Explicit chain homotopy for the Alexander-Whitney, Eilenberg-Zilber pair












4












$begingroup$


Let $A$ and $B$ be simplicial abelian groups, and let $N_ast(-)$ denote the normalized chain complex functor. Let
$$AW_{A,B}colon N_ast(Aotimes B)
longrightarrow N_ast(A)otimes N_ast(B)$$
and
$$ EZ_{A,B}colon N_ast(A)otimes N_ast(B)
longrightarrow N_ast(Aotimes B)$$
denote the Alexander-Whitney map
and the Eilenberg-Zilber map respectively.
Does anyone know of an explicit chain homotopy realizing
$$EZ_{A,B}circ AW_{A,B}sim Id_{N_ast(Aotimes B)}.$$



Motivation for its existence can be found in the comments of this question.






reference-request at.algebraic-topology homological-algebra simplicial-stuff abelian-categories







share|cite|improve this question











$endgroup$

















    4












    $begingroup$


    Let $A$ and $B$ be simplicial abelian groups, and let $N_ast(-)$ denote the normalized chain complex functor. Let
    $$AW_{A,B}colon N_ast(Aotimes B)
    longrightarrow N_ast(A)otimes N_ast(B)$$
    and
    $$ EZ_{A,B}colon N_ast(A)otimes N_ast(B)
    longrightarrow N_ast(Aotimes B)$$
    denote the Alexander-Whitney map
    and the Eilenberg-Zilber map respectively.
    Does anyone know of an explicit chain homotopy realizing
    $$EZ_{A,B}circ AW_{A,B}sim Id_{N_ast(Aotimes B)}.$$



    Motivation for its existence can be found in the comments of this question.






    reference-request at.algebraic-topology homological-algebra simplicial-stuff abelian-categories







    share|cite|improve this question











    $endgroup$















      4












      4








      4


      1



      $begingroup$


      Let $A$ and $B$ be simplicial abelian groups, and let $N_ast(-)$ denote the normalized chain complex functor. Let
      $$AW_{A,B}colon N_ast(Aotimes B)
      longrightarrow N_ast(A)otimes N_ast(B)$$
      and
      $$ EZ_{A,B}colon N_ast(A)otimes N_ast(B)
      longrightarrow N_ast(Aotimes B)$$
      denote the Alexander-Whitney map
      and the Eilenberg-Zilber map respectively.
      Does anyone know of an explicit chain homotopy realizing
      $$EZ_{A,B}circ AW_{A,B}sim Id_{N_ast(Aotimes B)}.$$



      Motivation for its existence can be found in the comments of this question.






      reference-request at.algebraic-topology homological-algebra simplicial-stuff abelian-categories







      share|cite|improve this question











      $endgroup$




      Let $A$ and $B$ be simplicial abelian groups, and let $N_ast(-)$ denote the normalized chain complex functor. Let
      $$AW_{A,B}colon N_ast(Aotimes B)
      longrightarrow N_ast(A)otimes N_ast(B)$$
      and
      $$ EZ_{A,B}colon N_ast(A)otimes N_ast(B)
      longrightarrow N_ast(Aotimes B)$$
      denote the Alexander-Whitney map
      and the Eilenberg-Zilber map respectively.
      Does anyone know of an explicit chain homotopy realizing
      $$EZ_{A,B}circ AW_{A,B}sim Id_{N_ast(Aotimes B)}.$$



      Motivation for its existence can be found in the comments of this question.






      reference-request at.algebraic-topology homological-algebra simplicial-stuff abelian-categories




      reference-request at.algebraic-topology homological-algebra simplicial-stuff abelian-categories






      share|cite|improve this question















      share|cite|improve this question













      share|cite|improve this question




      share|cite|improve this question








      edited 4 hours ago









      David Roberts

      17.2k462176




      17.2k462176










      asked 4 hours ago









      User371User371

      1686




      1686






















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          $begingroup$

          You have it in page 7 of this paper.






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            $begingroup$

            You have it in page 7 of this paper.






            share|cite|improve this answer









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              $begingroup$

              You have it in page 7 of this paper.






              share|cite|improve this answer









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                $begingroup$

                You have it in page 7 of this paper.






                share|cite|improve this answer









                $endgroup$



                You have it in page 7 of this paper.







                share|cite|improve this answer












                share|cite|improve this answer



                share|cite|improve this answer










                answered 3 hours ago









                Fernando MuroFernando Muro

                11.6k23363




                11.6k23363






























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