# Omega-spectrum

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

## 0.1 Introduction

In algebraic topology a spectrum ${\bf S}$ is defined as a sequence of topological spaces $[X_{0};X_{1};...X_{i};X_{i+1};...]$ 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 $\Omega$–spectrum

One can express the definition of an $\Omega$–spectrum in terms of a sequence of CW complexes, $K_{1},K_{2},...$ as follows.

###### Definition 0.1.

Let us consider $\Omega K$, 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 $\Omega$–spectrum $\left\{K_{n}\right\}$ is defined as a sequence $K_{1},K_{2},...$ of CW complexes together with weak homotopy equivalences (http://planetmath.org/WeakHomotopyEquivalence) ($\epsilon_{n}$):

 $\epsilon_{n}:\Omega K_{n}\to K_{n+1},$

with $n$ being an integer.

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

###### Definition 0.2.

An $\Omega$–spectrum, or Omega spectrum, is a spectrum ${\bf E}$ 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 $X_{i}\cong\Omega X_{i+1}$.

## 0.3 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 (http://planetmath.org/ProofOfCohomologyGroupTheorem) 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]).

## References

• 1 H. Masana. 2008. The Tate-Thomason Conjecture. http://www.math.uiuc.edu/K-theory/0919/TT.pdfSection 1.0.4. , on p.4.
• 2 M. F. Atiyah, “K-theory: lectures.”, Benjamin (1967).
• 3 H. Bass,“” , 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. http://www.math.cornell.edu/ 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 $\Omega$ 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 $\Omega$–spectrum Defines Omega spectrum Defines unit circle Defines cohomology group Defines category of spectra