Spanier-Whitehead duality(2)

Whitehead–Spanier duality

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

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

is a contravariant functor with many duality properties. E.g.,

i) ;

ii) ;

iii) ;

iv) ;

v) for a (generalized) homology theory there is a natural isomorphism between and .

In many ways is similar to the linear duality functor for finite-dimensional vector spaces over a field .

For , the -dimensional sphere, the classical Alexander duality theorem says that is isomorphic to , 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