Open subspaces of CW complexesCounterexamples in algebraic topology?Is the polynomial de Rham functor a...



Open subspaces of CW complexes


Counterexamples in algebraic topology?Is the polynomial de Rham functor a Quillen equivalence?When is the category of pro-objects a homotopy category?Question about topological monoid mapsAre infinite simplicial complexes all manifolds?What do absolute neighborhood retracts look like?In a topological space if there exists a loop that cannot be contracted to a point does there exist a simple loop that cannot be contracted also?













6












$begingroup$


I am looking at the paper




Covering homotopy properties of maps between CW complexes or ANRs
by
Mark Steinberger and James West




and a claim is made in the proof of their first main theorem that (slightly rephrased)




since $U$ is a contractible subspace of the CW complex $B$, $U$ "is" a CW complex




Question: Is it generally true that an open subspace of a CW complex can be given the structure of a CW complex? Is it true in general only for contractible subspaces? Why? Is there a reference?






at.algebraic-topology gn.general-topology homotopy-theory







share|cite|improve this question









$endgroup$














  • $begingroup$
    As John mentions, there are counter-examples to open subsets of CW-complexes being CW-complexes, but I believe under reasonable "finiteness" assumptions the result should be true. I have not looked at the counter-examples John cites but I imagine they are due to the CW-complex having fairly "bad" attaching maps. In my mind I'm imagining an adaptation of the proof that open subsets of $mathbb R^n$ admit CW-structures.
    $endgroup$
    – Ryan Budney
    6 hours ago






  • 1




    $begingroup$
    One of Cauty's counterexamples has only three cells, so the problem is not about finiteness, but rather about the attaching maps. Regular CW complexes can be triangulated, and open subsets of simplicial complexes can again be triangulated, so it is true that open subsets of regular CW complexes are CW spaces.
    $endgroup$
    – John Rognes
    6 hours ago
















6












$begingroup$


I am looking at the paper




Covering homotopy properties of maps between CW complexes or ANRs
by
Mark Steinberger and James West




and a claim is made in the proof of their first main theorem that (slightly rephrased)




since $U$ is a contractible subspace of the CW complex $B$, $U$ "is" a CW complex




Question: Is it generally true that an open subspace of a CW complex can be given the structure of a CW complex? Is it true in general only for contractible subspaces? Why? Is there a reference?






at.algebraic-topology gn.general-topology homotopy-theory







share|cite|improve this question









$endgroup$














  • $begingroup$
    As John mentions, there are counter-examples to open subsets of CW-complexes being CW-complexes, but I believe under reasonable "finiteness" assumptions the result should be true. I have not looked at the counter-examples John cites but I imagine they are due to the CW-complex having fairly "bad" attaching maps. In my mind I'm imagining an adaptation of the proof that open subsets of $mathbb R^n$ admit CW-structures.
    $endgroup$
    – Ryan Budney
    6 hours ago






  • 1




    $begingroup$
    One of Cauty's counterexamples has only three cells, so the problem is not about finiteness, but rather about the attaching maps. Regular CW complexes can be triangulated, and open subsets of simplicial complexes can again be triangulated, so it is true that open subsets of regular CW complexes are CW spaces.
    $endgroup$
    – John Rognes
    6 hours ago














6












6








6


1



$begingroup$


I am looking at the paper




Covering homotopy properties of maps between CW complexes or ANRs
by
Mark Steinberger and James West




and a claim is made in the proof of their first main theorem that (slightly rephrased)




since $U$ is a contractible subspace of the CW complex $B$, $U$ "is" a CW complex




Question: Is it generally true that an open subspace of a CW complex can be given the structure of a CW complex? Is it true in general only for contractible subspaces? Why? Is there a reference?






at.algebraic-topology gn.general-topology homotopy-theory







share|cite|improve this question









$endgroup$




I am looking at the paper




Covering homotopy properties of maps between CW complexes or ANRs
by
Mark Steinberger and James West




and a claim is made in the proof of their first main theorem that (slightly rephrased)




since $U$ is a contractible subspace of the CW complex $B$, $U$ "is" a CW complex




Question: Is it generally true that an open subspace of a CW complex can be given the structure of a CW complex? Is it true in general only for contractible subspaces? Why? Is there a reference?






at.algebraic-topology gn.general-topology homotopy-theory




at.algebraic-topology gn.general-topology homotopy-theory






share|cite|improve this question













share|cite|improve this question











share|cite|improve this question




share|cite|improve this question










asked 11 hours ago









Jeff StromJeff Strom

7,9292 gold badges31 silver badges61 bronze badges




7,9292 gold badges31 silver badges61 bronze badges















  • $begingroup$
    As John mentions, there are counter-examples to open subsets of CW-complexes being CW-complexes, but I believe under reasonable "finiteness" assumptions the result should be true. I have not looked at the counter-examples John cites but I imagine they are due to the CW-complex having fairly "bad" attaching maps. In my mind I'm imagining an adaptation of the proof that open subsets of $mathbb R^n$ admit CW-structures.
    $endgroup$
    – Ryan Budney
    6 hours ago






  • 1




    $begingroup$
    One of Cauty's counterexamples has only three cells, so the problem is not about finiteness, but rather about the attaching maps. Regular CW complexes can be triangulated, and open subsets of simplicial complexes can again be triangulated, so it is true that open subsets of regular CW complexes are CW spaces.
    $endgroup$
    – John Rognes
    6 hours ago


















  • $begingroup$
    As John mentions, there are counter-examples to open subsets of CW-complexes being CW-complexes, but I believe under reasonable "finiteness" assumptions the result should be true. I have not looked at the counter-examples John cites but I imagine they are due to the CW-complex having fairly "bad" attaching maps. In my mind I'm imagining an adaptation of the proof that open subsets of $mathbb R^n$ admit CW-structures.
    $endgroup$
    – Ryan Budney
    6 hours ago






  • 1




    $begingroup$
    One of Cauty's counterexamples has only three cells, so the problem is not about finiteness, but rather about the attaching maps. Regular CW complexes can be triangulated, and open subsets of simplicial complexes can again be triangulated, so it is true that open subsets of regular CW complexes are CW spaces.
    $endgroup$
    – John Rognes
    6 hours ago
















$begingroup$
As John mentions, there are counter-examples to open subsets of CW-complexes being CW-complexes, but I believe under reasonable "finiteness" assumptions the result should be true. I have not looked at the counter-examples John cites but I imagine they are due to the CW-complex having fairly "bad" attaching maps. In my mind I'm imagining an adaptation of the proof that open subsets of $mathbb R^n$ admit CW-structures.
$endgroup$
– Ryan Budney
6 hours ago




$begingroup$
As John mentions, there are counter-examples to open subsets of CW-complexes being CW-complexes, but I believe under reasonable "finiteness" assumptions the result should be true. I have not looked at the counter-examples John cites but I imagine they are due to the CW-complex having fairly "bad" attaching maps. In my mind I'm imagining an adaptation of the proof that open subsets of $mathbb R^n$ admit CW-structures.
$endgroup$
– Ryan Budney
6 hours ago




1




1




$begingroup$
One of Cauty's counterexamples has only three cells, so the problem is not about finiteness, but rather about the attaching maps. Regular CW complexes can be triangulated, and open subsets of simplicial complexes can again be triangulated, so it is true that open subsets of regular CW complexes are CW spaces.
$endgroup$
– John Rognes
6 hours ago




$begingroup$
One of Cauty's counterexamples has only three cells, so the problem is not about finiteness, but rather about the attaching maps. Regular CW complexes can be triangulated, and open subsets of simplicial complexes can again be triangulated, so it is true that open subsets of regular CW complexes are CW spaces.
$endgroup$
– John Rognes
6 hours ago










1 Answer
1






active

oldest

votes


















11













$begingroup$

It is not generally true that each open subset of a CW complex admits the structure of a CW complex. Counterexamples were given in



bib{MR1157891}{article}{

   author={Cauty, Robert},

   title={Sur les ouverts des CW-complexes et les fibr'{e}s de Serre},

   language={French},

   journal={Colloq. Math.},

   volume={63},

   date={1992},

   number={1},

   pages={1--7},

   issn={0010-1354},

   review={MR{1157891}},

   doi={10.4064/cm-63-1-1-7},

}


in response to the paper you mention by Steinberger and West. Here is the abstract:




M. Steinberger et J. West ont prouvé dans [7] qu'un fibré de Serre p:E
→ B entre CW-complexes a la propriété de relèvement des homotopies par
rapport aux k-espaces. Malheureusement, leur démonstration contient
une légère erreur. Ils affirment que certains ensembles (notés U et
p−1U×U) sont des CW-complexes car ce sont des ouverts de CW-complexes.
Ceci est généralement faux, et notre premier objectif dans cette note
est de donner des exemples d'ouverts de CW-complexes n'admettant
aucune décomposition CW. Malgré cela, le théorème de Steinberger et
West est vrai, et notre deuxième objectif est de montrer comment leur
démonstration peut être rectifiée.







share|cite|improve this answer









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    1 Answer
    1






    active

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    active

    oldest

    votes






    active

    oldest

    votes









    11













    $begingroup$

    It is not generally true that each open subset of a CW complex admits the structure of a CW complex. Counterexamples were given in



    bib{MR1157891}{article}{
    
   author={Cauty, Robert},
    
   title={Sur les ouverts des CW-complexes et les fibr'{e}s de Serre},
    
   language={French},
    
   journal={Colloq. Math.},
    
   volume={63},
    
   date={1992},
    
   number={1},
    
   pages={1--7},
    
   issn={0010-1354},
    
   review={MR{1157891}},
    
   doi={10.4064/cm-63-1-1-7},
    
}


    in response to the paper you mention by Steinberger and West. Here is the abstract:




    M. Steinberger et J. West ont prouvé dans [7] qu'un fibré de Serre p:E
    → B entre CW-complexes a la propriété de relèvement des homotopies par
    rapport aux k-espaces. Malheureusement, leur démonstration contient
    une légère erreur. Ils affirment que certains ensembles (notés U et
    p−1U×U) sont des CW-complexes car ce sont des ouverts de CW-complexes.
    Ceci est généralement faux, et notre premier objectif dans cette note
    est de donner des exemples d'ouverts de CW-complexes n'admettant
    aucune décomposition CW. Malgré cela, le théorème de Steinberger et
    West est vrai, et notre deuxième objectif est de montrer comment leur
    démonstration peut être rectifiée.







    share|cite|improve this answer









    $endgroup$




















      11













      $begingroup$

      It is not generally true that each open subset of a CW complex admits the structure of a CW complex. Counterexamples were given in



      bib{MR1157891}{article}{
      
   author={Cauty, Robert},
      
   title={Sur les ouverts des CW-complexes et les fibr'{e}s de Serre},
      
   language={French},
      
   journal={Colloq. Math.},
      
   volume={63},
      
   date={1992},
      
   number={1},
      
   pages={1--7},
      
   issn={0010-1354},
      
   review={MR{1157891}},
      
   doi={10.4064/cm-63-1-1-7},
      
}


      in response to the paper you mention by Steinberger and West. Here is the abstract:




      M. Steinberger et J. West ont prouvé dans [7] qu'un fibré de Serre p:E
      → B entre CW-complexes a la propriété de relèvement des homotopies par
      rapport aux k-espaces. Malheureusement, leur démonstration contient
      une légère erreur. Ils affirment que certains ensembles (notés U et
      p−1U×U) sont des CW-complexes car ce sont des ouverts de CW-complexes.
      Ceci est généralement faux, et notre premier objectif dans cette note
      est de donner des exemples d'ouverts de CW-complexes n'admettant
      aucune décomposition CW. Malgré cela, le théorème de Steinberger et
      West est vrai, et notre deuxième objectif est de montrer comment leur
      démonstration peut être rectifiée.







      share|cite|improve this answer









      $endgroup$


















        11














        11










        11







        $begingroup$

        It is not generally true that each open subset of a CW complex admits the structure of a CW complex. Counterexamples were given in



        bib{MR1157891}{article}{
        
   author={Cauty, Robert},
        
   title={Sur les ouverts des CW-complexes et les fibr'{e}s de Serre},
        
   language={French},
        
   journal={Colloq. Math.},
        
   volume={63},
        
   date={1992},
        
   number={1},
        
   pages={1--7},
        
   issn={0010-1354},
        
   review={MR{1157891}},
        
   doi={10.4064/cm-63-1-1-7},
        
}


        in response to the paper you mention by Steinberger and West. Here is the abstract:




        M. Steinberger et J. West ont prouvé dans [7] qu'un fibré de Serre p:E
        → B entre CW-complexes a la propriété de relèvement des homotopies par
        rapport aux k-espaces. Malheureusement, leur démonstration contient
        une légère erreur. Ils affirment que certains ensembles (notés U et
        p−1U×U) sont des CW-complexes car ce sont des ouverts de CW-complexes.
        Ceci est généralement faux, et notre premier objectif dans cette note
        est de donner des exemples d'ouverts de CW-complexes n'admettant
        aucune décomposition CW. Malgré cela, le théorème de Steinberger et
        West est vrai, et notre deuxième objectif est de montrer comment leur
        démonstration peut être rectifiée.







        share|cite|improve this answer









        $endgroup$



        It is not generally true that each open subset of a CW complex admits the structure of a CW complex. Counterexamples were given in



        bib{MR1157891}{article}{
        
   author={Cauty, Robert},
        
   title={Sur les ouverts des CW-complexes et les fibr'{e}s de Serre},
        
   language={French},
        
   journal={Colloq. Math.},
        
   volume={63},
        
   date={1992},
        
   number={1},
        
   pages={1--7},
        
   issn={0010-1354},
        
   review={MR{1157891}},
        
   doi={10.4064/cm-63-1-1-7},
        
}


        in response to the paper you mention by Steinberger and West. Here is the abstract:




        M. Steinberger et J. West ont prouvé dans [7] qu'un fibré de Serre p:E
        → B entre CW-complexes a la propriété de relèvement des homotopies par
        rapport aux k-espaces. Malheureusement, leur démonstration contient
        une légère erreur. Ils affirment que certains ensembles (notés U et
        p−1U×U) sont des CW-complexes car ce sont des ouverts de CW-complexes.
        Ceci est généralement faux, et notre premier objectif dans cette note
        est de donner des exemples d'ouverts de CW-complexes n'admettant
        aucune décomposition CW. Malgré cela, le théorème de Steinberger et
        West est vrai, et notre deuxième objectif est de montrer comment leur
        démonstration peut être rectifiée.








        share|cite|improve this answer












        share|cite|improve this answer



        share|cite|improve this answer










        answered 6 hours ago









        John RognesJohn Rognes

        5,03626 silver badges30 bronze badges




        5,03626 silver badges30 bronze badges

































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