Torus - Singh, Example 2.2.5 - Baffled by certain aspects

  • #1
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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
 

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  • #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.
 
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  • #3
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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
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.

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
 
  • #4
lavinia
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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
yes he could have defined C_1 as the bottom of the torus.
 
  • #5
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Thanks Lavinia ... Appreciate the help ...

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

Peter
 
  • #6
lavinia
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Thanks Lavinia ... Appreciate the help ...

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

Peter
yes. sorry
 
  • #7
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Thanks again Lavinia ... most helpful ...

Peter
 

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