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From: nullgraph on 18 Jul 2008 15:39
Hi, I'm trying to calculate the fundamental group of the Klein bottle.
I know I should be using Van Kampen and start with removing a point
from the Klein bottle. I'm thinking of breaking the Klein bottle into
2 parts, one is a circle without the removed point and the other is
everything except the removed point. How do I continue from there? I
read in a previous thread that the 2nd part (punctured Klein bottle)
is homotopic to the figure-eight, why is that?
Thank you.
From: Mariano Suárez-Alvarez on 18 Jul 2008 16:47
On Jul 18, 4:39 pm, nullgraph <nullgr...(a)gmail.com> wrote:
> Hi, I'm trying to calculate the fundamental group of the Klein bottle.
> I know I should be using Van Kampen and start with removing a point
> from the Klein bottle. I'm thinking of breaking the Klein bottle into
> 2 parts, one is a circle without the removed point and the other is
> everything except the removed point. How do I continue from there? I
> read in a previous thread that the 2nd part (punctured Klein bottle)
> is homotopic to the figure-eight, why is that?
> Thank you.

Consider the Klein bottle as being obtained
from a rectangle by doing identifications
along its edges appropriately (this is explained
in Wikipedia, for example) and remove the point
which corresponds to the center of the rectangle.

Now look at the picture.

-- m
From: nullgraph on 18 Jul 2008 17:01
On Jul 18, 4:47 pm, Mariano Suárez-Alvarez
<mariano.suarezalva...(a)gmail.com> wrote:
> On Jul 18, 4:39 pm, nullgraph <nullgr...(a)gmail.com> wrote:
>
> > Hi, I'm trying to calculate the fundamental group of the Klein bottle.
> > I know I should be using Van Kampen and start with removing a point
> > from the Klein bottle. I'm thinking of breaking the Klein bottle into
> > 2 parts, one is a circle without the removed point and the other is
> > everything except the removed point. How do I continue from there? I
> > read in a previous thread that the 2nd part (punctured Klein bottle)
> > is homotopic to the figure-eight, why is that?
> > Thank you.
>
> Consider the Klein bottle as being obtained
> from a rectangle by doing identifications
> along its edges appropriately (this is explained
> in Wikipedia, for example) and remove the point
> which corresponds to the center of the rectangle.
>
> Now look at the picture.
>
> -- m

Hmm, I'm looking at the identification, but I still don't see it.
Perhaps I need better glasses? In a related note, how come the
punctured torus has different identification but also is homotopic to
figure-eight?
From: Mariano Suárez-Alvarez on 18 Jul 2008 17:32
On Jul 18, 6:01 pm, nullgraph <nullgr...(a)gmail.com> wrote:
> On Jul 18, 4:47 pm, Mariano Suárez-Alvarez
>
>
>
> <mariano.suarezalva...(a)gmail.com> wrote:
> > On Jul 18, 4:39 pm, nullgraph <nullgr...(a)gmail.com> wrote:
>
> > > Hi, I'm trying to calculate the fundamental group of the Klein bottle..
> > > I know I should be using Van Kampen and start with removing a point
> > > from the Klein bottle. I'm thinking of breaking the Klein bottle into
> > > 2 parts, one is a circle without the removed point and the other is
> > > everything except the removed point. How do I continue from there? I
> > > read in a previous thread that the 2nd part (punctured Klein bottle)
> > > is homotopic to the figure-eight, why is that?
> > > Thank you.

Please do not top-post.

Forget about the Klein bottle and the torus.
Can you see why a punctured *rectangle* (with no
identifications) is homotopic to a circle?

-- m
From: nullgraph on 19 Jul 2008 13:18
On Jul 18, 5:32 pm, Mariano Suárez-Alvarez
<mariano.suarezalva...(a)gmail.com> wrote:
> On Jul 18, 6:01 pm, nullgraph <nullgr...(a)gmail.com> wrote:
>
> > On Jul 18, 4:47 pm, Mariano Suárez-Alvarez
>
> > <mariano.suarezalva...(a)gmail.com> wrote:
> > > On Jul 18, 4:39 pm, nullgraph <nullgr...(a)gmail.com> wrote:
>
> > > > Hi, I'm trying to calculate the fundamental group of the Klein bottle.
> > > > I know I should be using Van Kampen and start with removing a point
> > > > from the Klein bottle. I'm thinking of breaking the Klein bottle into
> > > > 2 parts, one is a circle without the removed point and the other is
> > > > everything except the removed point. How do I continue from there? I
> > > > read in a previous thread that the 2nd part (punctured Klein bottle)
> > > > is homotopic to the figure-eight, why is that?
> > > > Thank you.
>
> Please do not top-post.
>
> Forget about the Klein bottle and the torus.
> Can you see why a punctured *rectangle* (with no
> identifications) is homotopic to a circle?
>
> -- m

I'm confused... where and how did I top-post?

Yes, I can see why a punctured rectangle with no identification is
homotopic to a circle. I guess the identification is the part that
confuse me.
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