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Fundamental Group

★★★★★Graduate+

📖Definition

The fundamental group π₁(X) consists of homotopy equivalence classes of loops (closed paths) starting and ending at a point in space X.

π₁(X, x₀) = [γ] : γ(0) = γ(1) = x₀

Definition of fundamental group

π₁(S¹) ≅ ℤ

Fundamental group of circle

π₁(Sⁿ) = 0 for n ≥ 2

Fundamental group of higher spheres

예제 1

What is the fundamental group of plane minus a point?

예제 2

What is the fundamental group of a torus?

Discovered by: Henri Poincaré (1895)

Poincaré introduced fundamental groups while founding algebraic topology.

Space classification

Quantum field theory, topological defects

Path planning

#기본군#푸앵카레#fundamental group#homotopy