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Omega -spectrum (Topic)

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

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 $S1 \bigwedge X_i \to X_{i+1}$, where $S1$ is the unit circle (that is, a circle with a unit radius).

Omega–( or $\Omega$)–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 ( $\epsilon_n$):

$\displaystyle \epsilon_n: \Omega K_n \to K_{n + 1},$
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 are all weak equivalences $X_i \cong \Omega X_{i+1}$.

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 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]).

Bibliography

1
H. Masana. 2008. “The Tate-Thomason Conjecture”. Section 1.0.4. on p.4.
2
M. F. Atiyah, “K-theory: lectures.”, Benjamin (1967).
3
H. Bass,“Algebraic K-theory.” , 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. Algebraic Topology., Cambridge University Press; Cambridge, UK.



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Keywords:  Omega -spectrum

Cross-references: theorem, cohomology theory on CW complexes, functors, isomorphism, group, homotopy category of spectra, homotopy theory, category, topological, sequence of topological spaces, spectrum, algebraic topology, cohomology theories

This is version 3 of Omega -spectrum, born on 2009-02-04, modified 2009-02-20.
Object id is 497, canonical name is OmegaSpectrum2.
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Physics Classification: 00. (GENERAL)
 02. (Mathematical methods in physics)
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