Cochain Functors for General Cohomology Theories I.

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.

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Some Remarks on General Cohomology Theories.

MATHEMATICA SCANDINAVICA ◽

10.7146/math.scand.a-10887 ◽

1968 ◽

Vol 22 ◽

pp. 227 ◽

Cited By ~ 1

Author(s):

A. Deleanu ◽

P. J. Hilton

Keyword(s):

General Cohomology

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A note on Thom classes in general cohomology

Nagoya Mathematical Journal ◽

10.1017/s0027763000018936 ◽

1980 ◽

Vol 79 ◽

pp. 69-78

Author(s):

Ronald Kirk Haas ◽

Hiroshi Uehara

Keyword(s):

Direct Approach ◽

Seventieth Birthday ◽

Characteristic Classes ◽

Classical Approach ◽

Chern Classes ◽

Kunneth Formula ◽

Relative Version ◽

Set Up ◽

Entire Theory ◽

General Cohomology

This note is dedicated to the second author’s teacher, Professor Atsuo Komatsu, in celebration of his seventieth birthday.It is well known [4], [7] that the theory of characteristic classes in general cohomology is essentially based upon one theorem, the Leray-Hirsch Theorem. We further claim [8] that the entire theory could be developed merely from the Künneth Formula if under suitable conditions a truly relative version of the Meyer-Vietoris sequence exists in the general cohomology theory. In his lectures delivered at Aarhus in 1968, Dold [6] set up the necessary machinery including the Leray-Hirsch Theorem to define Chern Classes with values in general cohomology. However, he then stated [6, p. 47] that he “found a difficulty here in choosing adequate orientations (Thorn Classes) for the bundles involved”, and proceeded differently, discarding the “classical” approach used in both ordinary cohomology [9] and K-theory [3]. Later, he [7] published a more categorical work, although the approach to Chern Classes was basically unchanged from his previous work. Consequently, the possibility of the classical direct approach to the theory has remained open.

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