. Sympos. Pure Math., Vol. XXII, Univ. Wisconsin, Madison, Wis., 1970) Amer. Math. Soc., Providence, R.I., 1971, pp. 247–272. MR 0321079
- 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
Additional Information
- Paul Selick
- Affiliation: Department of Mathematics, University of Toronto, 100 St. George St., Toronto, Ontario, Canada M5S 1A1
- MR Author ID: 158410
- Email: selick@math.toronto.edu
- Received by editor(s): August 9, 1994
- Additional Notes: Research partially supported by a grant from NSERC
- © Copyright 1996 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 348 (1996), 3573-3589
- MSC (1991): Primary 55P99, 55P10
- DOI: https://doi.org/10.1090/S0002-9947-96-01517-6
- MathSciNet review: 1329539