Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



On compact cohomology theories and Pontrjagin duality

Author: Keith Johnson
Journal: Trans. Amer. Math. Soc. 279 (1983), 237-247
MSC: Primary 55N20; Secondary 55N15, 55S25
MathSciNet review: 704613
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Abstract: Cohomology theories taking values in the category of topological groups are examined and a representation theorem is established for those whose coefficient groups are compact and locally euclidean. A method for constructing unstable homology operations is developed using this theorem, and application is made to the case of complex $ K$-theory.

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  • [1] J. F. Adams, Stable homotopy and generalized homology, Chicago Lectures in Math., Univ. of Chicago Press, Chicago, Ill., 1970.
  • [2] D. W. Anderson, A universal coefficient theorem for $ K$-theory, unpublished seminar notes, MIT, 1970.
  • [3] M. Artin and B. Mazur, Etale homotopy, Lecture Notes in Mathematics, No. 100, Springer-Verlag, Berlin-New York, 1969. MR 0245577
  • [4] Edgar H. Brown Jr., Cohomology theories, Ann. of Math. (2) 75 (1962), 467–484. MR 0138104
  • [5] Edgar H. Brown Jr. and Michael Comenetz, Pontrjagin duality for generalized homology and cohomology theories, Amer. J. Math. 98 (1976), no. 1, 1–27. MR 0405403
  • [6] Samuel Eilenberg and Norman Steenrod, Foundations of algebraic topology, Princeton University Press, Princeton, New Jersey, 1952. MR 0050886
  • [7] László Fuchs, Infinite abelian groups. Vol. I, Pure and Applied Mathematics, Vol. 36, Academic Press, New York-London, 1970. MR 0255673
  • [8] Peter Hilton, Putting coefficients into a cohomology theory. I, Nederl. Akad. Wetensch. Proc. Ser. A 73=Indag. Math. 32 (1970), 196–209. MR 0266197
  • [9] C. R. F. Maunder, Cohomology operations and duality, Proc. Cambridge Philos. Soc. 64 (1968), 15–30. MR 0222888
  • [10] Sidney A. Morris, Pontryagin duality and the structure of locally compact abelian groups, Cambridge University Press, Cambridge-New York-Melbourne, 1977. London Mathematical Society Lecture Note Series, No. 29. MR 0442141
  • [11] Helmut H. Schaefer, Topological vector spaces, Springer-Verlag, New York-Berlin, 1971. Third printing corrected; Graduate Texts in Mathematics, Vol. 3. MR 0342978
  • [12] E. H. Spanier, Function spaces and duality, Ann. of Math. (2) 70 (1959), 338–378. MR 0107862
  • [13] N. E. Steenrod, Cohomology operations, Symposium internacional de topología algebraica International symposium on algebraic topology, Universidad Nacional Autónoma de México and UNESCO, Mexico City, 1958, pp. 165–185. MR 0098367
  • [14] Dennis Sullivan, Genetics of homotopy theory and the Adams conjecture, Ann. of Math. (2) 100 (1974), 1–79. MR 0442930
  • [15] Rainer Vogt, Boardman’s stable homotopy category, Lecture Notes Series, No. 21, Matematisk Institut, Aarhus Universitet, Aarhus, 1970. MR 0275431
  • [16] Zen-ichi Yosimura, Universal coefficient sequences for cohomology theories of CW-spectra. II, Osaka J. Math. 16 (1979), no. 1, 201–217. MR 527026
  • [17] S. Zdravkovska, Topological objects in homotopy theory, Glasnik Mat. Ser. III 12(32) (1977), no. 1, 175–190 (English, with Macedonian summary). MR 0461488

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