Homotopy theory MOC

Fundamental group

The fundamental group of a topological space with base point¹ is the automorphism group of in the Fundamental groupoid, i.e. the set of homotopy classes of continuous loops with base point together with the joining operation to form a group. homotopy

2. Associative
3. Identity
4. Inverse by reverse paths

The fundamental group is the first in a series of higher homotopy groups.

Functor

is a covariant functor from Category of pointed topological spaces to Category of groups. A basepoint-respecting continuous map is mapped as follows

Proof of functor

proof

Properties

• If is path-connected then all fundamental groups are isomorphic, so we write .
• A space is simply connected iff its fundamental group is trivial.
• Fundamental group respects products

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Footnotes

1. German die Fundamentalgruppe mit Aufpunkt ↩