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A Torus with two Möbius Strips

  1. Dec 16, 2016 #1
    Another zany homology question:

    Just let me know if I have my labeling scheme right so far.

    I have a torus. I have cut two holes into it and am attaching Möbius strips around the holes. (Clearly we are not in 3 dimensions?)

    My Torus is represented by a polygon with the labeling scheme ## a b a^{-1}b^{-1} ## and my strips are labeled ## a b c b ## as follows (I'll subscript them after pasting)

    Torusmobius.jpg

    I'll subscript the torus with 1s, and the 2 mobius strips will be subscripted with 2s and 3s.

    I think I cut a hole in the Torus just by cutting a corner off of the polygon. I'll attach the strips to attain the following monstrosity:

    torusmobius2.jpg


    I end up with a long labeling scheme:

    ##a_1 b_1 c_1 b_2 b_1 b_3 c_3 b_3 a_1^{-1} b_1^{-1} ##

    Before I say any further, is this what cutting a hole in a Torus and gluing a Möbius band even would look like? Obviously I can't picture this outside the polygons here. Thanks for your patience.

    -Dave K
     
  2. jcsd
  3. Dec 16, 2016 #2
    "Before I say any further, is this what cutting a hole in a Torus and gluing a Möbius band even would look like?"

    Do you mean "a" Möbius band, or "two" of them?
     
  4. Dec 16, 2016 #3
    Two.
     
  5. Dec 19, 2016 #4
    Sorry, that's not what I end up with. I get:


    ##a_1 b_2 c_2 b_2 b_1 b_3 c_3 b_3 a_1^{-1} b_1^{-1} ##

    Still, if anyone can let me know if I have even this preliminary step right I would appreciate it.

    Might as well keep going.

    When I relabel I get:

    ## a b c d a^{-1}b^{-1} ##

    Which still isn't something I know from the classification theorem.
     
  6. Dec 20, 2016 #5

    lavinia

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    I don't see how you diagram accomplishes what you want. For instance, once you cut the corner s off the identification of ##b1## no longer makes sense. Can you explain this better?
     
  7. Dec 22, 2016 #6
    Sorry, I didn't realize there'd been an update to this thread. Thought I was going to have to go to stackexchange. ::shudder::

    I admit that I am confused as to how to properly make this hole with regards to the polygonal region. I will revisit.

    -Dave K
     
  8. Dec 22, 2016 #7
    Nope. Don't know how to do it. I am thinking that cutting a hole in a torus equates to cutting a corner off the torus, which is really like adding a side. i.e. when Munkres glues two tori together:

    upload_2016-12-22_14-1-8.png

    So why doesn't my ## b_1 ## make sense?
     
  9. Dec 22, 2016 #8

    lavinia

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    It might make sense. I found it confusing because you cut off part of one ##b_{1}## that is identified to the other. So where does the identification go?
     
  10. Dec 22, 2016 #9
    Of course in a regular torus, the ##b_1## would go back to the other ##b_1##, The ##a_1## similarly. The Mobius strips would be joined up in their usual way, though I don't know how this effects the overall "shape". I certainly can't visualize it! But at least according to the question, this thing should be able to be classified, so it's an n-fold torus, n-fold projective plane or a sphere, although it certainly looks like something much weirder to me.

    -Dave K
     
  11. Dec 23, 2016 #10

    lavinia

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    If you mean the diagram for a torus with a hole in it that you showed here in post 7 then nothing is wrong with ##b_1##.

    The curve ##c## is a closed loop and you have attached a Mobius band to it using a half the boundary of the band which is not a closed loop. So the band gets folded and is no longer a surface with a boundary. Is that what you want?

    One thing that can be done is to attach the entire boundary of the Mobius band to the curve ##c##. This lines up two boundary circles so that the resulting glued together space is a closed surface without boundary. Can you redraw this space to make a polygon with edges identified?

    I think by Van Kampen's Theorem that the relation will be ##aba^{-1}b^{-1}c^{-2}##.
     
    Last edited: Dec 23, 2016
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