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.