Covering space

A covering map $p:\tilde{X}\to X$ is locally a disjoint union of homeomorphisms onto the same evenly covered open neighbourhood.

A covering map over a topological graph lifts its vertices to all their preimages and each open edge to disjoint open edges. Their endpoints are supplied by the path lifting theorem applied to each parametrizing path of a closed edge. The covering sheets over small vertex stars verify that the resulting graph topology agrees with the original topology upstairs. A constant finite degree of a covering map $d$ therefore multiplies both vertex and edge counts by $d$, proving the Euler characteristic under a finite covering formula for finite topological graphs.

An open neighbourhood $U$ of a point in the base of a covering map $p$ is an evenly covered neighbourhood if $p^{-1}(U)$ is a disjoint union of open subsets, each mapped onto all of $U$ by a homeomorphism obtained by restricting $p$.

A covering sheet over an evenly covered neighbourhood $U$ is one of the disjoint open subsets of $p^{-1}(U)$ mapped onto $U$ by a homeomorphism obtained by restricting the covering map.

For a covering map with nonempty connected base, the degree of a covering map is the common cardinality of its fibres. Each evenly covered neighbourhood makes this number locally constant, so connectedness makes it constant. The total covering space need not be connected. On a path-connected base the path lifting theorem gives a fibre bijection by path lifting. A disconnected base may have different degrees on different connected components.

A topological space is semilocally simply connected if every point has an open neighbourhood whose loops based at that point become null-homotopic in the whole space. The neighbourhood itself need not be simply connected. Together with path connectedness and local path connectedness, this condition gives the classical existence and classification of connected covering spaces by subgroups of the fundamental group.

A covering map between Riemannian manifolds that is a local isometry. The metric on the covering manifold is the pullback of the base metric; its deck transformations are isometries. This local metric agreement makes the Laplace-Beltrami operator commute with pullback of smooth functions.

For an $n$-sheeted covering space $f:X\to Y$, define the transfer of a singular simplex as the sum of its $n$ lifts. Restriction of these lifts to any face gives exactly the lifts of that face, so transfer is a chain map. Its induced map $\mathrm{tr}:H_i(Y;R)\to H_i(X;R)$ satisfies $f_{\ast }\mathrm{tr}=n\mathrm{id}$ for any commutative coefficient ring $R$. If $n$ is invertible in $R$, then $n^{-1}\mathrm{tr}$ is a right inverse of $f_{\ast }$. In particular a finite covering surjects onto the base's rational homology.

There are three connected two-sheeted covering spaces of the wedge of two circles, distinguished by the three nonzero homomorphisms from its fundamental group to $\mathbb{Z}/2\mathbb{Z}$. If exactly one circle exchanges sheets, the covering graph has two parallel joining edges and one loop at each vertex. If both circles exchange sheets, it has four parallel joining edges. These give two homeomorphism types of total spaces even though they give three inequivalent covers of the fixed base.

An oriented coordinate neighborhood of a smooth manifold gives two sheets distinguished by the sign of its orientation. On an overlap, the sheets agree where the two local orientation forms differ by a positive function. These open agreement sets define a two-sheeted smooth covering space, whose projection is locally a diffeomorphism.

For a connected smooth manifold, every component of the orientation double cover maps onto the base by the path lifting theorem. There can therefore be at most two components. A global orientation splits the cover into two copies of the base. Conversely, if there are two components, each meets every fibre once, so its inverse projection gives a smooth global choice of orientation.

At $(p,o)$ in the orientation double cover, use the derivative of the covering projection to transport the chosen orientation $o$ to the tangent space of the cover. In local oriented sheets this prescription is smooth, so it defines a global orientation. The involution $(p,o)\mapsto (p,-o)$ reverses it.

A homotopy into the base of a covering space has a unique continuous lift once its initial map is continuously lifted. Local inverse sheets construct the lift along each time interval, and compactness of that interval permits a finite subdivision locally around each point of the homotopy's parameter space. Uniqueness on overlapping intervals follows from path lifting theorem uniqueness. In a homotopy of paths with endpoints fixed, a chosen lifted endpoint remains fixed because it varies continuously in a discrete fiber.

A branched covering is a map that is a covering space away from a branch set and has a specified local covering model at that set. For a smooth two-fold covering branched over a codimension-two submanifold, the local model is $(z,u)\mapsto (z^2,u)$. A two-fold branched cover of a knot is the basic three-dimensional example.

For a finite group generated by $a,b$, take the regular covering of a thrice-punctured sphere with monodromy $a,b,(ab{)}^{-1}$ and fill each lifted puncture by a disk. Locally the projection is $z\mapsto z^m$, where $m$ is the order of its monodromy. The result is a closed oriented topological surface with an orientation-preserving group action. The nontrivial stabiliser subgroups are the conjugates of $⟨a⟩$, $⟨b⟩$ and $⟨ab⟩$ at the three types of ramification points. Counting lifted cells gives the displayed Euler characteristic. A subgroup acts freely exactly when it meets every conjugate of these three cyclic groups trivially. Its quotient is then a regular covering of degree equal to the subgroup order, and quotient Euler characteristic is divided by that order.

The connected covering space corresponding to $\ker \alpha$ has deck transformation group $\mathbb{Z}$. Choosing a deck generator makes its integral homology a module over the Laurent polynomial ring $\mathbb{Z}[t,t^{-1}]$.

Given an integral epimorphism $\alpha :{\pi }_1(X)\to \mathbb{Z}$, the Alexander module is the first integral homology of the corresponding infinite cyclic cover associated to an epimorphism, with $t$ acting by its chosen deck transformation. Its order is the greatest common divisor of maximal presentation matrix minors when it is finitely presented and torsion. For the meridional epimorphism of a knot exterior, this is the Alexander module of a knot.

For a connected space, this is the connected covering corresponding to the kernel of the homomorphism from its fundamental group to its abelianization. Its deck transformation group is $H_1(Y;\mathbb{Z})$. Lifted cells make its cellular chain complex a complex over the group ring $\mathbb{Z}[H_1(Y)]$.

For $C_2\ast \mathbb{Z}\to S_3$ taking the two generators to a transposition and a $3$-cycle, the kernel's six-sheeted regular covering space consists of three spheres and two circles. Each sphere has two marked points, and each circle meets one point on each sphere. It retracts onto a rank-four graph by replacing each sphere by a meridian arc, but admits no deformation retraction onto any graph because its degree-two homology is ${\mathbb{Z}}^3$.

For $n\ge 1$, mapby sending the projective-plane generator and the first circle generator to the two standard generators. The kernel defines a connected regular four-sheeted cover ${\tilde{X}}_n$. Its lifted cellular boundary in degree two is multiplication by $1+v$ on $\mathbb{Z}[C_2\times C_2]$, so

Let $p:\tilde{X}\to X$ be a covering map and let $Y$ be path-connected. If two lifts ${\tilde{f}}_1,{\tilde{f}}_2:Y\to \tilde{X}$ of the same map $f:Y\to X$ agree at one point, they agree everywhere. Join that point to any other by a path and apply uniqueness in the path lifting theorem.

Lifting loops based at $x_0$ permutes the fibre $p^{-1}(x_0)$ and defines the monodromy action of ${\pi }_1(X,x_0)$ on that fibre. A connected covering has transitive monodromy.

For a nondegenerate embedded smooth algebraic curve over the complex numbers, the cover that chooses one point of a transverse hyperplane section has transitive monodromy. The analogous incidence space for two distinct ordered points is irreducible, giving two-transitivity. A loop around an ordinary tangent hyperplane exchanges two intersection points. Two-transitivity conjugates this transposition to every transposition, so the monodromy is the full symmetric group. In particular its action on fixed-size subsets is transitive.

Lifting a based loop in a covering space from every point of its finite fibre gives a permutation of that fibre. Composition of lifted loops yields the monodromy action of a covering space. If a closed geodesic of length $\ell$ has a cycle of length $m$ in its monodromy permutation, the corresponding lift closes after $m$ traversals and has length $m\ell$. For a primitive base geodesic, these cycles describe the primitive lift components.

A deck transformation of a covering $p:\tilde{X}\to X$ is a homeomorphism $g:\tilde{X}\to \tilde{X}$ satisfying $p\circ g=p$. Every double covering has a canonical nonidentity deck transformation that exchanges the two points in each fibre.

The deck transformation group consists of all deck transformations under composition. For a connected regular covering corresponding to a normal subgroup $H◃{\pi }_1(X)$, it is isomorphic to ${\pi }_1(X)/H$.

If a finite group $G$ is generated by $h$ elements, send the $a$-generators of the genus-$h$ surface group to those generators and its $b$-generators to the identity. The surface relation is satisfied, so this is a surjective group homomorphism. Its kernel defines a connected regular covering space, on which the deck transformation group $G$ acts freely. The lifted orientation and multiplicativity of the Euler characteristic give genus $g=1+|G|(h-1)$.

Every finitely generated group acts freely and properly discontinuously by Riemannian isometries on a connected manifold in any dimension $d\ge 3$. Choose a surjection $F_r\to T$ and the regular covering of ${\#}_r(S^1\times S^{d-1})$ associated with its kernel. Pulling back a metric makes the deck transformation group isometric. Uniqueness of lifts makes its action free. The cover can be noncompact and need not be simply connected; no claim about realizing every group as a closed three-manifold fundamental group is involved.

For a connected covering, restriction to one fibre identifies the deck transformation group with the centralizer in the fibre's symmetric group of the monodromy action of a covering space. A deck transformation commutes with every lifted loop, and every commuting fibre permutation extends uniquely over the covering by path lifting.

Every two-sheeted covering of a path-connected base has a nonidentity deck transformation that exchanges the two sheets. In monodromy terms, every subgroup of $S_2$ is centralized by its unique transposition.

For a double covering $p:X\to Y$, the mod-two transfer sends a singular simplex of $Y$ to the sum of its two lifts. Together with the chain map $p_{\#}$, it gives a short exact sequence

A universal cover is a simply connected covering space. The universal cover of a cylinder is the plane, with the angular coordinate unwrapped to a real coordinate.

The universal cover of a connected closed manifold of dimension three with finite fundamental group is a closed simply connected three-manifold. It is orientable, since its orientation character has trivial domain. Its first integral homology vanishes, and the universal coefficient theorem for cohomology then gives $H^1=0$. Integral Poincare duality gives $H_2=H^1=0$, while $H_0$ and $H_3$ are $\mathbb{Z}$. Thus it is an integral homology sphere. This proof does not require classifying simply connected three-manifolds.

For a locally path-connected subspace $A\subseteq X$, a selected component of the inverse image in a universal cover covers the corresponding component $A_0$ of $A$. Its covering subgroup is $\ker ({\pi }_1(A,a_0)\to {\pi }_1(X,a_0))$; surjectivity onto all of $A$ additionally requires $A$ to be path connected.

For a path-connected locally path-connected base, any two based universal covers lift to one another. The two composites and the respective identity maps are based lifts of the same covering maps, so uniqueness of lifts makes the composites identities. The lift is therefore a homeomorphism.

Let $X$ have contractible universal cover, let $K$ be a path-connected simplicial complex, and let $i:K^1\hookrightarrow K$. A map $f:|K^1|\to X$ extends to $|K|$ exactly when its induced fundamental-group map factors through $i_{\ast }$. The factorization kills every 2-simplex boundary; all higher-dimensional boundary maps lift to the contractible universal cover and are null-homotopic.

Under the usual local hypotheses, based connected coverings of $X$ correspond to subgroups of ${\pi }_1(X)$; changing the point above the base point conjugates the subgroup. Normal subgroups correspond to regular coverings.

For a path-connected, locally path-connected, semilocally simply connected space $X$, base point $x_0$, and subgroup $H\le {\pi }_1(X,x_0)$, take paths starting at $x_0$ modulo the relation $\alpha \sim \beta$ when their endpoints agree and $[\alpha {\beta }^{-1}]\in H$. Projection sends a class to its endpoint. Over a path-connected neighbourhood whose loops die in $X$, classes $[\alpha \eta ]$, with $\eta$ inside that neighbourhood, form a sheet mapping homeomorphically to it. These sheets define a covering space. A loop lifts closed precisely when its class belongs to $H$, so the injective induced fundamental group map has image $H$.

A connected covering is regular when its deck transformation group acts transitively on every fibre. Under the classification of coverings, this is equivalent to the corresponding subgroup of the fundamental group being normal.

For a connected covering corresponding to $H\le {\pi }_1(X)$, every fibre has cardinality $[{\pi }_1(X):H]$.

If the base of a covering is path-connected, a path from $x_0$ to $x_1$ gives a bijectionby sending each initial lift point to the endpoint of its unique lifted path. Lifting the reversed path gives the inverse bijection.

For a presentation complex, the covering associated with a subgroup has one vertex for each coset. A generator gives directed edges according to its coset action, and every relator has one lifted 2-cell beginning at each vertex.

Choose permutations ${\alpha }_1,\dots ,{\alpha }_r\in S_n$. A covering of a wedge of $r$ oriented circles has fibre $\{1,\dots ,n\}$ and an edge labelled $j$ from $i$ to ${\alpha }_j(i)$ above the $j$th circle. It is connected exactly when the generated permutation group is transitive, and its deck group is the centralizer of that group in $S_n$.

For a connected regular covering corresponding to $H\le {\pi }_1(X,x_0)$, the subgroup $H$ is normal andThe quotient acts on a fibre by endpoints of lifted loops, and regularity makes this action transitive with kernel $H$.

A connected covering is normal, or regular, when its deck transformations act transitively on each fibre. Under the subgroup classification, this is equivalent to the associated subgroup of the fundamental group being normal.

A universal covering has simply connected total space, so its associated subgroup of the base fundamental group is trivial. The trivial subgroup is normal; hence every universal covering map is a normal covering map.

For a connected degree-$n$ cover of a closed orientable surface ${\Sigma }_g$, normality is forced when $n=1$, when $n=2$, or when $g=1$. For $g=0$, existence itself forces $n=1$. For every $g\ge 2$ and $n\ge 3$, nonnormal connected degree-$n$ covers exist.

For $g\ge 2$ and $n\ge 3$, sendwhere $\sigma =(12\cdots n)$ and $\tau =(12)$ in $S_n$, and send all remaining surface generators to the identity. The two commutators cancel, and $\sigma ,\tau$ generate $S_n$. The inverse image of a point stabilizer is a nonnormal subgroup of index $n$, hence defines a connected nonnormal degree-$n$ cover.

A path in the base of a covering has a unique lift after its initial point in the fibre is chosen. Subdivide its compact parameter interval so that each segment lies in an evenly covered set, then use the inverse sheet maps successively.

For a covering $p:(\tilde{X},{\tilde{x}}_0)\to (X,x_0)$ and a based map $f:(Y,y_0)\to (X,x_0)$ from a path-connected locally path-connected space, a based lift $\tilde{f}$ exists exactly when

An action of $G$ on $X$ is a covering space action when every point has a neighbourhood $U$ disjoint from $gU$ for every nonidentity $g$. The quotient map is then locally a covering map, although without stronger properness assumptions its quotient need not be Hausdorff.

If $\Gamma$ is a discrete subgroup of a topological group $G$, then left translation by $\Gamma$ is a covering-space action. Choose an identity neighbourhood $U$ with $U{U}^{-1}\cap \Gamma =\{1\}$; the translates $\gamma U$ are pairwise disjoint, and the quotient map $G\to \Gamma \backslash G$ restricts to a homeomorphism on each translate.

On ${\mathbb{R}}^2\setminus \{0\}$, the powers of $T(x,y)=(2x,y/2)$ act freely with locally disjoint translates. Nevertheless $(2^{-n},1)\to (0,1)$ while $T^n(2^{-n},1)\to (1,0)$, so the two distinct limiting orbits cannot be separated in the quotient.

Lifting the hyperbolic-scaling action to the universal cover of ${\mathbb{C}}^{\ast }$ preserves the two inseparable limiting orbits. The covering surface is biholomorphic to $\mathbb{C}$, so a simply connected Riemann surface can still have a non-Hausdorff quotient by a covering space action of homeomorphisms.