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From: Hodol on 11 Jan 2010 10:46
Hi..
I'm studying algebraic topology.
I know what is fundamental group, and what is free group.

And I know the fact that the fundamental group of torus is ZxZ(direct
sum of two cyclic groups, this is abelian!), and the fundamental group
of figure8 is Z*Z (free group with two generators, this is not
abelian).

However, I searching for what is the geomerical meaning of differences
between those two fundamental group.


Could somebody show me the light?? please, precisely.
Thanks. have a good time!


From: James Dolan on 12 Jan 2010 12:00
in article <ebc11d85-d498-40cf-9db9-4eb61cb6905d(a)f5g2000yqh.googlegroups.com>,
hodol <skyhos(a)gmail.com> wrote:

|Hi..
|I'm studying algebraic topology.
|I know what is fundamental group, and what is free group.
|
|And I know the fact that the fundamental group of torus is ZxZ(direct
|sum of two cyclic groups, this is abelian!), and the fundamental
|group of figure8 is Z*Z (free group with two generators, this is not
|abelian).
|
|However, I searching for what is the geomerical meaning of
|differences between those two fundamental group.
|
|
|Could somebody show me the light?? please, precisely.
|Thanks. have a good time!

i don't know what you're looking for- i don't know what you mean by
"geometric" here.

consider an infinite doubly periodic grid of horizontal and vertical
lines in the plane:

| | | |
-@-@-@-@-
| | | |
-@-@-@-@-
| | | |
-@-@-@-@-
| | | |
-@-@-@-@-
| | | |

(use a fixed font when viewing the picture.)

when you wrap the plane up into a torus so that all the @'s represent
the same point, then the grid lines form a figure-8 lying on the
torus. the original plane is the universal unwrapping (also known as
the "universal covering space") of the torus, which implies that you
can identify the fundamental group zXz of the torus with the @'s in
the picture. but consider two different paths for getting from one @
to another while sticking to the grid lines (that is, while staying on
the figure-8); for example "move right, then move down":

| | | |
-@-@-@-@-
| | | |
-@-***-@-
| | * |
-@-@-*-@-
| | | |
-@-@-@-@-
| | | |


versus "move down, then move right":

| | | |
-@-@-@-@-
| | | |
-@-*-@-@-
| * | |
-@-***-@-
| | | |
-@-@-@-@-
| | | |

although these two paths are homotopic on the torus they're not
homotopic on the figure-8 (because then you have to stick to the grid
lines without crossing the empty no-man's-land in between). so
whereas the fundamental group of the torus is the free commutative
group on the generators "move right" and "move down", in the
fundamental group of the figure-8 the generators "move right" and
"move down" fail to commute with each other, and so you get the
ordinary free group instead of the free commutative group.


--


jdolan(a)math.ucr.edu

From: Hodol on 12 Jan 2010 20:40
On 1¿ù13ÀÏ, ¿ÀÀü2½Ã00ºÐ, jdo...(a)math.UUCP (James Dolan) wrote:
> in article <ebc11d85-d498-40cf-9db9-4eb61cb69...(a)f5g2000yqh.googlegroups.com>,
>
> hodol <sky...(a)gmail.com> wrote:
>
> |Hi..
> |I'm studying algebraic topology.
> |I know what is fundamental group, and what is free group.
> |
> |And I know the fact that the fundamental group of torus is ZxZ(direct
> |sum of two cyclic groups, this is abelian!), and the fundamental
> |group of figure8 is Z*Z (free group with two generators, this is not
> |abelian).
> |
> |However, I searching for what is the geomerical meaning of
> |differences between those two fundamental group.
> |
> |
> |Could somebody show me the light?? please, precisely.
> |Thanks. have a good time!
>
> i don't know what you're looking for- i don't know what you mean by
> "geometric" here.
>
> consider an infinite doubly periodic grid of horizontal and vertical
> lines in the plane:
>
> | | | |
> -@-@-@-@-
> | | | |
> -@-@-@-@-
> | | | |
> -@-@-@-@-
> | | | |
> -@-@-@-@-
> | | | |
>
> (use a fixed font when viewing the picture.)
>
> when you wrap the plane up into a torus so that all the @'s represent
> the same point, then the grid lines form a figure-8 lying on the
> torus. the original plane is the universal unwrapping (also known as
> the "universal covering space") of the torus, which implies that you
> can identify the fundamental group zXz of the torus with the @'s in
> the picture. but consider two different paths for getting from one @
> to another while sticking to the grid lines (that is, while staying on
> the figure-8); for example "move right, then move down":
>
> | | | |
> -@-@-@-@-
> | | | |
> -@-***-@-
> | | * |
> -@-@-*-@-
> | | | |
> -@-@-@-@-
> | | | |
>
> versus "move down, then move right":
>
> | | | |
> -@-@-@-@-
> | | | |
> -@-*-@-@-
> | * | |
> -@-***-@-
> | | | |
> -@-@-@-@-
> | | | |
>
> although these two paths are homotopic on the torus they're not
> homotopic on the figure-8 (because then you have to stick to the grid
> lines without crossing the empty no-man's-land in between). so
> whereas the fundamental group of the torus is the free commutative
> group on the generators "move right" and "move down", in the
> fundamental group of the figure-8 the generators "move right" and
> "move down" fail to commute with each other, and so you get the
> ordinary free group instead of the free commutative group.
>
> --
>
> jdo...(a)math.ucr.edu

Oh! THANK YOU VERY MUCH!
Your answer is exactly what I am looking for.
Very HELPFUL!!

However, I wonder how to show that two generator loops in figure-8,
say [a],[b] is not commutative. ([] means homotopy class.)
I mean,

show that [a]*[b] != [b]*[a] in geometrical language (Not
algebraically...)

In torus, it's enough to show that there exists a homotopy F such that
F(*,0) = [a]*[b] and F(*,1) = [b]*[a].
This F give me a movie(?) : [a]*[b] varys contiuously [b]*[a].(Like
your answer!)

But in figure-8, I have to show that NOT exists any homotopy [a]*[b]
to [b]*[a].
Could you help me?



....with Thanks again for your great answer!
From: victor_meldrew_666 on 13 Jan 2010 06:16
On 13 Jan, 01:40, Hodol <sky...(a)gmail.com> wrote:

> However, I wonder how to show that two generator loops in figure-8,
> say [a],[b] is not commutative. ([] means homotopy class.)
> I mean,
>
> show that [a]*[b] != [b]*[a] in geometrical language (Not
> algebraically...)

The standard way of doing this is to look for a covering
space of the figure 8 in which the loops [a][b] and [b][a]
lift to paths with the same initial point but different
endpoints. The universal covering space, which is the
"free quadrivalent graph" certainly works, but there will be
be others too. Indeed there will be a covering space with
finitely many sheets that works....
From: Hodol on 15 Jan 2010 07:39
On 1¿ù13ÀÏ, ¿ÀÈÄ8½Ã16ºÐ, "victor_meldrew_...(a)yahoo.co.uk"
<victor_meldrew_...(a)yahoo.co.uk> wrote:
> On 13 Jan, 01:40, Hodol <sky...(a)gmail.com> wrote:
>
> > However, I wonder how to show that two generator loops in figure-8,
> > say [a],[b] is not commutative. ([] means homotopy class.)
> > I mean,
>
> > show that [a]*[b] != [b]*[a] in geometrical language (Not
> > algebraically...)
>
> The standard way of doing this is to look for a covering
> space of the figure 8 in which the loops [a][b] and [b][a]
> lift to paths with the same initial point but different
> endpoints. The universal covering space, which is the
> "free quadrivalent graph" certainly works, but there will be
> be others too. Indeed there will be a covering space with
> finitely many sheets that works....

Thanks. I get much about the problem. :)
 | 
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