Spanier-Whitehead duality(2)
Whitehead–Spanier duality
Let be a CW-spectrum (see Spectrum of spaces) and consider
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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 |
Spanier-Whitehead duality(2). Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Spanier-Whitehead_duality(2)&oldid=50023