Cobordism of $(k)$-framed manifolds
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- by E. Micha PDF
- Trans. Amer. Math. Soc. 291 (1985), 269-280 Request permission
Abstract:
The cobordism theories that arise by considering manifolds whose stable normal bundle has category $k$ are introduced. Using these theories, we define a new filtration of the homotopy groups of spheres. We study the filtration and obtain an upper bound for the filtration of elements in the stable $n$-stem.References
- J. F. Adams, The Kahn-Priddy theorem, Proc. Cambridge Philos. Soc. 73 (1973), 45–55. MR 310878, DOI 10.1017/s0305004100047459
- D. W. Anderson, E. H. Brown Jr., and F. P. Peterson, The structure of the Spin cobordism ring, Ann. of Math. (2) 86 (1967), 271–298. MR 219077, DOI 10.2307/1970690
- John Ewing and Larry Smith, Cobordism of hyperframed manifolds and the stable $J$-homomorphism, Manuscripta Math. 22 (1977), no. 2, 171–197. MR 458443, DOI 10.1007/BF01167860
- Morris W. Hirsch, Differential topology, Graduate Texts in Mathematics, No. 33, Springer-Verlag, New York-Heidelberg, 1976. MR 0448362, DOI 10.1007/978-1-4684-9449-5
- I. M. James, On category, in the sense of Lusternik-Schnirelmann, Topology 17 (1978), no. 4, 331–348. MR 516214, DOI 10.1016/0040-9383(78)90002-2
- K. Knapp, On the bi-stable $J$-homomorphism, Algebraic topology, Aarhus 1978 (Proc. Sympos., Univ. Aarhus, Aarhus, 1978), Lecture Notes in Math., vol. 763, Springer, Berlin, 1979, pp. 13–22. MR 561211 —, Some applications of $K$-theory to framed bordism: $E$ invariant and transfer, Habilitations-schrift, Bonn, 1979.
- R. Lashof, Poincaré duality and cobordism, Trans. Amer. Math. Soc. 109 (1963), 257–277. MR 156357, DOI 10.1090/S0002-9947-1963-0156357-4 E. Micha, On the Pontrjagin-Thom construction, Ph. D. Thesis, Oxford, 1981.
- John Milnor, Construction of universal bundles. II, Ann. of Math. (2) 63 (1956), 430–436. MR 77932, DOI 10.2307/1970012
- Nigel Ray, A geometrical observation on the Arf invariant of a framed manifold, Bull. London Math. Soc. 4 (1972), 163–164. MR 322866, DOI 10.1112/blms/4.2.163
- Larry Smith, On realizing complex bordism modules. Applications to the stable homotopy of spheres, Amer. J. Math. 92 (1970), 793–856. MR 275429, DOI 10.2307/2373397 A. S. Svarc, The genus of a fibre space, Amer. Math. Soc. Transl. 55 (1966), 49-140.
Additional Information
- © Copyright 1985 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 291 (1985), 269-280
- MSC: Primary 57R90; Secondary 55Q50, 57R15
- DOI: https://doi.org/10.1090/S0002-9947-1985-0797059-5
- MathSciNet review: 797059