Omega-spectrum

Omega-spectrum

This is a topic entry on

Ω

–spectra and their important role in reduced cohomology theories
on

CW complexes

.

0.1

Introduction

In

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

(

http://planetmath.org/ClassesOfAlgebras

)

S

⁢

1

⁢

⋀

X

i

→

X

i

+

1

, where

S

⁢

1

is the

unit circle

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

0.2

Ω

–spectrum

One can express the definition of an

Ω

–spectrum in terms of a sequence of CW
complexes,

K

1

,

K

2

,

…

as follows.

Definition 0.1

.

Let us consider

Ω

⁢

K

, the space of loops in a

C

⁢

W

complex

K

called
the

loopspace of

K

, which is topologized as a

subspace

of the space

K

I

of all maps

I

→

K

, where

K

I

is given the

compact-open topology

.
Then, an

Ω

–spectrum

{

K

n

}

is defined as a
sequence

K

1

,

K

2

,

…

of CW complexes together with

weak homotopy equivalences

(

http://planetmath.org/WeakHomotopyEquivalence

) (

ϵ

n

):

ϵ

n

:

Ω

⁢

K

n

→

K

n

+

1

,

with

n

being an integer.

An alternative definition of the

Ω

–spectrum can also be formulated as follows.

Definition 0.2

.

An

Ω

–spectrum

, or

Omega spectrum

, is a spectrum

𝐄

such that for every index

i

,
the topological space

X

i

is fibered, and also the

adjoints

of the structure mappings (

http://planetmath.org/ClassesOfAlgebras

) are all

weak equivalences

X

i

≅

Ω

⁢

X

i

+

1

.

0.3

The Role of

Ω

-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

Ω

–spectrum one is able to canonically define an associated

cohomology

theory; thus, one defines the

cohomology groups

(

http://planetmath.org/ProofOfCohomologyGroupTheorem

) of a

CW-complex

K

associated with the

Ω

–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 group structure by requiring that

(

ϵ

n

)

*

:

[

K

,

E

n

]

→

[

K

,

Ω

E

n

+

1

]

is an

isomorphism

which defines the

multiplication

in

[

K

,

E

n

]

induced by

ϵ

n

.

One can prove that if

{

K

n

}

is a an

Ω

-spectrum then the

functors

defined by the assignments

X

⟼

h

n

⁢

(

X

)

=

(

X

,

K

n

)

,

with

n

∈

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

Ω

-spectrum (the Brown representability theorem; p. 397 of

[

6

]

).

References

1

H. Masana. 2008.

The Tate-Thomason Conjecture

.

http://www.math.uiuc.edu/

K-theory

/0919/TT.pdf

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.

http://www.math.cornell.edu/ hatcher/AT/AT.pdf

Algebraic Topology., Cambridge University Press; Cambridge, UK.

Title

Omega-spectrum

Canonical name

Omegaspectrum

Date of creation

2013-03-22 18:24:01

Last modified on

2013-03-22 18:24:01

Owner

bci1 (20947)

Last modified by

bci1 (20947)

Numerical id

80

Author

bci1 (20947)

Entry type

Topic

Classification

msc 55T20

Classification

msc 55T25

Classification

msc 55T05

Synonym

Omega spectrum

Synonym

Ω

spectrum

Related topic

PointedTopologicalSpace

Related topic

CategoricalSequence

Related topic

ClassesOfAlgebras

Related topic

HomotopyCategory

Related topic

WeakHomotopyEquivalence

Related topic

WeakHomotopyDoubleGroupoid

Related topic

CohomologyGroupTheorem

Related topic

GroupCohomology

Related topic

ProofOfCohomologyGroupTheorem

Defines

equence of CW complexes

Defines

spectrum

Defines

Ω

–spectrum

Defines

Omega spectrum

Defines

unit circle

Defines

cohomology group

Defines

category of spectra

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