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A strong shape category <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120280/s1202801.png" /> has topological spaces with, or without, base points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120280/s1202802.png" />, respectively <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120280/s1202803.png" />, as objects and strong shape mappings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120280/s1202804.png" /> as morphisms (cf. also [[Category|Category]]; [[Topological space|Topological space]]). There are strong shape categories for arbitrary topological spaces, for compact metric spaces and (stably) for separable metric, finite-dimensional spaces (the compact-open strong shape morphisms are called <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120280/s1202806.png" />-morphisms). The main objective of strong shape theory is to perform [[Algebraic topology|algebraic topology]] (i.e. generalized homology and cohomology theory, (stable) homotopy theory, [[S-duality|<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120280/s1202807.png" />-duality]] and eventually fibre theory) for more exotic spaces in the same way as one did in ordinary homotopy theory for CW-spaces (cf. also [[Cohomology|Cohomology]]; [[Cohomology of a complex|Cohomology of a complex]]; [[CW-complex|CW-complex]]).
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Let, for example, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120280/s1202808.png" /> be a spectrum (cf. also [[Spectrum of spaces|Spectrum of spaces]]) and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120280/s1202809.png" /> a CW-space. Then the generalized (co)homology of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120280/s12028010.png" /> with coefficients in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120280/s12028011.png" /> is defined by
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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/s/s120/s120280/s12028012.png" /></td> </tr></table>
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A strong shape category $\bf K$ has topological spaces with, or without, base points $X = ( X , x _ { 0 } )$, respectively $X$, as objects and strong shape mappings $\overline { f } : X \rightarrow Y$ as morphisms (cf. also [[Category|Category]]; [[Topological space|Topological space]]). There are strong shape categories for arbitrary topological spaces, for compact metric spaces and (stably) for separable metric, finite-dimensional spaces (the compact-open strong shape morphisms are called <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120280/s1202806.png"/>-morphisms). The main objective of strong shape theory is to perform [[Algebraic topology|algebraic topology]] (i.e. generalized homology and cohomology theory, (stable) homotopy theory, [[S-duality|$S$-duality]] and eventually fibre theory) for more exotic spaces in the same way as one did in ordinary homotopy theory for CW-spaces (cf. also [[Cohomology|Cohomology]]; [[Cohomology of a complex|Cohomology of a complex]]; [[CW-complex|CW-complex]]).
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Let, for example, $\mathbf{E} = \{ E _ { n } , \sigma : \Sigma E _ { n } \rightarrow E _ { n  + 1} \}$ be a spectrum (cf. also [[Spectrum of spaces|Spectrum of spaces]]) and $X$ a CW-space. Then the generalized (co)homology of $X$ with coefficients in $\mathbf E$ is defined by
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\begin{equation*} {\bf E} _ { n } ( X ) = \operatorname { lim } _ { k } \pi _ { n + k } ( X \bigwedge E _ { k } ) = \pi _ { n } ^ { S } ( X \bigwedge {\bf E} ), \end{equation*}
  
 
respectively
 
respectively
  
<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/s120/s120280/s12028013.png" /></td> </tr></table>
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\begin{equation*} \mathbf{E} ^ { n } ( X ) = [ \Sigma ^ { k } X , E _ { n + k } ] , \quad n \in \mathbf{Z}. \end{equation*}
  
Suppose one is working with compact metric spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120280/s12028014.png" /> and replaces, in the previous definition of homology, continuous mappings by strong shape mappings. Then one obtains again a homology theory, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120280/s12028015.png" />, the [[Steenrod–Sitnikov homology|Steenrod–Sitnikov homology]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120280/s12028016.png" /> with coefficients in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120280/s12028017.png" />.
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Suppose one is working with compact metric spaces $X$ and replaces, in the previous definition of homology, continuous mappings by strong shape mappings. Then one obtains again a homology theory, $\overline { \mathbf{E} } * ( X )$, the [[Steenrod–Sitnikov homology|Steenrod–Sitnikov homology]] of $X$ with coefficients in $\mathbf E$.
  
 
For even more general spaces, a kind of homology theory which lives in a strong shape category is the so-called [[Strong homology|strong homology]]. It can be characterized by a set of axioms very similar to the Milnor axioms for Steenrod–Sitnikov homology (the [[Steenrod–Eilenberg axioms|Steenrod–Eilenberg axioms]] without a dimension axiom, but with a very strong excision axiom and a kind of continuity axiom (on a chain level), resembling Milnor's cluster axiom).
 
For even more general spaces, a kind of homology theory which lives in a strong shape category is the so-called [[Strong homology|strong homology]]. It can be characterized by a set of axioms very similar to the Milnor axioms for Steenrod–Sitnikov homology (the [[Steenrod–Eilenberg axioms|Steenrod–Eilenberg axioms]] without a dimension axiom, but with a very strong excision axiom and a kind of continuity axiom (on a chain level), resembling Milnor's cluster axiom).
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Steenrod–Sitnikov homology is the appropriate tool for establishing Alexander duality theorems [[#References|[a2]]] (cf. also [[Alexander duality|Alexander duality]]):
 
Steenrod–Sitnikov homology is the appropriate tool for establishing Alexander duality theorems [[#References|[a2]]] (cf. also [[Alexander duality|Alexander duality]]):
  
<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/s120/s120280/s12028018.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a1)</td></tr></table>
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\begin{equation} \tag{a1} \overline{ \mathbf{E}}_p ( X ) \approx \overline { \mathbf{E} } \square ^ { q } ( S ^ { n } \backslash X ) , p + q = n - 1, \end{equation}
  
for suitable (e.g. CW-) spectra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120280/s12028019.png" /> and spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120280/s12028020.png" />.
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for suitable (e.g. CW-) spectra $\mathbf E$ and spaces $X \subset S ^ { n}$.
  
 
The corresponding cohomology reveals itself as a Čech-type cohomology, cf. also [[Čech cohomology|Čech cohomology]].
 
The corresponding cohomology reveals itself as a Čech-type cohomology, cf. also [[Čech cohomology|Čech cohomology]].
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120280/s12028021.png" />-duality for arbitrary separable metric, finite-dimensional spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120280/s12028022.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120280/s12028023.png" /> guarantees the existence of a duality functor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120280/s12028024.png" /> satisfying
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$S$-duality for arbitrary separable metric, finite-dimensional spaces $X$, $Y$ guarantees the existence of a duality functor $D$ satisfying
  
<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/s120/s120280/s12028025.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a2)</td></tr></table>
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\begin{equation} \tag{a2} \{ X , Y \} \approx \{ D Y , D X \}, \end{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/s120/s120280/s12028026.png" /></td> </tr></table>
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\begin{equation*} D ^ { 2 } X \approx X. \end{equation*}
  
Here, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120280/s12028027.png" /> on both sides of (a2) are in fact <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120280/s12028028.png" />-morphisms between objects in an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120280/s12028029.png" />-category [[#References|[a3]]]. The associated result for ordinary shape and compact <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120280/s12028030.png" /> goes back to E. Lima [[#References|[a8]]], while there is a corresponding theorem for compacta and strong shape by Q. Haxhibeqiri and S. Novak [[#References|[a7]]]; both of them are, unlike (a2), not symmetrical, because the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120280/s12028031.png" />-dual <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120280/s12028032.png" /> of a compact space need no longer be compact.
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Here, $\{ \ldots \}$ on both sides of (a2) are in fact <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120280/s12028028.png"/>-morphisms between objects in an $S$-category [[#References|[a3]]]. The associated result for ordinary shape and compact $X$ goes back to E. Lima [[#References|[a8]]], while there is a corresponding theorem for compacta and strong shape by Q. Haxhibeqiri and S. Novak [[#References|[a7]]]; both of them are, unlike (a2), not symmetrical, because the $S$-dual $D X$ of a compact space need no longer be compact.
  
Since, unlike for ordinary shape, one has in a strong shape category access to individual shape morphisms (rather than merely to their homotopy classes), one can define a shape singular complex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120280/s12028033.png" />, a Kan-complex, which is achieved in the same way as the singular complex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120280/s12028034.png" /> of a space, but with individual strong shape morphisms replacing continuous mappings.
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Since, unlike for ordinary shape, one has in a strong shape category access to individual shape morphisms (rather than merely to their homotopy classes), one can define a shape singular complex $\overline { S } ( X )$, a Kan-complex, which is achieved in the same way as the singular complex $S ( X )$ of a space, but with individual strong shape morphisms replacing continuous mappings.
  
In particular, this allows the construction of a kind of shape singular homology <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120280/s12028035.png" />, which turns out to be the appropriate generalization of Borel–Moore homology.
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In particular, this allows the construction of a kind of shape singular homology $\mathbf{E}_{*} ( | \overline { S } ( X ) | )$, which turns out to be the appropriate generalization of Borel–Moore homology.
  
 
Strong shape theory has been independently discovered by A.D. Edwards and H.M. Hastings [[#References|[a5]]] and F.W. Bauer [[#References|[a1]]]. More recently, B. Guenther [[#References|[a6]]] introduced an ultimate model for a strong shape category for arbitrary topological spaces. Subsequently, numerous authors have contributed to strong shape theory, like S. Mardešić (notably his theory of ANR expansions of a space, cf. [[#References|[a6]]] for further references or [[Retract of a topological space|Retract of a topological space]]), J. Segal, Y. Kodama ( "fine shape" ) and others.
 
Strong shape theory has been independently discovered by A.D. Edwards and H.M. Hastings [[#References|[a5]]] and F.W. Bauer [[#References|[a1]]]. More recently, B. Guenther [[#References|[a6]]] introduced an ultimate model for a strong shape category for arbitrary topological spaces. Subsequently, numerous authors have contributed to strong shape theory, like S. Mardešić (notably his theory of ANR expansions of a space, cf. [[#References|[a6]]] for further references or [[Retract of a topological space|Retract of a topological space]]), J. Segal, Y. Kodama ( "fine shape" ) and others.
  
An ordinary shape mapping is a  "machine"  (in fact, of course, a [[Functor|functor]] between appropriate categories), assigning to a homotopy class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120280/s12028036.png" />, with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120280/s12028037.png" /> a  "good space"  (e.g. a polyhedron, a CW-space or an absolute neighbourhood retract), a homotopy class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120280/s12028038.png" />, respecting homotopy commutative diagrams (i.e. if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120280/s12028039.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120280/s12028040.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120280/s12028041.png" /> are given such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120280/s12028042.png" />, then the associated diagram for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120280/s12028043.png" /> is homotopy commutative (cf. also [[Diagram|Diagram]]).
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An ordinary shape mapping is a  "machine"  (in fact, of course, a [[Functor|functor]] between appropriate categories), assigning to a homotopy class $[ g ] : Y \rightarrow P$, with $P$ a  "good space"  (e.g. a polyhedron, a CW-space or an absolute neighbourhood retract), a homotopy class $\overline { f } ( [ g ] ) : X \rightarrow P$, respecting homotopy commutative diagrams (i.e. if $[ g _ { i } ] : Y \rightarrow P _ { i }$, $i = 1,2$ and $[ r ] : P _ { 1 } \rightarrow P _ { 2 }$ are given such that $| r g _ { 1 } | = [ g_{2} ]$, then the associated diagram for $X$ is homotopy commutative (cf. also [[Diagram|Diagram]]).
  
In strong shape theory one has to work with individual mappings (rather than with homotopy classes) and with specific homotopies <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120280/s12028044.png" />: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120280/s12028045.png" /> instead of the more unspecific statement that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120280/s12028046.png" /> is homotopic to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120280/s12028047.png" />. Since this involves higher homotopies of arbitrary high degree, one has to deal with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120280/s12028048.png" />-categories and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120280/s12028049.png" />-functors for arbitrary high <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120280/s12028050.png" />.
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In strong shape theory one has to work with individual mappings (rather than with homotopy classes) and with specific homotopies $\omega$: $r g _ { 1 } \simeq g_2$ instead of the more unspecific statement that $rg_1$ is homotopic to $g_{2}$. Since this involves higher homotopies of arbitrary high degree, one has to deal with $n$-categories and $n$-functors for arbitrary high $n$.
  
 
This is very similar to the approach taken by K. Sitnikov to (ordinary) Steenrod–Sitnikov homology, where one has (other than for Vietoris or Čech homology) to replace homology classes by individual cycles and homology relations by specifying individual connecting chains between cycles.
 
This is very similar to the approach taken by K. Sitnikov to (ordinary) Steenrod–Sitnikov homology, where one has (other than for Vietoris or Čech homology) to replace homology classes by individual cycles and homology relations by specifying individual connecting chains between cycles.
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====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  F.W. Bauer,  "A shape theory with singular homology"  ''Pacific J. Math.'' , '''62''' :  1  (1976)  pp. 25–65</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  F.W. Bauer,  "Duality in manifolds"  ''Ann. Mat. Pura Appl.'' , '''4''' :  136  (1984)  pp. 241–302</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  F.W. Bauer,  "A strong shape theory admitting an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120280/s12028051.png" />-dual"  ''Topol. Appl.'' , '''62'''  (1995)  pp. 207–232</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  F.W. Cathey,  "Strong shape theory" , ''Proc. Dubrovnik'' , ''Lecture Notes Math.'' , '''870''' , Springer  (1981)  pp. 215–238</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  D.A. Edwards,  H.M. Hastings,  "Cech and Steenrod homotopy theory with applications to geometric topology" , ''Lecture Notes Math.'' , '''542''' , Springer  (1976)</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  B. Guenther,  "Use of semi-simplicial complexes in strong shape theory"  ''Glascow Math.'' , '''27''' :  47  (1992)  pp. 101–144</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  Q. Haxhibeqiri,  S. Novak,  "Duality between stable strong shape morphisms and stable homotopy classes"  ''Glascow Math.''  (to appear)</TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top">  E. Lima,  "The Spanier–Whitehead duality in two new categories"  ''Summa Brasil. Math.'' , '''4'''  (1959)  pp. 91–148</TD></TR></table>
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<table><tr><td valign="top">[a1]</td> <td valign="top">  F.W. Bauer,  "A shape theory with singular homology"  ''Pacific J. Math.'' , '''62''' :  1  (1976)  pp. 25–65</td></tr><tr><td valign="top">[a2]</td> <td valign="top">  F.W. Bauer,  "Duality in manifolds"  ''Ann. Mat. Pura Appl.'' , '''4''' :  136  (1984)  pp. 241–302</td></tr><tr><td valign="top">[a3]</td> <td valign="top">  F.W. Bauer,  "A strong shape theory admitting an $S$-dual"  ''Topol. Appl.'' , '''62'''  (1995)  pp. 207–232</td></tr><tr><td valign="top">[a4]</td> <td valign="top">  F.W. Cathey,  "Strong shape theory" , ''Proc. Dubrovnik'' , ''Lecture Notes Math.'' , '''870''' , Springer  (1981)  pp. 215–238</td></tr><tr><td valign="top">[a5]</td> <td valign="top">  D.A. Edwards,  H.M. Hastings,  "Cech and Steenrod homotopy theory with applications to geometric topology" , ''Lecture Notes Math.'' , '''542''' , Springer  (1976)</td></tr><tr><td valign="top">[a6]</td> <td valign="top">  B. Guenther,  "Use of semi-simplicial complexes in strong shape theory"  ''Glascow Math.'' , '''27''' :  47  (1992)  pp. 101–144</td></tr><tr><td valign="top">[a7]</td> <td valign="top">  Q. Haxhibeqiri,  S. Novak,  "Duality between stable strong shape morphisms and stable homotopy classes"  ''Glascow Math.''  (to appear)</td></tr><tr><td valign="top">[a8]</td> <td valign="top">  E. Lima,  "The Spanier–Whitehead duality in two new categories"  ''Summa Brasil. Math.'' , '''4'''  (1959)  pp. 91–148</td></tr></table>

Revision as of 16:59, 1 July 2020

A strong shape category $\bf K$ has topological spaces with, or without, base points $X = ( X , x _ { 0 } )$, respectively $X$, as objects and strong shape mappings $\overline { f } : X \rightarrow Y$ as morphisms (cf. also Category; Topological space). There are strong shape categories for arbitrary topological spaces, for compact metric spaces and (stably) for separable metric, finite-dimensional spaces (the compact-open strong shape morphisms are called -morphisms). The main objective of strong shape theory is to perform algebraic topology (i.e. generalized homology and cohomology theory, (stable) homotopy theory, $S$-duality and eventually fibre theory) for more exotic spaces in the same way as one did in ordinary homotopy theory for CW-spaces (cf. also Cohomology; Cohomology of a complex; CW-complex).

Let, for example, $\mathbf{E} = \{ E _ { n } , \sigma : \Sigma E _ { n } \rightarrow E _ { n + 1} \}$ be a spectrum (cf. also Spectrum of spaces) and $X$ a CW-space. Then the generalized (co)homology of $X$ with coefficients in $\mathbf E$ is defined by

\begin{equation*} {\bf E} _ { n } ( X ) = \operatorname { lim } _ { k } \pi _ { n + k } ( X \bigwedge E _ { k } ) = \pi _ { n } ^ { S } ( X \bigwedge {\bf E} ), \end{equation*}

respectively

\begin{equation*} \mathbf{E} ^ { n } ( X ) = [ \Sigma ^ { k } X , E _ { n + k } ] , \quad n \in \mathbf{Z}. \end{equation*}

Suppose one is working with compact metric spaces $X$ and replaces, in the previous definition of homology, continuous mappings by strong shape mappings. Then one obtains again a homology theory, $\overline { \mathbf{E} } * ( X )$, the Steenrod–Sitnikov homology of $X$ with coefficients in $\mathbf E$.

For even more general spaces, a kind of homology theory which lives in a strong shape category is the so-called strong homology. It can be characterized by a set of axioms very similar to the Milnor axioms for Steenrod–Sitnikov homology (the Steenrod–Eilenberg axioms without a dimension axiom, but with a very strong excision axiom and a kind of continuity axiom (on a chain level), resembling Milnor's cluster axiom).

Steenrod–Sitnikov homology is the appropriate tool for establishing Alexander duality theorems [a2] (cf. also Alexander duality):

\begin{equation} \tag{a1} \overline{ \mathbf{E}}_p ( X ) \approx \overline { \mathbf{E} } \square ^ { q } ( S ^ { n } \backslash X ) , p + q = n - 1, \end{equation}

for suitable (e.g. CW-) spectra $\mathbf E$ and spaces $X \subset S ^ { n}$.

The corresponding cohomology reveals itself as a Čech-type cohomology, cf. also Čech cohomology.

$S$-duality for arbitrary separable metric, finite-dimensional spaces $X$, $Y$ guarantees the existence of a duality functor $D$ satisfying

\begin{equation} \tag{a2} \{ X , Y \} \approx \{ D Y , D X \}, \end{equation}

\begin{equation*} D ^ { 2 } X \approx X. \end{equation*}

Here, $\{ \ldots \}$ on both sides of (a2) are in fact -morphisms between objects in an $S$-category [a3]. The associated result for ordinary shape and compact $X$ goes back to E. Lima [a8], while there is a corresponding theorem for compacta and strong shape by Q. Haxhibeqiri and S. Novak [a7]; both of them are, unlike (a2), not symmetrical, because the $S$-dual $D X$ of a compact space need no longer be compact.

Since, unlike for ordinary shape, one has in a strong shape category access to individual shape morphisms (rather than merely to their homotopy classes), one can define a shape singular complex $\overline { S } ( X )$, a Kan-complex, which is achieved in the same way as the singular complex $S ( X )$ of a space, but with individual strong shape morphisms replacing continuous mappings.

In particular, this allows the construction of a kind of shape singular homology $\mathbf{E}_{*} ( | \overline { S } ( X ) | )$, which turns out to be the appropriate generalization of Borel–Moore homology.

Strong shape theory has been independently discovered by A.D. Edwards and H.M. Hastings [a5] and F.W. Bauer [a1]. More recently, B. Guenther [a6] introduced an ultimate model for a strong shape category for arbitrary topological spaces. Subsequently, numerous authors have contributed to strong shape theory, like S. Mardešić (notably his theory of ANR expansions of a space, cf. [a6] for further references or Retract of a topological space), J. Segal, Y. Kodama ( "fine shape" ) and others.

An ordinary shape mapping is a "machine" (in fact, of course, a functor between appropriate categories), assigning to a homotopy class $[ g ] : Y \rightarrow P$, with $P$ a "good space" (e.g. a polyhedron, a CW-space or an absolute neighbourhood retract), a homotopy class $\overline { f } ( [ g ] ) : X \rightarrow P$, respecting homotopy commutative diagrams (i.e. if $[ g _ { i } ] : Y \rightarrow P _ { i }$, $i = 1,2$ and $[ r ] : P _ { 1 } \rightarrow P _ { 2 }$ are given such that $| r g _ { 1 } | = [ g_{2} ]$, then the associated diagram for $X$ is homotopy commutative (cf. also Diagram).

In strong shape theory one has to work with individual mappings (rather than with homotopy classes) and with specific homotopies $\omega$: $r g _ { 1 } \simeq g_2$ instead of the more unspecific statement that $rg_1$ is homotopic to $g_{2}$. Since this involves higher homotopies of arbitrary high degree, one has to deal with $n$-categories and $n$-functors for arbitrary high $n$.

This is very similar to the approach taken by K. Sitnikov to (ordinary) Steenrod–Sitnikov homology, where one has (other than for Vietoris or Čech homology) to replace homology classes by individual cycles and homology relations by specifying individual connecting chains between cycles.

The strong shape homotopy category for compacta turns out to be equivalent to a quotient category of the category of compacta (with continuous mappings as morphisms) by converting all so-called SSDR-mappings into equivalences [a4]. Ordinary shape does not have this property.

See also Shape theory.

References

[a1] F.W. Bauer, "A shape theory with singular homology" Pacific J. Math. , 62 : 1 (1976) pp. 25–65
[a2] F.W. Bauer, "Duality in manifolds" Ann. Mat. Pura Appl. , 4 : 136 (1984) pp. 241–302
[a3] F.W. Bauer, "A strong shape theory admitting an $S$-dual" Topol. Appl. , 62 (1995) pp. 207–232
[a4] F.W. Cathey, "Strong shape theory" , Proc. Dubrovnik , Lecture Notes Math. , 870 , Springer (1981) pp. 215–238
[a5] D.A. Edwards, H.M. Hastings, "Cech and Steenrod homotopy theory with applications to geometric topology" , Lecture Notes Math. , 542 , Springer (1976)
[a6] B. Guenther, "Use of semi-simplicial complexes in strong shape theory" Glascow Math. , 27 : 47 (1992) pp. 101–144
[a7] Q. Haxhibeqiri, S. Novak, "Duality between stable strong shape morphisms and stable homotopy classes" Glascow Math. (to appear)
[a8] E. Lima, "The Spanier–Whitehead duality in two new categories" Summa Brasil. Math. , 4 (1959) pp. 91–148
How to Cite This Entry:
Strong shape theory. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Strong_shape_theory&oldid=50304
This article was adapted from an original article by F.W. Bauer (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article