This is a topic entry on
-spectra and their important role in reduced cohomology theories
on CW complexes.
with
An alternative definition of the
-spectrum can also be formulated as follows.
A category
of spectra (regarded as above as sequences) will provide a model category that enables one to construct a stable homotopy theory, so that the homotopy category of spectra
is canonically defined in the classical manner. Therefore, for any given construction of an
-spectrum one is able to canonically define an associated cohomology theory; thus, one defines the cohomology groups of a CW-complex
associated with the
-spectrum
by setting the rule:
The latter set when
is a CW complex can be endowed with a group
structure by requiring that
is an isomorphism which defines the multiplication
in
induced by
.
One can prove that if
is a an
-spectrum then the functors
defined by the assignments
with
define a reduced cohomology theory on the category of basepointed CW complexes and basepoint preserving maps; furthermore, every reduced cohomology theory on CW complexes
arises in this manner from an
-spectrum (the Brown representability theorem; p. 397 of [6]).
As of this snapshot date, this entry was owned by bci1.