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 3 hours ago









      David Roberts

      17.2k462176




      17.2k462176










      asked 4 hours ago









      User371User371

      1686




      1686






















          1 Answer
          1






          active

          oldest

          votes


















          2












          $begingroup$

          You have it in page 7 of this paper.






          share|cite|improve this answer









          $endgroup$













            Your Answer





            StackExchange.ifUsing("editor", function () {
            return StackExchange.using("mathjaxEditing", function () {
            StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
            StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
            });
            });
            }, "mathjax-editing");

            StackExchange.ready(function() {
            var channelOptions = {
            tags: "".split(" "),
            id: "504"
            };
            initTagRenderer("".split(" "), "".split(" "), channelOptions);

            StackExchange.using("externalEditor", function() {
            // Have to fire editor after snippets, if snippets enabled
            if (StackExchange.settings.snippets.snippetsEnabled) {
            StackExchange.using("snippets", function() {
            createEditor();
            });
            }
            else {
            createEditor();
            }
            });

            function createEditor() {
            StackExchange.prepareEditor({
            heartbeatType: 'answer',
            autoActivateHeartbeat: false,
            convertImagesToLinks: true,
            noModals: true,
            showLowRepImageUploadWarning: true,
            reputationToPostImages: 10,
            bindNavPrevention: true,
            postfix: "",
            imageUploader: {
            brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
            contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
            allowUrls: true
            },
            noCode: true, onDemand: true,
            discardSelector: ".discard-answer"
            ,immediatelyShowMarkdownHelp:true
            });


            }
            });














            draft saved

            draft discarded


















            StackExchange.ready(
            function () {
            StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmathoverflow.net%2fquestions%2f323966%2fexplicit-chain-homotopy-for-the-alexander-whitney-eilenberg-zilber-pair%23new-answer', 'question_page');
            }
            );

            Post as a guest















            Required, but never shown

























            1 Answer
            1






            active

            oldest

            votes








            1 Answer
            1






            active

            oldest

            votes









            active

            oldest

            votes






            active

            oldest

            votes









            2












            $begingroup$

            You have it in page 7 of this paper.






            share|cite|improve this answer









            $endgroup$


















              2












              $begingroup$

              You have it in page 7 of this paper.






              share|cite|improve this answer









              $endgroup$
















                2












                2








                2





                $begingroup$

                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






























                    draft saved

                    draft discarded




















































                    Thanks for contributing an answer to MathOverflow!


                    • Please be sure to answer the question. Provide details and share your research!

                    But avoid …



                    • Asking for help, clarification, or responding to other answers.

                    • Making statements based on opinion; back them up with references or personal experience.


                    Use MathJax to format equations. MathJax reference.


                    To learn more, see our tips on writing great answers.




                    draft saved


                    draft discarded














                    StackExchange.ready(
                    function () {
                    StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmathoverflow.net%2fquestions%2f323966%2fexplicit-chain-homotopy-for-the-alexander-whitney-eilenberg-zilber-pair%23new-answer', 'question_page');
                    }
                    );

                    Post as a guest















                    Required, but never shown





















































                    Required, but never shown














                    Required, but never shown












                    Required, but never shown







                    Required, but never shown

































                    Required, but never shown














                    Required, but never shown












                    Required, but never shown







                    Required, but never shown







                    -

                    Popular posts from this blog

                    404 Error Contact Form 7 ajax form submitting

                    Image
                    0 The contact form does not work, and affects loading after one try. The chrome console shows the error message: POST http://example.com/wp-json/contact-form-7/v1/contact-forms/50/feedback 404 (Not Found) send @ jquery.js?ver=1.12.4:4 ajax @ jquery.js?ver=1.12.4:4 wpcf7.submit @ scripts.js?ver=5.0.3:346 (anonymous) @ scripts.js?ver=5.0.3:53 dispatch @ jquery.js?ver=1.12.4:3 r.handle @ jquery.js?ver=1.12.4:3 How to solve this issue? thanks in advance javascript wordpress contact-form-7 share | improve this question edited Jul 17 '18 at 5:39 Kiran Shahi 4,745 3 16 43
                    Read more

                    How to know if a Active Directory user can login interactively

                    Image
                    1 I would like to know if and how is it possible to know if a AD user can login interactively (on a server) in a domain. I need to know if I can find it out using a LDAP search. Thanks. windows active-directory login user-permissions user-profile share | improve this question edited 1 hour ago yagmoth555 ♦ 11.6k 3 17 42 asked 1 hour ago Luigi Luigi 6 1 New contributor
                    Read more

                    TypeError: fit_transform() missing 1 required positional argument: 'X'

                    Image
                    0 I am trying to do Feature Scaling in a dataset, but I get an error and have no idea how to proceed: > Traceback (most recent call last): > > File "<ipython-input-10-71bea414b4d0>", line 22, in <module> > x_train = sc_X.fit_transform(x_train) > > TypeError: fit_transform() missing 1 required positional argument: 'X' and here is my code: import pandas as pd # Importing the dataset dataset = pd.read_csv('Data.csv') X = dataset.iloc[:, :-1].values y = dataset.iloc[:, 3].values # Taking care of missing data from sklearn.preprocessing import Imputer imputer = Imputer(missing_values="NaN", strategy="mean", axis=0) imputer = Imputer.fit(imputer,X[:,1:3]) X[:, 1:3] = Imputer.transform(imputer,X[
                    Read more