On the homotopy category of Moore spaces and the cohomology of the category of abelian groups

Hans-Joachim Baues; Manfred Hartl

Fundamenta Mathematicae (1996)

  • Volume: 150, Issue: 3, page 265-289
  • ISSN: 0016-2736

Abstract

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The homotopy category of Moore spaces in degree 2 represents a nontrivial cohomology class in the cohomology of the category of abelian groups. We describe various properties of this class. We use James-Hopf invariants to obtain explicitly the image category under the functor chain complex of the loop space.

How to cite

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Baues, Hans-Joachim, and Hartl, Manfred. "On the homotopy category of Moore spaces and the cohomology of the category of abelian groups." Fundamenta Mathematicae 150.3 (1996): 265-289. <http://eudml.org/doc/212177>.

@article{Baues1996,
abstract = {The homotopy category of Moore spaces in degree 2 represents a nontrivial cohomology class in the cohomology of the category of abelian groups. We describe various properties of this class. We use James-Hopf invariants to obtain explicitly the image category under the functor chain complex of the loop space.},
author = {Baues, Hans-Joachim, Hartl, Manfred},
journal = {Fundamenta Mathematicae},
keywords = {homotopy category; Moore spaces; cohomology; category of abelian groups; James-Hopf invariants},
language = {eng},
number = {3},
pages = {265-289},
title = {On the homotopy category of Moore spaces and the cohomology of the category of abelian groups},
url = {http://eudml.org/doc/212177},
volume = {150},
year = {1996},
}

TY - JOUR
AU - Baues, Hans-Joachim
AU - Hartl, Manfred
TI - On the homotopy category of Moore spaces and the cohomology of the category of abelian groups
JO - Fundamenta Mathematicae
PY - 1996
VL - 150
IS - 3
SP - 265
EP - 289
AB - The homotopy category of Moore spaces in degree 2 represents a nontrivial cohomology class in the cohomology of the category of abelian groups. We describe various properties of this class. We use James-Hopf invariants to obtain explicitly the image category under the functor chain complex of the loop space.
LA - eng
KW - homotopy category; Moore spaces; cohomology; category of abelian groups; James-Hopf invariants
UR - http://eudml.org/doc/212177
ER -

References

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  1. [1] J. F. Adams and P. J. Hilton, On the chain algebra of a loop space, Comment. Math. Helv. 30 (1956), 305-330. Zbl0071.16403
  2. [2] H.-J. Baues, Algebraic Homotopy, Cambridge Stud. Adv. Math. 15, Cambridge University Press, 1988. 
  3. [3] H.-J. Baues, Combinatorial Homotopy and 4-Dimensional Complexes, de Gruyter, Berlin, 1991. 
  4. [4] H.-J. Baues, Homotopy Type and Homology, Oxford Math. Monograph, Oxford University Press, 1996. Zbl0857.55001
  5. [5] H.-J. Baues, Commutator Calculus and Groups of Homotopy Classes, London Math. Soc. Lecture Note Ser. 50, Cambridge University Press, 1981. Zbl0473.55001
  6. [6] H.-J. Baues, Homotopy types, in: Handbook of Algebraic Topology, Chapter I, I. M. James (ed.), Elsevier, 1995, 1-72. 
  7. [7] H.-J. Baues, On the cohomology of categories, universal Toda brackets, and homotopy pairs, K-Theory, to appear. 
  8. [8] H.-J. Baues and W. Dreckmann, The cohomology of homotopy categories and the general linear group, K-Theory 3 (1989), 307-338. Zbl0701.18009
  9. [9] H.-J. Baues and G. Wirsching, The cohomology of small categories, J. Pure Appl. Algebra 38 (1985), 187-211. Zbl0587.18006
  10. [10] K. A. Hardie, On the category of homotopy pairs, Topology Appl. 14 (1982), 59-69. Zbl0499.55002
  11. [11] P. Hilton, Homotopy Theory and Duality, Gordon and Breach, 1965. 
  12. [12] I. M. James, Reduced product spaces, Ann. of Math. 62 (1955), 170-197. 
  13. [13] M. Jibladze and T. Pirashvili, Cohomology of algebraic theories, J. Algebra 137 (1991), 253-296. Zbl0724.18005
  14. [14] T. Pirashvili and F. Waldhausen, MacLane homology and topological Hochschild homology, J. Pure Appl. Algebra 82 (1992), 81-98. Zbl0767.55010
  15. [15] J. H. C. Whitehead, A certain exact sequence, Ann. of Math. 52 (1950), 51-110. Zbl0037.26101

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