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 sequencePlanetmathPlanetmath of topological spacesMathworldPlanetmath [X0;X1;Xi;Xi+1;] together with structureMathworldPlanetmath mappings ( S1XiXi+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 subspaceMathworldPlanetmath of the space KI of all maps IK , where KI is given the compact-open topologyMathworldPlanetmath. Then, an Ω–spectrum {Kn} 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 XiΩXi+1.

0.3 The Role of Ω-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 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,En].

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 isomorphismMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath which defines the multiplication in [K,En] induced by ϵn.

One can prove that if {Kn} is a an Ω-spectrum then the functorsMathworldPlanetmath defined by the assignments Xhn(X)=(X,Kn), with n define a reduced cohomology theory on the categoryMathworldPlanetmath 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]).


  • 1 H. Masana. 2008. The Tate-Thomason Conjecture. 1.0.4. , on p.4.
  • 2 M. F. Atiyah, “K-theory: lectures.”, Benjamin (1967).
  • 3 H. Bass,“Algebraic K-theoryMathworldPlanetmath.” , 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. hatcher/AT/AT.pdfAlgebraic Topology., Cambridge University Press; Cambridge, UK.
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 Ω 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 Ω–spectrum
Defines Omega spectrum
Defines unit circle
Defines cohomology group
Defines category of spectra