The generalized Boardman homomorphisms

Dominique Arlettaz

Open Mathematics (2004)

  • Volume: 2, Issue: 1, page 50-56
  • ISSN: 2391-5455

Abstract

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This paper provides universal upper bounds for the exponent of the kernel and of the cokernel of the classical Boardman homomorphism b n: π n(X)→H n(H;ℤ), from the cohomotopy groups to the ordinary integral cohomology groups of a spectrum X, and of its various generalizations π n(X)→E n(X), F n(X)→(E∧F)n(X), F n(X)→H n(X;π 0 F) and F n(X)→H n+t(X;π t F) for other cohomology theories E *(−) and F *(−). These upper bounds do not depend on X and are given in terms of the exponents of the stable homotopy groups of spheres and, for the last three homomorphisms, in terms of the order of the Postnikov invariants of the spectrum F.

How to cite

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Dominique Arlettaz. "The generalized Boardman homomorphisms." Open Mathematics 2.1 (2004): 50-56. <http://eudml.org/doc/268736>.

@article{DominiqueArlettaz2004,
abstract = {This paper provides universal upper bounds for the exponent of the kernel and of the cokernel of the classical Boardman homomorphism b n: π n(X)→H n(H;ℤ), from the cohomotopy groups to the ordinary integral cohomology groups of a spectrum X, and of its various generalizations π n(X)→E n(X), F n(X)→(E∧F)n(X), F n(X)→H n(X;π 0 F) and F n(X)→H n+t(X;π t F) for other cohomology theories E *(−) and F *(−). These upper bounds do not depend on X and are given in terms of the exponents of the stable homotopy groups of spheres and, for the last three homomorphisms, in terms of the order of the Postnikov invariants of the spectrum F.},
author = {Dominique Arlettaz},
journal = {Open Mathematics},
keywords = {Primary: 55 N 20; 55 Q 55; Secondary: 55 Q 45; 55 S 45},
language = {eng},
number = {1},
pages = {50-56},
title = {The generalized Boardman homomorphisms},
url = {http://eudml.org/doc/268736},
volume = {2},
year = {2004},
}

TY - JOUR
AU - Dominique Arlettaz
TI - The generalized Boardman homomorphisms
JO - Open Mathematics
PY - 2004
VL - 2
IS - 1
SP - 50
EP - 56
AB - This paper provides universal upper bounds for the exponent of the kernel and of the cokernel of the classical Boardman homomorphism b n: π n(X)→H n(H;ℤ), from the cohomotopy groups to the ordinary integral cohomology groups of a spectrum X, and of its various generalizations π n(X)→E n(X), F n(X)→(E∧F)n(X), F n(X)→H n(X;π 0 F) and F n(X)→H n+t(X;π t F) for other cohomology theories E *(−) and F *(−). These upper bounds do not depend on X and are given in terms of the exponents of the stable homotopy groups of spheres and, for the last three homomorphisms, in terms of the order of the Postnikov invariants of the spectrum F.
LA - eng
KW - Primary: 55 N 20; 55 Q 55; Secondary: 55 Q 45; 55 S 45
UR - http://eudml.org/doc/268736
ER -

References

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  1. [1] J.F. Adams: Stable homotopy and generalised homology, The University of Chicago Press, Chicago, 1974. Zbl0309.55016
  2. [2] D. Arlettaz: “The order of the differentials in the Atiyah-Hirzebruch spectral sequence”, K-Theory, Vol. 6, (1992), pp. 347–361. http://dx.doi.org/10.1007/BF00966117 Zbl0768.55012
  3. [3] D. Arlettaz: “Exponents for extraordinary homology groups”, Comment. Math. Helv., Vol. 68, (1993), pp. 653–672. Zbl0968.55003
  4. [4] D. Arlettaz: “The exponent of the homotopy groups of Moore spectra and the stable Hurewicz homomorphism”, Canad. J. Math., Vol. 48, (1996), pp. 483–495. Zbl0866.55003
  5. [5] C.R.F. Maunder: “The spectral sequence of an extraordinary cohomology theory”, Math. Proc. Cambridge Philos. Soc., Vol. 59, (1963), pp. 567–574 http://dx.doi.org/10.1017/S0305004100037245 Zbl0116.14603
  6. [6] R.M. Switzer: Algebraic topology-homotopy and homology, Die Grundlehren der mathematischen Wissenschaften, 1975. 
  7. [7] J.W. Vick: “Poincaré duality and Postnikov factors”, Rocky Mountain J. Math., Vol. 3, (1973), pp. 483–499 http://dx.doi.org/10.1216/RMJ-1973-3-3-483 Zbl0272.55011

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