Classification of surfacesconnected sum of torus with projective planeWhy can all surfaces with boundary be realized in $mathbbR^3$?2-cell embeddings of graphs in surfaces and Euler formulaAbout zeros of vector fields in compact surfacesClassification of orientable non-closed surfacesClosed, orientable surface whose genus is very hard to find intuitivelyNonorientable surfaces: genus or demigenus?Surface has Euler characteristic 2 iff equal to sphereClassification of surfaces theoremClassification of 2-dim topological manifold (not necessarily second countable)Connected sum of two non homeomorphic surfaces

bldc motor, esc and battery draw, nominal vs peak

Was there a shared-world project before "Thieves World"?

How do I deal with a coworker that keeps asking to make small superficial changes to a report, and it is seriously triggering my anxiety?

Who was the lone kid in the line of people at the lake at the end of Avengers: Endgame?

555 timer FM transmitter

Why didn't the Space Shuttle bounce back into space as many times as possible so as to lose a lot of kinetic energy up there?

Apply MapThread to all but one variable

Can I criticise the more senior developers around me for not writing clean code?

Mistake in years of experience in resume?

Pulling the rope with one hand is as heavy as with two hands?

What does ゆーか mean?

What are the steps to solving this definite integral?

Why does Mind Blank stop the Feeblemind spell?

Implications of cigar-shaped bodies having rings?

How to pronounce 'c++' in Spanish

How to display Aura JS Errors Lightning Out

Two field separators (colon and space) in awk

What is the most expensive material in the world that could be used to create Pun-Pun's lute?

Elements other than carbon that can form many different compounds by bonding to themselves?

On The Origin of Dissonant Chords

How to not starve gigantic beasts

Can an Area of Effect spell cast outside a Prismatic Wall extend inside it?

Can someone publish a story that happened to you?

can anyone help me with this awful query plan?



Classification of surfaces


connected sum of torus with projective planeWhy can all surfaces with boundary be realized in $mathbbR^3$?2-cell embeddings of graphs in surfaces and Euler formulaAbout zeros of vector fields in compact surfacesClassification of orientable non-closed surfacesClosed, orientable surface whose genus is very hard to find intuitivelyNonorientable surfaces: genus or demigenus?Surface has Euler characteristic 2 iff equal to sphereClassification of surfaces theoremClassification of 2-dim topological manifold (not necessarily second countable)Connected sum of two non homeomorphic surfaces













4












$begingroup$


The Classification Theorem for surfaces says that a compact connected surface $M$ is homeomorphic to $$S^2# (#_gT^2)# (#_b D^2)# (#_c mathbbRP^2),$$ so $g$ is the genus of the surface, $b$ the number of boundary components and $c$ the number of projective planes.



From there, it is easy to compute $chi(M)=2-2g-b-c$.



Nevertheless, I have read another statement of The Classification Theorem that states that a compact connected surface is determined by its orientability (yes/no), the number of boundary components and its Euler characteristic.



I do not understand how is it possible to know the decomposition of $M$ as a connected sum by knowing that. By knowing $b$, there are still two variables, $c$ and $g$ which have to be known from $chi(M)$, and orientability only tells us if $c=0$ or $cgeq 1$. Can someone help me, please?






manifolds surfaces orientation manifolds-with-boundary non-orientable-surfaces







share|cite|improve this question









$endgroup$











  • $begingroup$
    This can help you math.stackexchange.com/q/358724/654562
    $endgroup$
    – dcolazin
    2 hours ago















4












$begingroup$


The Classification Theorem for surfaces says that a compact connected surface $M$ is homeomorphic to $$S^2# (#_gT^2)# (#_b D^2)# (#_c mathbbRP^2),$$ so $g$ is the genus of the surface, $b$ the number of boundary components and $c$ the number of projective planes.



From there, it is easy to compute $chi(M)=2-2g-b-c$.



Nevertheless, I have read another statement of The Classification Theorem that states that a compact connected surface is determined by its orientability (yes/no), the number of boundary components and its Euler characteristic.



I do not understand how is it possible to know the decomposition of $M$ as a connected sum by knowing that. By knowing $b$, there are still two variables, $c$ and $g$ which have to be known from $chi(M)$, and orientability only tells us if $c=0$ or $cgeq 1$. Can someone help me, please?






manifolds surfaces orientation manifolds-with-boundary non-orientable-surfaces







share|cite|improve this question









$endgroup$











  • $begingroup$
    This can help you math.stackexchange.com/q/358724/654562
    $endgroup$
    – dcolazin
    2 hours ago













4












4








4





$begingroup$


The Classification Theorem for surfaces says that a compact connected surface $M$ is homeomorphic to $$S^2# (#_gT^2)# (#_b D^2)# (#_c mathbbRP^2),$$ so $g$ is the genus of the surface, $b$ the number of boundary components and $c$ the number of projective planes.



From there, it is easy to compute $chi(M)=2-2g-b-c$.



Nevertheless, I have read another statement of The Classification Theorem that states that a compact connected surface is determined by its orientability (yes/no), the number of boundary components and its Euler characteristic.



I do not understand how is it possible to know the decomposition of $M$ as a connected sum by knowing that. By knowing $b$, there are still two variables, $c$ and $g$ which have to be known from $chi(M)$, and orientability only tells us if $c=0$ or $cgeq 1$. Can someone help me, please?






manifolds surfaces orientation manifolds-with-boundary non-orientable-surfaces







share|cite|improve this question









$endgroup$




The Classification Theorem for surfaces says that a compact connected surface $M$ is homeomorphic to $$S^2# (#_gT^2)# (#_b D^2)# (#_c mathbbRP^2),$$ so $g$ is the genus of the surface, $b$ the number of boundary components and $c$ the number of projective planes.



From there, it is easy to compute $chi(M)=2-2g-b-c$.



Nevertheless, I have read another statement of The Classification Theorem that states that a compact connected surface is determined by its orientability (yes/no), the number of boundary components and its Euler characteristic.



I do not understand how is it possible to know the decomposition of $M$ as a connected sum by knowing that. By knowing $b$, there are still two variables, $c$ and $g$ which have to be known from $chi(M)$, and orientability only tells us if $c=0$ or $cgeq 1$. Can someone help me, please?






manifolds surfaces orientation manifolds-with-boundary non-orientable-surfaces




manifolds surfaces orientation manifolds-with-boundary non-orientable-surfaces






share|cite|improve this question













share|cite|improve this question











share|cite|improve this question




share|cite|improve this question










asked 3 hours ago









KarenKaren

1336




1336











  • $begingroup$
    This can help you math.stackexchange.com/q/358724/654562
    $endgroup$
    – dcolazin
    2 hours ago
















  • $begingroup$
    This can help you math.stackexchange.com/q/358724/654562
    $endgroup$
    – dcolazin
    2 hours ago















$begingroup$
This can help you math.stackexchange.com/q/358724/654562
$endgroup$
– dcolazin
2 hours ago




$begingroup$
This can help you math.stackexchange.com/q/358724/654562
$endgroup$
– dcolazin
2 hours ago










2 Answers
2






active

oldest

votes


















2












$begingroup$

If I read correctly you are trying to determine $c$ and $g$, given a compact connected surface $M$ for which you know $b$ the number of boundary components, $chi(M)$ the Euler characteristic the and whether or not $M$ is orientable.



This can't be done in a unique manner, as the connected sum of a tori and a projective plan is homeomorphic to the connected sum of three projective planes.



Please correct me if I misunderstood your question.






share|cite|improve this answer









$endgroup$




















    2












    $begingroup$

    Since $mathbbRP^2# mathbbRP^2#mathbbRP^2cong mathbbRP^2 # T^2$, $c$ and $g$ are not uniquely determined: if $cgeq 3$, you can subtract $2$ from $c$ and add $1$ to $g$ and get the same surface, or if $c,ggeq 1$, you can subtract $1$ from $g$ and add $2$ to $c$.



    Note, though, that the first operation can always be used to get a connected sum presentation where $cleq 2$. If you impose the additional restriction that $cleq 2$, then $c$ and $g$ can be uniquely determined and can be calculated from the data you mention. If the surface is orientable, then $c=0$ and then you can just solve for $g$. If the surface is not orientable, then you can determine whether $c=1$ or $c=2$ since $c$ must have the same parity as $chi(M)+b$. Once $c$ is determined, you can solve for $g$.






    share|cite|improve this answer









    $endgroup$













      Your Answer








      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%2f3203841%2fclassification-of-surfaces%23new-answer', 'question_page');

      );

      Post as a guest















      Required, but never shown

























      2 Answers
      2






      active

      oldest

      votes








      2 Answers
      2






      active

      oldest

      votes









      active

      oldest

      votes






      active

      oldest

      votes









      2












      $begingroup$

      If I read correctly you are trying to determine $c$ and $g$, given a compact connected surface $M$ for which you know $b$ the number of boundary components, $chi(M)$ the Euler characteristic the and whether or not $M$ is orientable.



      This can't be done in a unique manner, as the connected sum of a tori and a projective plan is homeomorphic to the connected sum of three projective planes.



      Please correct me if I misunderstood your question.






      share|cite|improve this answer









      $endgroup$

















        2












        $begingroup$

        If I read correctly you are trying to determine $c$ and $g$, given a compact connected surface $M$ for which you know $b$ the number of boundary components, $chi(M)$ the Euler characteristic the and whether or not $M$ is orientable.



        This can't be done in a unique manner, as the connected sum of a tori and a projective plan is homeomorphic to the connected sum of three projective planes.



        Please correct me if I misunderstood your question.






        share|cite|improve this answer









        $endgroup$















          2












          2








          2





          $begingroup$

          If I read correctly you are trying to determine $c$ and $g$, given a compact connected surface $M$ for which you know $b$ the number of boundary components, $chi(M)$ the Euler characteristic the and whether or not $M$ is orientable.



          This can't be done in a unique manner, as the connected sum of a tori and a projective plan is homeomorphic to the connected sum of three projective planes.



          Please correct me if I misunderstood your question.






          share|cite|improve this answer









          $endgroup$



          If I read correctly you are trying to determine $c$ and $g$, given a compact connected surface $M$ for which you know $b$ the number of boundary components, $chi(M)$ the Euler characteristic the and whether or not $M$ is orientable.



          This can't be done in a unique manner, as the connected sum of a tori and a projective plan is homeomorphic to the connected sum of three projective planes.



          Please correct me if I misunderstood your question.







          share|cite|improve this answer












          share|cite|improve this answer



          share|cite|improve this answer










          answered 2 hours ago









          Adam ChalumeauAdam Chalumeau

          48010




          48010





















              2












              $begingroup$

              Since $mathbbRP^2# mathbbRP^2#mathbbRP^2cong mathbbRP^2 # T^2$, $c$ and $g$ are not uniquely determined: if $cgeq 3$, you can subtract $2$ from $c$ and add $1$ to $g$ and get the same surface, or if $c,ggeq 1$, you can subtract $1$ from $g$ and add $2$ to $c$.



              Note, though, that the first operation can always be used to get a connected sum presentation where $cleq 2$. If you impose the additional restriction that $cleq 2$, then $c$ and $g$ can be uniquely determined and can be calculated from the data you mention. If the surface is orientable, then $c=0$ and then you can just solve for $g$. If the surface is not orientable, then you can determine whether $c=1$ or $c=2$ since $c$ must have the same parity as $chi(M)+b$. Once $c$ is determined, you can solve for $g$.






              share|cite|improve this answer









              $endgroup$

















                2












                $begingroup$

                Since $mathbbRP^2# mathbbRP^2#mathbbRP^2cong mathbbRP^2 # T^2$, $c$ and $g$ are not uniquely determined: if $cgeq 3$, you can subtract $2$ from $c$ and add $1$ to $g$ and get the same surface, or if $c,ggeq 1$, you can subtract $1$ from $g$ and add $2$ to $c$.



                Note, though, that the first operation can always be used to get a connected sum presentation where $cleq 2$. If you impose the additional restriction that $cleq 2$, then $c$ and $g$ can be uniquely determined and can be calculated from the data you mention. If the surface is orientable, then $c=0$ and then you can just solve for $g$. If the surface is not orientable, then you can determine whether $c=1$ or $c=2$ since $c$ must have the same parity as $chi(M)+b$. Once $c$ is determined, you can solve for $g$.






                share|cite|improve this answer









                $endgroup$















                  2












                  2








                  2





                  $begingroup$

                  Since $mathbbRP^2# mathbbRP^2#mathbbRP^2cong mathbbRP^2 # T^2$, $c$ and $g$ are not uniquely determined: if $cgeq 3$, you can subtract $2$ from $c$ and add $1$ to $g$ and get the same surface, or if $c,ggeq 1$, you can subtract $1$ from $g$ and add $2$ to $c$.



                  Note, though, that the first operation can always be used to get a connected sum presentation where $cleq 2$. If you impose the additional restriction that $cleq 2$, then $c$ and $g$ can be uniquely determined and can be calculated from the data you mention. If the surface is orientable, then $c=0$ and then you can just solve for $g$. If the surface is not orientable, then you can determine whether $c=1$ or $c=2$ since $c$ must have the same parity as $chi(M)+b$. Once $c$ is determined, you can solve for $g$.






                  share|cite|improve this answer









                  $endgroup$



                  Since $mathbbRP^2# mathbbRP^2#mathbbRP^2cong mathbbRP^2 # T^2$, $c$ and $g$ are not uniquely determined: if $cgeq 3$, you can subtract $2$ from $c$ and add $1$ to $g$ and get the same surface, or if $c,ggeq 1$, you can subtract $1$ from $g$ and add $2$ to $c$.



                  Note, though, that the first operation can always be used to get a connected sum presentation where $cleq 2$. If you impose the additional restriction that $cleq 2$, then $c$ and $g$ can be uniquely determined and can be calculated from the data you mention. If the surface is orientable, then $c=0$ and then you can just solve for $g$. If the surface is not orientable, then you can determine whether $c=1$ or $c=2$ since $c$ must have the same parity as $chi(M)+b$. Once $c$ is determined, you can solve for $g$.







                  share|cite|improve this answer












                  share|cite|improve this answer



                  share|cite|improve this answer










                  answered 2 hours ago









                  Eric WofseyEric Wofsey

                  194k14223354




                  194k14223354



























                      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%2f3203841%2fclassification-of-surfaces%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

                      Magento 2 duplicate PHPSESSID cookie when using session_start() in custom php scriptMagento 2: User cant logged in into to account page, no error showing!Magento duplicate on subdomainGrabbing storeview from cookie (after using language selector)How do I run php custom script on magento2Magento 2: Include PHP script in headerSession lock after using Cm_RedisSessionscript php to update stockMagento set cookie popupMagento 2 session id cookie - where to find it?How to import Configurable product from csv with custom attributes using php scriptMagento 2 run custom PHP script

                      Image
                      In the late 1940’s to early 1950’s what technology was available that could melt a LOT of ice? "One can do his homework in the library" Setting correct UTM zone Time travel short story where dinosaur doesn't taste like chicken Do Bugbears' arms literally get longer when it's their turn? Offered promotion but I'm leaving. Should I tell? Subset counting for even numbers The difference between 時　and 時は String reversal in Python Good allowance savings plan? Do f-stop and exposure time perfectly cancel? What are some noteworthy "mic-drop" moments in math? They call me Inspector Morse Do I really need to have a scientific explanation for my premise? Make a transparent 448*448 image Could a cubesat propel itself to Mars? What is the chance of making a successful appeal to dismissal decision from a PhD program after failing the qualifying exam in the 2nd attempt? Could a cubesat be propelled to the moon? the return of Stri
                      Read more

                      Can not update quote_id field of “quote_item” table magento 2Magento 2.1 - We can't remove the item. (Shopping Cart doesnt allow us to remove items before becomes empty)Add value for custom quote item attribute using REST apiREST API endpoint v1/carts/cartId/items always returns error messageCorrect way to save entries to databaseHow to remove all associated quote objects of a customer completelyMagento 2 - Save value from custom input field to quote_itemGet quote_item data using quote id and product id filter in Magento 2How to set additional data to quote_item table from controller in Magento 2?What is the purpose of additional_data column in quote_item table in magento2Set Custom Price to Quote item magento2 from controller

                      Image
                      iPad being using in wall mount battery swollen Should I cover my bicycle overnight while bikepacking? Do UK voters know if their MP will be the Speaker of the House? Why can't we play rap on piano? Plagiarism or not? Why didn't Miles's spider sense work before? How to tell a function to use the default argument values? Short story with a alien planet, government officials must wear exploding medallions If human space travel is limited by the G force vulnerability, is there a way to counter G forces? What about the virus in 12 Monkeys? Is it acceptable for a professor to tell male students to not think that they are smarter than female students? Avoiding the "not like other girls" trope? Apex Framework / library for consuming REST services Is there an expression that means doing something right before you will need it rather than doing it in case you might need it? Method Does Not Exist error message How to show a landlord what we have
                      Read more

                      How to solve knockout JS error in Magento 2 Planned maintenance scheduled April 23, 2019 at 23:30 UTC (7:30pm US/Eastern) Announcing the arrival of Valued Associate #679: Cesar Manara Unicorn Meta Zoo #1: Why another podcast?(Magento2) knockout.js:3012 Uncaught ReferenceError: Unable to process bindingUnable to process binding Knockout.js magento 2Cannot read property `scopeLabel` of undefined on Product Detail PageCan't get Customer Data on frontend in Magento 2Magento2 Order Summary - unable to process bindingKO templates are not loading in Magento 2.1 applicationgetting knockout js error magento 2Product grid not load -— Unable to process binding Knockout.js magento 2Product form not loaded in magento2Uncaught ReferenceError: Unable to process binding “if: function()return (isShowLegend()) ” magento 2

                      Image
                      Does the Pact of the Blade warlock feature allow me to customize the properties of the pact weapon I create? Why isn't everyone flabbergasted about Bran's "gift"? Paektu / Changbai border dispute — why does the BBC shade a large area around the mountain? Is it OK if I do not take the receipt in Germany? How to ask rejected full-time candidates to apply to teach individual courses? /bin/ls sorts differently than just ls What *exactly* is electrical current, voltage, and resistance? Coin Game with infinite paradox Diffie-Hellman with non-prime modulus Example of a central simple algebra Etymology of 見舞い Lights are flickering on and off after accidentally bumping into light switch A journey... into the MIND How to calculate density of unknown planet? How to produce a PS1 prompt in bash or ksh93 similar to tcsh What's the connection between Mr. Nancy and fried chicken? How can I delete rows in the text? How is it possible to implement
                      Read more