Build the fundamental group

Try

Start at a marked point x₀, follow a loop α, then follow another loop β. The result is again a loop at x₀. Different walking speeds should not change its essential route. We therefore work with endpoint-homotopy classes rather than literal parametrized paths. This quotient is what makes the group laws true.

Predict

Predict whether (α followed by β) followed by γ is the same function as α followed by (β followed by γ). The traversal order agrees, but the usual half-time convention gives different speeds. Predict also whether reversing a loop makes the concatenation literally constant. It does not; a homotopy cancels the outward and return portions.

Experiment

Compare time allocations for three loops.

Parenthesization	Time for α	Time for β	Time for γ
(αβ)γ	1/4	1/4	1/2
α(βγ)	1/2	1/4	1/4

Move the first breakpoint from 1/4 to 1/2 and the second from 1/2 to 3/4 at equal speed. They remain ordered and separated. Rescale each path to its current interval. This is an explicit associativity homotopy, not a claim that the functions were equal.

Understand

Fix x₀ in a space X. The fundamental group π₁(X,x₀) consists of endpoint-homotopy classes of loops at x₀. Our multiplication convention is first α, then β. The concatenated path uses α(2t) for t≤1/2 and β(2t−1) for t≥1/2. If representatives change through endpoint homotopies, concatenate those homotopies at each deformation time. The common fixed point makes their formulas agree at the seam, so multiplication is well defined.

Associativity follows from the breakpoint interpolation in the table. More generally, increasing piecewise-linear changes of speed are endpoint homotopic by interpolating the reparametrizing functions. Their endpoint values remain zero and one, and composing with the path gives continuity.

The identity is the constant loop. To remove a constant segment before or after a loop, continuously reduce the time allotted to the pause and rescale the moving segment. Continuity at the final zero-length pause follows from continuity of the original path at its endpoint; equivalently compose the path with a continuous family of parameter functions that changes a paused parameter to t.

The inverse of α is the reversed path ᾱ(t)=α(1−t). A concrete contraction of αᾱ is H(t,s)=α(2t(1−s)) on the first half and α(2(1−t)(1−s)) on the second. The two formulas agree at t=1/2, both endpoints are α(0), and at s=1 the whole loop is constant. Reversing the argument contracts ᾱα. Thus the classes satisfy all group axioms.

Nothing in these proofs makes arbitrary loop classes commute. Moving one loop past another is a separate geometric question. Later a two-circle graph will produce a nonabelian free group.

Apply

Take a loop in the plane and contract it to its basepoint by straight-line interpolation. Check every boundary condition and conclude that the plane’s fundamental group is trivial. Now try the same formula in a punctured plane and identify the potential forbidden point. Failure of this formula is only a warning; the eventual nontriviality proof will use winding numbers.

Continue

Keep three levels distinct: a parametrized loop, its endpoint-homotopy class, and an abstract group element representing that class. Algebraic cancellation is justified by the constructed homotopies. Next, continuous maps will turn these groups into comparable invariants, while changes of basepoint introduce conjugation.

Continue to Basepoints and induced homomorphisms.