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Crossed complexes and chain complexes with operators

Published online by Cambridge University Press:  24 October 2008

Ronald Brown  and
Philip J. Higgins
Ronald Brown
Affiliation:
School of Mathematics, University College of North Wales, Bangor, Gwynedd LL57 1UT, Wales
Philip J. Higgins
Affiliation:
Department of Mathematical Sciences, University of Durham, South Road, Durham DH1 3LE
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Extract

Chain complexes with a group of operators are a well known tool in algebraic topology, where they arise naturally as the chain complex of cellular chains of the universal cover of a reduced CW-complex X. The group of operators here is the fundamental group of X.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 107 , Issue 1 , January 1990 , pp. 33 - 57
Copyright
Copyright © Cambridge Philosophical Society 1990

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References

REFERENCES

[1]Baues, H. J.. Algebraic Homotopy (Cambridge University Press, 1988).Google Scholar
[2]Brown, R.. Coproducts of crossed P-modules: applications to the homology of groups and to second homotopy groups. Topology 23 (1984), 337–345.Google Scholar
[3]Brown, R.. Some non-Abelian methods in homotopy theory and homological algebra. In Proceedings International Conference Categorical Topology, Toledo, 1983 (ed. Bentley, H. L. et al. ), (Heldermann-Verlag, Berlin, 1984), pp. 108–146.Google Scholar
[4]Brown, R. and Golasinski, M.. A model structure for the homotopy category of crossed complexes. Cahiers Topologie Géom. Différentielle Catégoriques 30 (1989) (to appear).Google Scholar
[5]Brown, R. and Higgins, P. J.. The algebra of cubes. J. Pure Appl. Algebra 21 (1981), 233–260.Google Scholar
[6]Brown, R. and Higgins, P. J.. Colimit theorems for relative homotopy groups. J. Pure Appl. Algebra 22 (1981), 11–41.Google Scholar
[7]Brown, R. and Higgins, P. J.. Crossed complexes and non-Abelian extensions. In International Conference on Category Theory, Gummersbach (1981), Lecture Notes in Math, vol. 962 (Springer-Verlag), pp. 39–50.Google Scholar
[8]Brown, R. and Higgins, P. J.. Homotopies and tensor products for ω-groupoids and crossed complexes. J. Pure Appl. Algebra 47 (1987), 1–33.Google Scholar
[9]Brown, R. and Loday, J.-L.. Van Kampen Theorems for diagrams of spaces. Topology 26 (1987), 311–335.CrossRefGoogle Scholar
[10]Crowell, R. H.. Corresponding group and module sequences. Nagoya Math. J. 19 (1961), 27–40.Google Scholar
[11]Crowell, R. H.. The derived module of a homomorphism. Adv. in Math. 6 (1971), 210–238.Google Scholar
[12]Ehresmann, Ch.. Catégories topologiques et catégories differentiables. Cahiers Topologie Géom. Différentielle Catégoriques 25 (1983), 237–250.Google Scholar
[13]Ellis, G. J.. Homotopy classification the J. H. C. Whitehead way. Exposition. Math. 6 (1988), 97–110.Google Scholar
[14]Gilbert, N. D. and Higgins, P. J.. The non-Abelian tensor product of groups and related constructions. Glasgow Math. J. 31 (1989), 17–29.CrossRefGoogle Scholar
[15]Golasinski, M.. Homotopy groups of small categories and derived functors. Cahiers Topologie Géom. Différentielle Catégoriques 28 (1987), 89–98.Google Scholar
[16]Grothendieck, A.. Catégories cofibrées additif et complexes cotangent relatif. Lecture Notes in Math. vol. 79 (Springer-Verlag, 1968).Google Scholar
[17]Gruenberg, K. W. and Roggenkamp, K. W.. Extension categories of groups and modules II. Stem extensions. J. Algebra 67 (1980), 342–368.Google Scholar
[18]Howie, J.. Pullback functors and crossed complexes. Cahiers Topologie Géom. Différentielle Catégoriques 20 (1979), 284–296.Google Scholar
[19]Korkes, F. J. and Porter, T.. Continuous derivations, profinite crossed complexes and pseudocompact chain complexes. (U.C.N.W. Pure Math. Preprint No. 87.18.)Google Scholar
[20]Lomonaco, S. J. Jr. The homotopy groups of knots I. How to compute the algebraic 2-type. Pacific J. Math. 95 (1981), 343–390.CrossRefGoogle Scholar
[21]Lue, A. S.-T.. Cohomology of groups relative to a variety. J. Algebra 69 (1981), 155–174.CrossRefGoogle Scholar
[22]Mackenzie, K.. Lie Groupoids and Lie Algebroids in Differential Geometry. London Math. Soc. Lecture Notes Ser. no. 124 (Cambridge University Press, 1987).Google Scholar
[23]Maclane, S.. Homology (Springer-Verlag, 1963).Google Scholar
[24]Maclane, S.. Categories for the Working Mathematician. Graduate Texts in Math. no. 5 (Springer-Verlag, 1971).Google Scholar
[25]McPherson, J. M.. On the nullity and enclosure genus of wild knots. Trans. Amer. Math. Soc. 144 (1969), 545–555.Google Scholar
[26]Olum, P.. Mappings of manifolds and the notion of degree. Ann. of Math. 58 (1953), 458–480.CrossRefGoogle Scholar
[27]Plotnick, S. P. and Suciu, A. L.. k-invariants of knotted 2-spheres. Comment. Math. Helv. 60 (1985), 54–84.Google Scholar
[28]Porter, T.. Some categorical results in the theory of crossed modules in commutative algebras. J. Algebra 109 (1987), 415–420.Google Scholar
[29]Schellenberg, B.. The group of self-equivalences of some compact CW-complexes. Math. Ann. 200 (1973), 253–266.Google Scholar
[30]Taylor, J.. Group actions on ω-groupoids and crossed complexes, and the homotopy groups of orbit spaces. Ph.D. thesis, University of Durham (1982).Google Scholar
[31]Whitehead, J. H. C.. Combinatorial homotopy II. Bull. Amer. Math. Soc. 55 (1949), 453–496.Google Scholar