Covering Spaces Summary

Algebraic Topology · Axiom Academy

A comprehensive recap of covering space theory and the Galois correspondence Covering Space: A map p: X where every point has an evenly covered neighborhood U with p^ -1 (U) decomposing as disjoint copies of U Evenly Covered: The preimage consists of disjoint open sets (sheets), each mapping homeomorphically to U via p Fibers: All fibers p^ -1 (x) have the same cardinality, called the number of sheets Local Homeomorphism: The covering map p is locally a homeomorphism but globally can be highly nontrivial Fundamental Lifting Properties Path Lifting Property: Every path in X lifts uniquely to once you specify the starting point in the fiber Homotopy Lifting Property: Homotopies in X lift uniquely to homotopies in , given a lift of the initial map Consequence: Homotopic paths with fixed endpoints lift to homotopic paths with the same endpoints Key Insight: Lifting behavior is completely determined by homotopy classes, connecting covering spaces to _1(X) The Galois Correspondence (Classification Theorem) For path-connected, locally path-connected, semi-locally simply connected spaces, there is a bijection between connected covering spaces of X (up to isomorphism) and subgroups of _1(X, x_0) . The Map: A covering p: X corresponds to p_*( _1( )) _1(X) Injectivity: The induced map p_*: _1( ) _1(X) is always injective Order-Reversing: Smaller subgroups correspond to larger covering spaces (more sheets) Index Formula: Number of sheets = [ _1(X) : p_*( _1( ))]

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