Re: Very basic topology question
James wrote:
I am approached with some language and pictures in my textbook (Bredon's
Topology and Geometry) and they are never defined or explained. What is the
rigorous definition of "a square with opposite edges identified"? There are
two pictures in my book :
(1) A square whose right and left sides have arrows pointing upwards, and
whose top and bottom sides have arrows pointing rightwards
(2) A square whose right and left sides have arrows pointing upwards, and
whose top and bottom sides have arrows pointing leftwards
The first one is supposed to be the torus, the second the Klein bottle.
The trouble is I don't know what these arrows are supposed to mean. If I
take a square *** of paper, roll it up to get a cylinder, then attach the
ends of the cylinder to get a torus, I can see that I am identifying points,
but what does that have to do with the "arrows" I mention above? I guess
I'm just not sure what the arrows mean. What if the right and left sides
have arrows pointing downwards instead of upwards? What if one is up and
one is down?
The idea is this: if both arrows point upwards, you're identifying the
bottom point of the left side of the square with the bottom point of the
right side, you're identifying the top point of the left side with the
topo point of the right side and, more generally, you identify each
point of the left side with the point of the right side such that the
line that unites both points is parallel to the sides that have not
been mentioned yet.
If both arrows were pointing downwards, it would be the same thing.
If the left arrow points upwards and the right arrow points downwards,
then you're identifying each point of the left side with the point of
the right side which is obtained if you apply a rotation of pi radians
of the original point around the center of the square. In particular
the bottom point (respectively the top point) of the left side is
identified with the top point (resp. the bottom point) of the right
side.
Best regards,
Jose Carlos Santos
.
