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Proof of V-E+F=2-2p (Euler's 2nd formula).

  1. Oct 18, 2011 #1
    Hi. This post is about general topology and is Euler's second formula . Can you people help me by finding a proof for V-E+F=2-2p. :smile:
    p=genus of the surface.
    F=number of regions the surface is partitioned into.
    V=number of vertices.
    E=number of arcs.

    I'm currently reading this proof from a book 'Introduction to Modern Mathematics' by S.M Maskey(you won't be finding this book in the internet, we don't have a system of buying books from the internet in our country and most of the people are too lazy to make it into an ebook.And these books are just dumbed down(to simle english) versions of other good books ).Thus, this book has vague descriptions regarding the proving process. :mad:

    The proof here talks about considering spheres with p handles and removing them again.and i am not understanding the process written herein this book.

    Any links of this proof will be appreciated, or we could discuss . :approve:
  2. jcsd
  3. Oct 21, 2011 #2


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    V-E+F is the Euler characteristic of the surface. So is 2 - 2p
  4. Oct 21, 2011 #3
    Take a look at the wikipedia article for Euler Characteristic: http://en.wikipedia.org/wiki/Euler_characteristic

    Take a look at the wikipedia article on Handle: http://en.wikipedia.org/wiki/Handle_(mathematics)

    After reading through these pages click on the links for which you don't know what the term is, try not to digress too far from what it is you are actually trying to learn and then once you have a bit better of an understanding try re-reading your textbook and it might make more sense. In some situations I thought that a textbook had a bad explanation or was unclear, but it was simply a lack of the underlying meaning/geometric representation of some of the terminology that was holding me back from seeing the full picture.
  5. Oct 22, 2011 #4


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    To get you started.

    A sphere has no handles so p = 0.

    Triangulate the sphere as a tetrahedron. V - E + F = 2.

    So the theorem is true in this case.

    Try proving the rest by induction adding one handle at a time.
    Last edited: Oct 22, 2011
  6. Nov 2, 2011 #5
    Or, by removing one handle at a time. If you have a handle, you can cut it and cap the two boundary components you get with disks. (That is, you replace a cylinder S1 × [0,1] with two disks D2 × {0,1}.) This will reduce the genus by 1, and increase the Euler characteristic by 2. Repeat until you get no handles; you are left with a 2-sphere, which has genus 0 and Euler characteristic 2.

    Note that this formula is only true for closed surfaces. If your surface has genus g, b boundary components and r punctures, then the Euler characteristic is 2 - 2g - b - r.
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