Omega-spectrum

This is a topic entry on $\mathrm{\Omega }$–spectra and their important role in reduced cohomology theorieson CW complexes.

0.1 Introduction

In algebraic topology a spectrum $𝐒$ is defined as asequence of topological spaces $[X_0;X_1;\mathrm{\dots }{X}_i;X_{i+1};\mathrm{\dots }]$ together withstructure 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 CWcomplexes, $K_1,K_2,\mathrm{\dots }$ as follows.

Definition 0.1.

Let us consider $\mathrm{\Omega }K$, the space of loops in a $CW$ complex $K$ calledthe 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 asequence $K_1,K_2,\mathrm{\dots }$ of CW complexes together with weak homotopy equivalences (http://planetmath.org/WeakHomotopyEquivalence) (${ϵ}_n$):

$${ϵ}_n:\mathrm{\Omega }{K}_n\to K_{n+1},$$

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 $𝐄$ 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 $𝐄$ by setting the rule:$H^n(K;𝐄)=[K,E_n].$

The latter set when $K$ is a CW complex can be endowed with a group structure by requiring that$({ϵ}_n)*:[K,E_n]\to [K,\mathrm{\Omega }{E}_{n+1}]$ is an isomorphism which defines the multiplicationin $[K,E_n]$ induced by ${ϵ}_n$.

One can prove that if $\left\{K_n\right\}$ is a an $\mathrm{\Omega }$-spectrum then the functorsdefined by the assignments $X⟼h^n(X)=(X,K_n),$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]).