Omega-spectrum

This is a topic entry on $\mathrm{\Omega }$–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 $[X_0;X_1;\mathrm{\dots }{X}_i;X_{i+1};\mathrm{\dots }]$ together with structure 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 CW complexes, $K_1,K_2,\mathrm{\dots }$ as follows.

Definition 0.1.

Let us consider $\mathrm{\Omega }K$, the space of loops in a $CW$ complex $K$ 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 $\mathrm{\Omega }$–spectrum $\left\{K_n\right\}$ is defined as a sequence $K_1,K_2,\mathrm{\dots }$ 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 multiplication in $[K,E_n]$ induced by ${ϵ}_n$.

One can prove that if $\left\{K_n\right\}$ is a an $\mathrm{\Omega }$-spectrum then the functors defined 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]).

Title	Omega-spectrum
Canonical name	Omegaspectrum
Date of creation	2013-03-22 18:24:01
Last modified on	2013-03-22 18:24:01
Owner	bci1 (20947)
Last modified by	bci1 (20947)
Numerical id	80
Author	bci1 (20947)
Entry type	Topic
Classification	msc 55T20
Classification	msc 55T25
Classification	msc 55T05
Synonym	Omega spectrum
Synonym	$\mathrm{\Omega }$ spectrum
Related topic	PointedTopologicalSpace
Related topic	CategoricalSequence
Related topic	ClassesOfAlgebras
Related topic	HomotopyCategory
Related topic	WeakHomotopyEquivalence
Related topic	WeakHomotopyDoubleGroupoid
Related topic	CohomologyGroupTheorem
Related topic	GroupCohomology
Related topic	ProofOfCohomologyGroupTheorem
Defines	equence of CW complexes
Defines	spectrum
Defines	$\mathrm{\Omega }$–spectrum
Defines	Omega spectrum
Defines	unit circle
Defines	cohomology group
Defines	category of spectra