 Spectrum (Homotopy Theory)

# Spectrum (homotopy theory)

to get instant updates about 'Spectrum (Homotopy Theory)' on your MyPage. Meet other similar minded people. Its Free!

X

Description:
In algebraic topology, a branch of mathematics, a spectrum is an object representing a generalized cohomology theory. There are several different constructions of categories of spectra, any of which gives a context for the same stable homotopy theory.

## The definition of a spectrum

One definition of a spectrum is that a spectrum is a sequence [itex]E:= _ [/itex] of CW complexes together with cellular inclusions [itex] Sigma E_n to E_ [/itex].There are many variations of this definition.

## Functions, maps, and morphisms of spectra

There are three natural categories whose objects are spectra, whose morphisms are the functions, or maps, or morphisms defined below.

A function between two spectra E and F is a sequence of maps from E<sub>n</sub> to F<sub>n</sub> that commute with themaps &Sigma;E<sub>n</sub>&rarr;E<sub>n+1</sub> and &Sigma;F<sub>n</sub>&rarr;F<sub>n+1</sub>.

Given a spectrum [itex]E_n[/itex], a subspectrum [itex]F_n[/itex] is a sequence of subcomplexes that is also a spectrum. Noting that each i-cell in [itex]E_j[/itex] becomes an (i+1)-cell in [itex]E_[/itex], a cofinal subspectrum is a subspectrum for which each cell of the parent spectrum is eventually contained in the subspectrum after a finite number of suspensions. Spectra can then be turned into a category by defining a map of spectra...

No feeds found

All Posting your question. Please wait!...