Difference between revisions of "S-duality"
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''stationary duality, Spanier duality'' | ''stationary duality, Spanier duality'' | ||
− | A [[Duality|duality]] in homotopy theory which exists (in the absence of restrictions imposed on the dimensions of spaces) for the analogues of ordinary homotopy and cohomotopy groups in the suspension category — for the | + | A [[Duality|duality]] in homotopy theory which exists (in the absence of restrictions imposed on the dimensions of spaces) for the analogues of ordinary homotopy and cohomotopy groups in the suspension category — for the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083000/s0830002.png" />-homotopy and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083000/s0830003.png" />-cohomotopy groups or stationary homotopy and cohomotopy groups, forming extra-ordinary (generalized) homology and cohomology theories. A suspension category, or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083000/s0830005.png" />-category, is a [[Category|category]] whose objects are topological spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083000/s0830006.png" />, while its morphisms are classes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083000/s0830007.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083000/s0830009.png" />-homotopic mappings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083000/s08300010.png" /> from a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083000/s08300011.png" />-fold [[Suspension|suspension]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083000/s08300012.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083000/s08300013.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083000/s08300014.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083000/s08300015.png" /> being considered as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083000/s08300016.png" />-homotopic if there exists an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083000/s08300017.png" /> such that the suspensions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083000/s08300018.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083000/s08300019.png" /> are homotopic in the ordinary sense. The set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083000/s08300020.png" /> of such classes, which are known as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083000/s08300022.png" />-mappings, constitutes an Abelian group with respect to the so-called track addition [[#References|[1]]], [[#References|[2]]], [[#References|[4]]], [[#References|[5]]]. The group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083000/s08300023.png" /> is the limit of the direct spectrum of the sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083000/s08300024.png" /> of ordinary homotopy classes with suspension mappings as projections; if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083000/s08300025.png" /> is sufficiently large, it is a group spectrum with homomorphisms. There exists an isomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083000/s08300026.png" /> in which the corresponding elements are represented by one and the same mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083000/s08300027.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083000/s08300028.png" />. The <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083000/s08300030.png" />-dual polyhedron of the polyhedron <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083000/s08300031.png" /> in a sphere <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083000/s08300032.png" /> is an arbitrary polyhedron <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083000/s08300033.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083000/s08300034.png" /> which is an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083000/s08300035.png" />-deformation retract of the complement <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083000/s08300036.png" />, i.e. the morphism corresponding to the imbedding <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083000/s08300037.png" /> is an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083000/s08300038.png" />-equivalence. The polyhedron <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083000/s08300039.png" /> exists for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083000/s08300040.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083000/s08300041.png" /> may be considered as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083000/s08300042.png" />. |
− | homotopy and | ||
− | cohomotopy groups or stationary homotopy and cohomotopy groups, forming extra-ordinary (generalized) homology and cohomology theories. A suspension category, or | ||
− | category, is a [[Category|category]] whose objects are topological spaces | ||
− | while its morphisms are classes | ||
− | of | ||
− | homotopic mappings | ||
− | from a | ||
− | fold [[Suspension|suspension]] | ||
− | into | ||
− | |||
− | and | ||
− | being considered as | ||
− | homotopic if there exists an | ||
− | such that the suspensions | ||
− | and | ||
− | are homotopic in the ordinary sense. The set | ||
− | of such classes, which are known as | ||
− | mappings, constitutes an Abelian group with respect to the so-called track addition [[#References|[1]]], [[#References|[2]]], [[#References|[4]]], [[#References|[5]]]. The group | ||
− | is the limit of the direct spectrum of the sets | ||
− | of ordinary homotopy classes with suspension mappings as projections; if | ||
− | is sufficiently large, it is a group spectrum with homomorphisms. There exists an isomorphism | ||
− | in which the corresponding elements are represented by one and the same mapping | ||
− | |||
− | The | ||
− | dual polyhedron of the polyhedron | ||
− | in a sphere | ||
− | is an arbitrary polyhedron | ||
− | in | ||
− | which is an | ||
− | deformation retract of the complement | ||
− | i.e. the morphism corresponding to the imbedding | ||
− | is an | ||
− | equivalence. The polyhedron | ||
− | exists for all | ||
− | and | ||
− | may be considered as | ||
− | For any polyhedra | + | For any polyhedra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083000/s08300043.png" /> and any polyhedra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083000/s08300044.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083000/s08300045.png" /> which are dual to them, there exists a unique mapping |
− | and any polyhedra | ||
− | and | ||
− | which are dual to them, there exists a unique mapping | ||
− | + | <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/s/s083/s083000/s08300046.png" /></td> </tr></table> | |
− | |||
− | |||
− | |||
satisfying the following conditions: | satisfying the following conditions: | ||
− | a) It is an involutory contravariant functorial isomorphism, i.e. | + | a) It is an involutory contravariant functorial isomorphism, i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083000/s08300047.png" /> is a homomorphism such that if |
− | is a homomorphism such that if | ||
− | + | <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/s/s083/s083000/s08300048.png" /></td> </tr></table> | |
− | |||
− | |||
then | 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/s/s083/s083000/s08300049.png" /></td> </tr></table> | |
− | |||
− | |||
if | if | ||
− | + | <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/s/s083/s083000/s08300050.png" /></td> </tr></table> | |
− | |||
− | |||
− | |||
then | 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/s/s083/s083000/s08300051.png" /></td> </tr></table> | |
− | |||
− | |||
− | |||
− | if | + | if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083000/s08300052.png" /> is an element of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083000/s08300053.png" /> or of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083000/s08300054.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083000/s08300055.png" />. |
− | is an element of | ||
− | or of | ||
− | then | ||
b) The following relations are valid: | b) The following relations are valid: | ||
− | + | <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/s/s083/s083000/s08300056.png" /></td> </tr></table> | |
− | |||
− | |||
− | where | + | where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083000/s08300057.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083000/s08300058.png" /> are considered as polyhedra, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083000/s08300059.png" />-dual to polyhedra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083000/s08300060.png" /> and, correspondingly, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083000/s08300061.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083000/s08300062.png" /> this means that it does not depend on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083000/s08300063.png" /> and is stationary with respect to suspension. |
− | and | ||
− | are considered as polyhedra, | ||
− | dual to polyhedra | ||
− | and, correspondingly, | ||
− | |||
− | this means that it does not depend on | ||
− | and is stationary with respect to suspension. | ||
c) It satisfies the equation | c) It satisfies 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/s/s083/s083000/s08300064.png" /></td> </tr></table> | |
− | |||
− | |||
where | where | ||
− | + | <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/s/s083/s083000/s08300065.png" /></td> </tr></table> | |
− | |||
− | |||
and | and | ||
− | + | <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/s/s083/s083000/s08300066.png" /></td> </tr></table> | |
− | |||
− | |||
− | |||
− | are homomorphisms of the above homology and cohomology groups, induced by | + | are homomorphisms of the above homology and cohomology groups, induced by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083000/s08300067.png" />-mappings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083000/s08300068.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083000/s08300069.png" />, and |
− | mappings | ||
− | and | ||
− | and | ||
− | + | <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/s/s083/s083000/s08300070.png" /></td> </tr></table> | |
− | |||
− | |||
− | |||
− | is an isomorphism obtained from the isomorphism of [[Alexander duality|Alexander duality]] by replacing the set | + | is an isomorphism obtained from the isomorphism of [[Alexander duality|Alexander duality]] by replacing the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083000/s08300071.png" /> by its <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083000/s08300072.png" />-deformation retract <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083000/s08300073.png" />. |
− | by its | ||
− | deformation retract | ||
− | The construction of | + | The construction of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083000/s08300074.png" /> is based on the representation of a given mapping as the composition of an imbedding and an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083000/s08300075.png" />-deformation retract. |
− | is based on the representation of a given mapping as the composition of an imbedding and an | ||
− | deformation retract. | ||
− | The | + | The <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083000/s08300077.png" />-homotopy group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083000/s08300078.png" /> of a space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083000/s08300079.png" /> is the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083000/s08300080.png" />, and the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083000/s08300082.png" />-cohomotopy group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083000/s08300083.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083000/s08300084.png" /> is the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083000/s08300085.png" />. As in ordinary homotopy theory, one defines the homomorphisms |
− | homotopy group | ||
− | of a space | ||
− | is the group | ||
− | and the | ||
− | cohomotopy group | ||
− | of | ||
− | is the group | ||
− | As in ordinary homotopy theory, one defines the homomorphisms | ||
− | + | <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/s/s083/s083000/s08300086.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/s/s083/s083000/s08300087.png" /></td> </tr></table> | |
− | |||
− | |||
− | Regarding the spheres | + | Regarding the spheres <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083000/s08300088.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083000/s08300089.png" /> as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083000/s08300090.png" />-dual leads to the isomorphisms |
− | and | ||
− | as | ||
− | dual leads to the isomorphisms | ||
− | + | <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/s/s083/s083000/s08300091.png" /></td> </tr></table> | |
− | |||
− | |||
and to the commutative diagram | and to the commutative diagram | ||
− | + | <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/s/s083/s083000/s08300092.png" /></td> </tr></table> | |
− | Thus, the isomorphism | + | Thus, the isomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083000/s08300093.png" /> connects <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083000/s08300094.png" />-homotopy and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083000/s08300095.png" />-cohomotopy groups, just as the isomorphism of Alexander duality <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083000/s08300096.png" /> connects the homology and cohomology groups. Any duality in the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083000/s08300097.png" />-category entails a duality of ordinary homotopy classes if the conditions imposed on the space entail the existence of a one-to-one correspondence between the set of the above classes and the set of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083000/s08300098.png" />-homotopy classes. |
− | connects | ||
− | homotopy and | ||
− | cohomotopy groups, just as the isomorphism of Alexander duality | ||
− | connects the homology and cohomology groups. Any duality in the | ||
− | category entails a duality of ordinary homotopy classes if the conditions imposed on the space entail the existence of a one-to-one correspondence between the set of the above classes and the set of | ||
− | homotopy classes. | ||
− | Examples of dual assumptions in this theory include Hurewicz's isomorphism theorem and Hopf's classification theorem. | + | Examples of dual assumptions in this theory include Hurewicz's isomorphism theorem and Hopf's classification theorem. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083000/s08300099.png" /> converts one of these theorems into the other, which means that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083000/s083000100.png" />-homotopy groups are replaced by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083000/s083000101.png" />-cohomotopy groups, homology groups by cohomology groups, the mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083000/s083000102.png" /> by the mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083000/s083000103.png" />, the smallest dimension with a non-trivial homology group by the largest dimension with a non-trivial cohomology group, and vice versa. In ordinary homotopy theory the definition of an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083000/s083000104.png" />-cohomotopy group requires that the dimension of the space does not exceed <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083000/s083000105.png" /> (or, more generally, that the space be <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083000/s083000106.png" />-coconnected, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083000/s083000107.png" />), which impairs the perfectly general nature of duality. |
− | converts one of these theorems into the other, which means that | ||
− | homotopy groups are replaced by | ||
− | cohomotopy groups, homology groups by cohomology groups, the mapping | ||
− | by the mapping | ||
− | the smallest dimension with a non-trivial homology group by the largest dimension with a non-trivial cohomology group, and vice versa. In ordinary homotopy theory the definition of an | ||
− | cohomotopy group requires that the dimension of the space does not exceed | ||
− | or, more generally, that the space be | ||
− | coconnected, | ||
− | which impairs the perfectly general nature of duality. | ||
− | There are several trends of generalization of the theory: e.g. studies are made of spaces with the | + | There are several trends of generalization of the theory: e.g. studies are made of spaces with the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083000/s083000108.png" />-homotopy type of polyhedra, the relative case, a theory with supports, etc. [[#References|[3]]], [[#References|[5]]], , [[#References|[7]]]. The theory was one of the starting points in the development of stationary homotopy theory [[#References|[8]]]. |
− | homotopy type of polyhedra, the relative case, a theory with supports, etc. [[#References|[3]]], [[#References|[5]]], , [[#References|[7]]]. The theory was one of the starting points in the development of stationary homotopy theory [[#References|[8]]]. | ||
====References==== | ====References==== | ||
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> E.H. Spanier, "Duality and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083000/s083000109.png" />-theory" ''Bull. Amer. Math. Soc.'' , '''62''' (1956) pp. 194–203 {{MR|0085506}} {{ZBL|}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> E.H. Spanier, J.H.C. Whitehead, "Duality in homotopy theory" ''Mathematika'' , '''2''' : 3 (1955) pp. 56–80 {{MR|0074823}} {{ZBL|0064.17202}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> E.H. Spanier, J.H.C. Whitehead, "Duality in relative homotopy theory" ''Ann. of Math.'' , '''67''' : 2 (1958) pp. 203–238 {{MR|0105105}} {{ZBL|0092.15701}} </TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> M.G. Barratt, "Track groups 1; 2" ''Proc. London Math. Soc.'' , '''5''' (1955) pp. 71–106; 285–329</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top"> E.H. Spanier, J.H.C. Whitehead, "The theory of carriers and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083000/s083000110.png" />-theory" , ''Algebraic geometry and Topology (A Symp. in honor of S. Lefschetz)'' , Princeton Univ. Press (1957) pp. 330–360 {{MR|0084772}} {{ZBL|}} </TD></TR><TR><TD valign="top">[6a]</TD> <TD valign="top"> B. Eckmann, P.J. Hilton, "Groupes d'homotopie et dualité. Groupes absolus" ''C.R. Acad. Sci. Paris'' , '''246''' : 17 (1958) pp. 2444–2447 {{MR|0100261}} {{ZBL|0092.39901}} </TD></TR><TR><TD valign="top">[6b]</TD> <TD valign="top"> B. Eckmann, P.J. Hilton, "Groupes d'homotopie et dualité. Suites exactes" ''C.R. Acad. Sci. Paris'' , '''246''' : 18 (1958) pp. 2555–2558 {{MR|0100262}} {{ZBL|0092.40001}} </TD></TR><TR><TD valign="top">[6c]</TD> <TD valign="top"> B. Eckmann, P.J. Hilton, "Groupes d'homotopie et dualité. Coefficients" ''C.R. Acad. Sci. Paris'' , '''246''' : 21 (1958) pp. 2991–2993 {{MR|0100263}} {{ZBL|0092.40101}} </TD></TR><TR><TD valign="top">[6d]</TD> <TD valign="top"> B. Eckmann, P.J. Hilton, "Transgression homotopique et cohomologique" ''C.R. Acad. Sci. Paris'' , '''247''' : 6 (1958) pp. 620–623 {{MR|0100264}} {{ZBL|0092.40102}} </TD></TR><TR><TD valign="top">[6e]</TD> <TD valign="top"> B. Eckmann, P.J. Hilton, "Décomposition homologique d'un polyhèdre simplement connexe" ''C.R. Acad. Sci. Paris'' , '''248''' : 14 (1959) pp. 2054–2056</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top"> E.H. Spanier, "Algebraic topology" , McGraw-Hill (1966) {{MR|0210112}} {{MR|1325242}} {{ZBL|0145.43303}} </TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top"> G.W. Whitehead, "Recent advances in homotopy theory" , Amer. Math. Soc. (1970) {{MR|0309097}} {{ZBL|0217.48601}} </TD></TR></table> | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> E.H. Spanier, "Duality and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083000/s083000109.png" />-theory" ''Bull. Amer. Math. Soc.'' , '''62''' (1956) pp. 194–203 {{MR|0085506}} {{ZBL|}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> E.H. Spanier, J.H.C. Whitehead, "Duality in homotopy theory" ''Mathematika'' , '''2''' : 3 (1955) pp. 56–80 {{MR|0074823}} {{ZBL|0064.17202}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> E.H. Spanier, J.H.C. Whitehead, "Duality in relative homotopy theory" ''Ann. of Math.'' , '''67''' : 2 (1958) pp. 203–238 {{MR|0105105}} {{ZBL|0092.15701}} </TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> M.G. Barratt, "Track groups 1; 2" ''Proc. London Math. Soc.'' , '''5''' (1955) pp. 71–106; 285–329</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top"> E.H. Spanier, J.H.C. Whitehead, "The theory of carriers and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083000/s083000110.png" />-theory" , ''Algebraic geometry and Topology (A Symp. in honor of S. Lefschetz)'' , Princeton Univ. Press (1957) pp. 330–360 {{MR|0084772}} {{ZBL|}} </TD></TR><TR><TD valign="top">[6a]</TD> <TD valign="top"> B. Eckmann, P.J. Hilton, "Groupes d'homotopie et dualité. Groupes absolus" ''C.R. Acad. Sci. Paris'' , '''246''' : 17 (1958) pp. 2444–2447 {{MR|0100261}} {{ZBL|0092.39901}} </TD></TR><TR><TD valign="top">[6b]</TD> <TD valign="top"> B. Eckmann, P.J. Hilton, "Groupes d'homotopie et dualité. Suites exactes" ''C.R. Acad. Sci. Paris'' , '''246''' : 18 (1958) pp. 2555–2558 {{MR|0100262}} {{ZBL|0092.40001}} </TD></TR><TR><TD valign="top">[6c]</TD> <TD valign="top"> B. Eckmann, P.J. Hilton, "Groupes d'homotopie et dualité. Coefficients" ''C.R. Acad. Sci. Paris'' , '''246''' : 21 (1958) pp. 2991–2993 {{MR|0100263}} {{ZBL|0092.40101}} </TD></TR><TR><TD valign="top">[6d]</TD> <TD valign="top"> B. Eckmann, P.J. Hilton, "Transgression homotopique et cohomologique" ''C.R. Acad. Sci. Paris'' , '''247''' : 6 (1958) pp. 620–623 {{MR|0100264}} {{ZBL|0092.40102}} </TD></TR><TR><TD valign="top">[6e]</TD> <TD valign="top"> B. Eckmann, P.J. Hilton, "Décomposition homologique d'un polyhèdre simplement connexe" ''C.R. Acad. Sci. Paris'' , '''248''' : 14 (1959) pp. 2054–2056</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top"> E.H. Spanier, "Algebraic topology" , McGraw-Hill (1966) {{MR|0210112}} {{MR|1325242}} {{ZBL|0145.43303}} </TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top"> G.W. Whitehead, "Recent advances in homotopy theory" , Amer. Math. Soc. (1970) {{MR|0309097}} {{ZBL|0217.48601}} </TD></TR></table> |
Revision as of 14:53, 7 June 2020
stationary duality, Spanier duality
A duality in homotopy theory which exists (in the absence of restrictions imposed on the dimensions of spaces) for the analogues of ordinary homotopy and cohomotopy groups in the suspension category — for the -homotopy and
-cohomotopy groups or stationary homotopy and cohomotopy groups, forming extra-ordinary (generalized) homology and cohomology theories. A suspension category, or
-category, is a category whose objects are topological spaces
, while its morphisms are classes
of
-homotopic mappings
from a
-fold suspension
into
,
and
being considered as
-homotopic if there exists an
such that the suspensions
and
are homotopic in the ordinary sense. The set
of such classes, which are known as
-mappings, constitutes an Abelian group with respect to the so-called track addition [1], [2], [4], [5]. The group
is the limit of the direct spectrum of the sets
of ordinary homotopy classes with suspension mappings as projections; if
is sufficiently large, it is a group spectrum with homomorphisms. There exists an isomorphism
in which the corresponding elements are represented by one and the same mapping
,
. The
-dual polyhedron of the polyhedron
in a sphere
is an arbitrary polyhedron
in
which is an
-deformation retract of the complement
, i.e. the morphism corresponding to the imbedding
is an
-equivalence. The polyhedron
exists for all
, and
may be considered as
.
For any polyhedra and any polyhedra
and
which are dual to them, there exists a unique mapping
![]() |
satisfying the following conditions:
a) It is an involutory contravariant functorial isomorphism, i.e. is a homomorphism such that if
![]() |
then
![]() |
if
![]() |
then
![]() |
if is an element of
or of
, then
.
b) The following relations are valid:
![]() |
where and
are considered as polyhedra,
-dual to polyhedra
and, correspondingly,
,
this means that it does not depend on
and is stationary with respect to suspension.
c) It satisfies the equation
![]() |
where
![]() |
and
![]() |
are homomorphisms of the above homology and cohomology groups, induced by -mappings
and
, and
![]() |
is an isomorphism obtained from the isomorphism of Alexander duality by replacing the set by its
-deformation retract
.
The construction of is based on the representation of a given mapping as the composition of an imbedding and an
-deformation retract.
The -homotopy group
of a space
is the group
, and the
-cohomotopy group
of
is the group
. As in ordinary homotopy theory, one defines the homomorphisms
![]() |
![]() |
Regarding the spheres and
as
-dual leads to the isomorphisms
![]() |
and to the commutative diagram
![]() |
Thus, the isomorphism connects
-homotopy and
-cohomotopy groups, just as the isomorphism of Alexander duality
connects the homology and cohomology groups. Any duality in the
-category entails a duality of ordinary homotopy classes if the conditions imposed on the space entail the existence of a one-to-one correspondence between the set of the above classes and the set of
-homotopy classes.
Examples of dual assumptions in this theory include Hurewicz's isomorphism theorem and Hopf's classification theorem. converts one of these theorems into the other, which means that
-homotopy groups are replaced by
-cohomotopy groups, homology groups by cohomology groups, the mapping
by the mapping
, the smallest dimension with a non-trivial homology group by the largest dimension with a non-trivial cohomology group, and vice versa. In ordinary homotopy theory the definition of an
-cohomotopy group requires that the dimension of the space does not exceed
(or, more generally, that the space be
-coconnected,
), which impairs the perfectly general nature of duality.
There are several trends of generalization of the theory: e.g. studies are made of spaces with the -homotopy type of polyhedra, the relative case, a theory with supports, etc. [3], [5], , [7]. The theory was one of the starting points in the development of stationary homotopy theory [8].
References
[1] | E.H. Spanier, "Duality and ![]() |
[2] | E.H. Spanier, J.H.C. Whitehead, "Duality in homotopy theory" Mathematika , 2 : 3 (1955) pp. 56–80 MR0074823 Zbl 0064.17202 |
[3] | E.H. Spanier, J.H.C. Whitehead, "Duality in relative homotopy theory" Ann. of Math. , 67 : 2 (1958) pp. 203–238 MR0105105 Zbl 0092.15701 |
[4] | M.G. Barratt, "Track groups 1; 2" Proc. London Math. Soc. , 5 (1955) pp. 71–106; 285–329 |
[5] | E.H. Spanier, J.H.C. Whitehead, "The theory of carriers and ![]() |
[6a] | B. Eckmann, P.J. Hilton, "Groupes d'homotopie et dualité. Groupes absolus" C.R. Acad. Sci. Paris , 246 : 17 (1958) pp. 2444–2447 MR0100261 Zbl 0092.39901 |
[6b] | B. Eckmann, P.J. Hilton, "Groupes d'homotopie et dualité. Suites exactes" C.R. Acad. Sci. Paris , 246 : 18 (1958) pp. 2555–2558 MR0100262 Zbl 0092.40001 |
[6c] | B. Eckmann, P.J. Hilton, "Groupes d'homotopie et dualité. Coefficients" C.R. Acad. Sci. Paris , 246 : 21 (1958) pp. 2991–2993 MR0100263 Zbl 0092.40101 |
[6d] | B. Eckmann, P.J. Hilton, "Transgression homotopique et cohomologique" C.R. Acad. Sci. Paris , 247 : 6 (1958) pp. 620–623 MR0100264 Zbl 0092.40102 |
[6e] | B. Eckmann, P.J. Hilton, "Décomposition homologique d'un polyhèdre simplement connexe" C.R. Acad. Sci. Paris , 248 : 14 (1959) pp. 2054–2056 |
[7] | E.H. Spanier, "Algebraic topology" , McGraw-Hill (1966) MR0210112 MR1325242 Zbl 0145.43303 |
[8] | G.W. Whitehead, "Recent advances in homotopy theory" , Amer. Math. Soc. (1970) MR0309097 Zbl 0217.48601 |
S-duality. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=S-duality&oldid=48599