Omega -spectrum

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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