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Torus - Singh, Example 2.2.5 - Baffled by certain aspects

  1. Apr 17, 2015 #1
    I am baffled by some aspects of the torus ... I hope someone can help ...

    I am puzzled by some aspects of Singh's treatment of the torus in Example 2.2.5 ( Tej Bahadur Singh: Elements of Topology, CRC Press, 2013) ... ...

    Singh's Example 2.2.5 reads as follows:

    ?temp_hash=1d6597f8eee43d0cd15ffc346eef67d4.png

    My questions related to the above example of Singh's are as follows:


    Question 1

    [itex]C_1[/itex] as defined above seems to me to be a circle at 'height' [itex]z = 1[/itex], around the [itex]z[/itex]-axis ... why (for what reason?) did Singh choose [itex]C_1[/itex] to be at 'height' [itex]z = 1[/itex]?

    Why not choose [itex]C_1[/itex] as [itex]\{ (x,y, 0) \ | \ x^2 + y^2 = 4 \}[/itex]?

    Such a choice seems more natural if you are rotating [itex]C_2[/itex] around the [itex]z[/itex]-axis, since [itex]C_1[/itex] is at level [itex]z = 0[/itex] ... ...


    Question 2

    [itex]h \ : \ T \rightarrow C_1 \times C_2[/itex]

    maps [itex](x,y,z)[/itex] onto two three dimensional points in Euclidean [itex]3[/itex]-space and so essentially maps [itex](x,y,z)[/itex] into Euclidean [itex]6[/itex]-space ... ..

    ... BUT ...

    [itex]T[/itex] is homeomorphic to [itex]S^1 \times S^1[/itex] which is embedded in Euclidean [itex]4[/itex]-space ... ... how can this be ...


    I hope someone can clarify the above issues/questions ...

    Peter
     

    Attached Files:

    Last edited: Apr 17, 2015
  2. jcsd
  3. Apr 18, 2015 #2

    lavinia

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    Question 2: C1 and C2 are both circles in R^3 so the product is embedded in R^6.

    Question 1. The top of the torus,T, is a circle at height of 1 above the xy-plane. and has radius 2
    The circle C2 is the same circle.
     
    Last edited: Apr 18, 2015
  4. Apr 19, 2015 #3

    Thanks so much for your help Lavinia ... ...

    ... ... BUT .... for Question 1 ... do you mean "The circle C1 is the same circle"


    and ... if it is supposed to be C_1, then could Singh have alternatively, defined C_1 as

    [itex]C_1[/itex] as [itex]\{ (x,y, -1) \ | \ x^2 + y^2 = 4 \}[/itex] ...

    as this seems to define the bottom of the torus ...

    Hope you can help further ... ...

    Peter
     
  5. Apr 20, 2015 #4

    lavinia

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    yes he could have defined C_1 as the bottom of the torus.
     
  6. Apr 20, 2015 #5
    Thanks Lavinia ... Appreciate the help ...

    Just quickly ... In your answer to question 1, did you mean: The Circle C1 is the same circle?

    Peter
     
  7. Apr 20, 2015 #6

    lavinia

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    yes. sorry
     
  8. Apr 20, 2015 #7
    Thanks again Lavinia ... most helpful ...

    Peter
     
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