PlanetPhysics: $\Omega$-spectrum

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$\Omega$ -spectrum

(Definition)

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

is the unit circle (that is, a circle with a unit radius).

$\Omega$ -spectrum

One can express the definition of an $\Omega$ -spectrum in terms of a sequence of CW complexes,

as follows.

Definition 0.1 Let us consider $\Omega K$ , the space of loops in a

complex

called the loopspace of , which is topologized as a subspace of the space

of all maps $I \to K$ , where

is given the compact-open topology. Then, an $\Omega$ -spectrum $\left\{ K_n\right\}$ is defined as a sequence

of CW complexes together with weak homotopy equivalences ( $\epsilon_n$ ):

$\displaystyle \epsilon_n: \Omega K_n \to K_{n + 1},$

with

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

, the topological space

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 the sequences defined above) 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 associated with the $\Omega$ -spectrum ${\bf E}$ by setting the rule: $H^n(K;{\bf E}) = [K, E_n].$

The latter set when 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 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.

" $\Omega$ -spectrum" is owned by bci1.

(view preamble)

Also defines:

spectrum, homotopy category of spectra, cohomology theory on CW complexes, cohomology groups, sequence of topological spaces

Keywords:

Omega-spectrum, spectrum, homotopy category of spectra, categorical sequence, cohomology theory on CW complexes

Cross-references: theorem, functors, homotopy, category
There are 18 references to this object.

This is version 4 of $\Omega$ -spectrum, born on 2009-03-09, modified 2009-04-29.
Object id is 584, canonical name is OmegaSpectrum.
Accessed 558 times total.

Classification:

Physics Classification:	00. (GENERAL)
	02. (Mathematical methods in physics)
	03.65.Fd (Algebraic methods )

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Introduction

-spectrum

The Role of -spectra in Reduced Cohomology Theories

Bibliography

$\Omega$ -spectrum

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