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 7 hours ago









      David Roberts

      17.2k462176




      17.2k462176










      asked 7 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 6 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

                    How to label and detect the document text images

                    Image
                    1 $begingroup$ This is what I mean as document text image: I want to label the texts in image as separate blocks and my model should detect these labels as classes. NOTE: This is how the end result should be like: The labels like Block 1, Block 2, Block 3,.. should be Logo, Title, Date,.. Others, etc. Work done: First approach : I tried to implement this method via Object Detection, it didn't work. It didn't even detect any text. Second approach : Then I tried it using PixelLink. As this model is build for scene text detection, it detected each and every text in the image. But this method can detect multiple lines of text if the threshold values are increased. But I have no idea how do I add labels to the text blocks. PIXEL_CLS_WEIGHT_all_ones = 'PIXEL_CLS_WEIGHT_all_ones' PIXEL_C
                    Read more

                    Vallis Paradisi

                    Image
                    Coordinata: .mw-parser-output .geo-default,.mw-parser-output .geo-dms,.mw-parser-output .geo-dec{display:inline}.mw-parser-output .geo-nondefault,.mw-parser-output .geo-multi-punct{display:none}.mw-parser-output .longitude,.mw-parser-output .latitude{white-space:nowrap} 33°03′S 71°37′W  /  33.05°S 71.617°W  / -33.05; -71.617 Vallis Paradisi Res apud Vicidata repertae: Civitas : Chilia Numerus incolarum : 296 655 Zona horaria : UTC-4 Situs interretialis Nomen officiale : Valparaíso Gubernium Consilium : municipal council of Valparaíso Geographia Superficies : 401 600 000 Metrum quadratum Coniunctiones urbium Urbes gemellae : Ovetum , Badalona , Santa Fe , Ora longa , Sciamhaevum , Quantunia , Callao , Bat Yam , Melaka , Novorossiysk , Busan , Corduba , Odessa , Rosario , Malacca , Barcino , Soteropolis , Franciscopolis (California) , Guanajuato, Silao Tabula aut despectus Despectus in portum sinumque et urbem Vallis Paradisi Vallis Para
                    Read more

                    Tabula Rosettana

                    Image
                    Tabula multilinguis Rosettana in Museo Britannico ostenditur. Tabula Rosettana, [1] etiam titulo OGIS 90 agnita, est stela decreto de rebus sacris in Aegypto anno 196 a.C.n. lato inscripta. Tabula iuxta Rosettam Aegypti, urbem in delta Nili et ad oram maris Mediterranei iacentem, anno 1799 a milite Francico reperta est. Inventio stelae, linguis duabus scripturisque tribus inscriptae, eruditis Instituti Aegypti statim nuntiata est; ibi enim iussu imperatoris Napoleonis eruditi omnium scientiarum (sub aegide Commissionis Scientiarum et Artium) properaverant cum expeditione Francica. Qua a Britannis mox debellata, tabula Rosettana Londinium missa hodie apud Museum Britannicum iacet. Textus Graecus cito lectus interpretationi textuum Aegyptiorum (in formis hieroglyphica et demotica expressorum) gradatim adiuvit. Denique textum plene interpretatus est Ioannes Franciscus Champollion. Ab opere eruditorum cumulativo coepit hodiernus scripturae hieroglyphicae linguaeque Aegyptiae a
                    Read more