Hostname: page-component-848d4c4894-x5gtn Total loading time: 0 Render date: 2024-06-07T08:07:31.983Z Has data issue: false hasContentIssue false
  • English
  • Français

Homotopy classification of filtered complexes

Published online by Cambridge University Press:  09 April 2009

Ross Street
Ross Street
Affiliation:
School of Mathematics and Physics Macquarie UniversityNorth Ryde, N.S.W. 2113, Australia
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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’.

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 15 , Issue 3 , May 1973 , pp. 298 - 318
Copyright
Copyright © Australian Mathematical Society 1973

References

[1]Cartan, H. and Eilenberg, S., Homological Algebra (Princeton, 1956).Google Scholar
[2]Eilenberg, S. and Moore, J. C., ‘Foundations of relative homological algebra,’ Mem. Amer. Math. Soc. 55 (1965), 1–39.Google Scholar
[3]Hartshorne, R., Residues and Duality (Lecture Notes in Math. 20Springer-Verlag, Berlin, 1966).CrossRefGoogle Scholar
[4]Kelly, G. M., ‘Complete functors in homology II. The exact homology sequence.’ Proc. Camb. Phil. Soc. 60 (1964), 737–749.CrossRefGoogle Scholar
[5]Kelly, G. M., ‘Chain maps inducing zero homology maps.’ Proc. Camb. Soc. 61 (1965), 846–854.Google Scholar
[6]Puppe, D., ‘Stabile Homotopietheorie I’, Math. Ann. 169 (1967), 243–274.CrossRefGoogle Scholar
[7]Street, R., ‘Projective diagrams of interlocking sequences.’ Illinois J. Math. 15 (1971), 429–441.CrossRefGoogle Scholar
[8]Wall, C. T. C., ‘On the exactness of interlocking sequences’ L'Enseign. Math. (2), 12 (1966), 95–100.Google Scholar