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Homotopy classification of filtered complexes

  • Ross Street (a1)
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The homology functor from the category of free abelian chain complexes and homotopy classes of maps to that of graded abelian groups is full and replete (surjective on objects up to isomorphism) and reflects isomorphisms. Thus such a complex is determined to within homotopy equivalence (although not a unique homotopy equivalence) by its homology. The homotopy classes of maps between two such complexes should therefore be expressible in terms of the homology groups, and such an expression is in fact provided by the Künneth formula for Hom, sometimes called ‘the homotopy classification theorem’.

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References
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[1]Cartan, H. and Eilenberg, S., Homological Algebra (Princeton, 1956).
[2]Eilenberg, S. and Moore, J. C., ‘Foundations of relative homological algebra,’ Mem. Amer. Math. Soc. 55 (1965), 139.
[3]Hartshorne, R., Residues and Duality (Lecture Notes in Math. 20Springer-Verlag, Berlin, 1966).
[4]Kelly, G. M., ‘Complete functors in homology II. The exact homology sequence.’ Proc. Camb. Phil. Soc. 60 (1964), 737749.
[5]Kelly, G. M., ‘Chain maps inducing zero homology maps.’ Proc. Camb. Soc. 61 (1965), 846854.
[6]Puppe, D., ‘Stabile Homotopietheorie I’, Math. Ann. 169 (1967), 243274.
[7]Street, R., ‘Projective diagrams of interlocking sequences.’ Illinois J. Math. 15 (1971), 429441.
[8]Wall, C. T. C., ‘On the exactness of interlocking sequencesL'Enseign. Math. (2), 12 (1966), 95100.
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Journal of the Australian Mathematical Society
  • ISSN: 1446-7887
  • EISSN: 1446-8107
  • URL: /core/journals/journal-of-the-australian-mathematical-society
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