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Localized stable homotopy of some classifying spaces

  • Victor Snaith (a1)
Extract

In this note I will give new, simplified proofs of some of the results announced in (18) and proved in ((19), I, § 4, II, §§ 2, 9).These results mostly date from 1975/6 at which time they were proved using my crude stable decomposition of ωn σnX (20) and differential-geometric techniques with the Becker-Gottlieb transfer. Since then more systematic approaches have been developed towards the stable decompositions ((4); (5); (6); (10); (12)) and towards the transfer ((7), I and II; (8)). Actually, the system-atization of the stable decompositions was already latent in (15) if only I had realized it!

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COPYRIGHT: © Cambridge Philosophical Society 1981
References
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(1)Atiyah, M. F.K-theory (Benjamin, 1968).
(2)Becker, J. C. and Gottlieb, D. H.The transfer map and fibre bundles; Topology 14 (1975), 1–12.
(3)Becker, J. C. and Gottlieb, D. H.Transfer maps for fibrations and duality; Compositio Math. (To appear.)
(4)Caruso, J., Cohen, F. R., May, J. P. and Taylor, L. R. James maps, Segal maps and the Kahn–Priddy theorem. (To appear.)
(5)Cohen, F. R., May, J. P. and Taylor, L. R.Splitting of certain spaces CX. Math. Proc. Cambridge Philos. Soc. 84 (1978), 465–496.
(6)Cohen, F. R., May, J. P. and Taylor, L. R.Splitting of more spaces. Math. Proc. Cambridge Philos. Soc. 86 (1979), 227–236.
(7)Feshbach, M. The transfer and compact Lie groups. 1: Thesis Stanford (1976), II: Trans. Amer. Math. Soc. 251 (1979), 139–169.
(8)Feshbach, M.The transfer and compact Lie groups. Bull. Amer. Math. Soc. 83 (1977), 372–374.
(9)Harris, B. and Segal, G. B.K1 groups of algebraic integers. Annals of Math. 101 (1975), 20–33.
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(11)Mahowald, M.A new infinite family in . Topology 16 (1977), 249–256.
(12)May, J. P. Applications and generalisations of the approximation theorem. Conference Proceedings Aarhus (1978).
(13)May, J. P.Geometry of iterated loopspaces. Lecture Notes in Mathematics, no. 271, Springer-Verlag (1972).
(14)Mosher, R. E.Some stable homotopy of complex projective space. Topology 7 (1968), 179–193.
(15)Segal, G. B.Operations in stable homotopy. London Math. Soc. Lecture Notes, no. 11 (1974), 105–110.
(16)Segal, G. B.The stable homotopy of complex projective space. Quart. J. Math. (Oxford) 24 (1973), 1–5.
(17)Snaith, V. P.Stable decomposition of ωnSnX. J. London Math. Soc. (2) 7 (1974), 577–583.
(18)Snaith, V. P.Towards algebraic cobordism. Bull. Amer. Math. Soc. 83, no. 3 (1977).
(19)Snaith, V. P.Algebraic cobordism and K-theory. Mem. Amer. Math. Soc. no. 221 (1979).
(20) Problem session. Amer. Soc. Summer Institute, Stanford (1976).
(21)Toda, H.Mem. Coll. Sci. Univ. Kyoto 32 (1959), 103–119.
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Mathematical Proceedings of the Cambridge Philosophical Society
  • ISSN: 0305-0041
  • EISSN: 1469-8064
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