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A course in some aspects of classical homotopy theory

Part of the Lecture Notes in Mathematics book series (LNM,volume 1286)

Keywords

  • Hopf Algebra
  • Homotopy Group
  • Loop Space
  • Homotopy Equivalent
  • Serre Spectral Sequence

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References

  1. [A1] J.F. Adams, On the nonexistence of elements of Hopf invariant one, Ann. of Math., 72(1960), 20–104.

    CrossRef  MathSciNet  MATH  Google Scholar 

  2. [A2] J.F. Adams, On the structure and applications of the Steenrod algebra, Comment. Math. Helv. 32(1958), 180–214.

    CrossRef  MathSciNet  MATH  Google Scholar 

  3. S. Araki and T. Kudo, Topology of Hn-spaces and H-squaring operations, Mem. Fac. Sci. Kyūsyū Univ. Ser. A., 1956, 85–120.

    Google Scholar 

  4. M.G. Barratt, Spaces of finite characteristic, Quart. J. Math., 11(1960), 124–136.

    CrossRef  MathSciNet  MATH  Google Scholar 

  5. M.G. Barratt, J.D.S. Jones, and M.E. Mahowald, The Kervaire invariant problem, Contemp. Math., 19(1983), 9–22.

    CrossRef  MathSciNet  MATH  Google Scholar 

  6. R. Bott and H. Samelson, On the Pontrjagin product in spaces of paths, Comment. Math. Helv., 27(1953), 320–337.

    CrossRef  MathSciNet  MATH  Google Scholar 

  7. E. H. Brown and F.P. Peterson, Whitehead products and cohomology operations, Quart. J. Math., 15(1964), 116–120.

    CrossRef  MathSciNet  MATH  Google Scholar 

  8. W. Browder, Homology operations and loop spaces, Ill. J. Math., 4(1960), 347–357.

    MathSciNet  MATH  Google Scholar 

  9. H.E.A. Campbell, F.P. Peterson, and P.S. Selick, Self-maps of loop spaces I, to appear.

    Google Scholar 

  10. H.E.A. Campbell, F.R. Cohen, F.P. Peterson, and P.S. Selick, The space of maps of Moore spaces to spheres, to appear.

    Google Scholar 

  11. [C1] F.R. Cohen, Two-primary analogues of Selick's theorem and the Kahn-Priddy theorem for the 3-sphere, Topology, 23(1984), 401–421.

    CrossRef  MathSciNet  MATH  Google Scholar 

  12. [C2] F.R. Cohen, The unstable decomposition of θ2Σ2X and its applications, Math. Zeit., 182(1983), 553–568.

    CrossRef  Google Scholar 

  13. F.R. Cohen, T.J. Lada, and J.P. May, The homology of iterated loop spaces, L.N.M. v.533, Springer-Verlag, Berlin and New York, 1976.

    MATH  Google Scholar 

  14. F.R. Cohen and M.E. Mahowald, Unstable properties of θnSn+k, Contemp. Math., 12(1982), 81–90.

    CrossRef  MathSciNet  MATH  Google Scholar 

  15. [CMN1] F.R. Cohen, J.C. Moore, and J.A. Neisendorfer, Torsion in homotopy groups, Ann. of Math., 109(1979), 121–168.

    CrossRef  MathSciNet  MATH  Google Scholar 

  16. [CMN2] F.R. Cohen, J.C. Moore, and J.A. Neisendorfer, The double suspension and exponents of the homotopy groups of spheres, Ann. of Math, 110 (1979), 549–565.

    CrossRef  MathSciNet  MATH  Google Scholar 

  17. F.R. Cohen, J.C. Moore, and J.A. Neisendorfer, Exponents in homotopy theory, to appear.

    Google Scholar 

  18. F.R. Cohen and F.P. Peterson, Suspensions of Stiefel manifolds, Quart. J. Math. Oxford, 35(1984), 115–119.

    CrossRef  MathSciNet  MATH  Google Scholar 

  19. F.R. Cohen and P.S. Selick, Splitting of two function spaces, to appear.

    Google Scholar 

  20. L.E.J. Dickson, On quaternions and their generalizations and the history of the eight square theorem, Ann. of Math., 20(1919), 155–71.

    CrossRef  MathSciNet  MATH  Google Scholar 

  21. E. Dyer and R. Lashof, Homology of iterated loop spaces, Amer. J. Math., 1962, 35–88.

    Google Scholar 

  22. T. Ganea, A generalization of the homology and homotopy suspension, Commentarii Math. Helvetici, 39(1965), 295–322.

    CrossRef  MathSciNet  MATH  Google Scholar 

  23. [G1] B. Gray, A note on the Hilton-Milnor theorem, Topology, 10(1971), 199–201.

    CrossRef  MathSciNet  MATH  Google Scholar 

  24. B. Gray, On the double suspension, to appear.

    Google Scholar 

  25. [H1] P.J. Hilton, On the homotopy groups of a union of spheres, J. Lond. Math. Soc., 30(1955), 154–172.

    CrossRef  MathSciNet  MATH  Google Scholar 

  26. [H2] P.J. Hilton, Note on the Jacobi identity for Whitehead products, Proc. Camb. Phil. Soc., 57(1961), 180–182.

    CrossRef  MathSciNet  MATH  Google Scholar 

  27. [J1] I.M. James, Reduced product spaces, Ann. of Math., 62(1955), 170–197.

    CrossRef  MathSciNet  MATH  Google Scholar 

  28. [J2] I.M. James, The suspension triad of a sphere, Ann. of Math., 63(1956), 407–429.

    CrossRef  MathSciNet  MATH  Google Scholar 

  29. [J3] I.M. James, On the suspension triad, Ann. of Math., 63(1956), 191–247.

    CrossRef  MathSciNet  MATH  Google Scholar 

  30. [J4] I.M. James, On the suspension sequence, Ann. of Math., 65(1957), 74–107.

    CrossRef  MathSciNet  MATH  Google Scholar 

  31. [J5] I.M. James, The Topology of Stiefel Manifolds, London Math. Soc. Lecture Note Series, no.24, Cambridge Univ. Press, Cambridge, 1976.

    MATH  Google Scholar 

  32. D.S. Kahn and S.B. Priddy, The transfer and stable homotopy theory, Math. Proc. Camb. Phil. Soc., 83(1978), 103–111.

    CrossRef  MathSciNet  MATH  Google Scholar 

  33. M.E. Mahowald, Some remarks on the Kervaire invariant from the homotopy point of view, Proc. of Symposia in Pure Maht., XXII, Providence R.I., 1971, 165–169.

    CrossRef  MathSciNet  MATH  Google Scholar 

  34. W.S. Massey and H. Uehara, The Jacobi identity for Whitehead products, Algebraic Geometry and Topology, A Symposium in honor of S. Lefschetz, (361–377), Princeton Mathematical series, no.12, Princeton Univ. Press, Princeton, N.J., 1957.

    Google Scholar 

  35. J.P. May, The Geometry of Iterated Loop Spaces, L.N.M. v.268, Springer-Verlag, Berlin and New York, 1972.

    CrossRef  MATH  Google Scholar 

  36. R.J. Milgram, Iterated loop spaces, Ann. of Math., 84(1966), 386–403.

    CrossRef  MathSciNet  MATH  Google Scholar 

  37. J.W. Milnor, The construction FK, Algebraic Topology, A Student's Guide by J.F. Adams, 119–135. London Math. Soc. Lecture Notes, no.4, Cambridge Univ. Press, Cambridge, 1972.

    Google Scholar 

  38. J.W. Milnor and J.C. Moore, On the structure of Hopf algebras, Ann. of Math., 81(1965), 211–264.

    CrossRef  MathSciNet  MATH  Google Scholar 

  39. J.A. Neisendorfer, Primary Homotopy Theory, Memoirs of the A.M.S., 25(1980).

    Google Scholar 

  40. J.A. Neisendorfer, The exponent of a Moore space, to appear.

    Google Scholar 

  41. [N3] J.A. Neisendorfer, 3-primary exponents, Proc. Camb. Phil. Soc., 90(1981), 63–83.

    CrossRef  MathSciNet  MATH  Google Scholar 

  42. M. Nakaoka and H. Toda, On the Jacobi identity for Whitehead products, J. Inst. Polytech. Osaka City Univ., Ser. A (1954), 1–13.

    Google Scholar 

  43. G. Nishida, Cohomology operations in iterated loop spaces, Proc. Japan Acad., 44(1968), 104–109.

    CrossRef  MathSciNet  MATH  Google Scholar 

  44. G.J. Porter, The homotopy groups of wedges of suspensions, Amer. J. Math., 88(1966), 655–663.

    CrossRef  MathSciNet  MATH  Google Scholar 

  45. H. Samelson, A connection between the Whitehead product and the Pontrjagin product, Amer. J. Math., 28(1954), 278–287.

    MathSciNet  Google Scholar 

  46. R. Schafer, On the algebras formed by the Cayley-Dickson process, Amer. J. Math., 76(1954), 435–446.

    CrossRef  MathSciNet  MATH  Google Scholar 

  47. [S1] P.S. Selick, Odd primary torsion in πkS3, Topology, 17(1978), 407–412.

    CrossRef  MathSciNet  MATH  Google Scholar 

  48. [S2] P.S. Selick, A decomposition of π*(S2p+1; ℤ/p), Topology, 20(1981), 175–177.

    CrossRef  MathSciNet  Google Scholar 

  49. [S3] P.S. Selick, A spectral sequence concerning the double suspension, Invent. Math., 64(1981), 15–24.

    CrossRef  MathSciNet  MATH  Google Scholar 

  50. [S4] P.S. Selick, A reformulation of the Arf invariant one mod p problem and applications to atomic spaces, Pac. J. Math., 108(1983), 431–450.

    CrossRef  MathSciNet  MATH  Google Scholar 

  51. [S5] P.S. Selick, 2-primary exponents for the homotopy groups of spheres, Topology, 23(1984), 97–99.

    CrossRef  MathSciNet  MATH  Google Scholar 

  52. V.P. Snaith, A stable decomposition for θnΣnX, J. London Math. Soc., 7(1974), 577–583.

    CrossRef  MathSciNet  MATH  Google Scholar 

  53. N.E. Steenrod, A convenient category of topological spaces, Mich. Math. J., 14(1967), 133–152.

    CrossRef  MathSciNet  MATH  Google Scholar 

  54. [T1] H. Toda, Composition Methods in the Homotopy Groups of Spheres, Ann. of Math. Studies, v.49, Princeton Univ. Press, Princeton, N.J., 1962.

    MATH  Google Scholar 

  55. [T2] H. Toda, On the double suspension E2, J. Inst. Polytech. Osaka City Univ., Ser.A, 7(1956), 103–145.

    MathSciNet  Google Scholar 

  56. [T3] H. Toda, Non-existence of mappings of S31 into S16 with Hopf invariant one, J. Inst. Polytech Osaka City Univ., Ser.A, 8(1957), 31–34.

    MathSciNet  Google Scholar 

  57. [T4] H. Toda, Complex of standard paths and n-ad homotopy groups, J. Inst. Polytech Osaka City Univ., Ser.A, 6(1955), 101–120.

    MathSciNet  MATH  Google Scholar 

  58. D. Waggoner, Thesis, Univ. of Kentucky, 1985.

    Google Scholar 

  59. [W1] G.W. Whitehead, Elements of Homotopy Theory, Graduate Texts in Mathematics, Springer-Verlag, Berlin and New York, 1978.

    CrossRef  Google Scholar 

  60. [W2] G.W. Whitehead, On mappings into group-like spaces, Comment. Math. Helv., 28(1954), 320–328.

    CrossRef  MathSciNet  Google Scholar 

  61. J.H.C. Whitehead, On adding relations to homotopy groups, Ann. of Math., 42(1941), 409–428.

    CrossRef  MathSciNet  MATH  Google Scholar 

  62. H. Whitney, The self-intersections of a smooth n-manifold in 2n space, Ann. of Math., 45(1944), 220–246.

    CrossRef  MathSciNet  MATH  Google Scholar 

  63. H. Zassenhaus, The Theory of Groups, Chelsea, 1958.

    Google Scholar 

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Cohen, F.R. (1987). A course in some aspects of classical homotopy theory. In: Miller, H.R., Ravenel, D.C. (eds) Algebraic Topology. Lecture Notes in Mathematics, vol 1286. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0078738

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  • DOI: https://doi.org/10.1007/BFb0078738

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