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 EilenbergSteenrod
axioms. Those homology and cohomology theories
satisfying the EilenbergSteenrod axioms except the
dimension axiom are called generalized homology and
cohomology theories.
Classically they are deﬁned as functors from
the category of
CWcomplexes
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 CWcomplexes (CWspectra)

axioms of generalized (co)homology theories on a
model category.
A good reference for generalized (co)homology
theories on the category of CWcomplexes (or
CWspectra) 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
AtiyahHirzebruch 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
.
[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 Einﬁnity 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 Ktheory 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 bordismtype 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.
SpringerVerlag, Berlin, 1998.
[Swi75]
Robert M. Switzer.
Algebraic topology—homotopy and
homology
. SpringerVerlag, New York, 1975.