The topology of the torus
The torus is officially
T
=S
1
x S
1
={(u,v,x,y) in
R
4
| u
2
+ v
2
= x
2
+ y
2
= 1}
={(cos(
q
), sin(
q
), cos(
f
), sin(
f
)) |
q
,
f
in
R
}
This picture illustrates the map f:T->
R
3
given by
f(u,v,x,y)
= (2u,2v,0) + x(u,v,0) + y(0,0,1)
=((2+x)u, (2+x)v, y)
The point P in the diagram is
f(cos(
q
), sin(
q
), cos(
f
), sin(
f
)),
where
q
is the angle shown in yellow, and
f
is the angle shown in green.
This animation shows how the torus can be constructed by gluing together opposite edges of a square.
This animation shows that the torus with one point removed is homotopy equivalent to a figure eight.
If we collapse the figure eight down to a point, then the resulting space is homemorphic to the sphere S
2
This animation shows that the loop
aba
-1
b
-1
is nullhomotopic, illustrating the fact that the fundamental group of the torus is abelian.