Generalized cohomology theories

extraordinary cohomology theories

A class of special functors from the category of pairs of spaces into the category of graded Abelian groups.

A generalized cohomology theory is a pair , where is a functor from the category of pairs of topological spaces into the category of graded Abelian groups (that is, to each pair of spaces corresponds a graded Abelian group and to each continuous mapping a set of homomorphisms ), and is a set of homomorphisms

given for each pair and natural in the sense that for any continuous the following equation holds:

and the following three axioms must be satisfied.

1) The homotopy axiom. If two mappings are homotopic, then the homomorphisms and are the same for all .

2) The exactness axiom. For any pair the sequence

is exact; here and are the obvious inclusions.

3) The excision axiom. Let be a pair of spaces and let be such that . Then the inclusion induces, for all , isomorphisms

For a cofibration it follows from the axioms that the projection , where is a space consisting of a single point, induces an isomorphism

Often one simply writes instead of and instead of . The group is called the -dimensional (generalized) cohomology group of the pair , and the graded group is called the coefficient group of the generalized cohomology theory.

In the definition of a generalized cohomology theory the category can be replaced by the category of pairs of cofibrations or by the category of pairs of CW-complexes or by the category of pairs of finite CW-complexes (here, in the excision axiom one must require that the pair is isomorphic to an object in the appropriate category). In these cases one says that the generalized cohomology theory is defined on the category (respectively, ).

The choice of the term "generalized cohomology theory" is justified by the following circumstances. It was proved in [2] that any functor satisfying axioms 1)–3) and the so-called dimension axiom (which states that for ) is the usual cohomology theory with coefficients in . Later it was noticed that many useful constructions in algebraic topology (for example, cobordism; -theory) satisfy axioms 1)–3) and that the effectivity of these constructions depends to a significant extent on properties which follow formally from these axioms. This led to the acceptance of the concept of generalized cohomology theories, which had been formulated earlier.

Let be a pointed space and let be its basepoint. The reduced generalized cohomology group of is defined by putting

There is an obvious splitting

and this splitting is canonical, noting that the inclusion is induced by the mapping . It is clear that . Also, it follows from 1)–3) that for a cofibration there is an isomorphism (see [2], [3]), so that . Here, as usual, for .

If is a cofibration, then it follows from the axioms that the sequence

(*)

is exact (it is natural in the category of cofibrations). Here and are the obvious mappings and is the composition

In particular, if is the cone on (cf. Mapping-cone construction), then (the homotopy axiom), and is the suspension of ; the exactness of the sequence (*) implies that there is a suspension isomomorphism , natural with respect to . Here, the isomorphism allows one to reconstruct (see [2], [3]); this is done by means of the so-called Puppe sequence. Applying the functor , as , to the latter sequence gives the exactness of (*). Thus, the generalized cohomology theory can be completely reconstructed in terms of the reduced theory .

A generalized cohomology theory is called multiplicative if for any pairs of spaces , in there is given a natural pairing

satisfying the conditions of graded commutativity and associativity (see [4], [5]). In this case, for , the group is a graded (commutative, associative) ring with respect to the multiplication

where

is the diagonal mapping, and the induced mappings are ring homomorphisms. More generally, pairings of two generalized cohomology theories into a third may be defined [5].

The ordinary cohomology can be defined as the group of homotopy classes of continuous mappings of into the Eilenberg–MacLane space . This can be extended to generalized cohomology theories as follows. A sequence of spaces and continuous mappings , where is the suspension of , is called a spectrum of spaces. For a space the group is defined by the equation

Here, the mapping

is defined as the composition

The suspension isomorphisms are constructed in the obvious way. Thus, each spectrum of spaces gives a certain generalized cohomology theory and, hence, an unreduced generalized cohomology theory .

If, given a generalized cohomology theory , there exists a spectrum from which it is obtained by the above method, then one says that this spectrum represents , or that the theory is representable by this spectrum. It is known that any generalized cohomology theory on the category is representable by a spectrum [5].

If is representable by a ringed spectrum of spaces, then it is multiplicative [5]. For a generalized cohomology theory given on the category the converse is also true.

Let be a Serre fibration. For any generalized cohomology theory and any , the groups form a local system of groups on . There exists the Dold–Atiyah–Hirzebruch spectral sequence , with initial term . If is a finite CW-complex, then this spectral sequence converges and its limit term is associated to (see [1]). In particular, if , then one obtains the spectral sequence , (sometimes) allowing the group to be computed in terms of and .

With each generalized cohomology theory one can associate a dual generalized homology theory , whose axioms are analogous to those for a generalized cohomology theory except that homology is a covariant functor [4]. Here, if the spaces and are -dual (see -duality) then . Also, if is representable by the spectrum , then

Here, for a multiplicative generalized cohomology theory there is an intersection pairing :

The most important examples of generalized cohomology theories are -theory and the various cobordism theories. The generalized homology theories dual to cobordisms are the bordisms (cf. Bordism).

Let be an -dimensional vector bundle over , orientable (see Orientation) in a generalized cohomology theory , and let be its Thom space. In this case the generalized Thom isomorphism holds (see [1]). From this (and the Milnor–Spanier–Atiyah duality theorem [7]) follows the generalized Poincaré duality: Let be a Poincaré space of formal dimension (for example, a closed -dimensional manifold) whose normal bundle is orientable in . Then for any integer one has . Let be the -dimensional normal bundle over and let be its Thom space. The spaces and are -dual (the relation called -duality in the article -duality is often called -duality). Therefore

The element in corresponding to the identity under this isomorphism is called the fundamental class of in the theory ; this generalizes the classical concept of a fundamental class. It can be shown that the isomorphism is given by "intersection with the fundamental class" , that is, it has the form (see [4]).

Let be one of the fields or , or the skew-field of quaternions . A multiplicative generalized cohomology theory is called -orientable if all -vector bundles are orientable in . It turns out that for any -orientable theory and any -vector bundle over one can define the generalized characteristic classes (cf. Characteristic class) of a fibration with values in the group ; here, if is equal to , or , and if one uses the ordinary cohomology theory (or for ), then one obtains the Stiefel, Chern or Borel classes, respectively. In this context the theory of -cobordism (see Cobordism) is a universal -orientable generalized cohomology theory. This is also clear from the existence of the spectral sequence connecting with and , where is either or . In addition, a formal group over the ring can be associated with each -orientable generalized cohomology theory , and the universality of cobordisms is reflected in the fact that the formal group of the theory of unitary cobordism is universal (purely algebraically) in the class of all formal groups. Moreover, the formal group of the theory carries quite a lot of information on .

It often becomes necessary to extend a generalized cohomology theory from a subcategory to the whole category. For example, it may be necessary to extend a theory given on the category to the whole category .

First method: A spectrum representing (on ) is chosen, and by means of it the theory is extended to the whole of .

Second method: Let the theory be given on and let ; suppose that is an exhausting family of finite CW-subspaces of and set

Then is a functor on satisfying all the axioms for a generalized cohomology theory except the exactness axiom (the functor does not preserve exactness). Thus, for any and any generalized cohomology theory extending , the natural homomorphism

is epimorphic.

In the general case the spectral sequence appears, where ; here the are the higher derived functions of , see [10].

For a generalized homology theory , given on , the functor

satisfies the exactness axiom, and hence is always an extension of from to .

The third method is an analogue of the Aleksandrov–Čech method and depends on using the construction of a nerve (cf. Nerve of a family of sets).

Generalized cohomology theories can also be extended to the category of spectra. Let be a spectrum of spaces. The group is defined by the relation

and the mappings

have the form

The resulting functor on the category of spectra satisfies all the axioms for a reduced generalized cohomology theory (when transferred properly to the category of spectra) (see [5]).

There is a natural problem of "comparing" different generalized cohomology theories, and, in particular, the problem of expressing one cohomology theory in terms of another. The solution of the latter problem can be regarded as a far-reaching generalization of the universal coefficients formula. Spectral sequences of Adams type are the most powerful tool here. One such example has already been mentioned: The spectral sequences "from cobordisms to oriented generalized cohomology theories" . Another example: Let and be two generalized cohomology theories. Assume further that is the ring of cohomology operations (cf. Cohomology operation) of , that is a spectrum representing , and that is some spectrum (in particular, a space). Then (for "good X, Y and h*" , see [6]) there exists a spectral sequence with initial term , and with limit term conjugate with . There are also other spectral sequences (see [8], [9]) connecting one generalized cohomology theory with others.

It would be useful to learn how to treat a generalized cohomology theory as a cohomology functor, that is, to split into the composition , where is a canonical functor (not depending on ) into an Abelian category . One way of realizing this is outlined in [8].

References

[1]	A. Dold, "Relations between ordinary and extraordinary homology" , Colloq. Algebraic Topology, August 1–10, 1962 , Inst. Math. Aarhus Univ. (1962) pp. 2–9 Zbl 0145.20104
[2]	S. Eilenberg, N.E. Steenrod, "Foundations of algebraic topology" , Princeton Univ. Press (1952) MR0050886 Zbl 0047.41402
[3]	P.E. Conner, E.E. Floyd, "Differentiable periodic maps" , Springer (1964) MR0176478 Zbl 0125.40103
[4]	G.W. Whitehead, "Recent advances in homotopy theory" , Amer. Math. Soc. (1970) MR0309097 Zbl 0217.48601
[5]	R.M. Switzer, "Algebraic topology - homotopy and homology" , Springer (1975) MR0385836 Zbl 0305.55001
[6]	S.P. Novikov, "The method of algebraic topology from the viewpoint of cobordism theories" Math. USSR-Izv. , 1 (1967) pp. 827–913 Izv. Akad. Nauk SSSR Ser. Mat. , 31 : 4 (1967) pp. 855–951
[7]	M.F. Atiyah, "Thom complexes" Proc. London Math. Soc. , 11 (1961) pp. 291–310 MR0131880 Zbl 0124.16301
[8]	J.F. Adams, "Stable homotopy and generalised homology" , Univ. Chicago Press (1974) MR0402720 Zbl 0309.55016
[9]	E. Dyer, D. Kahn, "Some spectral sequences associated with fibrations" Trans. Amer. Math. Soc. , 145 (1969) pp. 397–437 MR0254840 Zbl 0191.53906
[10]	S. Araki, Z. Yosimura, "A spectral sequence associated with a cohomology theory of infinite CW-complexes" Osaka J. Math. , 9 : 3 (1972) pp. 351–365 MR0326733 Zbl 0253.55013
[11]	E. Dyer, "Cohomology theories" , Benjamin (1969) MR0268883 Zbl 0182.57002

Comments

For the Puppe sequence cf. the third part of the article Cone.