Van Kampen's theorem and fundamental groups

In summary, Van Kampen's theorem allows for the calculation of the fundamental group of a space by decomposing it into open, path-connected sets and using the free product of the fundamental groups of the individual sets modulo the relations imposed by the overlap. The normal subgroup N is found by identifying the loops in the overlap and imposing the corresponding relations. This method is demonstrated through examples, including the torus and the double torus. However, for more complicated spaces, it may be easier to use other methods, such as cutting open the surface into a polygon, as done by Riemann.
  • #1
sparkster
153
0
I didn't see a topology forum, so I thought I'd post this question here. Can anyone give any pointers on using van Kampen's theorem? I understand the basic way it works, decompose a space X into open, path-connected sets, say U and V.

Then pi1(U) * pi1(V) = pi1(X)/N, where N is a normal subgroup. The N is what gives me trouble. How do calculate what the normal subgroup is?

Any help is appreciated!
 
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  • #2
It depends on what the decomposition is. It is more complicated to describe than do. Let me recap a bit:

Effectively you decompose T into two smaller sets U and V such that you know the fund. grps of U and V and UnV. Obviously, any loop in UnV defines a loop in U and a loop in V.

Then the fund grp of T is the (free) product of the fundamental groups of U and V modulo the relations imposed by the fact that loops in the overlap are identified.

Those relations are the "modulo N" part.

I guess this isn't a direct answer, but then I wouldn't calculate N, i'd describe explicitly the relations on the overlap, that implicitly tell you what N is.

Let's do an example where we already know the answer: the torus.

Let U be the torus less one point - the punctured torus.

Let V be a disc around that puncture point.

U can then be deformation retracted to the bouquet of two circles, thus its fundamental group is the free group on two generators F_2. Let the gens be g and h, we'll explain why in a second.

V has trivial fundamental group

UnV is homotopic to a circle. It's fundamental group is Z; let t be some (homotopy class of) loop generating this group. In V this is sent to the identity, as V is contractible. In U this loop is sent to the path ghg^{-1}h^{-1}. To see this, imagine the torus as the square with opposite sides identified. U is then this square with, the centre missing, thus the punctred torus retracts to the boundary of the square, which identifies to give two circles joined at a point. A loop around the hole retracts to be a path that goes along the top edge of the square, down the side along the bottom and up the other side again, right? So it is a path around one circle, then around the other, back along the first circle in the opposite direction, and then along the second circle, again in the opp. direction, ie ghg^{-1}h^{-1} in the fundamental group.


Phew, this is complicated to write, but draw a diagram to see what's going on.

Anyway, by the van Kampen Theorem, the fundamental group is

(F_2)*(e) = F_2 modulo the relation that ghg^{-1}h^{-1}=e; we identify those loops we described.

This means that we make gh=hg, ie abelianize F_2=ZxZ, so N in this case would be the commutator subgroup of F_2.


It sounds more complicated than it is, honest.

N is the subgroup generated by the relations we impose because of the overlap.
 
  • #3
Sorry to resurrect an old thead. Thanks for the above explanation, it does help some. I tried copied your method for the double torus.

Let U=double torus - {point}
let V=Disk around the point.

Everything is open and connected like it should be and U u V = X = double torus.

U deformation retracts to a wedge of 4 circles--it's fundamental group is a free group with 4 generators, a1, b1, a2, b2, say. V is contractible.

U n V is homotopic to a circle. So tracing along the circle, we get a_1b_1a_1(^-1)b_1(^-1)a_2b_2a_2(^-1)b_2^(-1).

Okay, here's where its gets iffy for me. Since V is contractible, we equate the above string to e?

Then I get a_1b_1a_1(^-1)b_1(^-1)a_2b_2a_2(^-1)b_2^(-1)=e

a_1b_1a_1(^-1)b_1(^-1)=b_2a_2b_2^(-1)a_2^(-1)

So what is this?

From Hatcher's book I see that the presentation should be <a1,b1,a2,b2|[a1,b1][a2,b2]>

I guess my problem is figuring out what the relations should be.

Also, If V was not contractible, how would I proceed?


ETA:
[a1b1][a2b2] in the presentation just means multiplying the commutators together? If so, then I think I'm starting to get it. Still not sure what to do if fundamental group of V isn't the identity, though.
 
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  • #4
all compact connect oriented surfaces of genus g are most easily computed as riemann did, by cutting open the surface into a 4g sided polygon.

for a geuns one torus we get generators a,b, and one relation going around the outside of the rectangel, namely the commutator a b a^-1 b^-1, as matt said

in general, the edges of the polygon seem to give the one relation
a1 b1 a1^(-1) b1^(-1)...ag bg ag^-1 bg^-1. not what you have above. ah yes you misread hatcher page 51.

as often happens this problem is easier done without the machinery of van kampens theorem, although topologists love this theorem and their are no doubt many situations where it is essential.

actually hatcher is essentially using riemanns presentation of the manifold via a 2 cell attached to a wedge of 2g circles.
 
  • #5
mathwonk said:
in general, the edges of the polygon seem to give the one relation
a1 b1 a1^(-1) b1^(-1)...ag bg ag^-1 bg^-1. not what you have above. ah yes you misread hatcher page 51.
Unless I'm have a very stupid moment, isn't that what I wrote?
 
  • #6
my mistake, that is what you wrote. then what is your question?

i.e.hatcher's presentation is what a_1b_1a_1(^-1)b_1(^-1)a_2b_2a_2(^-1)b_2^(-1)=e
means.
 
  • #7
If the intersection of the decomposed spaces is a simple space, like S^1, and I know how to construct the original space from a polygon, like a genus 1 or 2 torus, then I think I can apply van kampen. That takes care of all orientable and nonorientable surfaces.

But what if it isn't so easy? One problem in Hatcher that I've been working on is show the fundamental group of R^2 - Q^2 is uncountable.
 
  • #8
That's a straight forward result, and doesn't need Van Kampen.

Pick irrational numbers r<s<t, and another irrational x consider the square loops round the points

(r,x) to (s,x) to (s,x+1) to (r,x+1) to (r,x)

and the same with s replaced by t. These loops are not homotopic, and thus we can easily find an uncountable number of non-homotopic loops.

As mathwonk said, Van Kampen is all well and good, but is almost always useless except in the nice cases.
 

Related to Van Kampen's theorem and fundamental groups

What is Van Kampen's theorem?

Van Kampen's theorem is a fundamental result in algebraic topology that relates the fundamental groups of different spaces. It states that if a space can be written as the union of two path-connected open sets, then the fundamental group of the space can be calculated from the fundamental groups of the two open sets and their intersection.

What is a fundamental group?

A fundamental group is a mathematical object that measures the connectivity of a topological space. It is a group that consists of all the possible closed loops in the space, up to homotopy. The fundamental group is an important tool in algebraic topology for distinguishing between topological spaces.

How do you calculate the fundamental group of a space?

The fundamental group of a space can be calculated by using Van Kampen's theorem or other algebraic topology techniques such as the Seifert-van Kampen theorem or the Mayer-Vietoris sequence. In general, the process involves breaking down the space into simpler pieces, calculating the fundamental groups of those pieces, and then combining them to find the fundamental group of the original space.

What is the importance of Van Kampen's theorem?

Van Kampen's theorem is an essential tool in algebraic topology for calculating fundamental groups. It allows us to break down complicated spaces into simpler pieces and use the knowledge of the fundamental groups of those pieces to find the fundamental group of the original space. This theorem has many applications in mathematics, including in the study of knot theory, surface classification, and the classification of 3-manifolds.

Are there any limitations to using Van Kampen's theorem?

Van Kampen's theorem is a powerful tool, but it does have some limitations. It only applies to spaces that can be written as the union of two path-connected open sets. Additionally, the theorem only applies to the fundamental group and does not provide information about other higher homotopy groups. In some cases, other techniques may be needed to calculate the fundamental group of a space.

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