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

Introduction

$\Omega$ -spectrum

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

The Role of $\Omega$ -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 $\Omega$ -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 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 [6]).

Bibliography

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.