PlanetMath (more info)
 Math for the people, by the people.
Encyclopedia | Requests | Forums | Docs | Wiki | Random | RSS  
Login
create new user
name:
pass:
forget your password?
Main Menu
sections
Encyclopædia
Papers
Books
Expositions

meta
Requests (231)
Orphanage
Unclass'd
Unproven (514)
Corrections (25)
Classification

talkback
Polls
Forums
Feedback
Bug Reports

downloads
Snapshots
PM Book

information
News
Docs
Wiki
ChangeLog
TODO List
Legalese
About
Owner confidence rating: High Entry average rating: No information on entry rating
$\Omega$-spectrum (Definition)
Definition 0.1   An $ \Omega$-spectrum $ E$ is defined as a sequence of $ n$ distinct based spaces $ E_n$, together with based weak homotopy equivalences $ \epsilon_n: \Omega E_n \to E_{n + 1}$, with $ n$ being an integer.

Note: For a given construction of an $ \Omega$-spectrum one is able to 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$'.



"$\Omega$-spectrum" is owned by bci1.
(view preamble | get metadata)

View style:

See Also: weak homotopy equivalence, weak homotopy double groupoid, cohomology group theorem for connected CW complexes, group cohomology, Eilenberg-MacLane space, derivation of cohomology group theorem for connected CW complexes

Also defines:  based weak homotopy equivalences, cohomology group
Keywords:  omega-spectrum, sequence of based spaces, weak homotopy equivalences
Log in to rate this entry.
(view current ratings)

Cross-references: induced, multiplication, isomorphism, structure, group, CW-complex, theory, cohomology, integer, sequence
There are 7 references to this entry.

This is version 11 of $\Omega$-spectrum, born on 2008-09-19, modified 2008-12-21.
Object id is 11046, canonical name is OmegaSpectrum.
Accessed 539 times total.

Classification:
AMS MSC: 55T05 (Algebraic topology :: Spectral sequences :: General)
 55T25 (Algebraic topology :: Spectral sequences :: Generalized cohomology)
 55T20 (Algebraic topology :: Spectral sequences :: Eilenberg-Moore spectral sequences)
 55T99 (Algebraic topology :: Spectral sequences :: Miscellaneous)
Pending Errata and Addenda
None.
[ View all 1 ]
Discussion
Style: Expand: Order:
forum policy

No messages.

Interact
post | correct | update request | add derivation | add example | add (any)