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''Whitehead–Spanier duality''
 
''Whitehead–Spanier duality''
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130440/s1304401.png" /> be a CW-spectrum (see [[Spectrum of spaces|Spectrum of spaces]]) and consider
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Let $X$ be a CW-spectrum (see [[Spectrum of spaces|Spectrum of spaces]]) and consider
  
<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/s130/s130440/s1304402.png" /></td> </tr></table>
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\begin{equation*} [ W \bigwedge X , S ] _ { 0 }, \end{equation*}
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130440/s1304403.png" /> is another CW-spectrum, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130440/s1304404.png" /> is the smash product of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130440/s1304405.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130440/s1304406.png" /> (see [[#References|[a2]]], Sect. III.4), <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130440/s1304407.png" /> is the sphere spectrum, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130440/s1304408.png" /> denotes stable homotopy classes of mappings of spectra. With <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130440/s1304409.png" /> fixed, this is a contravariant functor of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130440/s13044010.png" /> which satisfies the axioms of E.H. Brown (see [[#References|[a1]]]) and which is hence representable by a spectrum <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130440/s13044011.png" />, the Spanier–Whitehead dual of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130440/s13044012.png" />.
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where $W$ is another CW-spectrum, $W \wedge X$ is the smash product of $W$ and $X$ (see [[#References|[a2]]], Sect. III.4), $S$ is the sphere spectrum, and $[ , ] _ { 0 }$ denotes stable homotopy classes of mappings of spectra. With $X$ fixed, this is a contravariant functor of $W$ which satisfies the axioms of E.H. Brown (see [[#References|[a1]]]) and which is hence representable by a spectrum $D X$, the Spanier–Whitehead dual of $X$.
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130440/s13044013.png" /> is a contravariant functor with many duality properties. E.g.,
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$X \mapsto D X$ is a contravariant functor with many duality properties. E.g.,
  
i) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130440/s13044014.png" />;
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i) $[ W , Z \wedge D X ] * \simeq [ W \wedge X , Z ] *$;
  
ii) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130440/s13044015.png" />;
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ii) $\pi_{ *} ( D X \wedge Y ) \simeq [ X , Y ]_* $;
  
iii) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130440/s13044016.png" />;
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iii) $[ X , Y ]_{ *} \simeq [ D Y , D X ] _{ *}$;
  
iv) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130440/s13044017.png" />;
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iv) $D D X \simeq X$;
  
v) for a (generalized) homology theory <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130440/s13044018.png" /> there is a natural isomorphism between <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130440/s13044019.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130440/s13044020.png" />.
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v) for a (generalized) homology theory $E _ { * }$ there is a natural isomorphism between $E _ { k } ( X )$ and $E ^ { - k } ( D X )$.
  
In many ways <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130440/s13044021.png" /> is similar to the linear duality functor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130440/s13044022.png" /> for finite-dimensional vector spaces over a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130440/s13044023.png" />.
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In many ways $X \mapsto D X$ is similar to the linear duality functor $V \mapsto \operatorname { Hom } _ { k } ( V , k )$ for finite-dimensional vector spaces over a field $k$.
  
For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130440/s13044024.png" />, the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130440/s13044025.png" />-dimensional sphere, the classical Alexander duality theorem says that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130440/s13044026.png" /> is isomorphic to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130440/s13044027.png" />, and this forms the basic intuitive geometric idea behind Spanier–Whitehead duality.
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For $X \subset S ^ { N }$, the $N$-dimensional sphere, the classical Alexander duality theorem says that $H _ { k } ( X )$ is isomorphic to $H ^ { N - 1 - k } ( S ^ { n } \backslash X )$, and this forms the basic intuitive geometric idea behind Spanier–Whitehead duality.
  
 
For more details, see [[#References|[a2]]], Sect. II.5, and [[#References|[a3]]], Sect. 5.2. For an equivariant version, see [[#References|[a4]]], p. 300ff.
 
For more details, see [[#References|[a2]]], Sect. II.5, and [[#References|[a3]]], Sect. 5.2. For an equivariant version, see [[#References|[a4]]], p. 300ff.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  E.H. Brown,  "Cohomology theories"  ''Ann. of Math.'' , '''75'''  (1962)  pp. 467–484</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  J.F. Adams,  "Stable homotopy and generalised homology" , Chicago Univ. Press  (1974)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  D.C. Ravenel,  "The stable homotopy theory of finite complexes"  I.M. James (ed.) , ''Handbook of Algebraic Topology'' , Elsevier  (1995)  pp. 325–396</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  J.P.C. Greenlees,  J.P. May,  "Equivariant stable homotopy theory"  I.M. James (ed.) , ''Handbook of Algebraic Topology'' , Elsevier  (1995)  pp. 227–324</TD></TR></table>
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<table><tr><td valign="top">[a1]</td> <td valign="top">  E.H. Brown,  "Cohomology theories"  ''Ann. of Math.'' , '''75'''  (1962)  pp. 467–484</td></tr><tr><td valign="top">[a2]</td> <td valign="top">  J.F. Adams,  "Stable homotopy and generalised homology" , Chicago Univ. Press  (1974)</td></tr><tr><td valign="top">[a3]</td> <td valign="top">  D.C. Ravenel,  "The stable homotopy theory of finite complexes"  I.M. James (ed.) , ''Handbook of Algebraic Topology'' , Elsevier  (1995)  pp. 325–396</td></tr><tr><td valign="top">[a4]</td> <td valign="top">  J.P.C. Greenlees,  J.P. May,  "Equivariant stable homotopy theory"  I.M. James (ed.) , ''Handbook of Algebraic Topology'' , Elsevier  (1995)  pp. 227–324</td></tr></table>

Latest revision as of 16:46, 1 July 2020

Whitehead–Spanier duality

Let $X$ be a CW-spectrum (see Spectrum of spaces) and consider

\begin{equation*} [ W \bigwedge X , S ] _ { 0 }, \end{equation*}

where $W$ is another CW-spectrum, $W \wedge X$ is the smash product of $W$ and $X$ (see [a2], Sect. III.4), $S$ is the sphere spectrum, and $[ , ] _ { 0 }$ denotes stable homotopy classes of mappings of spectra. With $X$ fixed, this is a contravariant functor of $W$ which satisfies the axioms of E.H. Brown (see [a1]) and which is hence representable by a spectrum $D X$, the Spanier–Whitehead dual of $X$.

$X \mapsto D X$ is a contravariant functor with many duality properties. E.g.,

i) $[ W , Z \wedge D X ] * \simeq [ W \wedge X , Z ] *$;

ii) $\pi_{ *} ( D X \wedge Y ) \simeq [ X , Y ]_* $;

iii) $[ X , Y ]_{ *} \simeq [ D Y , D X ] _{ *}$;

iv) $D D X \simeq X$;

v) for a (generalized) homology theory $E _ { * }$ there is a natural isomorphism between $E _ { k } ( X )$ and $E ^ { - k } ( D X )$.

In many ways $X \mapsto D X$ is similar to the linear duality functor $V \mapsto \operatorname { Hom } _ { k } ( V , k )$ for finite-dimensional vector spaces over a field $k$.

For $X \subset S ^ { N }$, the $N$-dimensional sphere, the classical Alexander duality theorem says that $H _ { k } ( X )$ is isomorphic to $H ^ { N - 1 - k } ( S ^ { n } \backslash X )$, and this forms the basic intuitive geometric idea behind Spanier–Whitehead duality.

For more details, see [a2], Sect. II.5, and [a3], Sect. 5.2. For an equivariant version, see [a4], p. 300ff.

References

[a1] E.H. Brown, "Cohomology theories" Ann. of Math. , 75 (1962) pp. 467–484
[a2] J.F. Adams, "Stable homotopy and generalised homology" , Chicago Univ. Press (1974)
[a3] D.C. Ravenel, "The stable homotopy theory of finite complexes" I.M. James (ed.) , Handbook of Algebraic Topology , Elsevier (1995) pp. 325–396
[a4] J.P.C. Greenlees, J.P. May, "Equivariant stable homotopy theory" I.M. James (ed.) , Handbook of Algebraic Topology , Elsevier (1995) pp. 227–324
How to Cite This Entry:
Spanier-Whitehead duality(2). Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Spanier-Whitehead_duality(2)&oldid=50023
This article was adapted from an original article by M. Hazewinkel (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article