Omega-spectrum
This is a topic entry on $\mathrm{\Omega}$–spectra and their important role in reduced cohomology theories on CW complexes.
0.1 Introduction
In algebraic topology a spectrum $\mathbf{S}$ is defined as a sequence^{} of topological spaces^{} $[{X}_{0};{X}_{1};\mathrm{\dots}{X}_{i};{X}_{i+1};\mathrm{\dots}]$ together with structure^{} mappings (http://planetmath.org/ClassesOfAlgebras) $S1\bigwedge {X}_{i}\to {X}_{i+1}$, where $S1$ is the unit circle (that is, a circle with a unit radius).
0.2 $\mathrm{\Omega}$–spectrum
One can express the definition of an $\mathrm{\Omega}$–spectrum in terms of a sequence of CW complexes, ${K}_{1},{K}_{2},\mathrm{\dots}$ as follows.
Definition 0.1.
Let us consider $\mathrm{\Omega}K$, the space of loops in a $CW$ complex $K$ called the loopspace of $K$, which is topologized as a subspace^{} of the space ${K}^{I}$ of all maps $I\to K$ , where ${K}^{I}$ is given the compact-open topology^{}. Then, an $\mathrm{\Omega}$–spectrum $\left\{{K}_{n}\right\}$ is defined as a sequence ${K}_{1},{K}_{2},\mathrm{\dots}$ of CW complexes together with weak homotopy equivalences (http://planetmath.org/WeakHomotopyEquivalence) (${\u03f5}_{n}$):
$${\u03f5}_{n}:\mathrm{\Omega}{K}_{n}\to {K}_{n+1},$$ |
with $n$ being an integer.
An alternative definition of the $\mathrm{\Omega}$–spectrum can also be formulated as follows.
Definition 0.2.
An $\mathrm{\Omega}$–spectrum, or Omega spectrum, is a spectrum $\mathbf{E}$ such that for every index $i$, the topological space ${X}_{i}$ is fibered, and also the adjoints of the structure mappings (http://planetmath.org/ClassesOfAlgebras) are all weak equivalences ${X}_{i}\cong \mathrm{\Omega}{X}_{i+1}$.
0.3 The Role of $\mathrm{\Omega}$-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 $\mathrm{\Omega}$–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 $K$ associated with the $\mathrm{\Omega}$–spectrum $\mathbf{E}$ by setting the rule: ${H}^{n}(K;\mathbf{E})=[K,{E}_{n}].$
The latter set when $K$ is a CW complex can be endowed with a group structure by requiring that $({\u03f5}_{n})*:[K,{E}_{n}]\to [K,\mathrm{\Omega}{E}_{n+1}]$ is an isomorphism^{} which defines the multiplication in $[K,{E}_{n}]$ induced by ${\u03f5}_{n}$.
One can prove that if $\left\{{K}_{n}\right\}$ is a an $\mathrm{\Omega}$-spectrum then the functors^{} defined by the assignments $X\u27fc{h}^{n}(X)=(X,{K}_{n}),$ with $n\in \mathbb{Z}$ 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 $\mathrm{\Omega}$-spectrum (the Brown representability theorem; p. 397 of [6]).
References
- 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.
Title | Omega-spectrum |
Canonical name | Omegaspectrum |
Date of creation | 2013-03-22 18:24:01 |
Last modified on | 2013-03-22 18:24:01 |
Owner | bci1 (20947) |
Last modified by | bci1 (20947) |
Numerical id | 80 |
Author | bci1 (20947) |
Entry type | Topic |
Classification | msc 55T20 |
Classification | msc 55T25 |
Classification | msc 55T05 |
Synonym | Omega spectrum |
Synonym | $\mathrm{\Omega}$ spectrum |
Related topic | PointedTopologicalSpace |
Related topic | CategoricalSequence |
Related topic | ClassesOfAlgebras |
Related topic | HomotopyCategory |
Related topic | WeakHomotopyEquivalence |
Related topic | WeakHomotopyDoubleGroupoid |
Related topic | CohomologyGroupTheorem |
Related topic | GroupCohomology |
Related topic | ProofOfCohomologyGroupTheorem |
Defines | equence of CW complexes |
Defines | spectrum |
Defines | $\mathrm{\Omega}$–spectrum |
Defines | Omega spectrum |
Defines | unit circle |
Defines | cohomology group |
Defines | category of spectra |