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Simplicial and Homotopical Cohomology of Polyhedra

Published online by Cambridge University Press:  20 November 2018

Emil Stamm
Emil Stamm*
Affiliation:
University of Toronto
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It is well known that, on the category of finite polyhedra, any two cohomology theories, satisfying the Eilenberg-Steenrod axioms, are isomorphic. Examples of such theories are simplicial cohomology and homotopical cohomology (the latter is defined by means of homotopy classes of maps into Eilenberg-MacLane spaces). In the case of polyhedra, using triple sequences and spectral sequences, one obtains a deep insight into the relationship between general cohomology theories (without the dimension axiom) and ordinary simplicial cohomology (1, p. 66). As a corollary the abovementioned uniqueness of cohomology theories satisfying the dimension axiom is obtained.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 18 , 1966 , pp. 413 - 423
Copyright
Copyright © Canadian Mathematical Society 1966

References

1. Eckmann, B., Homotopy and cohomology theory, Proc. Int. Congr. of Mathematicians (Stockholm, 1962), pp. 59–73.Google Scholar
2. Eckmann, B. et Hilton, P. J., Transgression homotopique et cohomologique, C. R. Acad. Sci. Paris, 247 (1958), 620–623.Google Scholar
3. Hilton, P. J. and Wylie, S., Homology theory (Cambridge, 1960).Google Scholar
4. Hu, S. T., The homotopy addition theorem, Ann. of Math., 58 (1953), 108–122.Google Scholar
5. MacLane, S., Homology (Heidelberg, 1963).Google Scholar
6. Stamm, E., Ueber die Homotopiegruppen gewisser Faserungen, Math. Z., 86 (1964), 215–254.Google Scholar