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Generalized Homology and Cohomology

Singular homology and cohomology were axiomatized by Eilenberg and Steenrod in the famous book [ ES52 ] .

The discoveries of K -theory and cobordism suggested to modify (weaken) the Eilenberg-Steenrod axioms. Those homology and cohomology theories satisfying the Eilenberg-Steenrod axioms except the dimension axiom are called generalized homology and cohomology theories.

Classically they are defined as functors from the category of CW-complexes to the category of graded Abelian groups (or the category of graded modules over a graded ring). Nowadays we may consider homology and cohomology theories defined on model categories .

  • axioms of generalized (co)homology theories on the category of CW-complexes (CW-spectra)
  • axioms of generalized (co)homology theories on a model category.

A good reference for generalized (co)homology theories on the category of CW-complexes (or CW-spectra) is Adams’s book [ Ada74 ] . Another old book is Swizter’s [ Swi75 ] . These books are still useful, but I recommend to read the book [ May99 ] by Peter May first. A more recent book is Rudyak’s [ Rud98 ] .

A set of axioms for (co)homology theories on a model category was defined in [ BM ] by Basterra and Mandell, although their purpose is to study cohomology theories on the category of E -ring spectra.

By the suspension isomorphism, homology theories captures stable homotopic informations. It is natural to consider homology theories as functors from a stable model category or a triangulated category to an Abelian category . See Biedermann’s [ Bie ] , for example.

It is important to know good examples. The following two are fundamental.

One of the most important among cobordism groups is the complex cobordism.

The notion of spectra is very closely related to generalized (co)homology theories. By the Brown’s representability theorem [ Bro62 ] , we can express a cohomology theory as a (graded) homotopy set in the category of spectra.

For the category of “classical spectra”, see Adams’s book [ Ada74 ] . The category of “classical spectra” has many deficits. For example, it is not easy to define a monoidal structure. Fortunately, new categories of spectra were proposed in 90s.

Such an approach to (co)homology theories looks nice. But we need more geometric interpretations of cohomology theories for applications to geometric problems.

For K -homology theory , we have a construction by Baum and Douglas based on cobordisms. Their construction can be generalized to other homology theories. There are bivariant versions similar to Kasparov’s KK -theory .

Jakob’s construction is used by Chataur [ Cha ] in his study of string topology

One of the most fundamental tools to compute generalized (co)homology theories is the Atiyah-Hirzebruch spectral sequence.

Of course, there are many other spectral sequences for generalized (co)homology theories.

Cohomology operations are also useful for generalized cohomology theories.

We have equivariant generalized (co)homology theories for spaces with group actions .

References

[Ada74]     J. F. Adams. Stable homotopy and generalised homology . University of Chicago Press, Chicago, Ill., 1974.

[Bie]     Georg Biedermann. Interpolation categories for homology theories, arXiv:math.AT/0412388 .

[BM]     Maria Basterra and Michael A. Mandell. Homology and Cohomology of E-infinity Ring Spectra, arXiv:math.AT/0407209 .

[Bro62]     Edgar H. Brown, Jr. Cohomology theories. Ann. of Math. (2) , 75:467–484, 1962. Correction in Ann. of Math., vol. 78 (1963), p. 201.

[Cha]     David Chataur. A bordism approach to string topology, arXiv:math.AT/0306080 .

[EKMM97]     A. D. Elmendorf, I. Kriz, M. A. Mandell, and J. P. May. Rings, modules, and algebras in stable homotopy theory , volume 47 of Mathematical Surveys and Monographs . American Mathematical Society, Providence, RI, 1997.

[EM]     Heath Emerson and Ralf Meyer. Bivariant K-theory via correspondences, arXiv:0812.4949 .

[ES52]     Samuel Eilenberg and Norman Steenrod. Foundations of algebraic topology . Princeton University Press, Princeton, New Jersey, 1952.

[HSS00]     Mark Hovey, Brooke Shipley, and Jeff Smith. Symmetric spectra. J. Amer. Math. Soc. , 13(1):149–208, 2000.

[Jak98]     Martin Jakob. A bordism-type description of homology. Manuscripta Math. , 96(1):67–80, 1998.

[Jak02]     Martin Jakob. Bivariant theories for smooth manifolds. Appl. Categ. Structures , 10(3):279–290, 2002. Papers in honour of the seventieth birthday of Professor Heinrich Kleisli (Fribourg, 2000).

[May99]     J. P. May. A concise course in algebraic topology . Chicago Lectures in Mathematics. University of Chicago Press, Chicago, IL, 1999.

[MM02]     M. A. Mandell and J. P. May. Equivariant orthogonal spectra and S -modules. Mem. Amer. Math. Soc. , 159(755):x 108, 2002.

[Rud98]     Yuli B. Rudyak. On Thom spectra, orientability, and cobordism . Springer Monographs in Mathematics. Springer-Verlag, Berlin, 1998.

[Swi75]     Robert M. Switzer. Algebraic topology—homotopy and homology . Springer-Verlag, New York, 1975.