This is a topic entry on –spectra and their important role in reduced cohomology theories on CW complexes.
One can express the definition of an –spectrum in terms of a sequence of CW complexes, as follows.
Let us consider , the space of loops in a complex called the loopspace of , which is topologized as a subspace of the space of all maps , where is given the compact-open topology. Then, an –spectrum is defined as a sequence of CW complexes together with weak homotopy equivalences (http://planetmath.org/WeakHomotopyEquivalence) ():
with being an integer.
An alternative definition of the –spectrum can also be formulated as follows.
0.3 The Role of -spectra in Reduced Cohomology Theories
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 (http://planetmath.org/ProofOfCohomologyGroupTheorem) of a CW-complex associated with the –spectrum by setting the rule:
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 ).
- 1 H. Masana. 2008. The Tate-Thomason Conjecture. http://www.math.uiuc.edu/K-theory/0919/TT.pdfSection 1.0.4. , on p.4.
- 2 M. F. Atiyah, “K-theory: lectures.”, Benjamin (1967).
- 3 H. Bass,“Algebraic K-theory.” , Benjamin (1968)
- 4 R. G. Swan, “Algebraic K-theory.” , Springer (1968)
- 5 C. B. Thomas (ed.) and R.M.F. Moss (ed.) , “Algebraic K-theory and its geometric applications.”, Springer (1969)
- 6 Hatcher, A. 2001. http://www.math.cornell.edu/ hatcher/AT/AT.pdfAlgebraic Topology., Cambridge University Press; Cambridge, UK.
|Date of creation||2013-03-22 18:24:01|
|Last modified on||2013-03-22 18:24:01|
|Last modified by||bci1 (20947)|
|Defines||equence of CW complexes|
|Defines||category of spectra|