Klein Bottle question

[Tony]

Let a,b be two points of the Klein bottle K. Let X be the identification
space we get by identifying a and b. How can I calculate the fundamental
group of X? I am thinking Seifert Van Kampen theorem, but not sure how I
could apply it. Does it matter which two points I pick?

Tony

[AmmonRa]

Let a,b be two points of the Klein bottle K. Let X be >the identification
space we get by identifying a and b. How can I >calculate the fundamental
group of X? I am thinking Seifert Van Kampen theorem, >but not sure how I
could apply it. Does it matter which two points I pick?
Tony

I fail to see why you need to pick two points in the Klein bottle to calculate its fundamental group.

To compute the fundamental group of the Klien bottle, you certainly can use Van Kampen Theorem.

There are at least 2 ways I can think of breaking up the Klien bottle:

1) remove a trivial disc in the Klien bottle, the punctured Klien bottle is homotopic to the figure eight

2) A klein bottle is also the union of two mobius bands, just "glue" the two mobius bands by their boundries, and apply Van Kampen.

AmmonRa

[Tony]

Post by AmmonRa

Let a,b be two points of the Klein bottle K. Let X be >the identification
space we get by identifying a and b. How can I >calculate the fundamental
group of X? I am thinking Seifert Van Kampen theorem, >but not sure how I
could apply it. Does it matter which two points I pick?
Tony

I fail to see why you need to pick two points in the Klein bottle to
calculate its fundamental group.
To compute the fundamental group of the Klien bottle, you certainly can
use Van Kampen Theorem.
1) remove a trivial disc in the Klien bottle, the punctured Klien bottle
is homotopic to the figure eight
2) A klein bottle is also the union of two mobius bands, just "glue" the
two mobius bands by their boundries, and apply Van Kampen.
AmmonRa

I know what the fundamental group of the Klein bottle is, but that wasn't my
question.

My question is to find the fundamental group of X, where X is the
identification space of a Klein bottle with identifying two points of the
Klein bottle.

[W. Dale Hall]

Post by Tony
Let a,b be two points of the Klein bottle K. Let X be the identification
space we get by identifying a and b. How can I calculate the fundamental
group of X? I am thinking Seifert Van Kampen theorem, but not sure how I
could apply it. Does it matter which two points I pick?
Tony

The direct usage of Seifert & van Kampen's theorem might be a bit messy,
because the obvious decomposition of this space (take a closed
neighborhood of the identification point, and the closure of the
complement) produces a non-connected intersection.

Instead, note that the identification space, which I'll call K', is
homotopy equivalent to the one-point union (otherwise called the
"wedge") of K with a circle:

K' ~ K v S^1.

The details are simple, but messy. Here's the visualization:

Imagine the point [ab] in K' corresponding to the
identification of a and b is a gob of glue that
hasn't dried, and pull them apart slowly. The glue
will produce a thread joining those points.

Now, do it properly: Let K" denote the union of K with an arc I_ab
joining a and b (otherwise disjoint from K). There is an obvious
map of K" to K' that collapses I_ab to [ab]. The other direction is
less obvious to the beginner, and somewhat tedious: take small discs
surrounding a and b and map the one-point union (identifying a and b)
to the segment I_ab using the radial coordinate; take up the slack by
expanding the radial coordinate in small annular regions around the
original discs, and leave the rest of K' alone. Details and proof that
you get a homotopy equivalence are left to you.

Now that you've got K' ~ K", note that you can move the endpoints of
the arc I_ab without changing the homotopy type of K", and in fact
you can put the endpoints at the same location in K, still within
that homotopy type.

This gives you K' ~ K v S^1, and you can surely use the above-mentioned
theorem, but it's really overkill.

Dale.

[Lee Rudolph]

Let a,b be two points of the Klein bottle K. Let X be
the identification space we get by identifying a and b. How can I
calculate the fundamental group of X? I am thinking
Seifert Van Kampen theorem, but not sure how I
could apply it. Does it matter which two points I pick?

Yes, the van Kampen Theorem will work, and no, it doesn't
matter which two points you pick. Assuming this latter
statement (you should try to prove it, of course), take some
closed 2-disk D in K containing a and b, and consider
the identification W space formed from the disjoint union
of K and [0,1] by identifying a to 0 and b to 1. By
van Kampen, the fundamental group of W is the free
product of the fundamental group of K and the fundamental
group of C, the identification space formed from the
dijoint union of D and [0,1] by the same identification,
amalgamated over the free group of the intersection of
K and C, which is of course D; so the fundamental group
of W is the free product, period. On the other hand,
X is obviously (check it...) homeomorphic to the
identification space obtained from W by identifying
(the image in W of) [0,1] to a single point. Finally,
W is obviously (check it...) homotopy equivalent to S^1.
So...

Lee Rudolph