Fundamental group

The fundamental group $𝜋 1 (𝑋, 𝑥 0)$ of a topological space $𝑋$ with base point¹ $𝑥 0 \in 𝑋$ is the automorphism group of $𝑥 0$ in the Fundamental groupoid, i.e. the set of homotopy classes of continuous loops with base point $𝑥 0$ together with the joining operation to form a group. #m/def/homotopy

Associative $[𝛼] ([𝛽] [𝛾]) = ([𝛼] [𝛽]) [𝛾]$
Identity $[𝑐 𝑥 0 𝑇]$
Inverse by reverse paths

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

Functor

$𝜋 1 : 𝖳 𝗈 𝗉 • \to 𝖦 𝗋 𝗉$ is a covariant functor from $𝖳 𝗈 𝗉 ∙$ to $𝖦 𝗋 𝗉$ . A basepoint-respecting continuous map $𝑓 \in 𝖳 𝗈 𝗉 • ((𝑋, 𝑥 0), (𝑌, 𝑦 0))$ is mapped as follows

𝜋 1 (𝑓) : 𝜋 1 (𝑋, 𝑥 0) \to 𝜋 1 (𝑌, 𝑦 0) [𝛼] \mapsto [𝑓 \circ 𝛼]

Proof of functor

#missing/proof

Properties

If $𝑋$ is path-connected then all fundamental groups are isomorphic, so we write $𝜋 1 (𝑋)$ .
A space is simply connected iff its fundamental group is trivial.
Fundamental group respects products

#state/develop | #lang/en | #SemBr

German die Fundamentalgruppe mit Aufpunkt $𝑥 0$ ↩

Fundamental group

Functor

Properties

Footnotes