View source for 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043780/g0437801.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043780/g0437802.png" /> is a functor from the category <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043780/g0437803.png" /> of pairs of topological spaces into the category <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043780/g0437804.png" /> of graded Abelian groups (that is, to each pair of spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043780/g0437805.png" /> corresponds a graded Abelian group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043780/g0437806.png" /> and to each continuous mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043780/g0437807.png" /> a set of homomorphisms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043780/g0437808.png" />), and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043780/g0437809.png" /> is a set of homomorphisms

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given for each pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043780/g04378011.png" /> and natural in the sense that for any continuous <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043780/g04378012.png" /> the following equation holds:

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and the following three axioms must be satisfied.

1) The homotopy axiom. If two mappings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043780/g04378014.png" /> are homotopic, then the homomorphisms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043780/g04378015.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043780/g04378016.png" /> are the same for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043780/g04378017.png" />.

2) The exactness axiom. For any pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043780/g04378018.png" /> the sequence

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is exact; here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043780/g04378021.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043780/g04378022.png" /> are the obvious inclusions.

3) The excision axiom. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043780/g04378023.png" /> be a pair of spaces and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043780/g04378024.png" /> be such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043780/g04378025.png" />. Then the inclusion <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043780/g04378026.png" /> induces, for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043780/g04378027.png" />, isomorphisms

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For a [[Cofibration|cofibration]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043780/g04378029.png" /> it follows from the axioms that the projection <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043780/g04378030.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043780/g04378031.png" /> is a space consisting of a single point, induces an isomorphism

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Often one simply writes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043780/g04378033.png" /> instead of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043780/g04378034.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043780/g04378035.png" /> instead of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043780/g04378036.png" />. The group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043780/g04378037.png" /> is called the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043780/g04378040.png" />-dimensional (generalized) cohomology group of the pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043780/g04378041.png" />, and the graded group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043780/g04378042.png" /> is called the coefficient group of the generalized cohomology theory.

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 "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

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There is an obvious splitting

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and this splitting is canonical, noting that the inclusion <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043780/g04378062.png" /> is induced by the mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043780/g04378063.png" />. It is clear that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043780/g04378064.png" />. Also, it follows from 1)–3) that for a cofibration <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043780/g04378065.png" /> there is an isomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043780/g04378066.png" /> (see [[#References|[2]]], [[#References|[3]]]), so that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043780/g04378067.png" />. Here, as usual, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043780/g04378068.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043780/g04378069.png" />.

If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043780/g04378070.png" /> is a cofibration, then it follows from the axioms that the sequence

<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/g04378071.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>

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is exact (it is natural in the category of cofibrations). Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043780/g04378073.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043780/g04378074.png" /> are the obvious mappings and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043780/g04378075.png" /> is the composition

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In particular, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043780/g04378077.png" /> is the cone <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043780/g04378078.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043780/g04378079.png" /> (cf. [[Mapping-cone construction|Mapping-cone construction]]), then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043780/g04378080.png" /> (the homotopy axiom), and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043780/g04378081.png" /> is the [[Suspension|suspension]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043780/g04378082.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043780/g04378083.png" />; the exactness of the sequence (*) implies that there is a suspension isomomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043780/g04378084.png" />, natural with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043780/g04378085.png" />. Here, the isomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043780/g04378086.png" /> allows one to reconstruct <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043780/g04378087.png" /> (see [[#References|[2]]], [[#References|[3]]]); this is done by means of the so-called Puppe sequence. Applying the functor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043780/g04378088.png" />, as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043780/g04378089.png" />, to the latter sequence gives the exactness of (*). Thus, the generalized cohomology theory <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043780/g04378090.png" /> can be completely reconstructed in terms of the reduced theory <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043780/g04378091.png" />.

A generalized cohomology theory <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043780/g04378092.png" /> is called multiplicative if for any pairs of spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043780/g04378093.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043780/g04378094.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043780/g04378095.png" /> there is given a natural pairing

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satisfying the conditions of graded commutativity and associativity (see [[#References|[4]]], [[#References|[5]]]). In this case, for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043780/g04378097.png" />, the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043780/g04378098.png" /> is a graded (commutative, associative) ring with respect to the multiplication

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<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/g043780100.png" /></td> </tr></table>

where

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is the diagonal mapping, and the induced mappings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043780/g043780102.png" /> are ring homomorphisms. More generally, pairings of two generalized cohomology theories into a third may be defined [[#References|[5]]].

The ordinary cohomology <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043780/g043780103.png" /> can be defined as the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043780/g043780104.png" /> of homotopy classes of continuous mappings of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043780/g043780105.png" /> into the [[Eilenberg–MacLane space|Eilenberg–MacLane space]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043780/g043780106.png" />. This can be extended to generalized cohomology theories as follows. A sequence of spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043780/g043780107.png" /> and continuous mappings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043780/g043780108.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043780/g043780109.png" /> is the suspension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043780/g043780110.png" />, is called a [[Spectrum of spaces|spectrum of spaces]]. For a space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043780/g043780111.png" /> the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043780/g043780112.png" /> is defined by the equation

<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/g043780113.png" /></td> </tr></table>

Here, the mapping

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is defined as the composition

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The suspension isomorphisms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043780/g043780116.png" /> are constructed in the obvious way. Thus, each spectrum of spaces gives a certain generalized cohomology theory <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043780/g043780117.png" /> and, hence, an unreduced generalized cohomology theory <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043780/g043780118.png" />.

If, given a generalized cohomology theory <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043780/g043780119.png" />, there exists a spectrum from which it is obtained by the above method, then one says that this spectrum represents <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043780/g043780120.png" />, or that the theory <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043780/g043780121.png" /> is representable by this spectrum. It is known that any generalized cohomology theory on the category <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043780/g043780122.png" /> is representable by a spectrum [[#References|[5]]].

If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043780/g043780123.png" /> is representable by a ringed spectrum of spaces, then it is multiplicative [[#References|[5]]]. For a generalized cohomology theory given on the category <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043780/g043780124.png" /> the converse is also true.

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043780/g043780125.png" /> be a [[Serre fibration|Serre fibration]]. For any generalized cohomology theory <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043780/g043780126.png" /> and any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043780/g043780127.png" />, the groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043780/g043780128.png" /> form a local system of groups on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043780/g043780129.png" />. There exists the Dold–Atiyah–Hirzebruch spectral sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043780/g043780130.png" />, with initial term <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043780/g043780131.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043780/g043780132.png" /> is a finite CW-complex, then this spectral sequence converges and its limit term is associated to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043780/g043780133.png" /> (see [[#References|[1]]]). In particular, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043780/g043780134.png" />, then one obtains the spectral sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043780/g043780135.png" />, (sometimes) allowing the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043780/g043780136.png" /> to be computed in terms of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043780/g043780137.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043780/g043780138.png" />.

With each generalized cohomology theory <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043780/g043780139.png" /> one can associate a dual generalized homology theory <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043780/g043780140.png" />, whose axioms are analogous to those for a generalized cohomology theory except that homology is a covariant functor [[#References|[4]]]. Here, if the spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043780/g043780141.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043780/g043780142.png" /> are <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043780/g043780143.png" />-dual (see [[S-duality|<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043780/g043780144.png" />-duality]]) then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043780/g043780145.png" />. Also, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043780/g043780146.png" /> is representable by the spectrum <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043780/g043780147.png" />, then

<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/g043780148.png" /></td> </tr></table>

Here, for a multiplicative generalized cohomology theory <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043780/g043780149.png" /> there is an intersection pairing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043780/g043780150.png" />:

<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/g043780151.png" /></td> </tr></table>

The most important examples of generalized cohomology theories are [[K-theory|<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043780/g043780152.png" />-theory]] and the various [[Cobordism|cobordism]] theories. The generalized homology theories dual to cobordisms are the bordisms (cf. [[Bordism|Bordism]]).

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043780/g043780153.png" /> be an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043780/g043780154.png" />-dimensional vector bundle over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043780/g043780155.png" />, orientable (see [[Orientation|Orientation]]) in a generalized cohomology theory <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043780/g043780156.png" />, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043780/g043780157.png" /> be its [[Thom space|Thom space]]. In this case the generalized [[Thom isomorphism|Thom isomorphism]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043780/g043780158.png" /> holds (see [[#References|[1]]]). From this (and the Milnor–Spanier–Atiyah duality theorem [[#References|[7]]]) follows the generalized [[Poincaré duality|Poincaré duality]]: Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043780/g043780159.png" /> be a [[Poincaré space|Poincaré space]] of formal dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043780/g043780160.png" /> (for example, a closed <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043780/g043780161.png" />-dimensional manifold) whose normal bundle is orientable in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043780/g043780162.png" />. Then for any integer <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043780/g043780163.png" /> one has <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043780/g043780164.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043780/g043780165.png" /> be the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043780/g043780166.png" />-dimensional normal bundle over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043780/g043780167.png" /> and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043780/g043780168.png" /> be its Thom space. The spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043780/g043780169.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043780/g043780170.png" /> are <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043780/g043780171.png" />-dual (the relation called <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043780/g043780172.png" />-duality in the article [[S-duality|<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043780/g043780173.png" />-duality]] is often called <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043780/g043780174.png" />-duality). Therefore

<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 "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" />.

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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043780/g043780220.png" /> given on the category <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043780/g043780221.png" /> to the whole category <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043780/g043780222.png" />.

First method: A spectrum representing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043780/g043780223.png" /> (on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043780/g043780224.png" />) is chosen, and by means of it the theory is extended to the whole of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043780/g043780225.png" />.

Second method: Let the theory <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043780/g043780226.png" /> be given on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043780/g043780227.png" /> and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043780/g043780228.png" />; suppose that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043780/g043780229.png" /> is an exhausting family of finite CW-subspaces of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043780/g043780230.png" /> and set

<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/g043780231.png" /></td> </tr></table>

Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043780/g043780232.png" /> is a functor on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043780/g043780233.png" /> satisfying all the axioms for a generalized cohomology theory except the exactness axiom (the functor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043780/g043780234.png" /> does not preserve exactness). Thus, for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043780/g043780235.png" /> and any generalized cohomology theory <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043780/g043780236.png" /> extending <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043780/g043780237.png" />, the natural homomorphism

<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/g043780238.png" /></td> </tr></table>

is epimorphic.

In the general case the spectral sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043780/g043780239.png" /> appears, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043780/g043780240.png" />; here the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043780/g043780241.png" /> are the higher derived functions of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043780/g043780242.png" />, see [[#References|[10]]].

For a generalized homology theory <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043780/g043780243.png" />, given on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043780/g043780244.png" />, the functor

<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/g043780245.png" /></td> </tr></table>

satisfies the exactness axiom, and hence is always an extension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043780/g043780246.png" /> from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043780/g043780247.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043780/g043780248.png" />.

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|Nerve of a family of sets]]).

Generalized cohomology theories can also be extended to the category of spectra. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043780/g043780249.png" /> be a spectrum of spaces. The group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043780/g043780250.png" /> is defined by the relation

<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/g043780251.png" /></td> </tr></table>

and the mappings

<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/g043780252.png" /></td> </tr></table>

have the form

<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/g043780253.png" /></td> </tr></table>

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 "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]]].

====References====
<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>



====Comments====
For the Puppe sequence cf. the third part of the article [[Cone|Cone]].