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%%% Primary Title: Omega -spectrum
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\begin{document}
This is a topic entry on $\Omega$--spectra and their important role in reduced \htmladdnormallink{cohomology theories}{http://planetphysics.us/encyclopedia/NoncommutativeGeometry4.html} on CW complexes.
\subsection{Introduction}
In \htmladdnormallink{algebraic topology}{http://planetphysics.us/encyclopedia/ModuleAlgebraic.html} a \emph{\htmladdnormallink{spectrum}{http://planetphysics.us/encyclopedia/CohomologyTheoryOnCWComplexes.html}} ${\bf S}$ is defined as a
\htmladdnormallink{sequence of topological spaces}{http://planetphysics.us/encyclopedia/CohomologyTheoryOnCWComplexes.html} $[X_0;X_1;... X_i;X_{i+1};... ]$ together with
structure mappings $S1 \bigwedge X_i \to X_{i+1}$, where $S1$ is the \emph{unit circle} (that is, a circle with a unit radius).
\subsection{Omega--( or $\Omega$)--spectrum}
One can express the definition of an $\Omega$--spectrum in terms of a sequence of CW
complexes, $K_1,K_2,...$ as follows.
\begin{definition}
Let us consider $\Omega K$, the space of loops in a $CW$ complex $K$ called
the {\em 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 \emph{$\Omega$--spectrum} $\left\{ K_n\right\}$ is defined as a
sequence $K_1,K_2,...$ of CW complexes together with weak homotopy equivalences ($\epsilon_n$):
$$\epsilon_n: \Omega K_n \to K_{n + 1},$$ with $n$ being an integer.
\end{definition}
An alternative definition of the $\Omega$--spectrum can also be formulated as follows.
\begin{definition}
An \emph{$\Omega$--spectrum}, or \emph{Omega spectrum}, is a spectrum ${\bf E}$ such that for every index $i$,
the \htmladdnormallink{topological}{http://planetphysics.us/encyclopedia/CoIntersections.html} space $X_i$ is fibered, and also the adjoints of the structure mappings are all weak equivalences $X_i \cong \Omega X_{i+1}$.
\end{definition}
\subsection{The Role of Omega-spectra in Reduced Cohomology Theories}
A \htmladdnormallink{category}{http://planetphysics.us/encyclopedia/Cod.html} of spectra (regarded as above as sequences) will provide a model category that enables one to construct a stable \htmladdnormallink{homotopy theory}{http://planetphysics.us/encyclopedia/CubicalHigherHomotopyGroupoid.html}, so that the \htmladdnormallink{homotopy category of spectra}{http://planetphysics.us/encyclopedia/CohomologyTheoryOnCWComplexes.html} is canonically defined in the classical manner. Therefore, for any given construction of an $\Omega$--spectrum one is able to canonically define an associated cohomology theory; thus, one defines the cohomology groups of a CW-complex $K$ associated with the $\Omega$--spectrum ${\bf E}$ by setting the rule:
$H^n(K;{\bf E}) = [K, E_n].$
The latter set when $K$ is a CW complex can be endowed with a \htmladdnormallink{group}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html} structure by requiring that
$(\epsilon_n)* : [K, E_n] \to [K, \Omega E_{n+1}]$ is an \htmladdnormallink{isomorphism}{http://planetphysics.us/encyclopedia/IsomorphicObjectsUnderAnIsomorphism.html} which defines the multiplication
in $[K, E_n]$ induced by $\epsilon_n$.
One can prove that if $\left\{ K_n\right\}$ is a an $\Omega$-spectrum then the \htmladdnormallink{functors}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html} defined by the assignments $X \longmapsto h^n(X) = (X,K_n),$
with $n \in \mathbb{Z}$ define a reduced cohomology theory on the category of basepointed CW complexes and basepoint preserving maps; furthermore, every reduced \htmladdnormallink{cohomology theory on CW complexes}{http://planetphysics.us/encyclopedia/CohomologyTheoryOnCWComplexes.html} arises in this manner from an $\Omega$-spectrum (the Brown representability \htmladdnormallink{theorem}{http://planetphysics.us/encyclopedia/Formula.html}; p. 397 of \cite{AllenHatcher2k1}).
\begin{thebibliography}{9}
\bibitem{SM}
H. Masana. 2008. ``{\em The Tate-Thomason Conjecture}''.
\htmladdnormallink{Section 1.0.4.}{http://www.math.uiuc.edu/K-theory/0919/TT.pdf}
on p.4.
\bibitem{MFA67}
M. F. Atiyah, ``{\em K-theory: lectures.}'', Benjamin (1967).
\bibitem{HB68}
H. Bass,``{\em Algebraic K-theory.}'' , Benjamin (1968)
\bibitem{RGS68}
R. G. Swan, ``{\em Algebraic K-theory.}'' , Springer (1968)
\bibitem{CBT69}
C. B. Thomas (ed.) and R.M.F. Moss (ed.) , ``{\em Algebraic K-theory and its geometric applications.}'' , Springer (1969)
\bibitem{AllenHatcher2k1}
Hatcher, A. 2001. \htmladdnormallink{Algebraic Topology.}{http://www.math.cornell.edu/~hatcher/AT/AT.pdf}, Cambridge University Press; Cambridge, UK.
\end{thebibliography}
\end{document}
