# Torus - Singh, Example 2.2.5 - Baffled by certain aspects

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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:

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

Question 1

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

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

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

Question 2

$h \ : \ T \rightarrow C_1 \times C_2$

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

... BUT ...

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

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

Peter

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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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Gold Member
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:

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

Question 1

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

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

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

Question 2

$h \ : \ T \rightarrow C_1 \times C_2$

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

... BUT ...

$T$ is homeomorphic to $S^1 \times S^1$ which is embedded in Euclidean $4$-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

$C_1$ as $\{ (x,y, -1) \ | \ x^2 + y^2 = 4 \}$ ...

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

Hope you can help further ... ...

Peter

lavinia
Gold Member
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

$C_1$ as $\{ (x,y, -1) \ | \ x^2 + y^2 = 4 \}$ ...

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.

Gold Member
Thanks Lavinia ... Appreciate the help ...

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

Peter

lavinia
Gold Member
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

Gold Member
Thanks again Lavinia ... most helpful ...

Peter