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.