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Non-Abelian cohomology and the homotopy classification of maps. (English) Zbl 0548.55018

Homotopie algébrique et algèbre locale, Journ. Luminy/France 1982, Astérisque 113-114, 167-172 (1984).
[For the entire collection see Zbl 0535.00017.]
This paper announces results of joint work of the author and P. J. Higgins. They have shown in previous work [J. Pure Appl. Alg. 21, 233- 260 (1981; Zbl 0468.55007); ibid. 22, 11-41 (1981; Zbl 0475.55009)] that the standard relative homotopy theory could be formulated in terms of the homotopy crossed complex \(\pi \underset \tilde{} X\) of a filtered space \(\underset \tilde{} X\), and that \(\pi\) satisfies a van Kampen type theorem, from which results such as the relative Hurewicz theorem could be deduced, as well as other new results in homotopy theory. A part of the proof uses an equivalence of categories \(\lambda\) : (crossed complexes)\(\to (\omega\)- groupoids), the latter objects being cubical complexes with extra structure. Forgetting this structure on \(\lambda\) C gives a cubical complex NC for any crossed complex C; the geometric realization of NC is defined to be the classifying space BC of C.
Theorem 1 states essentially that if \(\underset \tilde{} X\) is a CW-complex with its skeletal filtration, then there is a natural bijection of homotopy classes [X,BC]\(\cong [\pi\underset \tilde{} X,C]\). This generalises a classical theorem of Eilenberg-MacLane, and also includes local coefficient results. The set [\(\pi \underset \tilde{} X,C]\) is regarded as a cohomology set \(H^ 0(X;C)\), which explains the title of the paper.
J. H. C. Whitehead in his little understood paper [Bull. Am. Math. Soc. 55, 453-496 (1949; Zbl 0040.388)] gave homotopy classification results involving \(C_*(\tilde X)\), the cellular chains of the universal cover of a reduced CW-complex X, namely giving sufficient conditions for the natural map of homotopy classes \([X,Y]\to [C_*(\tilde X),C_*(\tilde Y)]\) to be a bijection. It is shown how Theorem 1 implies and strengthens Whitehead’s results, which the reviewer feels should be regarded as basic facts in homotopy theory. For example, as shown in the Bangor M. Sc. thesis of G. J. Ellis (1982), they imply results of Olum, Jajodia and others on maps into non-simply connected spaces.

MSC:

55S37 Classification of mappings in algebraic topology
55Q05 Homotopy groups, general; sets of homotopy classes
55N25 Homology with local coefficients, equivariant cohomology
55N10 Singular homology and cohomology theory
18G55 Nonabelian homotopical algebra (MSC2010)
55R35 Classifying spaces of groups and \(H\)-spaces in algebraic topology