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Numerical Invariants in Homotopical Algebra, I

  • K. Varadarajan (a1)

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Classically CW-complexes were found to be the best suited objects for studying problems in homotopy theory. Certain numerical invariants associated to a CW-complex X such as the Lusternik-Schnirelmann Category of X, the index of nilpotency of ᘯ(X), the cocategory of X, the index of conilpotency of ∑ (X) have been studied by Eckmann, Hilton, Berstein and Ganea, etc. Recently D. G. Quillen [6] has developed homotopy theory for categories satisfying certain axioms. In the axiomatic set up of Quillen the duality observed in classical homotopy theory becomes a self-evident phenomenon, the axioms being so formulated.

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References

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1. Ganea, T., Lusternik-Schnirelmann category and cocategory, Proc. Lond. Math. Soc. 10 (1960), 623639.
2. Ganea, T., Fibrations and cocategory, Comment. Math. Helv. 35 (1961). 1524,
3. Ganea, T., Sur quelques invariants numériques du type d'homotopie. Cahiers de topologie et géométrie différentielle, Ehresmann Seminar, Paris, 1962.
4. Hilton, P. J., Homotopy theory and duality, Lecture Notes, Cornell University, 1959.
5. Maclane, S., Categories for the working mathematician, (Springer-Verlag, Berlin, 1971).
6. Quillen, D. G., Homotopical algebra, Springer Lecture Notes 43, 1967.
7. G. W., Whitehead, The homology suspension, Colloque de Topologie Algébrique, Louivan 1956, pp. 8995.
8. Whitehead, J. H. C., Combinatorial homotopy, I, Bull. Amer. Math. Soc. 55 (1949), 213245.
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Numerical Invariants in Homotopical Algebra, I

  • K. Varadarajan (a1)

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