Algebraic & Geometric Topology

Connective $\mathrm{Im}(J)$–theory for cyclic groups

Karlheinz Knapp

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We study connective Im(J)–theory for the classifying space Bpa of a finite cyclic p–group and compute the Im(J)–cohomology groups completely. We also compute the Im(J)–homology groups, with the exception of a finite range of dimensions.

Article information

Algebr. Geom. Topol., Volume 7, Number 2 (2007), 797-828.

Received: 18 July 2006
Accepted: 20 March 2007
First available in Project Euclid: 20 December 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 55N35: Other homology theories 19L64: Computations, geometric applications
Secondary: 19D99: None of the above, but in this section 19L20: $J$-homomorphism, Adams operations [See also 55Q50]

$\mathrm{Im}(J)$–theory cyclic groups


Knapp, Karlheinz. Connective $\mathrm{Im}(J)$–theory for cyclic groups. Algebr. Geom. Topol. 7 (2007), no. 2, 797--828. doi:10.2140/agt.2007.7.797.

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  • J F Adams, Lectures on generalised cohomology, from: “Category Theory, Homology Theory and their Applications, III (Battelle Institute Conference, Seattle, Wash., 1968, Vol. Three)”, Springer, Berlin (1969) 1–138
  • J F Adams, Stable homotopy and generalised homology, University of Chicago Press, Chicago, Ill. (1974) Chicago Lectures in Mathematics
  • A K Bousfield, The localization of spectra with respect to homology, Topology 18 (1979) 257–281
  • A K Bousfield, On the homotopy theory of $K$-local spectra at an odd prime, Amer. J. Math. 107 (1985) 895–932
  • M C Crabb, K Knapp, Applications of nonconnective ${\rm Im}(J)$-theory, from: “Handbook of algebraic topology”, North-Holland, Amsterdam (1995) 463–503
  • D Eisenbud, Commutative algebra, Graduate Texts in Mathematics 150, Springer, New York (1995) With a view toward algebraic geometry
  • S Hashimoto, A note on some periodicity of ${\rm Ad}$-cohomology groups of lens spaces, Osaka J. Math. 23 (1986) 307–312
  • A Jankowski, Splitting of $K$-theory and $g\sb\ast $ characteristic numbers, from: “Studies in algebraic topology”, Adv. in Math. Suppl. Stud. 5, Academic Press, New York (1979) 189–212
  • S Jäschke, Stabil sphärische Elemente in der Bild-$J$-Homologie des klassifizierenden Raumes der zyklischen Gruppe mit $p$ Elementen, PhD thesis, Wuppertal (1992)
  • K Knapp, Connective $Im(J)$-theory for torsion-free spaces, the complex projective space as an example, preprint
  • K Knapp, On the $K$-homology of classifying spaces, Math. Ann. 233 (1978) 103–124
  • K Knapp, Stably spherical classes in the $K$-homology of a finite group, Comment. Math. Helv. 63 (1988) 414–449
  • K Knapp, Introduction to nonconnective ${\rm Im}(J)$-theory, from: “Handbook of algebraic topology”, North-Holland, Amsterdam (1995) 425–461
  • K Knapp, Anderson duality in $K$-theory and ${\rm Im}(J)$-theory, $K$-Theory 18 (1999) 137–159
  • T Kobayashi, S Murakami, M Sugawara, Note on $J$-groups of lens spaces, Hiroshima Math. J. 7 (1977) 387–409
  • P Leiverkus, Diplomarbeit, Wuppertal
  • R Weth, Zur $A$-Theorie von $B\mathbb{Z}/p^{2}$, Diplomarbeit, Wuppertal (1992)