View source for Spanier-Whitehead duality(2)

<!--This article has been texified automatically. Since there was no Nroff source code for this article, 
the semi-automatic procedure described at https://encyclopediaofmath.org/wiki/User:Maximilian_Janisch/latexlist
was used.
If the TeX and formula formatting is correct, please remove this message and the {{TEX|semi-auto}} category.

Out of 27 formulas, 27 were replaced by TEX code.-->

{{TEX|semi-auto}}{{TEX|done}}
''Whitehead–Spanier duality''

Let $X$ be a CW-spectrum (see [[Spectrum of spaces|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 [[#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 $X$.

$X \mapsto D X$ 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 [[#References|[a2]]], Sect. II.5, and [[#References|[a3]]], Sect. 5.2. For an equivariant version, see [[#References|[a4]]], p. 300ff.

====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>