PlanetPhysics/Omega Spectrum

This is a topic entry on $\mathrm{\Omega }$--spectra and their important role in reduced [[../NoncommutativeGeometry4/|cohomology theories]] on CW complexes.

Introduction

In [[../CubicalHigherHomotopyGroupoid/|algebraic topology]] a spectrum $𝐒$ is defined as a sequence of topological spaces $[X_0;X_1;...X_i;X_{i+1};...]$ together with structure mappings $S1\bigwedge X_i\to X_{i+1}$, where $S1$ is the unit circle (that is, a circle with a unit radius).

$\mathrm{\Omega }$--spectrum

One can express the definition of an $\mathrm{\Omega }$--spectrum in terms of a sequence of CW complexes, $K_1,K_2,...$ as follows.

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 $K_1,K_2,...$ of CW complexes together with weak homotopy equivalences (${ϵ}_n$):

${ϵ}_n:\mathrm{\Omega }{K}_n\to K_{n+1},$ with $n$ being an integer.

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

An $\mathrm{\Omega }$--spectrum , or Omega spectrum , is a spectrum $𝐄$ such that for every index $i$, the [[../CoIntersections/|topological]] space $X_i$ is fibered, and also the adjoints of the structure mappings are all weak equivalences $X_i\cong \mathrm{\Omega }{X}_{i+1}$.

The Role of $\mathrm{\Omega }$-spectra in Reduced Cohomology Theories

A [[../Cod/|category]] of spectra (regarded as the sequences defined above) will provide a model category that enables one to construct a stable [[../CubicalHigherHomotopyGroupoid/|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 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 [[../TrivialGroupoid/|group]] structure by requiring that $({ϵ}_n)*:[K,E_n]\to [K,\mathrm{\Omega }{E}_{n+1}]$ is an [[../IsomorphicObjectsUnderAnIsomorphism/|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 [[../TrivialGroupoid/|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 [[../Formula/|theorem]]; p. 397 of ).