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# 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 $S1\bigwedge X_{i}\to X_{{i+1}}$, 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 ($\epsilon_{n}$):

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

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 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 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. 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.

## Mathematics Subject Classification

55T20*no label found*55T25

*no label found*55T05

*no label found*

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