Classify connected covers

Try

The real line covers the circle by t↦exp(2πit). Keeping the full real endpoint remembers how many turns were made; reducing that memory modulo d produces a connected d-sheeted circle cover. Predict how a construction using paths instead of real numbers might work on a more general space.

Choose a basepoint b₀. Two paths starting there will represent the same point upstairs when they have the same endpoint and are homotopic with endpoints fixed. This retains more information than simply recording the endpoint downstairs.

Predict

Does path connectedness alone make this path-class set a covering space? Small neighborhoods must prevent hidden loop changes from merging different sheets. Distinguish a neighborhood in which every loop contracts inside the neighborhood from one in which its loops contract in the whole base. The latter condition is sufficient for the construction below.

Experiment

For S¹, list the connected based covers corresponding to these subgroups of Z.

Subgroup	Cover	Number of sheets
Z	Identity circle cover	1
4Z	z↦z⁴	4
Zero subgroup	R→S¹	Infinite

The zero subgroup is not a “zero-sheeted cover.” Its index is infinite. For 4Z, trace four successive positive loops and identify when a starting point first returns. Compare the subgroup with its index rather than treating them as the same number.

Understand

Assume B is path connected, locally path connected and semilocally simply connected. The last condition means that each point has a neighborhood whose based loops become trivial in π₁(B); local path connectedness lets us choose that neighborhood open and path connected. It need not itself be simply connected.

Form the set of endpoint-preserving homotopy classes [γ] of paths starting at b₀, with endpoint map P([γ])=γ(1). Near [γ], ending at b, choose such a neighborhood U of b. A basic neighborhood consists of [γδ] for paths δ in U starting at b. For each endpoint in U, the class is independent of the chosen δ because the loop made from two choices contracts in B. Hence P maps this neighborhood bijectively onto U. Declare these sets to be a basis; smaller suitable neighborhoods around an endpoint refine intersections. If two such lifts of the same U meet, concatenating paths inside U shows that they agree throughout U. Thus the full inverse image of U splits into disjoint sheets, each homeomorphic to U. This proves the covering property rather than merely asserting it.

The path-class space is path connected: the initial segments of γ lift γ from the constant-path class to [γ]. It is simply connected as well. A loop upstairs starting at the constant class projects to a loop γ whose lifted endpoint is [γ]. Closure says [γ] is the constant class, so γ is null-homotopic. The injectivity of the induced fundamental-group map for a covering then makes the original loop trivial. This is the universal cover.

For a subgroup H of G=π₁(B,b₀), modify the equivalence relation: paths γ and δ with the same endpoint are equivalent when the class of γδ⁻¹ lies in H. The same small-neighborhood construction gives a connected cover with based subgroup H; fiber points are right cosets H g. Its sheet count is the index [G:H]. Every connected cover gives such an H, and path lifting defines an isomorphism to this constructed cover. Two based connected covers are isomorphic exactly when their subgroups agree.

If the chosen point in the fiber is forgotten, its stabilizer changes from H to g⁻¹Hg when it is moved by g. Consequently unbased isomorphism classes correspond to conjugacy classes of subgroups. Equal index alone does not imply isomorphic coverings. The universal cover corresponds to the zero or trivial subgroup, not to the full group.

Apply

For the torus, G=Z². The subgroup generated by (2,0) and (0,3) has index six and gives a six-sheeted regular cover; its deck group is Z/2Z × Z/3Z. Explain why all subgroups are normal here and why that fact does not extend to every fundamental group. If semilocal simple connectedness is absent, the displayed path-class construction need not be an ordinary covering; do not apply the classification theorem without checking its hypotheses.

Continue

This bridge establishes loop, presentation and covering methods, not a full algebraic topology curriculum. Homology, cohomology and higher homotopy require further development. Compare each construction with the hypotheses of Hatcher, Section 1.3, and write a complete subgroup-to-cover argument before claiming classification from a list of examples.