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On the cohomology of stable two stage Postnikov systems


Author: John R. Harper
Journal: Trans. Amer. Math. Soc. 152 (1970), 375-388
MSC: Primary 55.50
DOI: https://doi.org/10.1090/S0002-9947-1970-0268892-2
MathSciNet review: 0268892
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Abstract: We study the cohomology of certain fibre spaces. The spaces are the total spaces of stable two stage Postnikov systems. We study their cohomology as Hopf algebras over the Steenrod algebra. The first theorem determines the cohomology as a Hopf algebra over the ground field, the algebra structure being known previously. The second theorem relates the action of the Steenrod algebra to the Hopf algebra structure and other available structures. The work is in the direction of explicit computations of these structures but is not quite complete with regard to the action of the Steenrod algebra. The ideas of Massey and Peterson [7], Mem. Amer. Math. Soc. No. 74, are used extensively, and $ \bmod 2$ cohomology is used throughout.


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  • [1] J. F. Adams, On the non-existence of elements of Hopf invariant one, Ann. of Math. (2) 72 (1960), 20-104. MR 25 #4530. MR 0141119 (25:4530)
  • [2] -, On the structure and applications of the Steenrod algebra, Comment Math. Helv. 32 (1958), 180-214. MR 20 #2711. MR 0096219 (20:2711)
  • [3] L. Kristensen, On the cohomology of spaces with two non-vanishing homotopy groups, Math. Scand. 12 (1963), 83-105. MR 28 #2551. MR 0159334 (28:2551)
  • [4] -, On secondary cohomology operation. II, Conference on Algebraic Topology, University of Illinois at Chicago Circle, pp. 117-133 (mimeographed). MR 0250300 (40:3539)
  • [5] A. L. Liulevicius, The factorization of cyclic reduced powers by secondary cohomology operations, Proc. Nat. Acad. Sci. U. S. A. 46 (1960), 978-981. MR 24 #A2383. MR 0132543 (24:A2383)
  • [6] W. S. Massey and F. P. Peterson, The cohomology structure of certain fibre spaces. I, Topology 4 (1965), 47-65. MR 32 #6459. MR 0189032 (32:6459)
  • [7] -, The $ \bmod 2$ cohomology structure of certain fibre spaces, Mem. Amer. Math. Soc. No. 74 (1967). MR 37 #2226.
  • [8] R. J. Milgram, The structure over the Steenrod algebra of some $ 2$-stage Postnikov systems, Quart. J. Math. Oxford Ser. (2) 20 (1969), 161-169. MR 0248811 (40:2061)
  • [9] -, Steenrod squares and higher Massey products, Bol. Soc. Mat. Mexicana (2) 13 (1968), 32-57. MR 0263074 (41:7679)
  • [10] F. P. Peterson, A note on $ H$-spaces, Bol. Soc. Mat. Mexicana (2) 13 (1968), 30-31. MR 0120644 (22:11393)
  • [11] L. Smith, The cohomology of stable two stage Postnikov systems, Illinois J. Math. 11 (1967), 310-329. MR 34 #8406. MR 0208597 (34:8406)
  • [12] N. E. Steenrod, Cohomology operations, Lectures by N. E. Steenrod and revised by D. B. A. Epstein, Ann. of Math. Studies, no. 50, Princeton Univ. Press, Princeton, N. J., 1962. MR 26 #3056. MR 0145525 (26:3056)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1970-0268892-2
Keywords: Fibration, fibre space, Postnikov system, Hopf algebra, Steenrod algebra, stable $ k$-invariant, Massey-Peterson fundamental sequence
Article copyright: © Copyright 1970 American Mathematical Society

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