How to abelianizing the fundamental group?

[kakarotyjn]

There is a theorem:If |K| is connected,abelianizing its fundamental group gives the first homotopy group of K.

How to abelianize a group? And how to understand this theorem more obviously?Can anyone show me an example to see it?

I myself will think this problem for more time because I learn it just now and haven't think it much.

Thank you!

 

[Landau]

How to abelianize a group?

Mod out by [G,G], its commutator subgroup. E.g. see here.

 

[lavinia]

There is a theorem:If |K| is connected,abelianizing its fundamental group gives the first homotopy group of K.

How to abelianize a group? And how to understand this theorem more obviously?Can anyone show me an example to see it?

I myself will think this problem for more time because I learn it just now and haven't think it much.

Thank you!

The fundamental group is the first homotopy group. Abelianized, it is the first homology group with Z coefficients.

The abelianization, as Landau said, is the quotient group modulo the commutator subgroup.

Example. The Euclidean plane minus 2 points. Its fundamental group is the free group on two generators. It first homology group is the free abelian group on two generators.

 

[kakarotyjn]

Thank you! I haven't learned commutator subgroups,but I will pick it up now to understand it.

 

[blue_raver22]

I am happy to provide a response to your inquiry about abelianizing the fundamental group. First, let me explain the concept of abelianizing a group. Abelianizing a group means taking a group and modifying it in a way that makes it into an abelian group, which is a group where the order of operations does not matter. This can be done by taking the commutator subgroup of the original group, which is the subgroup generated by all elements of the form aba^-1b^-1, where a and b are elements of the original group.

Now, let's look at the theorem you mentioned. It states that if K is a connected space, then the abelianization of its fundamental group (which is the group of all possible loops in K up to homotopy) is isomorphic to the first homotopy group of K. This means that by abelianizing the fundamental group, we are essentially capturing the same information as the first homotopy group, which is the group of all possible paths in K up to homotopy.

To understand this theorem more clearly, let's consider an example. Let K be the unit circle in the complex plane, which is a connected space. The fundamental group of K is isomorphic to the integers, where the generator represents a loop around the circle in the counterclockwise direction. However, the abelianization of this fundamental group is isomorphic to the trivial group, which only has one element. This is because in an abelian group, all elements commute, so any loop around the circle can be deformed into a single point without changing the group structure.

In summary, abelianizing the fundamental group is a way to simplify the group structure and capture the same information as the first homotopy group. I hope this helps clarify the concept and the theorem for you. Keep exploring and learning, and feel free to ask for further clarification if needed.