Spanier-Whitehead duality(2)

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