note / modern mathematics

Session 09 — Fundamental Group

Fundamental Group

Central Question

How do loops become a group that records information about a space?

Why This Matters

The fundamental group is a first substantial example of translating geometric behaviour into an algebraic invariant.

Learning Prompt

Motivate the fundamental group as an algebraic detector of holes. Construct $\pi_1(X,x_0)$ from loops and homotopy classes and explain why loop composition produces a group. Explore $\mathbb R^2$ , $S^1$ and the torus.

Ideas

Examples

Questions

Connections

One Thing That Surprised Me

Further Rabbit Holes

Session 08 — From Automorphisms to Groups Session 07 — Quotient Spaces Session 08 — Homotopy Session 10 — Why the Circle Produces the Integers Session 11 — A Glimpse of Algebraic Topology Week 02 — Topology

Referenced by

Session 08 — From Automorphisms to Groups Session 04 — Homeomorphism Session 07 — Quotient Spaces Session 08 — Homotopy Session 10 — Why the Circle Produces the Integers Session 11 — A Glimpse of Algebraic Topology Session 12 — Week 2 Synthesis Week 02 — Topology Connections Exploration Map