In the paper we construct and develop a fiber strong shape theory for
arbitrary spaces over fixed metrizable space $\Bo$. Our approach is based on
the method of Marde\v{s}i\'{c}-Lisica and instead of resolutions, introduced by
Marde\v{s}i\'{c}, their fiber preserving analogues are used. The fiber strong
shape theory yields the classification of spaces over $\Bo$ which is coarser
than the classification of spaces over $\Bo$ induced by fiber homotopy theory,
but is finer than the classification of spaces over $\Bo$ given by usual fiber
shape theory

Baladze, Vladimer

Tsinaridze, Ruslan

English

arXiv

The purpose of this paper is the construction and investigation of fiber strong shape theory for compact metrizable spaces over a fixed base space B0 , using the fiber versions of cotelescop, fibrant space and SSDR-map. In the paper obtained results containing the characterizations of fiber strong shape equivalences, based on the notion of double mapping cylinder over a fixed space B0. Besides, in the paper we construct and develop a fiber strong shape theory for arbitrary spaces over fixed metrizable space B0. Our approach is based on the method of Mardešić-Lisica and instead of resolutions, introduced by Mardešić, their fiber preserving analogues are used.
The fiber strong shape theory yields the classification of spaces over B0 which is coarser than the classification of spaces over B0 induced by fiber homotopy theory, but is finer than the classification of spaces over B0 given by usual fiber shape theory

University of Dayton

University of DaytoneCommonsSummer Conference on Topology and ItsApplications Department of Mathematics6-2017Fiber Strong Shape Theory for Topological SpacesRuslan TsinaridzeBatumi Shota Rustaveli State University, r.tsinaridze@bsu.edu.geVladimer BaladzeBatumi Shota Rustaveli State University, vbaladze@gmail.comFollow this and additional works at: http://ecommons.udayton.edu/topology_confPart of the Geometry and Topology Commons, and the Special Functions CommonsThis Topology + Foundations is brought to you for free and open access by the Department of Mathematics at eCommons. It has been accepted forinclusion in Summer Conference on Topology and Its Applications by an authorized administrator of eCommons. For more information, pleasecontact frice1@udayton.edu, mschlangen1@udayton.edu.eCommons CitationTsinaridze, Ruslan and Baladze, Vladimer, "Fiber Strong Shape Theory for Topological Spaces" (2017). Summer Conference on Topologyand Its Applications. 57.http://ecommons.udayton.edu/topology_conf/57The authors supported in part by grant FR/233/5-103/14 from Shota Rustaveli National Science Foundation (SRNSF) Mathematics Fiber Strong Shape Theory for Topological Spaces Ruslan Tsinaridze Department of Mathematics, Batumi Shota Rustaveli State University ABSTRACT. In the paper we construct and develop a fiber strong shape theory for arbitrary spaces over fixed metrizable space 0B . Our approach is based on the method of Mardešic'-Lisica and instead of resolutions, introduced by Mardešic', their fiber preserving analogues are used. The fiber strong shape theory yields the classification of spaces over 0B  which is coarser than the classification of spaces over 0B  induced by fiber homotopy theory, but is finer than the classification of spaces over 0B  given by usual fiber shape theory. Math. Sub. Class.:54C55, 54C56, 55P55. Keywords and Phrases: Fiber shape, Fiber homotopy, Fiber resolution, Fiber shape expansion, Fiber strong expansion, B0A(N)R -space, B0A(N)E -space.                                                       1  Resolution and Strong Expansions of Spaces over 0B   An inverse system of the category B0Top  is a collection '= ( , , )X p X A  of space X  over 0B  indexed by a directed set A  and f.p. maps ' ':p X X   , '  , such that ' ' ' ' =p p p      and = 1Xp ,   A . A morphism '( , ) : = ( , , )f Y q   X Y B  of inverse systems of the category B0Top  consists of a function : B A  and of f.p. maps ( ):f X Y    ,   B , such that whenever '  , then there is an index '( ), ( )      for which ( ) ' ' '( ) =   f p q f p        . Two morphisms ( , ), ( , ) :f g   X Y  are said to be equivalent, 0Bf g , provided for each   B  there is an   A , ( ), ( )     , such that ( ) ( ) =  f p g p        . Let B0pro -Top  be a category, whose objects are the inverse systems X  of the category B0Top  and whose morphisms are the equivalence classes f  of morphisms ,( ) :f  X Y  with respect to relation 0B. A morphism '= ( ) : = ( , , )p X X p  p X A  from a rudimentary system ( )X  to an inverse system X  consists of the f.p. maps :p X X  ,  A , such that ' '=  p p p   , '  . Definition 1.1  Let X  be a space over 0B  and let '= ( , , )X p X A  be an inverse system of the category B0Top . We say that : X p X  is a resolution over 0B  or fiber resolution of the space X  2  Vladimer Baladze, Ruslan Tsinaridze over 0B  provided it satisfies the following two conditions: B0R 1).  Let B0ANRP , let  be an open covering of P  and let :h X P  be a f.p. map. Then there exist an index    and a f.p. map :f X P   such that h  and  f p  are -near. B0R 2).  Let B0ANRP  and let  be an open covering of P . Then there is an open cover '  of P  with the following property: if    and ', :f f X P  are f.p. maps such that the f.p. maps  f p  and ' f p  are ' -near, then there is an index '   such that the f.p. maps ' f p  and '' f p are -near. If in a fiber resolution ': = ( , , )X X p p X A  of the space X  over 0B  each X  is an B0ANR , then we say that p  is a fiber B0ANR -resolution.  The next theorem is essential in the construction of the fiber shape category for arbitrary spaces over 0B . Theorem 1.2  Every space X  over a metrizable space 0B admits an B0ANR -resolution over 0B .      In the proof of Theorem 1.2 we shall need the following lemma.  Lemma 1.3  Let :f X Y  be a f.p. map from the topological space X  over 0B  to an B0ANR -space Y . Then there exists an B0ANR -space Z  of weight 0 0( ) { ( ), ( ), )}w Z max w X w B   and f.p. maps :g X Z  and :h Z Y  such that  =f h g .  Definition 1.4  Let X  be a topological space over 0B , '= ( , , )X p X A  an inverse system in B0Top  and = ( ) :p X p X  a morphism of B0pro -Top . We call p  an expansion over 0B  of the space X  over 0B  provided it has the following properties: B0E 1).  For every B0ANR -space P  over 0B  and f.p. map :f X P  there is an index    and a f. p. map :h X P   such that 0 Bh p f . B0E 2).  If ', :f f X P   are f. p. maps, B0ANRP  and '0  Bf p f p  , then there is an index '   such that '' '0  Bf p f p .  Definition 1.5  A morphism ': ( , , )X X p p A  is called a strong expansion over 0B  provided it satisfies condition E01)B  and the following condition: B0SE 2).  Let P  be an B0ANR -space, let 0 1, :f f X P  ,   A  be f.p. maps and let :F X I P   be a f.p. homotopy such that   0( ,0) = ( ),     ,S x f p x x X    1( ,1) = ( ),     .S x f p x x X   Then there exists a '   and a f.p. homotopy ':H X I P   , such that   0 ' '( ,0) = ( ),     ,H x f p z z X    1 ' '( ,1) = ( ),     ,H x f p z z X    '0( 1 ) (rel( )).IBH p S X I   It is clear that, every strong expansion over 0B  is an expansion over 0B . If all B0ANRX  , then p  is called an B0ANR -expansion and strong B0ANR -expansion, Fiber Strong Shape Theory for Topological Spaces  3  respectively. The main result of section 1 is the following theorem. Theorem 1.6  Let X  be a topological space over 0B . Then every resolution :p X X  over 0B  induces a strong B0ANR -expansion.                                                                                                       Corollary 1.7  Every B0ANR -resolution over 0B  induces B0ANR -expansion.                                     Corollary 1.8  Every space X  over 0B  admits a cofinite strong B0ANR -expansion.                           In the proof of Theorem 1.6 we need the following lemma.  Lemma 1.9  Let X  be a topological space over metrizamble space 0B , let ',P P  be B0ANR -spaces, let ':f X P , '0 1, :h h P P  be f.p. maps and let :S X I P   be a f.p. homotopy such that   0( ,0) = ( ),      ,S x h f x x X   1( ,1) = ( ),      .S x h f x x X  Then there exists an B0ANR -space 'P  , f.p. maps ' ':f X P  , ' ':h P P   and a f.p. homotopy ':K P I P   such that   ' = ,h f f   '0( ,0) = ( ),     K z h h z z P   '1( ,1) = ( ),     K z h h z z P   '( 1 ) = .IK f S  Lemma 1.10  Let :p X X  be a resolution over 0B  and let 0 1, , ,P f f  and F  be as in 02)BSE . Then for every open covering  of P , there exist a '   and a f.p. homotopy ':H X I P    such that   0 ' '( ,0) =  ( ),         H y f p y y X    1 ' '( ,1) =  ( ),         H y f p y y X    '( , (1 ))S H p    2  On Fiber Strong Shape Category   The objects of category 0BSSH  are all topological spaces over 0B . The morphisms of category 0BSSH  are defined by the following way.  Let X p X  and Y q Y  be an 0BANR -resolutions of X  and Y , respectively. Let [ ]f  X Y  be a some morphism of category 0BCPHTop . Let X  p X , Y  q Y ,[ ]f    X Y  be another triple of fiber resolutions of spaces X  and Y  over 0B  and morphism of category0BCPHTop .  Now define the following equivalence relation. We say the triples ( [ ]f p q ) and ( [ ]f   p q ) are equivalent if   [ ][ ] [ ][ ]f i j f    where [ ]i  X X  and [ ]j  Y Y  are isomorphisms of category 0BCPHTop .  The fiber strong shape morphisms F X Y   are the equivalence classes of triples ( [ ]f p q ) with respect to the above defined relation .  Let F X Y   and G Y Z   be the fiber strong shape morphisms, defined by triples ( [ ])f p q  4  Vladimer Baladze, Ruslan Tsinaridze and ( [ ])g  p q , where Y  p Y , Z q Z  and [ ]g  Y Z .  As we know there exists an unique morphism [ ]h  Y Y  of category 0BCPHTop  such that [ ][ ] [ ]h q q . Note that   [ ][ ] [ ] [ ][ ]j q q h q    Hence, [ ] [ ]j h . Besides, [ ][ ] [ ][ ][1 ]g j g hZ.  Thus, we can assume that the morphisms F  and G  are given by triples ( [ ])f p q  and ( [ ])g  p q .  Consequently, we can define the composition GF X Z   as the morphism given by triple ( [ ][ ])g f p r .  In the role an identity morphism X X I  we can take the morphism defined by triple( , [1 ])Xp p .  The obtained category 0BSSH call the fiber strong shape category.  Let 0( )X ob BSSH . By symbol 0Bssh ( )X  denote the equivalence class of topological space X  and call the fiber strong shape of X .  For each f.p. map X Y    choose 0BANR -resolutions X p X  and Y q Y . There exists a unique morphism [ ]f  X Y  of category 0BCPHTop  such that [ ][ ] [ ][ ]q f p  .  We can define a functor 00 0BSS   BBTop SSH . By definition,   0 0BSS ( ) ( )X X X ob   BTop  and   0 0BSS ( ) ( )Mor X Y      BTop  Here   is a fiber strong shape morphism defined by triple ( [ ])f p q .  As in [L-M] we can prove that functor 0BSS  induces a functor00 0BSS   BBHTop SSH , which we call the fiber strong shape functor. By definition,   0 0BSS ( ) ( )X X X ob  BHTop  and   0 0 0 0 0B B B BSS ([ ] ) SS ( ) [ ] ( )Mor X Y      BHTop  Let us define a functor 0S  0B BSHSSH . Assume that S( )X X  for each object 0( )X ob BSSH . Let F X Y   be a fiber strong shape morphism given by a triple [ ])f (p q .  Consider the morphism E([ ])f  as an image of [ ]f  with respect the functor 0E   BB 0CPH pro -HTopTop . The triple (H H E[ ])f p q  generates a fiber shape morphism, which we denote by S(F) X Y  .  Now we can formulate the following  Theorem 2.5  There exists the following commutative diagram   where B0S  is V.Baladze fiber shape functor 4[ ]B .                                                                                 Corollary 2.6  Let X  and Y  be topological spaces over 0B . If B B0 0ssh ( ) = ssh ( )X Y , then B B0 0sh ( ) = sh ( )X Y .                                                                                                                                     Remark 2.7  Using the methods developed in this paper and papers ([B 6 ], [L-M],[M 1 ], [M 2 ]) it is possible to construct fiber strong shape theory for category of arbitrary continuous maps.               Fiber Strong Shape Theory for Topological Spaces  5    REFERENCES  [B 1 ]  V. 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Fiber Strong Shape Theory for Topological Spaces

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