REFERENCE SHEET

The fundamental group

Encode the ways a based loop can wind around holes.

Before this Topological spaces & homotopy, Groups & subgroups

π₁(S¹) ≅ ℤ; π₁(S¹×S¹) ≅ ℤ²

Definitions

Based loop

A continuous path γ:[0,1]→X with γ(0)=γ(1)=x₀.

Fundamental group

π₁(X,x₀) consists of loops based at x₀ modulo homotopies fixing both endpoints. Multiplication concatenates loops.

Simply connected

A path-connected space with trivial fundamental group. Merely being connected is not enough.

Key theorems

Functoriality

A based continuous map f induces a homomorphism f*([γ])=[f∘γ]. Composition and identity maps induce composition and identity homomorphisms.

Circle and products

Winding number identifies π₁(S¹) with ℤ. For based spaces X,Y, π₁(X×Y)≅π₁(X)×π₁(Y).

Basepoint qualification

In a path-connected space, a path between basepoints induces an isomorphism of their fundamental groups. Different path choices can change it by an inner automorphism.

Notable applications

Obstructions

Nonisomorphic fundamental groups rule out homotopy equivalence.

Winding

Planar loops around a missing point connect topology to contour integration.

Work it out

Try before revealing

01 / Compute

Find π₁(ℝ² without the origin).

Hint

Use radial deformation retraction.

Solution

The punctured plane deformation retracts onto S¹, so its fundamental group is ℤ.

02 / Compute

On a torus, a loop winds twice in the first circle direction and −3 times in the second. What is its class and inverse?

Hint

Use π₁(S¹×S¹)≅ℤ².