Why Is the Fundamental Group of a Torus Described as Z+Z Instead of ZxZ?

In summary, the author has been using Seifert-Van Kampen to calculate the fundamental group of the torus, but is having difficulty because he does not know group theory. They say that the free product of Z*Z is isomorphic to the direct sum of Z+Z, but they do not know why this is the case.
  • #1
andlook
33
0
Hi

So I've been using Seifert-Van Kampen (SVK) to calculate the fundamental group of the torus. I haven't done any formal group theory, hence my problem ...

I have T^2=(S^1)x(S^1)

If A= S^1, B=S^1, A intersection B is 0. And T^2 = union of A and B.

Then fundamental group of (A intersection B) = 0

And I already have fundamental group of (S^1) = Z

Then using SVK the fundamental group of the torus is the free product of S^1 with S^1 over 0.

Which I think is isomorphic to the ZxZ.

In the literature this is written as the direct sum of Z+Z. Why is this the direct sum and not the cross product?

Since I don't know group theory better I don't know if it is possible to just ask the simpler: Why is the free product of Z*Z is isomorphic to direct sum Z+Z not the product of ZxZ.
 
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  • #2
Three comments:

1) If you have a finite collection of abelian groups, then their direct product and direct sum are the same.

2) The free product of Z*Z is not isomorphic to either Z[itex]\oplus[/itex]Z or ZxZ; in fact, Z*Z is nonabelian (as most free products are).

3) SVK is unnecessary for computing pi_1 of the torus. Instead one can just use the fact that pi_1(XxY) = pi_1(X) x pi_1(Y), which is easy to prove to directly. If you insist on using SVK, then you must be doing something wrong: pi_1 of the torus isn't Z*Z -- it's Z*Z modulo a certain normal subgroup.
 
  • #3
I just noticed that you provided your working for how you applied SVK. What do you mean when you say "A= S^1, B=S^1"? I can't put any meaning to this assignment that gives us "T^2 = union of A and B".

In any case, SVK needs a special open cover of T^2, not an arbitrary cover.
 
  • #4
thanks for your help on this,

I will work on this and reply later
 

Related to Why Is the Fundamental Group of a Torus Described as Z+Z Instead of ZxZ?

1. What is the fundamental group?

The fundamental group is a mathematical concept in algebraic topology that measures the connectivity of a topological space. It is a group of homotopy classes of loops in the space, with the group operation being concatenation of loops. It is denoted by the symbol π1.

2. How is the fundamental group calculated?

The fundamental group can be calculated using various techniques, such as the Van Kampen theorem or the Seifert-van Kampen theorem. It involves breaking down the space into smaller, simpler pieces and then using algebraic operations to combine the fundamental groups of these pieces to get the fundamental group of the whole space.

3. What is a torus?

A torus is a surface in three-dimensional space that looks like a doughnut or a tire. It can be defined as the product of two circles, and is an example of a closed, compact, orientable surface. It has a rich mathematical structure and is often studied in topology and geometry.

4. How is the fundamental group related to the torus?

The fundamental group of a torus is isomorphic to the group ℤ₀ x ℤ₀, where ℤ is the integers and ₀ is the integers modulo 2. This means that the fundamental group of a torus has two generators and two relations. This is a fundamental result in algebraic topology and has many applications in other areas of mathematics.

5. What is the significance of the Seifert-van Kampen theorem?

The Seifert-van Kampen theorem is a powerful tool in algebraic topology that allows us to calculate the fundamental group of a space by breaking it down into smaller, simpler pieces. It is used extensively in the study of the fundamental group of the torus as well as other topological spaces. It also has applications in other areas such as knot theory and homotopy theory.

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