Difference between revisions of "Generalized cohomology theories"
(Importing text file) |
Ulf Rehmann (talk | contribs) m (MR/ZBL numbers added) |
||
Line 35: | Line 35: | ||
In the definition of a generalized cohomology theory the category <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043780/g04378043.png" /> can be replaced by the category of pairs of cofibrations or by the category <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043780/g04378044.png" /> of pairs of CW-complexes or by the category <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043780/g04378045.png" /> of pairs of finite CW-complexes (here, in the excision axiom one must require that the pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043780/g04378046.png" /> is isomorphic to an object in the appropriate category). In these cases one says that the generalized cohomology theory <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043780/g04378047.png" /> is defined on the category <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043780/g04378048.png" /> (respectively, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043780/g04378049.png" />). | In the definition of a generalized cohomology theory the category <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043780/g04378043.png" /> can be replaced by the category of pairs of cofibrations or by the category <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043780/g04378044.png" /> of pairs of CW-complexes or by the category <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043780/g04378045.png" /> of pairs of finite CW-complexes (here, in the excision axiom one must require that the pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043780/g04378046.png" /> is isomorphic to an object in the appropriate category). In these cases one says that the generalized cohomology theory <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043780/g04378047.png" /> is defined on the category <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043780/g04378048.png" /> (respectively, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043780/g04378049.png" />). | ||
− | The choice of the term | + | The choice of the term "generalized cohomology theory" is justified by the following circumstances. It was proved in [[#References|[2]]] that any functor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043780/g04378050.png" /> satisfying axioms 1)–3) and the so-called dimension axiom (which states that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043780/g04378051.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043780/g04378052.png" />) is the usual [[Cohomology|cohomology]] theory <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043780/g04378053.png" /> with coefficients in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043780/g04378054.png" />. Later it was noticed that many useful constructions in algebraic topology (for example, [[Cobordism|cobordism]]; [[K-theory|<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043780/g04378055.png" />-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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043780/g04378056.png" /> be a pointed space and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043780/g04378057.png" /> be its basepoint. The reduced generalized cohomology group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043780/g04378058.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043780/g04378059.png" /> is defined by putting | Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043780/g04378056.png" /> be a pointed space and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043780/g04378057.png" /> be its basepoint. The reduced generalized cohomology group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043780/g04378058.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043780/g04378059.png" /> is defined by putting | ||
Line 109: | Line 109: | ||
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043780/g043780175.png" /></td> </tr></table> | <table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043780/g043780175.png" /></td> </tr></table> | ||
− | The element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043780/g043780176.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043780/g043780177.png" /> corresponding to the identity <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043780/g043780178.png" /> under this isomorphism is called the fundamental class of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043780/g043780179.png" /> in the theory <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043780/g043780180.png" />; this generalizes the classical concept of a [[Fundamental class|fundamental class]]. It can be shown that the isomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043780/g043780181.png" /> is given by | + | The element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043780/g043780176.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043780/g043780177.png" /> corresponding to the identity <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043780/g043780178.png" /> under this isomorphism is called the fundamental class of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043780/g043780179.png" /> in the theory <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043780/g043780180.png" />; this generalizes the classical concept of a [[Fundamental class|fundamental class]]. It can be shown that the isomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043780/g043780181.png" /> is given by "intersection with the fundamental class" , that is, it has the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043780/g043780182.png" /> (see [[#References|[4]]]). |
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043780/g043780183.png" /> be one of the fields <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043780/g043780184.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043780/g043780185.png" />, or the skew-field of quaternions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043780/g043780186.png" />. A multiplicative generalized cohomology theory <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043780/g043780187.png" /> is called <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043780/g043780191.png" />-orientable if all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043780/g043780192.png" />-vector bundles are orientable in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043780/g043780193.png" />. It turns out that for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043780/g043780194.png" />-orientable theory <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043780/g043780195.png" /> and any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043780/g043780196.png" />-vector bundle over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043780/g043780197.png" /> one can define the generalized characteristic classes (cf. [[Characteristic class|Characteristic class]]) of a fibration <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043780/g043780198.png" /> with values in the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043780/g043780199.png" />; here, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043780/g043780200.png" /> is equal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043780/g043780201.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043780/g043780202.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043780/g043780203.png" />, and if one uses the ordinary cohomology theory <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043780/g043780204.png" /> (or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043780/g043780205.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043780/g043780206.png" />), then one obtains the Stiefel, Chern or Borel classes, respectively. In this context the theory of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043780/g043780207.png" />-cobordism (see [[Cobordism|Cobordism]]) is a universal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043780/g043780208.png" />-orientable generalized cohomology theory. This is also clear from the existence of the spectral sequence connecting <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043780/g043780209.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043780/g043780210.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043780/g043780211.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043780/g043780212.png" /> is either <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043780/g043780213.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043780/g043780214.png" />. In addition, a [[Formal group|formal group]] over the ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043780/g043780215.png" /> can be associated with each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043780/g043780216.png" />-orientable generalized cohomology theory <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043780/g043780217.png" />, 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043780/g043780218.png" /> carries quite a lot of information on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043780/g043780219.png" />. | Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043780/g043780183.png" /> be one of the fields <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043780/g043780184.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043780/g043780185.png" />, or the skew-field of quaternions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043780/g043780186.png" />. A multiplicative generalized cohomology theory <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043780/g043780187.png" /> is called <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043780/g043780191.png" />-orientable if all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043780/g043780192.png" />-vector bundles are orientable in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043780/g043780193.png" />. It turns out that for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043780/g043780194.png" />-orientable theory <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043780/g043780195.png" /> and any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043780/g043780196.png" />-vector bundle over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043780/g043780197.png" /> one can define the generalized characteristic classes (cf. [[Characteristic class|Characteristic class]]) of a fibration <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043780/g043780198.png" /> with values in the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043780/g043780199.png" />; here, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043780/g043780200.png" /> is equal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043780/g043780201.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043780/g043780202.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043780/g043780203.png" />, and if one uses the ordinary cohomology theory <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043780/g043780204.png" /> (or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043780/g043780205.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043780/g043780206.png" />), then one obtains the Stiefel, Chern or Borel classes, respectively. In this context the theory of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043780/g043780207.png" />-cobordism (see [[Cobordism|Cobordism]]) is a universal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043780/g043780208.png" />-orientable generalized cohomology theory. This is also clear from the existence of the spectral sequence connecting <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043780/g043780209.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043780/g043780210.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043780/g043780211.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043780/g043780212.png" /> is either <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043780/g043780213.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043780/g043780214.png" />. In addition, a [[Formal group|formal group]] over the ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043780/g043780215.png" /> can be associated with each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043780/g043780216.png" />-orientable generalized cohomology theory <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043780/g043780217.png" />, 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043780/g043780218.png" /> carries quite a lot of information on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043780/g043780219.png" />. | ||
Line 151: | Line 151: | ||
The resulting functor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043780/g043780254.png" /> on the category of spectra satisfies all the axioms for a reduced generalized cohomology theory (when transferred properly to the category of spectra) (see [[#References|[5]]]). | The resulting functor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043780/g043780254.png" /> on the category of spectra satisfies all the axioms for a reduced generalized cohomology theory (when transferred properly to the category of spectra) (see [[#References|[5]]]). | ||
− | There is a natural problem of | + | 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043780/g043780255.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043780/g043780256.png" /> be two generalized cohomology theories. Assume further that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043780/g043780257.png" /> is the ring of cohomology operations (cf. [[Cohomology operation|Cohomology operation]]) of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043780/g043780258.png" />, that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043780/g043780259.png" /> is a spectrum representing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043780/g043780260.png" />, and that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043780/g043780261.png" /> is some spectrum (in particular, a space). Then (for "good X, Y and h*" , see [[#References|[6]]]) there exists a spectral sequence with initial term <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043780/g043780262.png" />, and with limit term conjugate with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043780/g043780263.png" />. There are also other spectral sequences (see [[#References|[8]]], [[#References|[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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043780/g043780264.png" /> into the composition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043780/g043780265.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043780/g043780266.png" /> is a canonical functor (not depending on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043780/g043780267.png" />) into an [[Abelian category|Abelian category]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043780/g043780268.png" />. One way of realizing this is outlined in [[#References|[8]]]. | It would be useful to learn how to treat a generalized cohomology theory as a cohomology functor, that is, to split <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043780/g043780264.png" /> into the composition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043780/g043780265.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043780/g043780266.png" /> is a canonical functor (not depending on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043780/g043780267.png" />) into an [[Abelian category|Abelian category]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043780/g043780268.png" />. One way of realizing this is outlined in [[#References|[8]]]. | ||
====References==== | ====References==== | ||
− | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> | + | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> A. Dold, "Relations between ordinary and extraordinary homology" , ''Colloq. Algebraic Topology, August 1–10, 1962'' , Inst. Math. Aarhus Univ. (1962) pp. 2–9 {{MR|}} {{ZBL|0145.20104}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> S. Eilenberg, N.E. Steenrod, "Foundations of algebraic topology" , Princeton Univ. Press (1952) {{MR|0050886}} {{ZBL|0047.41402}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> P.E. Conner, E.E. Floyd, "Differentiable periodic maps" , Springer (1964) {{MR|0176478}} {{ZBL|0125.40103}} </TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> G.W. Whitehead, "Recent advances in homotopy theory" , Amer. Math. Soc. (1970) {{MR|0309097}} {{ZBL|0217.48601}} </TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top"> R.M. Switzer, "Algebraic topology - homotopy and homology" , Springer (1975) {{MR|0385836}} {{ZBL|0305.55001}} </TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top"> 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</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top"> M.F. Atiyah, "Thom complexes" ''Proc. London Math. Soc.'' , '''11''' (1961) pp. 291–310 {{MR|0131880}} {{ZBL|0124.16301}} </TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top"> J.F. Adams, "Stable homotopy and generalised homology" , Univ. Chicago Press (1974) {{MR|0402720}} {{ZBL|0309.55016}} </TD></TR><TR><TD valign="top">[9]</TD> <TD valign="top"> E. Dyer, D. Kahn, "Some spectral sequences associated with fibrations" ''Trans. Amer. Math. Soc.'' , '''145''' (1969) pp. 397–437 {{MR|0254840}} {{ZBL|0191.53906}} </TD></TR><TR><TD valign="top">[10]</TD> <TD valign="top"> 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 {{MR|0326733}} {{ZBL|0253.55013}} </TD></TR><TR><TD valign="top">[11]</TD> <TD valign="top"> E. Dyer, "Cohomology theories" , Benjamin (1969) {{MR|0268883}} {{ZBL|0182.57002}} </TD></TR></table> |
Revision as of 17:32, 31 March 2012
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.
Generalized cohomology theories. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Generalized_cohomology_theories&oldid=24075