How to abelianizing the fundamental group?

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SUMMARY

Abelianizing the fundamental group involves taking the quotient of the group by its commutator subgroup, denoted as [G,G]. This process transforms the fundamental group into the first homology group with integer coefficients. For instance, the fundamental group of the Euclidean plane minus two points is a free group on two generators, while its abelianization results in a free abelian group on the same two generators. This theorem is applicable when the space |K| is connected, establishing a direct relationship between the fundamental group and the first homotopy group.

PREREQUISITES
  • Understanding of fundamental groups in algebraic topology
  • Familiarity with homotopy groups and homology groups
  • Knowledge of commutator subgroups in group theory
  • Basic concepts of quotient groups
NEXT STEPS
  • Study the properties of commutator subgroups in group theory
  • Learn about the relationship between fundamental groups and homology groups
  • Explore examples of abelianization in various topological spaces
  • Investigate the implications of the first homotopy group in algebraic topology
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Mathematicians, particularly those specializing in algebraic topology, students learning about fundamental groups, and anyone interested in the concepts of homotopy and homology in topological spaces.

kakarotyjn
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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!:smile:
 
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kakarotyjn said:
How to abelianize a group?
Mod out by [G,G], its commutator subgroup. E.g. see here.
 
kakarotyjn said:
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!:smile:

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

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