Fundamental groups and based loops

Fix $x_0\in X$. A based loop is a continuous map $\gamma:I\to X$ with both endpoints equal to $x_0$. Two loops are equivalent when a homotopy fixes both endpoints at every time. Multiply classes by traversing the first loop and then the second, each at twice its original speed. The constant loop represents the identity, and reversing a loop represents its inverse. These claims hold for homotopy classes: concatenating a loop with a constant path does not literally preserve its parametrization, but a continuous reparametrization homotopes it back.

Associativity similarly comes from changing the time allotted to three consecutive paths. The cancellation homotopy contracts a loop followed by its reversal by gradually shortening the excursion. These constructions show that the operation is well-defined and satisfies the group axioms. We write the resulting group as $\pi_1(X,x_0)$. No local path-connectivity hypothesis is needed for its definition. Such hypotheses will become important when groups are used to construct or classify covering spaces, rather than merely to record loops already present.

The map $q:\mathbb R\to S^1$, $q(s)=e^{2\pi i s}$, unwraps circle paths. A loop based at one has a unique lift starting at zero; its endpoint is an integer because q of that endpoint is one. Endpoint-fixed homotopies preserve this integer, and concatenation adds endpoints after translating the second lifted path. Conversely, loops with the same endpoint integer are based-homotopic, using the contractibility of the real line and projection. This identifies the circle's fundamental group with the additive integers, with positive orientation fixing the sign convention.

A based continuous map $f:(X,x_0)\to(Y,y_0)$ induces a homomorphism by composition with loops. For the power map $z\mapsto z^m$, the loop of winding k becomes one of winding mk. A based homotopy between maps gives the same induced homomorphism. The degree m may be positive, negative, or zero; a zero-degree circle map is nullhomotopic, while degree minus one reverses the generator. This algebra measures deformation classes, not the speed or length with which a representative loop is traversed.

If c is a path from $x_0$ to $x_1$, a loop gamma based at $x_1$ becomes the loop $c*\gamma*\overline c$ at $x_0$. This gives an isomorphism between the two fundamental groups when the points lie in the same path component. Choosing a different connecting path can change that isomorphism by an inner automorphism. Thus a path-connected space has a fundamental group well-defined up to isomorphism, but identifying individual elements at different basepoints requires the connecting path. An unbased homotopy of maps also carries a basepoint path that must be accounted for.

Free homotopy of loops allows the basepoint to move, and in a path-connected space the resulting classes correspond to conjugacy classes in the fundamental group. This does not mean every two elements with equal abelianized winding data are freely homotopic. In a free group, a commutator can have zero exponent sum in every generator while remaining nontrivial. Distinguish based classes, conjugacy classes, and abelianization: they discard different amounts of information. The covering classification later reflects exactly the same distinction between a specified lift of the basepoint and an unpointed covering.