Cell Structure and Computing Homology Groups












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I am computing homology group of space $X=S^{2}cup T^{2}/sim$ such that $S^{2}$ and $T^{2}$ gluing by there points on 2-sphere $s_{1}, s_{2}, s_{3}$ and three points on 2-torus $t_{1}, t_{2}, t_{3}$ where $s_{i}sim t_{i}$. How can i found cell structure of this space and homology groups?



It looks like $S^{2}lor S^{1}lor S^{1}lor S^{1}lor S^{1}$ but I am not sure, maybe I should add one more $S^{2}$, I couldn't imagine it! I totally confused.



Thank you for your help!






algebraic-topology







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    $begingroup$


    I am computing homology group of space $X=S^{2}cup T^{2}/sim$ such that $S^{2}$ and $T^{2}$ gluing by there points on 2-sphere $s_{1}, s_{2}, s_{3}$ and three points on 2-torus $t_{1}, t_{2}, t_{3}$ where $s_{i}sim t_{i}$. How can i found cell structure of this space and homology groups?



    It looks like $S^{2}lor S^{1}lor S^{1}lor S^{1}lor S^{1}$ but I am not sure, maybe I should add one more $S^{2}$, I couldn't imagine it! I totally confused.



    Thank you for your help!






    algebraic-topology







    share|cite|improve this question











    $endgroup$















      0












      0








      0





      $begingroup$


      I am computing homology group of space $X=S^{2}cup T^{2}/sim$ such that $S^{2}$ and $T^{2}$ gluing by there points on 2-sphere $s_{1}, s_{2}, s_{3}$ and three points on 2-torus $t_{1}, t_{2}, t_{3}$ where $s_{i}sim t_{i}$. How can i found cell structure of this space and homology groups?



      It looks like $S^{2}lor S^{1}lor S^{1}lor S^{1}lor S^{1}$ but I am not sure, maybe I should add one more $S^{2}$, I couldn't imagine it! I totally confused.



      Thank you for your help!






      algebraic-topology







      share|cite|improve this question











      $endgroup$




      I am computing homology group of space $X=S^{2}cup T^{2}/sim$ such that $S^{2}$ and $T^{2}$ gluing by there points on 2-sphere $s_{1}, s_{2}, s_{3}$ and three points on 2-torus $t_{1}, t_{2}, t_{3}$ where $s_{i}sim t_{i}$. How can i found cell structure of this space and homology groups?



      It looks like $S^{2}lor S^{1}lor S^{1}lor S^{1}lor S^{1}$ but I am not sure, maybe I should add one more $S^{2}$, I couldn't imagine it! I totally confused.



      Thank you for your help!






      algebraic-topology




      algebraic-topology






      share|cite|improve this question















      share|cite|improve this question













      share|cite|improve this question




      share|cite|improve this question








      edited Nov 25 '18 at 17:59









      amWhy

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      asked Nov 25 '18 at 17:49









      user619499user619499

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          $begingroup$

          You can use the Mayer Vietoris sequence with $X_1 = S^2$, $X_2 = T^2$, $X_1 cap X_2 = {a, b, c}$ and $X = X_1 cup X_2$.






          share|cite|improve this answer









          $endgroup$













          • $begingroup$
            What about cell structure, how can I found it?
            $endgroup$
            – user619499
            Nov 25 '18 at 18:04








          • 1




            $begingroup$
            Try to find a way of realizing $S^2$ and $T^2$ with $3$ $0$-simplices each. With the torus you can start from the familiar square with sides identified, adding extra simplices on the edges.
            $endgroup$
            – Lukas Kofler
            Nov 25 '18 at 18:09










          • $begingroup$
            I am really sorry I couldn't do it. Can you explain more? Thank you.
            $endgroup$
            – user619499
            Nov 25 '18 at 18:18






          • 1




            $begingroup$
            For $S^2$ you can take a triangle with $2$ $2$-simplexes; a top and a bottom one. For the torus you could take a square with additional vertex in the middle of each edge: after identifying edges, you will be left with $3$ vertices.
            $endgroup$
            – Lukas Kofler
            Nov 25 '18 at 18:26











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          1 Answer
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          active

          oldest

          votes









          1












          $begingroup$

          You can use the Mayer Vietoris sequence with $X_1 = S^2$, $X_2 = T^2$, $X_1 cap X_2 = {a, b, c}$ and $X = X_1 cup X_2$.






          share|cite|improve this answer









          $endgroup$













          • $begingroup$
            What about cell structure, how can I found it?
            $endgroup$
            – user619499
            Nov 25 '18 at 18:04








          • 1




            $begingroup$
            Try to find a way of realizing $S^2$ and $T^2$ with $3$ $0$-simplices each. With the torus you can start from the familiar square with sides identified, adding extra simplices on the edges.
            $endgroup$
            – Lukas Kofler
            Nov 25 '18 at 18:09










          • $begingroup$
            I am really sorry I couldn't do it. Can you explain more? Thank you.
            $endgroup$
            – user619499
            Nov 25 '18 at 18:18






          • 1




            $begingroup$
            For $S^2$ you can take a triangle with $2$ $2$-simplexes; a top and a bottom one. For the torus you could take a square with additional vertex in the middle of each edge: after identifying edges, you will be left with $3$ vertices.
            $endgroup$
            – Lukas Kofler
            Nov 25 '18 at 18:26
















          1












          $begingroup$

          You can use the Mayer Vietoris sequence with $X_1 = S^2$, $X_2 = T^2$, $X_1 cap X_2 = {a, b, c}$ and $X = X_1 cup X_2$.






          share|cite|improve this answer









          $endgroup$













          • $begingroup$
            What about cell structure, how can I found it?
            $endgroup$
            – user619499
            Nov 25 '18 at 18:04








          • 1




            $begingroup$
            Try to find a way of realizing $S^2$ and $T^2$ with $3$ $0$-simplices each. With the torus you can start from the familiar square with sides identified, adding extra simplices on the edges.
            $endgroup$
            – Lukas Kofler
            Nov 25 '18 at 18:09










          • $begingroup$
            I am really sorry I couldn't do it. Can you explain more? Thank you.
            $endgroup$
            – user619499
            Nov 25 '18 at 18:18






          • 1




            $begingroup$
            For $S^2$ you can take a triangle with $2$ $2$-simplexes; a top and a bottom one. For the torus you could take a square with additional vertex in the middle of each edge: after identifying edges, you will be left with $3$ vertices.
            $endgroup$
            – Lukas Kofler
            Nov 25 '18 at 18:26














          1












          1








          1





          $begingroup$

          You can use the Mayer Vietoris sequence with $X_1 = S^2$, $X_2 = T^2$, $X_1 cap X_2 = {a, b, c}$ and $X = X_1 cup X_2$.






          share|cite|improve this answer









          $endgroup$



          You can use the Mayer Vietoris sequence with $X_1 = S^2$, $X_2 = T^2$, $X_1 cap X_2 = {a, b, c}$ and $X = X_1 cup X_2$.







          share|cite|improve this answer












          share|cite|improve this answer



          share|cite|improve this answer










          answered Nov 25 '18 at 17:57









          Lukas KoflerLukas Kofler

          1,2632519




          1,2632519












          • $begingroup$
            What about cell structure, how can I found it?
            $endgroup$
            – user619499
            Nov 25 '18 at 18:04








          • 1




            $begingroup$
            Try to find a way of realizing $S^2$ and $T^2$ with $3$ $0$-simplices each. With the torus you can start from the familiar square with sides identified, adding extra simplices on the edges.
            $endgroup$
            – Lukas Kofler
            Nov 25 '18 at 18:09










          • $begingroup$
            I am really sorry I couldn't do it. Can you explain more? Thank you.
            $endgroup$
            – user619499
            Nov 25 '18 at 18:18






          • 1




            $begingroup$
            For $S^2$ you can take a triangle with $2$ $2$-simplexes; a top and a bottom one. For the torus you could take a square with additional vertex in the middle of each edge: after identifying edges, you will be left with $3$ vertices.
            $endgroup$
            – Lukas Kofler
            Nov 25 '18 at 18:26


















          • $begingroup$
            What about cell structure, how can I found it?
            $endgroup$
            – user619499
            Nov 25 '18 at 18:04








          • 1




            $begingroup$
            Try to find a way of realizing $S^2$ and $T^2$ with $3$ $0$-simplices each. With the torus you can start from the familiar square with sides identified, adding extra simplices on the edges.
            $endgroup$
            – Lukas Kofler
            Nov 25 '18 at 18:09










          • $begingroup$
            I am really sorry I couldn't do it. Can you explain more? Thank you.
            $endgroup$
            – user619499
            Nov 25 '18 at 18:18






          • 1




            $begingroup$
            For $S^2$ you can take a triangle with $2$ $2$-simplexes; a top and a bottom one. For the torus you could take a square with additional vertex in the middle of each edge: after identifying edges, you will be left with $3$ vertices.
            $endgroup$
            – Lukas Kofler
            Nov 25 '18 at 18:26
















          $begingroup$
          What about cell structure, how can I found it?
          $endgroup$
          – user619499
          Nov 25 '18 at 18:04






          $begingroup$
          What about cell structure, how can I found it?
          $endgroup$
          – user619499
          Nov 25 '18 at 18:04






          1




          1




          $begingroup$
          Try to find a way of realizing $S^2$ and $T^2$ with $3$ $0$-simplices each. With the torus you can start from the familiar square with sides identified, adding extra simplices on the edges.
          $endgroup$
          – Lukas Kofler
          Nov 25 '18 at 18:09




          $begingroup$
          Try to find a way of realizing $S^2$ and $T^2$ with $3$ $0$-simplices each. With the torus you can start from the familiar square with sides identified, adding extra simplices on the edges.
          $endgroup$
          – Lukas Kofler
          Nov 25 '18 at 18:09












          $begingroup$
          I am really sorry I couldn't do it. Can you explain more? Thank you.
          $endgroup$
          – user619499
          Nov 25 '18 at 18:18




          $begingroup$
          I am really sorry I couldn't do it. Can you explain more? Thank you.
          $endgroup$
          – user619499
          Nov 25 '18 at 18:18




          1




          1




          $begingroup$
          For $S^2$ you can take a triangle with $2$ $2$-simplexes; a top and a bottom one. For the torus you could take a square with additional vertex in the middle of each edge: after identifying edges, you will be left with $3$ vertices.
          $endgroup$
          – Lukas Kofler
          Nov 25 '18 at 18:26




          $begingroup$
          For $S^2$ you can take a triangle with $2$ $2$-simplexes; a top and a bottom one. For the torus you could take a square with additional vertex in the middle of each edge: after identifying edges, you will be left with $3$ vertices.
          $endgroup$
          – Lukas Kofler
          Nov 25 '18 at 18:26


















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