PlanetPhysics/Omega Spectrum

This is a topic entry on $$\Omega$$--spectra and their important role in reduced cohomology theories on CW complexes.

Introduction
In algebraic topology a spectrum $${\mathbf S}$$ is defined as a sequence of topological spaces $$[X_0;X_1;... X_i;X_{i+1};... ]$$ together with structure mappings $$S1 \bigwedge X_i \to X_{i+1}$$, where $$S1$$ is the unit circle (that is, a circle with a unit radius).

$$\Omega$$--spectrum
One can express the definition of an $$\Omega$$--spectrum in terms of a sequence of CW complexes, $$K_1,K_2,...$$ as follows.

Let us consider $$\Omega K$$, 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 $$\Omega$$--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.

An alternative definition of the $$\Omega$$--spectrum can also be formulated as follows.

An $$\Omega$$--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 are all weak equivalences $$X_i \cong \Omega X_{i+1}$$.

The Role of $$\Omega$$-spectra in Reduced Cohomology Theories
A category of spectra (regarded as the sequences defined above) 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 $$\Omega$$--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 $${\mathbf E}$$ 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 $$(\epsilon_n)* : [K, E_n] \to [K, \Omega E_{n+1}]$$ is an isomorphism 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 functors 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 cohomology theory on CW complexes arises in this manner from an $$\Omega$$-spectrum (the Brown representability theorem; p. 397 of ).