Is the concept of a “numerable” fiber bundle really useful or an empty generalization? The Next CEO of Stack OverflowNon trivial vector bundle over non-paracompact contractible spaceExample of fiber bundle that is not a fibrationFiber bundle = principal bundle + fiber?Numerable covers from the point of view of Grothendieck topologiesGlobal sections for torus fiber bundleAre there analogs of smooth partitions of unity and good open covers for PL-manifolds?Two natural maps asssociated with the nerve of a coverDescent theory, fibrations, and bundlesIn which sense are Euler-Lagrange PDE's on fiber bundles quasi-linear?What is the local structure of a fibration?Complete proof of Homotopy invariance of a numerable fiber bundle based on CHPLocally trivial fibration over a suspension

Is the concept of a “numerable” fiber bundle really useful or an empty generalization?



The Next CEO of Stack OverflowNon trivial vector bundle over non-paracompact contractible spaceExample of fiber bundle that is not a fibrationFiber bundle = principal bundle + fiber?Numerable covers from the point of view of Grothendieck topologiesGlobal sections for torus fiber bundleAre there analogs of smooth partitions of unity and good open covers for PL-manifolds?Two natural maps asssociated with the nerve of a coverDescent theory, fibrations, and bundlesIn which sense are Euler-Lagrange PDE's on fiber bundles quasi-linear?What is the local structure of a fibration?Complete proof of Homotopy invariance of a numerable fiber bundle based on CHPLocally trivial fibration over a suspension










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Numerable fiber bundles are defined by Dold (DOLD 1962 - Partitions of Unity in theory of Fibrations) as a generalization of fiber bundles over a paracompact space : the trivialization cover of the base admit a subordinate partition of unity (locally finite). He proves in this paper that almost all important theorems for fiber bundles over paracompact spaces are also valid for numerable bundles.



But is it really an interesting generalization ? Are there examples of "natural" or "useful" numerable fiber bundles that are not paracompact?






at.algebraic-topology fibre-bundles







share|cite|improve this question











$endgroup$







  • 5




    $begingroup$
    Really it means that on the category of all topological spaces, it is the Grothendieck topology generated by numerable open covers for which the usual prestack of principal bundles is a stack. On this category, the Grothendieck topology with all open covers gives an inequivalent site (there are too many covers). This is a fine distinction that isn't usually observed outside of algebraic geometry, where the nicest classes of spaces of interest already give many inequivalent sites.
    $endgroup$
    – David Roberts
    8 hours ago






  • 3




    $begingroup$
    I am not a homotopy theorist but thought the point is that "numerable" is the right notion which makes the theory of fiber bundles work. Paracompactness of the base is not relevant, and it may be unavailable or hard to check.
    $endgroup$
    – Igor Belegradek
    6 hours ago















9












$begingroup$


Numerable fiber bundles are defined by Dold (DOLD 1962 - Partitions of Unity in theory of Fibrations) as a generalization of fiber bundles over a paracompact space : the trivialization cover of the base admit a subordinate partition of unity (locally finite). He proves in this paper that almost all important theorems for fiber bundles over paracompact spaces are also valid for numerable bundles.



But is it really an interesting generalization ? Are there examples of "natural" or "useful" numerable fiber bundles that are not paracompact?






at.algebraic-topology fibre-bundles







share|cite|improve this question











$endgroup$







  • 5




    $begingroup$
    Really it means that on the category of all topological spaces, it is the Grothendieck topology generated by numerable open covers for which the usual prestack of principal bundles is a stack. On this category, the Grothendieck topology with all open covers gives an inequivalent site (there are too many covers). This is a fine distinction that isn't usually observed outside of algebraic geometry, where the nicest classes of spaces of interest already give many inequivalent sites.
    $endgroup$
    – David Roberts
    8 hours ago






  • 3




    $begingroup$
    I am not a homotopy theorist but thought the point is that "numerable" is the right notion which makes the theory of fiber bundles work. Paracompactness of the base is not relevant, and it may be unavailable or hard to check.
    $endgroup$
    – Igor Belegradek
    6 hours ago













9












9








9


2



$begingroup$


Numerable fiber bundles are defined by Dold (DOLD 1962 - Partitions of Unity in theory of Fibrations) as a generalization of fiber bundles over a paracompact space : the trivialization cover of the base admit a subordinate partition of unity (locally finite). He proves in this paper that almost all important theorems for fiber bundles over paracompact spaces are also valid for numerable bundles.



But is it really an interesting generalization ? Are there examples of "natural" or "useful" numerable fiber bundles that are not paracompact?






at.algebraic-topology fibre-bundles







share|cite|improve this question











$endgroup$




Numerable fiber bundles are defined by Dold (DOLD 1962 - Partitions of Unity in theory of Fibrations) as a generalization of fiber bundles over a paracompact space : the trivialization cover of the base admit a subordinate partition of unity (locally finite). He proves in this paper that almost all important theorems for fiber bundles over paracompact spaces are also valid for numerable bundles.



But is it really an interesting generalization ? Are there examples of "natural" or "useful" numerable fiber bundles that are not paracompact?






at.algebraic-topology fibre-bundles




at.algebraic-topology fibre-bundles






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited 5 hours ago









David White

12.9k462103




12.9k462103










asked 9 hours ago









ychemamaychemama

51129




51129







  • 5




    $begingroup$
    Really it means that on the category of all topological spaces, it is the Grothendieck topology generated by numerable open covers for which the usual prestack of principal bundles is a stack. On this category, the Grothendieck topology with all open covers gives an inequivalent site (there are too many covers). This is a fine distinction that isn't usually observed outside of algebraic geometry, where the nicest classes of spaces of interest already give many inequivalent sites.
    $endgroup$
    – David Roberts
    8 hours ago






  • 3




    $begingroup$
    I am not a homotopy theorist but thought the point is that "numerable" is the right notion which makes the theory of fiber bundles work. Paracompactness of the base is not relevant, and it may be unavailable or hard to check.
    $endgroup$
    – Igor Belegradek
    6 hours ago












  • 5




    $begingroup$
    Really it means that on the category of all topological spaces, it is the Grothendieck topology generated by numerable open covers for which the usual prestack of principal bundles is a stack. On this category, the Grothendieck topology with all open covers gives an inequivalent site (there are too many covers). This is a fine distinction that isn't usually observed outside of algebraic geometry, where the nicest classes of spaces of interest already give many inequivalent sites.
    $endgroup$
    – David Roberts
    8 hours ago






  • 3




    $begingroup$
    I am not a homotopy theorist but thought the point is that "numerable" is the right notion which makes the theory of fiber bundles work. Paracompactness of the base is not relevant, and it may be unavailable or hard to check.
    $endgroup$
    – Igor Belegradek
    6 hours ago







5




5




$begingroup$
Really it means that on the category of all topological spaces, it is the Grothendieck topology generated by numerable open covers for which the usual prestack of principal bundles is a stack. On this category, the Grothendieck topology with all open covers gives an inequivalent site (there are too many covers). This is a fine distinction that isn't usually observed outside of algebraic geometry, where the nicest classes of spaces of interest already give many inequivalent sites.
$endgroup$
– David Roberts
8 hours ago




$begingroup$
Really it means that on the category of all topological spaces, it is the Grothendieck topology generated by numerable open covers for which the usual prestack of principal bundles is a stack. On this category, the Grothendieck topology with all open covers gives an inequivalent site (there are too many covers). This is a fine distinction that isn't usually observed outside of algebraic geometry, where the nicest classes of spaces of interest already give many inequivalent sites.
$endgroup$
– David Roberts
8 hours ago




3




3




$begingroup$
I am not a homotopy theorist but thought the point is that "numerable" is the right notion which makes the theory of fiber bundles work. Paracompactness of the base is not relevant, and it may be unavailable or hard to check.
$endgroup$
– Igor Belegradek
6 hours ago




$begingroup$
I am not a homotopy theorist but thought the point is that "numerable" is the right notion which makes the theory of fiber bundles work. Paracompactness of the base is not relevant, and it may be unavailable or hard to check.
$endgroup$
– Igor Belegradek
6 hours ago










1 Answer
1






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9












$begingroup$

In algebraic topology, it is often more convenient to know that a map is a fibration (has the homotopy lifting property with respect to all spaces) than a fibre bundle, because then calculational tools such as long exact sequences of homotopy groups and Serre spectral sequences of (co)homology groups become available.



It is easy to cook up examples of fibrations which are not fibre bundles (the projection of a $2$-simplex onto one of its edges being the easiest example I know). It is somewhat harder to find examples of fibre bundles which are not fibrations, but they do exist; see here and here.



Numerability is precisely the extra condition on fibre bundles which makes them into fibrations. Of course this means that any fibre bundle over a paracompact base is a fibration.



The homotopy lifting property is used extensively when proving the homotopy classification of principal $G$-bundles, i.e. that isomorphism classes of principal $G$-bundles a given base space $B$ are in one-to-one correspondence with homotopy classes $[B,BG]$. To obtain such a result for arbitrary base spaces $B$ you had better therefore restrict to numerable bundles.



This doesn't really answer your question, in that I haven't given you a natural example of a numerable bundle over a non-paracompact base. But hopefully it indicates why numerable bundles are a useful concept in homotopy theory.






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    9












    $begingroup$

    In algebraic topology, it is often more convenient to know that a map is a fibration (has the homotopy lifting property with respect to all spaces) than a fibre bundle, because then calculational tools such as long exact sequences of homotopy groups and Serre spectral sequences of (co)homology groups become available.



    It is easy to cook up examples of fibrations which are not fibre bundles (the projection of a $2$-simplex onto one of its edges being the easiest example I know). It is somewhat harder to find examples of fibre bundles which are not fibrations, but they do exist; see here and here.



    Numerability is precisely the extra condition on fibre bundles which makes them into fibrations. Of course this means that any fibre bundle over a paracompact base is a fibration.



    The homotopy lifting property is used extensively when proving the homotopy classification of principal $G$-bundles, i.e. that isomorphism classes of principal $G$-bundles a given base space $B$ are in one-to-one correspondence with homotopy classes $[B,BG]$. To obtain such a result for arbitrary base spaces $B$ you had better therefore restrict to numerable bundles.



    This doesn't really answer your question, in that I haven't given you a natural example of a numerable bundle over a non-paracompact base. But hopefully it indicates why numerable bundles are a useful concept in homotopy theory.






    share|cite|improve this answer











    $endgroup$

















      9












      $begingroup$

      In algebraic topology, it is often more convenient to know that a map is a fibration (has the homotopy lifting property with respect to all spaces) than a fibre bundle, because then calculational tools such as long exact sequences of homotopy groups and Serre spectral sequences of (co)homology groups become available.



      It is easy to cook up examples of fibrations which are not fibre bundles (the projection of a $2$-simplex onto one of its edges being the easiest example I know). It is somewhat harder to find examples of fibre bundles which are not fibrations, but they do exist; see here and here.



      Numerability is precisely the extra condition on fibre bundles which makes them into fibrations. Of course this means that any fibre bundle over a paracompact base is a fibration.



      The homotopy lifting property is used extensively when proving the homotopy classification of principal $G$-bundles, i.e. that isomorphism classes of principal $G$-bundles a given base space $B$ are in one-to-one correspondence with homotopy classes $[B,BG]$. To obtain such a result for arbitrary base spaces $B$ you had better therefore restrict to numerable bundles.



      This doesn't really answer your question, in that I haven't given you a natural example of a numerable bundle over a non-paracompact base. But hopefully it indicates why numerable bundles are a useful concept in homotopy theory.






      share|cite|improve this answer











      $endgroup$















        9












        9








        9





        $begingroup$

        In algebraic topology, it is often more convenient to know that a map is a fibration (has the homotopy lifting property with respect to all spaces) than a fibre bundle, because then calculational tools such as long exact sequences of homotopy groups and Serre spectral sequences of (co)homology groups become available.



        It is easy to cook up examples of fibrations which are not fibre bundles (the projection of a $2$-simplex onto one of its edges being the easiest example I know). It is somewhat harder to find examples of fibre bundles which are not fibrations, but they do exist; see here and here.



        Numerability is precisely the extra condition on fibre bundles which makes them into fibrations. Of course this means that any fibre bundle over a paracompact base is a fibration.



        The homotopy lifting property is used extensively when proving the homotopy classification of principal $G$-bundles, i.e. that isomorphism classes of principal $G$-bundles a given base space $B$ are in one-to-one correspondence with homotopy classes $[B,BG]$. To obtain such a result for arbitrary base spaces $B$ you had better therefore restrict to numerable bundles.



        This doesn't really answer your question, in that I haven't given you a natural example of a numerable bundle over a non-paracompact base. But hopefully it indicates why numerable bundles are a useful concept in homotopy theory.






        share|cite|improve this answer











        $endgroup$



        In algebraic topology, it is often more convenient to know that a map is a fibration (has the homotopy lifting property with respect to all spaces) than a fibre bundle, because then calculational tools such as long exact sequences of homotopy groups and Serre spectral sequences of (co)homology groups become available.



        It is easy to cook up examples of fibrations which are not fibre bundles (the projection of a $2$-simplex onto one of its edges being the easiest example I know). It is somewhat harder to find examples of fibre bundles which are not fibrations, but they do exist; see here and here.



        Numerability is precisely the extra condition on fibre bundles which makes them into fibrations. Of course this means that any fibre bundle over a paracompact base is a fibration.



        The homotopy lifting property is used extensively when proving the homotopy classification of principal $G$-bundles, i.e. that isomorphism classes of principal $G$-bundles a given base space $B$ are in one-to-one correspondence with homotopy classes $[B,BG]$. To obtain such a result for arbitrary base spaces $B$ you had better therefore restrict to numerable bundles.



        This doesn't really answer your question, in that I haven't given you a natural example of a numerable bundle over a non-paracompact base. But hopefully it indicates why numerable bundles are a useful concept in homotopy theory.







        share|cite|improve this answer














        share|cite|improve this answer



        share|cite|improve this answer








        edited 3 hours ago

























        answered 4 hours ago









        Mark GrantMark Grant

        22.2k657134




        22.2k657134



























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