This is a topic entry on Ω–spectra and their important role in reduced cohomology theories on CW
complexes.
0.1 Introduction
In algebraic topology a spectrum S is defined as a sequence of topological spaces
[X0; X1; ...Xi; Xi+1; ...] together with structure mappings S1 ∧
Xi → Xi+1, where S1 is the unit
circle (that is, a circle with a unit radius).
0.2 Ω–spectrum
One can express the definition of an Ω–spectrum in terms of a sequence of CW complexes,
K1,K2,... as follows.
Definition 0.1. Let us consider ΩK, the space of loops in a CW complex K called the
loopspace of K, which is topologized as a subspace of the space KI of all maps I → K ,
where KI is given the compact-open topology. Then, an Ω–spectrum 
is defined as a
sequence K1,K2,... of CW complexes together with weak homotopy equivalences (𝜖n):
with n being an integer.
An alternative definition of the Ω–spectrum can also be formulated as follows.
Definition 0.2. An Ω–spectrum, or Omega spectrum, is a spectrum E such that for every
index i, the topological space Xi is fibered, and also the adjoints of the structure mappings
are all weak equivalences Xi
ΩXi+1.
0.3 The Role of Ω-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 Ω–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 Ω–spectrum E by setting the rule:
Hn(K; E) = [K,E
n].
The latter set when K is a CW complex can be endowed with a group structure by requiring that
(𝜖n)∗ : [K,En] → [K, ΩEn+1] is an isomorphism which defines the multiplication in [K,En] induced
by 𝜖n.
One can prove that if 
is a an Ω-spectrum then the functors defined by the assignments
X
hn(X) = (X,K
n), with n ∈ 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 Ω-spectrum (the Brown
representability theorem; p. 397 of [6]).
References
[1] H. Masana. 2008. “The Tate-Thomason Conjecture”. Section 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. Algebraic Topology., Cambridge University Press; Cambridge, UK.
