Some questions about specific topological embeddings












1












$begingroup$


It often happens in algebraic topology that we have a construction on, say, a space $X$, usually obtained by taking a product and then a quotient. Then we may have a "natural" injection of $X$ in the constructed space. My question is: under these circumstances, is there a general method to prove that this injection is indeed a (topological) embedding?



Let me list some examples related to mapping cones and cylinders:




  1. Consider the unreduced cone $C'X:=Xtimes I/Xtimes1$. Then the canonical injection $Xhookrightarrow C'X, xmapsto[x,0]$ is an embedding. This I can prove directly.

  2. More generally, consider the reduced mapping cone on a pointed space $(X,*)$ defined by $CX:=Xtimes I/(Xtimes1cup*times I)$. The canonical injection $Xhookrightarrow CX, xmapsto[x,0]$ is also an embedding. I cannot seem to find a proof of this...

  3. Another situation arises in cofibrations. Let $i:Asubseteq X$. The mapping cylinder is $M(i):=Atimes Icup_i X$ obtained by identifying $(a,0)sim i(a)$. We have an injection $M(i)hookrightarrow Xtimes I$ defined in the obvious way. Is this map an embedding?


Explicitly, my question is: (1) How do I prove the above 2 and 3? (2) Is there a general method to tackle these questions?










share|cite|improve this question









$endgroup$












  • $begingroup$
    In 3., do you assume that $i$ is a cofibration?
    $endgroup$
    – Paul Frost
    Dec 21 '18 at 13:49
















1












$begingroup$


It often happens in algebraic topology that we have a construction on, say, a space $X$, usually obtained by taking a product and then a quotient. Then we may have a "natural" injection of $X$ in the constructed space. My question is: under these circumstances, is there a general method to prove that this injection is indeed a (topological) embedding?



Let me list some examples related to mapping cones and cylinders:




  1. Consider the unreduced cone $C'X:=Xtimes I/Xtimes1$. Then the canonical injection $Xhookrightarrow C'X, xmapsto[x,0]$ is an embedding. This I can prove directly.

  2. More generally, consider the reduced mapping cone on a pointed space $(X,*)$ defined by $CX:=Xtimes I/(Xtimes1cup*times I)$. The canonical injection $Xhookrightarrow CX, xmapsto[x,0]$ is also an embedding. I cannot seem to find a proof of this...

  3. Another situation arises in cofibrations. Let $i:Asubseteq X$. The mapping cylinder is $M(i):=Atimes Icup_i X$ obtained by identifying $(a,0)sim i(a)$. We have an injection $M(i)hookrightarrow Xtimes I$ defined in the obvious way. Is this map an embedding?


Explicitly, my question is: (1) How do I prove the above 2 and 3? (2) Is there a general method to tackle these questions?










share|cite|improve this question









$endgroup$












  • $begingroup$
    In 3., do you assume that $i$ is a cofibration?
    $endgroup$
    – Paul Frost
    Dec 21 '18 at 13:49














1












1








1


1



$begingroup$


It often happens in algebraic topology that we have a construction on, say, a space $X$, usually obtained by taking a product and then a quotient. Then we may have a "natural" injection of $X$ in the constructed space. My question is: under these circumstances, is there a general method to prove that this injection is indeed a (topological) embedding?



Let me list some examples related to mapping cones and cylinders:




  1. Consider the unreduced cone $C'X:=Xtimes I/Xtimes1$. Then the canonical injection $Xhookrightarrow C'X, xmapsto[x,0]$ is an embedding. This I can prove directly.

  2. More generally, consider the reduced mapping cone on a pointed space $(X,*)$ defined by $CX:=Xtimes I/(Xtimes1cup*times I)$. The canonical injection $Xhookrightarrow CX, xmapsto[x,0]$ is also an embedding. I cannot seem to find a proof of this...

  3. Another situation arises in cofibrations. Let $i:Asubseteq X$. The mapping cylinder is $M(i):=Atimes Icup_i X$ obtained by identifying $(a,0)sim i(a)$. We have an injection $M(i)hookrightarrow Xtimes I$ defined in the obvious way. Is this map an embedding?


Explicitly, my question is: (1) How do I prove the above 2 and 3? (2) Is there a general method to tackle these questions?










share|cite|improve this question









$endgroup$




It often happens in algebraic topology that we have a construction on, say, a space $X$, usually obtained by taking a product and then a quotient. Then we may have a "natural" injection of $X$ in the constructed space. My question is: under these circumstances, is there a general method to prove that this injection is indeed a (topological) embedding?



Let me list some examples related to mapping cones and cylinders:




  1. Consider the unreduced cone $C'X:=Xtimes I/Xtimes1$. Then the canonical injection $Xhookrightarrow C'X, xmapsto[x,0]$ is an embedding. This I can prove directly.

  2. More generally, consider the reduced mapping cone on a pointed space $(X,*)$ defined by $CX:=Xtimes I/(Xtimes1cup*times I)$. The canonical injection $Xhookrightarrow CX, xmapsto[x,0]$ is also an embedding. I cannot seem to find a proof of this...

  3. Another situation arises in cofibrations. Let $i:Asubseteq X$. The mapping cylinder is $M(i):=Atimes Icup_i X$ obtained by identifying $(a,0)sim i(a)$. We have an injection $M(i)hookrightarrow Xtimes I$ defined in the obvious way. Is this map an embedding?


Explicitly, my question is: (1) How do I prove the above 2 and 3? (2) Is there a general method to tackle these questions?







general-topology algebraic-topology homotopy-theory






share|cite|improve this question













share|cite|improve this question











share|cite|improve this question




share|cite|improve this question










asked Dec 21 '18 at 2:52









ColescuColescu

3,26011136




3,26011136












  • $begingroup$
    In 3., do you assume that $i$ is a cofibration?
    $endgroup$
    – Paul Frost
    Dec 21 '18 at 13:49


















  • $begingroup$
    In 3., do you assume that $i$ is a cofibration?
    $endgroup$
    – Paul Frost
    Dec 21 '18 at 13:49
















$begingroup$
In 3., do you assume that $i$ is a cofibration?
$endgroup$
– Paul Frost
Dec 21 '18 at 13:49




$begingroup$
In 3., do you assume that $i$ is a cofibration?
$endgroup$
– Paul Frost
Dec 21 '18 at 13:49










1 Answer
1






active

oldest

votes


















1












$begingroup$

Question (1):



2.) Let $p : X times I to CX$ denote the quotient map. It identifies $X' = X times { 1} cup { *} times I$ to a point.The canonical injection $i : X to CX$ is continuous. To see that $i$ is an embedding, we consider a closed $A subset X$ and show that $i(A)$ is closed in $i(X)$.



Case 1. $* in A$. Then $B = X times { 1 } cup A times I$ is closed in $X times I$. Since $X' subset B$, we have $p^{-1}(p(B)) = B$. Hence $p(B)$ is closed in $CX$ because $p$ is a quotient map. But $p(B) cap i(X) = i(A)$.



Case 2. $* notin A$. Then $B = A times [0,1/2] $ is closed in $X times I$. Since $X' cap B = emptyset$, we have $p^{-1}(p(B)) = B$ whence $p(B)$ is closed. Again $p(B) cap i(X) = i(A)$.



Note, however, that $i(X)$ is closed in $CX$ if and only if ${ ast }$ is closed in $X$. In contrast, in the unreduced cone $i(X)$ is always closed.



3.) Let $j : M(i) to X times I$ denote the unique map such that $j([x]) = (x,0)$ for $x in X$ and $j([a,t]) = (a,t)$ for $(a,t) in A times I$. It is always an injection whose image is $M' = j(M(i)) = X times { 0 } cup A times I$.
Moreover, it is an embedding if (a) $A$ is closed or (b) $i : A to X$ is a cofibration. (a) is trivial. The highly nontrivial proof of (b) can be found in



Strøm, Arne. "Note on cofibrations II." Mathematica Scandinavica 22.1 (1969): 130-142.



See Lemma 3 of this paper and the comment following the proof. Note that there are cofibrations $i : A to X$ where $A$ is not closed,
but only with non-Hausdorff $X$.



If neither $A$ is closed nor $i$ is a cofibration, we get counterexamples like $X = I, A = [0,1)$.



Question (2):



I do not see a general method.






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: "69"
    };
    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%2fmath.stackexchange.com%2fquestions%2f3048136%2fsome-questions-about-specific-topological-embeddings%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









    1












    $begingroup$

    Question (1):



    2.) Let $p : X times I to CX$ denote the quotient map. It identifies $X' = X times { 1} cup { *} times I$ to a point.The canonical injection $i : X to CX$ is continuous. To see that $i$ is an embedding, we consider a closed $A subset X$ and show that $i(A)$ is closed in $i(X)$.



    Case 1. $* in A$. Then $B = X times { 1 } cup A times I$ is closed in $X times I$. Since $X' subset B$, we have $p^{-1}(p(B)) = B$. Hence $p(B)$ is closed in $CX$ because $p$ is a quotient map. But $p(B) cap i(X) = i(A)$.



    Case 2. $* notin A$. Then $B = A times [0,1/2] $ is closed in $X times I$. Since $X' cap B = emptyset$, we have $p^{-1}(p(B)) = B$ whence $p(B)$ is closed. Again $p(B) cap i(X) = i(A)$.



    Note, however, that $i(X)$ is closed in $CX$ if and only if ${ ast }$ is closed in $X$. In contrast, in the unreduced cone $i(X)$ is always closed.



    3.) Let $j : M(i) to X times I$ denote the unique map such that $j([x]) = (x,0)$ for $x in X$ and $j([a,t]) = (a,t)$ for $(a,t) in A times I$. It is always an injection whose image is $M' = j(M(i)) = X times { 0 } cup A times I$.
    Moreover, it is an embedding if (a) $A$ is closed or (b) $i : A to X$ is a cofibration. (a) is trivial. The highly nontrivial proof of (b) can be found in



    Strøm, Arne. "Note on cofibrations II." Mathematica Scandinavica 22.1 (1969): 130-142.



    See Lemma 3 of this paper and the comment following the proof. Note that there are cofibrations $i : A to X$ where $A$ is not closed,
    but only with non-Hausdorff $X$.



    If neither $A$ is closed nor $i$ is a cofibration, we get counterexamples like $X = I, A = [0,1)$.



    Question (2):



    I do not see a general method.






    share|cite|improve this answer











    $endgroup$


















      1












      $begingroup$

      Question (1):



      2.) Let $p : X times I to CX$ denote the quotient map. It identifies $X' = X times { 1} cup { *} times I$ to a point.The canonical injection $i : X to CX$ is continuous. To see that $i$ is an embedding, we consider a closed $A subset X$ and show that $i(A)$ is closed in $i(X)$.



      Case 1. $* in A$. Then $B = X times { 1 } cup A times I$ is closed in $X times I$. Since $X' subset B$, we have $p^{-1}(p(B)) = B$. Hence $p(B)$ is closed in $CX$ because $p$ is a quotient map. But $p(B) cap i(X) = i(A)$.



      Case 2. $* notin A$. Then $B = A times [0,1/2] $ is closed in $X times I$. Since $X' cap B = emptyset$, we have $p^{-1}(p(B)) = B$ whence $p(B)$ is closed. Again $p(B) cap i(X) = i(A)$.



      Note, however, that $i(X)$ is closed in $CX$ if and only if ${ ast }$ is closed in $X$. In contrast, in the unreduced cone $i(X)$ is always closed.



      3.) Let $j : M(i) to X times I$ denote the unique map such that $j([x]) = (x,0)$ for $x in X$ and $j([a,t]) = (a,t)$ for $(a,t) in A times I$. It is always an injection whose image is $M' = j(M(i)) = X times { 0 } cup A times I$.
      Moreover, it is an embedding if (a) $A$ is closed or (b) $i : A to X$ is a cofibration. (a) is trivial. The highly nontrivial proof of (b) can be found in



      Strøm, Arne. "Note on cofibrations II." Mathematica Scandinavica 22.1 (1969): 130-142.



      See Lemma 3 of this paper and the comment following the proof. Note that there are cofibrations $i : A to X$ where $A$ is not closed,
      but only with non-Hausdorff $X$.



      If neither $A$ is closed nor $i$ is a cofibration, we get counterexamples like $X = I, A = [0,1)$.



      Question (2):



      I do not see a general method.






      share|cite|improve this answer











      $endgroup$
















        1












        1








        1





        $begingroup$

        Question (1):



        2.) Let $p : X times I to CX$ denote the quotient map. It identifies $X' = X times { 1} cup { *} times I$ to a point.The canonical injection $i : X to CX$ is continuous. To see that $i$ is an embedding, we consider a closed $A subset X$ and show that $i(A)$ is closed in $i(X)$.



        Case 1. $* in A$. Then $B = X times { 1 } cup A times I$ is closed in $X times I$. Since $X' subset B$, we have $p^{-1}(p(B)) = B$. Hence $p(B)$ is closed in $CX$ because $p$ is a quotient map. But $p(B) cap i(X) = i(A)$.



        Case 2. $* notin A$. Then $B = A times [0,1/2] $ is closed in $X times I$. Since $X' cap B = emptyset$, we have $p^{-1}(p(B)) = B$ whence $p(B)$ is closed. Again $p(B) cap i(X) = i(A)$.



        Note, however, that $i(X)$ is closed in $CX$ if and only if ${ ast }$ is closed in $X$. In contrast, in the unreduced cone $i(X)$ is always closed.



        3.) Let $j : M(i) to X times I$ denote the unique map such that $j([x]) = (x,0)$ for $x in X$ and $j([a,t]) = (a,t)$ for $(a,t) in A times I$. It is always an injection whose image is $M' = j(M(i)) = X times { 0 } cup A times I$.
        Moreover, it is an embedding if (a) $A$ is closed or (b) $i : A to X$ is a cofibration. (a) is trivial. The highly nontrivial proof of (b) can be found in



        Strøm, Arne. "Note on cofibrations II." Mathematica Scandinavica 22.1 (1969): 130-142.



        See Lemma 3 of this paper and the comment following the proof. Note that there are cofibrations $i : A to X$ where $A$ is not closed,
        but only with non-Hausdorff $X$.



        If neither $A$ is closed nor $i$ is a cofibration, we get counterexamples like $X = I, A = [0,1)$.



        Question (2):



        I do not see a general method.






        share|cite|improve this answer











        $endgroup$



        Question (1):



        2.) Let $p : X times I to CX$ denote the quotient map. It identifies $X' = X times { 1} cup { *} times I$ to a point.The canonical injection $i : X to CX$ is continuous. To see that $i$ is an embedding, we consider a closed $A subset X$ and show that $i(A)$ is closed in $i(X)$.



        Case 1. $* in A$. Then $B = X times { 1 } cup A times I$ is closed in $X times I$. Since $X' subset B$, we have $p^{-1}(p(B)) = B$. Hence $p(B)$ is closed in $CX$ because $p$ is a quotient map. But $p(B) cap i(X) = i(A)$.



        Case 2. $* notin A$. Then $B = A times [0,1/2] $ is closed in $X times I$. Since $X' cap B = emptyset$, we have $p^{-1}(p(B)) = B$ whence $p(B)$ is closed. Again $p(B) cap i(X) = i(A)$.



        Note, however, that $i(X)$ is closed in $CX$ if and only if ${ ast }$ is closed in $X$. In contrast, in the unreduced cone $i(X)$ is always closed.



        3.) Let $j : M(i) to X times I$ denote the unique map such that $j([x]) = (x,0)$ for $x in X$ and $j([a,t]) = (a,t)$ for $(a,t) in A times I$. It is always an injection whose image is $M' = j(M(i)) = X times { 0 } cup A times I$.
        Moreover, it is an embedding if (a) $A$ is closed or (b) $i : A to X$ is a cofibration. (a) is trivial. The highly nontrivial proof of (b) can be found in



        Strøm, Arne. "Note on cofibrations II." Mathematica Scandinavica 22.1 (1969): 130-142.



        See Lemma 3 of this paper and the comment following the proof. Note that there are cofibrations $i : A to X$ where $A$ is not closed,
        but only with non-Hausdorff $X$.



        If neither $A$ is closed nor $i$ is a cofibration, we get counterexamples like $X = I, A = [0,1)$.



        Question (2):



        I do not see a general method.







        share|cite|improve this answer














        share|cite|improve this answer



        share|cite|improve this answer








        edited Dec 21 '18 at 15:31

























        answered Dec 21 '18 at 15:12









        Paul FrostPaul Frost

        12.6k31035




        12.6k31035






























            draft saved

            draft discarded




















































            Thanks for contributing an answer to Mathematics Stack Exchange!


            • 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%2fmath.stackexchange.com%2fquestions%2f3048136%2fsome-questions-about-specific-topological-embeddings%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

            Plaza Victoria

            Puebla de Zaragoza

            Musa