Fundamental groups and coverings

Fix a point x₀ in a space X. A based loop is a continuous map from [0,1] to X whose two endpoints both equal x₀. Two loops represent the same element of π₁(X,x₀) if a continuous homotopy deforms one into the other while keeping both endpoints at x₀ throughout. The operation concatenates loops, with a change of speed to fit them into the unit interval. Associativity and the identity laws hold for homotopy classes even though parametrized concatenation is not literally associative as a formula.

The constant loop represents the identity and reversing a loop gives the inverse. The group need not be commutative: going around two distinct holes in opposite orders can produce different classes. If X is path-connected, changing basepoint along a chosen path produces an isomorphism of fundamental groups, but generally not a canonical one independent of the path. A simply connected space is path-connected with trivial fundamental group; mere connectedness is not enough to use that definition.

A covering map p:E→X is a continuous surjection such that every point of X has an open neighborhood U whose preimage is a disjoint union of open sets, each mapped homeomorphically onto U. The real line covers the circle by p(t)=exp(2πit). Small arcs have separated interval copies above them, even though the entire circle cannot be assigned one continuous real-valued angle. A map that merely has several preimages is not automatically a covering; the uniform local sheet condition is essential.

A path in X has a unique lift once its starting point in E is chosen. For a loop on the circle based at 1, a lift starting at zero ends at an integer. That integer is the winding number and is unchanged by based homotopies. Concatenation adds endpoint displacements, giving π₁(S¹,1) isomorphic to the integers. The covering real line is simply connected, so lifted paths cannot hide a further winding obstruction there.

For a path-connected, locally path-connected, semilocally simply connected space, a simply connected universal covering exists. Semilocal simple connectivity means that each point has a neighborhood whose loops become trivial when regarded as loops in X. Under these hypotheses, connected based covering spaces correspond to subgroups of π₁(X,x₀). If basepoints are forgotten, the correspondence uses conjugacy classes of subgroups. These qualifications matter for spaces with infinitely accumulating small holes, where the familiar covering-space picture can fail.

For the circle, the subgroup nℤ corresponds to the connected n-sheeted circle cover, and the zero subgroup corresponds to its real-line universal cover. A covering map is not necessarily a one-to-one coordinate chart, and its total space need not be a subset of the base. Coverings instead provide a controlled way to replace global identifications by extra sheets while preserving local topology. This distinction is useful for periodic variables, branch choices, and global phases, provided the actual domain and map satisfy the covering definition.