A Homotopy Relation on the Category of Inverse and Direct Spectra of Topological Spaces
by
Cigdem Aras
University of Kocaeli
Coauthors: Sadi Bayramov

In this article, A homotopy relation is introduced on the category of inverse and direct spectra of topological spaces. This relation is a generalization of the usual homotopy relation in the class of topological spaces. It is proved that the given relation is an equivalence relation and the composition operation is invariant with respect to this relation. Later a link between the homotopy relation and the relation of limit spaces of inverse spectra is stated.

Consider the inverse spectra:

X=( { X\alpha}\alpha in A  ,   { P\alpha\alpha':   X\alpha' --> X\alpha}\alpha < \alpha') , Y=( { Y\beta}\beta in B  ,   { q\beta\beta':   Y\beta' --> Y\beta}\beta < \beta') .

For the morphisms of the above inverse spectra

f=( \pi: B --> A,   { f\beta: X\pi(\beta) --> Y\beta}\beta in B ), g=( \rho: B --> A,   { g\beta: X\rho(\beta) --> Y\beta}\beta in B ).

Definition 1. If for all\beta in B     exists\alpha in A satisfying \alpha > \pi(\beta), \rho(\beta) and mappings

f\beta o P\pi(\beta)\alpha   and    g\beta o P\rho(\beta)\alpha    ( f\beta o P\pi(\beta)\alpha=g\beta o P\rho(\beta)\alpha )

are homotopic then the morphisms f, g: X --> Y are said to be spectrally homotopic morphisms (canonically homotopic).

Similarly, we define the concept of homotopy in a category of direct spectra of topological spaces.

Theorem 1.a) The spectral homotopy relation of Inv(Top)   (Dir(Top)) in the category of inverse ( direct ) spectra of topological spaces is an equivalance relation.
                b) The composition operation in the categories Inv(Top) and Dir(Top) are invariant with respect to the spectral homotopy relation.

Let F : Top --> C be any covariant (contravariant) functor. It is abvious that the functor F in the categories Inv(Top) and Dir(Top) induces the covariant (contravariant) functors;

F*: Inv(Top) --> Inv(C)          ( F*: Inv(Top) --> Dir(C)) F*: Dir(Top) --> Dir(C)         ( F*: Dir(Top) --> Inv(C))

Theorem 2. Let F: Top --> C be a homotopy invariant functor. Then, the composition of the induced functor F_*(F^*) and one of the suitable functors lim _<-- or lim _--> will be a spectral homotopy invariant functor.

Theorem 3. If lim _<-- f,   lim _<-- g : lim _<-- X --> lim _<-- Y are homotopic maps and for each \alpha in A,   p_\alpha: X --> X_\alpha is a cofibration, then f, g : X --> Y of the inverse spectra are spectrally homotopic.

Theorem 4. If lim _<-- f,   lim _<-- g : lim _<-- X --> lim _<-- Y are homotopic in the class of homotopy types, then the morphism of the inverse spectra f, g : X --> Y are spectrally homotopic.

Theorem 5. In the category Inv[Top], the limits of spectrally homotopic morphisms are homotopic.

Date received: July 5, 2001

Copyright © 2001 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cagx-31.