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Covering Spaces

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Covering Space

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A covering space of a topological space XX is a topological space CC together with a continuous surjective map p:CXp : C \to X such that for every xXx \in X, there exists an open neighborhood UU of xx where p1(U)p^{-1}(U) is a disjoint union of open sets in CC, each of which is homeomorphic to UU by pp.

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Fundamental Group

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The fundamental group of a topological space XX, denoted π1(X,x0)\pi_1(X, x_0), is the group of loops based at a point x0x_0 up to homotopy, with the group operation given by the concatenation of loops.

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Lifting Property

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Given a covering space p:CXp : C \to X and a map f:YXf : Y \to X from a path-connected and locally path-connected space YY, a lift of ff is a map f~:YC\tilde{f} : Y \to C such that pf~=fp \circ \tilde{f} = f. The lifting property states that such a lift exists if and only if f(π1(Y,y0))p(π1(C,c0))f_*(\pi_1(Y, y_0)) \subseteq p_*(\pi_1(C, c_0)).

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Path Lifting Property

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The path lifting property states that for each path f:[0,1]Xf: [0,1] \to X in a topological space XX and a point cc in a covering space CC such that p(c)=f(0)p(c) = f(0), there exists a unique path f~:[0,1]C\tilde{f}: [0,1] \to C in CC starting at cc such that pf~=fp \circ \tilde{f} = f.

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Deck Transformation

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A deck transformation, or covering transformation, for a covering space p:CXp : C \to X is a homeomorphism d:CCd : C \to C such that pd=pp \circ d = p. The set of all deck transformations forms a group under composition, called the Deck transformation group or group of covering transformations.

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Monodromy Action

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The monodromy action of the fundamental group π1(X,x0)\pi_1(X, x_0) on a fiber p1(x0)p^{-1}(x_0) is a group action where each loop in the base space starting and ending at x0x_0 determines a permutation of the points in the fiber above x0x_0 via lifting and following paths.

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Universal Cover

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A universal cover of a space XX is a covering space p:CXp : C \to X where CC is simply connected (i.e., its fundamental group is trivial). The universal cover is unique up to homeomorphism and covers any other covering space of XX.

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Semi-locally Simply Connected

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A space XX is semi-locally simply connected if every point xXx \in X has a neighborhood UU such that every loop in UU when included in XX is null-homotopic (can be continuously shrunk to a point in XX). This property is necessary for the space to have a universal cover.

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Simply Connected

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A space XX is simply connected if it is both path-connected and every loop in XX can be continuously shrunk to a point (i.e., it has trivial fundamental group). Simply connected spaces are often used as universal covers for other spaces.

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Regular Covering

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A covering space p:CXp : C \to X is a regular (or normal) covering if the group of deck transformations acts transitively on each fiber. Equivalently, the covering is regular if p(π1(C,c0))p_*(\pi_1(C, c_0)) is a normal subgroup of π1(X,p(c0))\pi_1(X, p(c_0)).

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Deck Transformation Group

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The Deck transformation group (also known as the group of covering transformations) of a covering space p:CXp : C \to X is the group consisting of all homeomorphisms of CC that make pp commute, i.e., p=pdp = p \circ d for each deck transformation dd. This group is isomorphic to the quotient of the fundamental groups π1(X,x0)/p(π1(C,c0))\pi_1(X, x_0)/p_*(\pi_1(C, c_0)) when CC is path-connected.

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Homotopy Lifting Property

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The homotopy lifting property asserts that given a homotopy H:Y×[0,1]XH: Y \times [0,1] \to X and a map f~:YC\tilde{f}: Y \to C such that pf~=H(,0)p \circ \tilde{f} = H(-, 0), there exists a unique homotopy H~:Y×[0,1]C\tilde{H}: Y \times [0,1] \to C starting with f~\tilde{f} such that pH~=Hp \circ \tilde{H} = H.

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