A covering map satisfies the local triviality condition. Intuitively, such maps locally project a "stack of pancakes" above an open region, U, onto U.

In mathematics, more specifically algebraic topology, a covering map is a continuous surjective function p ^a[›] from a topological space, C, to a topological space, X, such that each point in X has a neighbourhood evenly covered by p. This means that for each point x in X, there is associated an ordered pair, (K, U), where U is a neighborhood of x and where K is a collection of disjoint open sets in C, each of which gets mapped homeomorphically, via p, to U (as shown in the image). In particular, this means that every covering map is necessarily a local homeomorphism. Under this definition, C is called a covering space of X.

Covering spaces play an important role in homotopy theory, harmonic analysis, Riemannian geometry and differential topology. In Riemannian geometry for example, ramification is a generalization of the notion of covering maps. Covering spaces are also deeply intertwined with the study of homotopy groups and, in particular, the fundamental group. An important application comes from the result that, if X is a "sufficiently good" topological space, there is a bijection from the collection of all isomorphism classes of connected coverings of X and subgroups of the fundamental group of X.

Formal definition[link]

Let X be a topological space. A covering space of X is a space C together with a continuous surjective map

Failed to parse (Missing texvc executable; please see math/README to configure.): p \colon C \to X\,

such that for every x ∈ X, there exists an open neighborhood U of x, such that p⁻¹(U) (the inverse image of U under p) is a disjoint union of open sets in C, each of which is mapped homeomorphically onto U by p.

The map p is called the covering map: the space X is often called the base space of the covering and the space C is called the total space of the covering. For any point x in the base the inverse image of x in C is necessarily a discrete space called the fiber over x.

The special open neighborhoods U of x given in the definition are called evenly-covered neighborhoods. The evenly-covered neighborhoods form an open cover of the space X. The homeomorphic copies in C of an evenly-covered neighborhood U are called the sheets over U. One generally pictures C as "hovering above" X, with p mapping "downwards", the sheets over U being horizontally stacked above each other and above U, and the fiber over x consisting of those points of C that lie "vertically above" x. In particular, covering maps are locally trivial. This means that locally, each covering map is 'isomorphic' to a projection in the sense that there is a homeomorphism from the pre-image of an evenly covered neighbourhood U, to U X F, where F is the fiber, satisfying the local trivialization condition. That is, if we project this homeomorphism onto U (and thus the composition of the projection with this homeomorphism will be a map from the pre-image of U to U), the derived composition will equal p.

Alternative definitions[link]

Many authors impose some connectivity conditions on the spaces X and C in the definition of a covering map. In particular, many authors require both spaces to be path-connected and locally path-connected.^{[citation needed]} This can prove helpful because many theorems hold only if the spaces in question have these properties. Some authors omit the assumption of surjectivity, for if X is connected and C is nonempty then surjectivity of the covering map actually follows from the other axioms.

Examples[link]

Consider the unit circle Failed to parse (Missing texvc executable; please see math/README to configure.): S^1 \subseteq \mathbb{R}^2 . The map Failed to parse (Missing texvc executable; please see math/README to configure.): p : \mathbb{R} \rightarrow S^1

with 

Failed to parse (Missing texvc executable; please see math/README to configure.): p (t) = ( \cos t, \sin t )

is a cover where each point of Failed to parse (Missing texvc executable; please see math/README to configure.): S^1

is covered infinitely often.

As another example, take the complex plane with the origin removed (denoted by Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbb{C}^* ), and pick a non-zero integer Failed to parse (Missing texvc executable; please see math/README to configure.): n . Then Failed to parse (Missing texvc executable; please see math/README to configure.): q_n : \mathbb{C}^* \rightarrow \mathbb{C}^*

given by 

Failed to parse (Missing texvc executable; please see math/README to configure.): q_n (z) = z^n

is a cover. Here, every fiber has Failed to parse (Missing texvc executable; please see math/README to configure.): n

elements. The map Failed to parse (Missing texvc executable; please see math/README to configure.): q_n
leaves the unit circle Failed to parse (Missing texvc executable; please see math/README to configure.): S^1
invariant, and the restriction of this map to Failed to parse (Missing texvc executable; please see math/README to configure.): S^1
is an Failed to parse (Missing texvc executable; please see math/README to configure.): n

-fold cover of the circle by itself.

In fact, Failed to parse (Missing texvc executable; please see math/README to configure.): S^1

and Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbb{R}
are the only connected covering spaces of the circle. To prove this, we first note that the fundamental group of the circle is isomorphic to the additive group of integers Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbb{Z}

. As follows from the correspondence between equivalence classes of connected coverings and conjugacy classes of subgroups of the fundamental group of the base space discussed below, a connected covering Failed to parse (Missing texvc executable; please see math/README to configure.): f : C \rightarrow S^1

is determined by a subgroup Failed to parse (Missing texvc executable; please see math/README to configure.): f_\# ( \pi_1 (C) )
of Failed to parse (Missing texvc executable; please see math/README to configure.): \pi_1 (S^1) = \mathbb{Z}

, where Failed to parse (Missing texvc executable; please see math/README to configure.): f_\#

is the induced homomorphism. The group Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbb{Z}
is abelian and it only has two kinds of subgroups: the trivial subgroup (which has infinite subgroup index in Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbb{Z}

) and the subgroups Failed to parse (Missing texvc executable; please see math/README to configure.): H_n = \{ kn \mid k \in \mathbb{Z} \}

for Failed to parse (Missing texvc executable; please see math/README to configure.): n = 1, 2, 3, \ldots

, where Failed to parse (Missing texvc executable; please see math/README to configure.): H_n

has index Failed to parse (Missing texvc executable; please see math/README to configure.): n
in Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbb{Z}

. Each of the subgroups Failed to parse (Missing texvc executable; please see math/README to configure.): H_n

of Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbb{Z}
is realized by the covering Failed to parse (Missing texvc executable; please see math/README to configure.): q_n : S^1 \rightarrow S^1
since one can check that Failed to parse (Missing texvc executable; please see math/README to configure.): (q_n)_\# : \mathbb{Z} \rightarrow \mathbb{Z}
maps an integer Failed to parse (Missing texvc executable; please see math/README to configure.): k
to Failed to parse (Missing texvc executable; please see math/README to configure.): kn
and hence, Failed to parse (Missing texvc executable; please see math/README to configure.): (q_n)_\# (\mathbb{Z}) = H_n

. The trivial subgroup of Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbb{Z}

is realized by the covering Failed to parse (Missing texvc executable; please see math/README to configure.): p : \mathbb{R} \rightarrow S^1

, since Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbb{R}

is simply connected and has trivial fundamental group and hence Failed to parse (Missing texvc executable; please see math/README to configure.): p_\# (\pi_1 (\mathbb{R}) ) = \{ 0 \}

, the trivial subgroup of Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbb{Z} . Since the total space of the coverings Failed to parse (Missing texvc executable; please see math/README to configure.): q_n

is Failed to parse (Missing texvc executable; please see math/README to configure.): S^1
and the total space of the covering Failed to parse (Missing texvc executable; please see math/README to configure.): p
is Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbb{R}

, this shows that every connected cover of Failed to parse (Missing texvc executable; please see math/README to configure.): S^1

is either Failed to parse (Missing texvc executable; please see math/README to configure.): S^1
or Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbb{R}

.

A further example, originating from physics (see quantum mechanics), is the special orthogonal group Failed to parse (Missing texvc executable; please see math/README to configure.): SO (3) of rotations of Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbb{R}^3 , which has the "double" covering group Failed to parse (Missing texvc executable; please see math/README to configure.): SU (2) of unitary rotations of Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbb{C}^2

(in quantum mechanics acting as the group of spinor rotations). Both groups have identical Lie algebras, but only Failed to parse (Missing texvc executable; please see math/README to configure.): SU (2)
is simply connected.

Properties[link]

Common local properties: Every cover p : C → X is a local homeomorphism (i.e. to every Failed to parse (Missing texvc executable; please see math/README to configure.): c\in C

there exists an open set A in C containing c and an open set B in X such that the restriction of p to A yields a homeomorphism between A and B). This implies that C and X share all local properties. If X is simply connected and C is connected, then this holds globally as well, and the covering p is a homeomorphism.

Homeomorphism of the fibres: For every x in X, the fiber over x is a discrete subset of C. On every connected component of X, the fibers are homeomorphic.

If X is connected, there is a discrete space F such that for every x in X the fiber over x is homeomorphic to F and, moreover, for every x in X there is a neighborhood U of x such that its full pre-image p⁻¹(U) is homeomorphic to U x F. In particular, the cardinality of the fiber over x is equal to the cardinality of F and it is called the degree of the cover p : C → X. Thus, if every fiber has n elements, we speak of an n-fold covering (for the case n = 1, the covering is trivial; when n = 2, the covering is a double cover; when n = 3, the covering is a triple cover and so on).

The lifting property: If p : C → X is a cover and γ is a path in X (i.e. a continuous map from the unit interval [0,1] into X) and Failed to parse (Missing texvc executable; please see math/README to configure.): c\in C

is a point "lying over" γ(0) (i.e. p(c) = γ(0)), then there exists a unique path Γ in C lying over γ (i.e. p o Γ = γ) and with Γ(0) = c.  The curve Γ is called the lift of γ. If x and y are two points in X connected by a path, then that path furnishes a bijection between the fiber over x and the fiber over y via the lifting property. 

A more general lifting property is described as follows:

Let p : C → X be a cover and let f be a continuous map from Z to X where Z is path connected and locally path connected. Let z in Z be a base-point, let x = f(z) and let c in C be in the fiber over x, that is such that p(c)=x.

Then there exists a lift of f (that is, a continuous map g : Z → C such that p o g = f and g(z)=c) if and only if for the induced homomorphisms at the level of the fundamental groups we have

Failed to parse (Missing texvc executable; please see math/README to configure.): f_\#(\pi_1(Z,z))\subset p_\#(\pi_1(C,c)).

    (♠)

Moreover, if such a lift g of f exists, it is unique.

In particular, if the space Z is assumed to be simply connected (so that π₁(Z,z) = 1), condition (♠) is automatically satisfied and every continuous map from Z to X can be lifted. Since the unit interval [0,1] is simply connected, the lifting property for paths is a special case of the lifting property for maps stated above.

If p : C → X is a covering and c ∈ C and x ∈ X are such that p(c) = x, then the induced homomorphism p_# : π₁(C,c) → π₁(X,x) is injective and the induced homomorphisms p_# : π_n(C,c) → π_n(X,x) are isomorphisms for all n ≥ 2. Both of these statements can be deduced from the lifting property for continuous maps. Surjectivity of p_# for n ≥ 2 follows from the fact that for n ≥ 2 the sphere Sⁿ is simply connected and hence every continuous map from Sⁿ to X can be lifted to C.

Equivalence: Let p₁ : C₁ → X and p₂ : C₂ → X be two coverings. One says that the two coverings p₁ and p₂ are equivalent if there exists a homeomorphism p₂₁ : C₂ → C₁ and such that p₂ = p₁ o p₂₁. Equivalence classes of coverings correspond to conjugacy classes of subgroups of the fundamental group of X, as discussed below. If p₂₁ : C₂ → C₁ is a covering (rather than a homeomorphism) and p₂ = p₁ o p₂₁, then one says that p₂ dominates p₁.

Since coverings are local homeomorphisms, a covering of a topological n-manifold is an n-manifold. However a space covered by an n-manifold may be a non-Hausdorff manifold. An example is given by letting C be the plane with the origin deleted and X the quotient space obtained by identifying every point (x,y) with (2x, y/2). If p:C → X is the quotient map then it is a covering since the action of Z on C generated by f(x,y) = (2x,y/2) is properly discontinuous. The points p(1,0) and p(0,1) do not have disjoint neighborhoods in X.

Any covering space of a differentiable manifold may be equipped with a (natural) differentiable structure that turns p (the covering map in question) into a local diffeomorphism – a map with constant rank n.

Universal covers[link]

A connected covering space is a universal cover if it is simply connected. The name universal cover comes from the following important property: if the mapping q : D → X is a universal cover of the space X and the mapping p : C → X is any cover of the space X where the covering space C is connected, then there exists a covering map f : D → C such that p ◦ f = q. This can be phrased as

The universal cover of the space X covers all connected covers of the space X.

The map f is unique in the following sense: if we fix a point x in the space X and a point d in the space D with q(d) = x and a point c in the space C with p(c) = x, then there exists a unique covering map f : D → C such that p ◦ f = q and f(d) = c.

If the space X has a universal cover then that universal cover is essentially unique: if the mappings q₁ : D₁ → X and q₂ : D₂ → X are two universal covers of the space X then there exists a homeomorphism f : D₁ → D₂ such that q₂ ◦ f = q₁.

The space X has a universal cover if it is connected, locally path-connected and semi-locally simply connected. The universal cover of the space X can be constructed as a certain space of paths in the space X.

The example R → S¹ given above is a universal cover. The map S³ → SO(3) from unit quaternions to rotations of 3D space described in quaternions and spatial rotation is also a universal cover.

If the space X carries some additional structure, then its universal cover normally inherits that structure:

• if the space X is a manifold, then so is its universal cover D
• if the space X is a Riemann surface, then so is its universal cover D, and p is a holomorphic map
• if the space X is a Lorentzian manifold, then so is its universal cover. Furthermore, suppose the subset p⁻¹(U) is a disjoint union of open sets each of which is diffeomorphic with U by the mapping p. If the space X contains a closed timelike curve, then the space X is timelike multiply connected (no CTC can be timelike homotopic to a point, as that point would not be causally well-behaved), its universal (diffeomorphic) cover is timelike simply connected (it does not contain a CTC).
• if X is a Lie group (as in the two examples above), then so is its universal cover D, and the mapping p is a homomorphism of Lie groups. In this case the universal cover is also called the universal covering group. This has particular application to representation theory and quantum mechanics, since ordinary representations of the universal covering group (D) are projective representations of the original (classical) group (X).

The universal cover first arose in the theory of analytic functions as the natural domain of an analytic continuation.

G-coverings[link]

Let G be a discrete group acting on the topological space X. It is natural to ask under what conditions the projection, Failed to parse (Missing texvc executable; please see math/README to configure.): X \to X/G , from X to the orbit space, X/G, is a covering map. This is not always true since the action may have fixed points. An example for this is the cyclic group of order 2 acting on a product Failed to parse (Missing texvc executable; please see math/README to configure.): X \times X

by the twist action where the non-identity element acts by Failed to parse (Missing texvc executable; please see math/README to configure.): (x, y) \mapsto (y, x) 

. Thus the study of the relation between the fundamental groups of X and X/G is not so straightforward.

However the group G does act on the fundamental groupoid of X, and so the study is best handled by considering groups acting on groupoids, and the corresponding orbit groupoids. The theory for this is set down in Chapter 11 of the book Topology and groupoids referred to below. The main result is that for discontinuous actions of a group G on a Hausdorff space X which admits a universal cover, then the fundamental groupoid of the orbit space X/G is isomorphic to the orbit groupoid of the fundamental groupoid of X, i.e. the quotient of that groupoid by the action of the group G. This leads to explicit computations, for example of the fundamental group of the symmetric square of a space.

[edit] Deck transformation group, regular covers

A deck transformation or automorphism of a cover p : C → X is a homeomorphism f : C → C such that p o f = p. The set of all deck transformations of p forms a group under composition, the deck transformation group Aut(p). Deck transformations are also called covering transformations. Every deck transformation permutes the elements of each fiber. This defines a group action of the deck transformation group on each fiber. Note that by the unique lifting property, if f is not the identity and C is path connected, then f has no fixed points.

Now suppose p : C → X is a covering map and C (and therefore also X) is connected and locally path connected. The action of Aut(p) on each fiber is free. If this action is transitive on some fiber, then it is transitive on all fibers, and we call the cover regular. Every such regular cover is a principal G-bundle, where G = Aut(p) is considered as a discrete topological group.

Every universal cover p : D → X is regular, with deck transformation group being isomorphic to the fundamental group Failed to parse (Missing texvc executable; please see math/README to configure.): \pi_1(X) .

The example p : C^× → C^× with p(z) = zⁿ from above is a regular cover. The deck transformations are multiplications with n-th roots of unity and the deck transformation group is therefore isomorphic to the cyclic group C_n.

Another example: Failed to parse (Missing texvc executable; please see math/README to configure.): \,p:\mathbf{C}^*\rightarrow\mathbf{C}^*

with Failed to parse (Missing texvc executable; please see math/README to configure.): p(z)\,=\,z^{n!}
from above is regular.  Here one has a hierarchy of deck transformation groups.  In fact Cx! is a subgroup of Cy!, for Failed to parse (Missing texvc executable; please see math/README to configure.): \,1 \le x \le y \le n

.

Monodromy action[link]

Again suppose p : C → X is a covering map and C (and therefore also X) is connected and locally path connected. If x is in X and c belongs to the fiber over x (i.e. p(c) = x), and γ:[0,1]→X is a path with γ(0)=γ(1)=x, then this path lifts to a unique path in C with starting point c. The end point of this lifted path need not be c, but it must lie in the fiber over x. It turns out that this end point only depends on the class of γ in the fundamental group Failed to parse (Missing texvc executable; please see math/README to configure.): \pi_1(X, x) , and in this fashion we obtain a right group action of Failed to parse (Missing texvc executable; please see math/README to configure.): \pi_1(X, x)

on the fiber over x. This is known as the monodromy action.

So there are two actions on the fiber over x: Aut(p) acts on the left and Failed to parse (Missing texvc executable; please see math/README to configure.): \pi_1(X, x)

acts on the right. These two actions are compatible in the following sense:

f.(c.γ) = (f.c).γ

for all f in Aut(p), c in p ^-1(x) and γ in Failed to parse (Missing texvc executable; please see math/README to configure.): \pi_1(X, x) .

If p is a universal cover, then Aut(p) can be naturally identified with the opposite group of Failed to parse (Missing texvc executable; please see math/README to configure.): \pi_1(X, x)

so that the left action of the opposite group of Failed to parse (Missing texvc executable; please see math/README to configure.): \pi_1(X, x)
coincides with the action of Aut(p) on the fiber over x. Note that  Aut(p) and Failed to parse (Missing texvc executable; please see math/README to configure.): \pi_1(X, x)
are naturally isomorphic in this case (as a group is always naturally isomorphic to its opposite through Failed to parse (Missing texvc executable; please see math/README to configure.): g\mapsto g^{-1}
).

If p is a regular cover, then Aut(p) is naturally isomorphic to a quotient of Failed to parse (Missing texvc executable; please see math/README to configure.): \pi_1(X, x) .

In general (for good spaces), Aut(p) is naturally isomorphic to the quotient of the normalizer of Failed to parse (Missing texvc executable; please see math/README to configure.): p_{\ast}(\pi_1(C, c))

in Failed to parse (Missing texvc executable; please see math/README to configure.): \pi_1(X, x)
over Failed to parse (Missing texvc executable; please see math/README to configure.): p_{\ast}(\pi_1(C, c))

, where Failed to parse (Missing texvc executable; please see math/README to configure.): p(c)=x .

More on the group structure[link]

Let p : C → X be a covering map where both X and C are path-connected. Let x ∈ X be a basepoint of X and let c ∈ C be one of its pre-images in C, that is p(c) = x. There is an induced homomorphism of fundamental groups p_# : π₁(C,c) → π₁(X,x) which is injective by the lifting property of coverings. Specifically if γ is a closed loop at c such that p_#([γ]) = 1, that is p o γ is null-homotopic in X, then consider a null-homotopy of p o γ as a map f : D² → X from the 2-disc D² to X such that the restriction of f to the boundary S¹ of D² is equal to p o γ. By the lifting property the map f lifts to a continuous map g : D² → C such that the restriction of f to the boundary S¹ of D² is equal to γ. Therefore γ is null-homotopic in C, so that the kernel of p_# : π₁(C,c) → π₁(X,x) is trivial and thus p_# : π₁(C,c) → π₁(X,x) is an injective homomorphism.

Therefore π₁(C,c) is isomorphic to the subgroup p_# (π₁(C,c)) of π₁(X,x). If c₁ ∈ C is another pre-image of x in C then the subgroups p_# (π₁(C,c)) and p_# (π₁(C,c₁)) are conjugate in π₁(X,x) by p-image of a curve in C connecting c to c₁. Thus a covering map p : C → X defines a conjugacy class of subgroups of π₁(X,x) and one can show that equivalent covers of X define the same conjugacy class of subgroups of π₁(X,x).

For a covering p : C → X the group p_#(π₁(C,c)) can also be seen to be equal to

Failed to parse (Missing texvc executable; please see math/README to configure.): \Gamma_p(c) = \{ [\gamma] : \gamma_C \mbox{ is a closed curve in } C \mbox { passing through } c\in C \}

, the set of homotopy classes of those closed curves γ based at x whose lifts γ_C in C, starting at c, are closed curves at c. If X and C are path-connected, the degree of the cover p (that is, the cardinality of any fiber of p) is equal to the index [π₁(X,x):p_# (π₁(C,c))] of the subgroup p_# (π₁(C,c)) in π₁(X,x).

A key result of the covering space theory says that for a "sufficiently good" space X (namely, if X is path-connected, locally path-connected and semi-locally simply connected) there is in fact a bijection between equivalence classes of path-connected covers of X and the conjugacy classes of subgroups of the fundamental group π₁(X,x). The main step in proving this result is establishing the existence of a universal cover, that is a cover corresponding to the trivial subgroup of π₁(X,x). Once the existence of a universal cover C of X is established, if H ≤ π₁(X,x) is an arbitrary subgroup, the space C/H is the covering of X corresponding to H. One also needs to check that two covers of C corresponding to the same (conjugacy class of) subgroup of π₁(X,x) are equivalent. Connected cell complexes and connected manifolds are examples of "sufficiently good" spaces.

Let N(Γ_p) be the normalizer of Γ_p in π₁(X,x). The deck transformation group Aut(p) is isomorphic to the quotient group N(Γ_p)/Γ_p. If p is a universal covering, then Γ_p is the trivial group, and Aut(p) is isomorphic to π₁(X).

Let us reverse this argument. Let N be a normal subgroup of π₁(X,x). By the above arguments, this defines a (regular) covering p : C → X . Let c₁ in C be in the fiber of x. Then for every other c₂ in the fiber of x, there is precisely one deck transformation that takes c₁ to c₂. This deck transformation corresponds to a curve g in C connecting c₁ to c₂.

Relations with groupoids[link]

One of the ways of expressing the algebraic content of the theory of covering spaces is using groupoids and the fundamental groupoid. The latter functor gives an equivalence of categories

Failed to parse (Missing texvc executable; please see math/README to configure.): \pi_1: \mathrm{TopCov}(X) \to \mathrm{GpdCov}(\pi_1 X)

between the category of covering spaces of a reasonably nice space X and the category of groupoid covering morphisms of Failed to parse (Missing texvc executable; please see math/README to configure.): \pi_1 X . Thus a particular kind of map of spaces is well modelled by a particular kind of morphism of groupoids. The category of covering morphisms of a groupoid G is also equivalent to the category of actions of G on sets, and this allows the recovery of more traditional classifications of coverings. Proofs of these facts are given in the book `Topology and Groupoids' referenced below.

Relations with classifying spaces and group cohomology[link]

If X is a connected cell complex with homotopy groups π_n(X) =0 for all n ≥ 2, then the universal covering space T of X is contractible, as follows from applying the Whitehead theorem to T. In this case X is a classifying space or K(G,1) for G = π₁(X).

Moreover, for every n ≥ 0 the group of cellular n-chains C_n(T) (that is, a free abelian group with basis given by n-cells in T) also has a natural ZG-module structure. Here for an n-cell σ in T and for g in G the cell g σ is exactly the translate of σ by a covering transformation of T corresponding to g. Moreover, C_n(T) is a free ZG-module with free ZG-basis given by representatives of G-orbits of n-cells in T. In this case the standard topological chain complex

Failed to parse (Missing texvc executable; please see math/README to configure.): \cdots \overset{\partial}{\to} C_n(T)\overset{\partial}{\to} C_{n-1}(T)\overset{\partial}{\to} \cdots \overset{\partial}{\to} C_0(T)\overset{\varepsilon}{\to} \mathbb Z,

where ε is the augmentation map, is a free ZG-resolution of Z (where Z is equipped with the trivial ZG-module structure, g m = m for every g ∈ G and every m ∈ Z). This resolution can be used to compute group cohomology of G with arbitrary coefficients.

Generalizations[link]

As a homotopy theory, the notion of covering spaces works well when the deck transformation group is discrete, or, equivalently, when the space is locally path-connected. However, when the deck transformation group is a topological group whose topology is not discrete, difficulties arise. Some progress has been made for more complex spaces, such as the Hawaiian earring; see the references there for further information.

Applications[link]

Gimbal lock occurs because the map T³ → RP³ is not a covering space. This animation shows a set of three gimbals mounted together to allow three degrees of freedom. When all three gimbals are lined up (in the same plane), the system can only move in two dimensions from this configuration, not three, and is in gimbal lock. In this case it can pitch or yaw, but not roll (rotate in the plane that the axes all lie in).

An important practical application of covering spaces occurs in charts on SO(3), the rotation group. This group occurs widely in engineering, due to 3-dimensional rotations being heavily used in navigation, nautical engineering, and aerospace engineering, among many other uses. Topologically, SO(3) is the real projective space RP³, with fundamental group Z/2, and only (non-trivial) covering space the hypersphere S³, which is the group Spin(3), and represented by the unit quaternions. Thus quaternions are a preferred method for representing spatial rotations – see quaternions and spatial rotation.

However, it is often desirable to represent rotations by a set of three numbers, known as Euler angles (in numerous variants), both because this is conceptually simpler, and because one can build a combination of three gimbals to produce rotations in three dimensions. Topologically this corresponds to a map from the 3-torus T³ of three angles to the real projective space RP³ of rotations, and the resulting map has imperfections due to this map being unable to be a covering map. Specifically, the failure of the map to be a local homeomorphism at certain points is referred to as gimbal lock, and is demonstrated in the animation at the right – at some points (when the axes are coplanar) the rank of the map is 2, rather than 3, meaning that only 2 dimensions of rotations can be realized from that point by changing the angles. This causes problems in applications, and is formalized by the notion of a covering space.

See also[link]

• Bethe lattice is the universal cover of a Cayley graph
• Covering graph, a covering space for an undirected graph, and its special case the bipartite double cover.
• Covering group
• Galois connection

Notes[link]

^ a: Some authors do not require covering maps to be surjective; see above for more details.

References[link]

• Farkas, Hershel M.; Irwin Kra (1980). Riemann Surfaces (2nd ed. ed.). New York: Springer. ISBN 0-387-90465-4.  See chapter 1 for a simple review.
• Hatcher, Allen (2002). Algebraic Topology. Cambridge University Press. ISBN 0-521-79540-0. http://www.math.cornell.edu/~hatcher/AT/ATpage.html. 
• Jost, Jurgen (2002). Compact Riemann Surfaces. New York: Springer. ISBN 3-540-43299-X.  See section 1.3
• Massey, William (1991). A Basic Course in Algebraic Topology. New York: Springer. ISBN 0-387-97430-X.  See chapter 5.
• Brown, Ronald (2006). Topology and Groupoids. Charleston, S. Carolina: Booksurge LLC. ISBN 1-4196-2722-8. http://www.bangor.ac.uk/r.brown/topgpds.html.  See chapter 10.
• Categories and groupoids, P.J. Higgins, downloadable reprint of van Nostrand Notes in Mathematics, 1971, which deal with applications of groupoids in group theory and topology.

This article needs additional citations for verification. Please help improve this article by adding citations to reliable sources. Unsourced material may be challenged and removed. (April 2012)

Alice Cooper
Cooper at the Scream Awards, 2007
Background information
Birth name	Vincent Damon Furnier
Born	(1948-02-04) February 4, 1948 (age 64) Detroit, Michigan, United States
Genres	Rock, hard rock, heavy metal, shock rock
Occupations	Singer-songwriter, actor, DJ
Instruments	Vocals, harmonica, guitar
Years active	1964–present
Labels	Straight, Warner Bros., Atlantic, MCA, Epic, Spitfire, Eagle, New West
Website	www.alicecooper.com

Alice Cooper (born Vincent Damon Furnier; February 4, 1948)^[1] is an American rock singer, songwriter, and musician whose career spans more than four decades. With a stage show that features guillotines, electric chairs, fake blood, boa constrictors, and baby dolls, Cooper has drawn equally from horror movies, vaudeville, and garage rock to pioneer a grandly theatrical and violent brand of heavy metal designed to shock.^[2]

Originating in Detroit in the late 1960s, Alice Cooper was originally a band consisting of Furnier on vocals and harmonica, lead guitarist Glen Buxton, Michael Bruce on rhythm guitar, Dennis Dunaway on bass guitar, and drummer Neal Smith. The original Alice Cooper band broke into the international music mainstream with the 1971 hit "I'm Eighteen" from the album Love It to Death, which was followed by the even bigger single "School's Out" in 1972. The band reached their commercial peak with the 1973 album Billion Dollar Babies.

Furnier's solo career as Alice Cooper, adopting the band's name as his own name, began with the 1975 concept album Welcome to My Nightmare; in 2011 he released Welcome 2 My Nightmare, his 19th album as a solo artist, and his 26th album in total. Expanding from his Detroit rock roots, in his career Cooper has experimented with a number of musical styles, including conceptual rock, art rock, hard rock, New Wave, pop rock, experimental rock and industrial rock.

Alice Cooper is known for his social and witty persona offstage; The Rolling Stone Album Guide has called him the world's most "beloved heavy metal entertainer".^[3] Cooper is credited with helping to shape the sound and look of heavy metal, and he is regarded as being the artist who "first introduced horror imagery to rock'n'roll, and whose stagecraft and showmanship have permanently transformed the genre".^[4] Away from music, Cooper is a film actor, a golfing celebrity, a restaurateur and, since 2004, a popular radio DJ with his classic rock show Nights with Alice Cooper.

In 2011 the original Alice Cooper band was inducted into The Rock and Roll Hall of Fame.^[5]

Early life[link]

Cooper was born as Vincent Damon Furnier in Detroit, Michigan, the son of Ella Mae (née McCart) and Ether Moroni Furnier. His father was a lay preacher in the Church of Jesus Christ (also known as the Bickertonite Church) which, historically, is a branch of the Latter Day Saint movement.^[6] He has French Huguenot, Sioux Native American, English, Scottish, and Irish ancestry,^[7] and was named after one of his uncles (Vincent Collier Furnier) and the writer Damon Runyon.^[8] His paternal grandfather, Thurman Sylvester Furnier, was an apostle in the Church of Jesus Christ (Bickertonite). Vincent Furnier was active in his church at the ages of 11 and 12.^[9][10]

While growing up in Detroit, Furnier attended Washington Elementary School, then Nankin Mills Jr. High, now Lutheran High School Westland. Following a series of childhood illnesses, Furnier moved with his family to Phoenix, Arizona, where he attended Cortez High School in north Phoenix.^[11] Vincent had gained admission into the University of Arizona, University of Colorado and University of California-Davis – he declined all these offers.^{[citation needed]}

Recording career[link]

1960s[link]

In 1964 sixteen year-old Furnier was eager to take part in the local annual letterman's talent show, so he gathered fellow cross-country teammates to form a group for the show.^[12] They named themselves The Earwigs. Because they did not know how to play any instruments at the time, they dressed up like The Beatles and mimed their performance to Beatles songs. As a result of winning the talent show and loving the experience of being onstage, the group immediately proceeded to learn how to play instruments they acquired from a local pawn shop. They soon renamed themselves The Spiders, featuring Furnier on vocals, Glen Buxton on lead guitar, John Tatum on rhythm guitar, Dennis Dunaway on bass guitar and John Speer on drums.^{[citation needed]} Musically, the group was inspired by artists such as The Beatles, The Rolling Stones, The Who, The Kinks, The Doors and The Yardbirds. For the next year the band performed regularly around the Phoenix area with a huge black spider's web as their backdrop, the group's first stage prop. In 1965 they recorded their first single, "Why Don't You Love Me" (originally performed by The Blackwells), with Furnier learning the harmonica for the song.^{[citation needed]}

In 1966 The Spiders graduated from high school, and after North High School footballer Michael Bruce replaced John Tatum on rhythm guitar, the band scored a local No. 1 radio hit with "Don't Blow Your Mind", an original composition from their second single release. By 1967 the band had begun to make regular road trips to Los Angeles to play shows. They soon renamed themselves The Nazz and released the single "Wonder Who's Lovin' Her Now", backed with future Alice Cooper track "Lay Down And Die, Goodbye". At around this time drummer John Speer was replaced by Neal Smith. By the end of the year the band had relocated to Los Angeles.^{[citation needed]}

In 1968 upon learning that Todd Rundgren also had a band called Nazz, the band was again in need of another stage name. Believing that the group needed a gimmick to succeed, and that other bands were not exploiting the showmanship potential of the stage, Furnier chose "Alice Cooper" as the band's name and adopted this stage name as his own. Cooper later stated that the name change was one of his most important and successful career moves.^[13]

Nonetheless, at the time Cooper and the band realized that the concept of a male playing the role of a villain, a woman killer, in tattered women's clothing and wearing make-up, would have the potential to cause considerable social controversy and grab headlines. In 2007 in his book Alice Cooper, Golf Monster Cooper stated that his look was inspired in part by film. One of the band's all-time favorite movies was What Ever Happened to Baby Jane starring Bette Davis. "In the movie, Bette wears disgusting caked makeup smeared on her face and underneath her eyes, with deep, dark, black eyeliner." Another movie the band watched over and over was Barbarella. "When I saw Anita Pallenberg playing the Great Tyrant in that movie in 1968, wearing long black leather gloves with switchblades coming out of them, I thought, 'That's what Alice should look like'. That, and a little bit of Emma Peel from The Avengers".^[14]

The classic Alice Cooper group line-up consisted of singer Alice Cooper (Vincent Furnier), lead guitarist Glen Buxton, rhythm guitarist Michael Bruce, bassist Dennis Dunaway and drummer Neal Smith. With the exception of Smith, who graduated from Camelback High School (which is referred to in the song "Alma Mater" on the album School's Out), all of the band members were on the Cortez High School cross-country team, and many of Cooper's stage effects were inspired by their cross-country coach, Emmett Smith^[15] (one of Smith's class projects was to build a working guillotine for slicing watermelons). Cooper, Buxton and Dunaway were also art students, and their admiration for the works of surrealist artists such as Salvador Dalí would further inspire their future stage antics.^{[citation needed]}

One night after an unsuccessful gig at the Cheetah club in Venice, California, where the band emptied the entire room of patrons after playing just ten minutes, they were approached and enlisted by music manager Shep Gordon, who ironically saw the band's negative impact that night as a force that could be turned in a more productive direction. Shep then arranged an audition for the band with composer and renowned record producer, Frank Zappa, who was looking to sign bizarre music acts to his new record label, Straight Records. For the audition Zappa told them to come to his house "at 7 o'clock." The band mistakenly assumed he meant 7 o'clock in the morning. Being woken up by a band willing to play that particular brand of psychedelic rock at seven in the morning impressed Zappa enough to sign them to a three-album deal. Another Zappa-signed act, the all-female GTOs, who liked to "dress the Cooper boys up like full size Barbie dolls," played a major role in developing the band's early onstage look.^[16][17]

Cooper's first album Pretties for You (released in 1969) had a slight psychedelic feel. Although it touched the US charts for one week at No. 193, it was ultimately a critical and commercial failure.

Alice Cooper's "shock rock" reputation apparently developed almost by accident at first. An unrehearsed stage routine involving Cooper, a feather pillow and a live chicken garnered attention from the press; the band decided to capitalize on the tabloid sensationalism, creating in the process a new subgenre, shock rock. Cooper claims that the infamous "Chicken Incident" at the Toronto Rock and Roll Revival concert in September 1969 was an accident. A chicken somehow made its way onto the stage into the feathers of a feather pillow they would open during Cooper's performance, and not having any experience around farm animals, Cooper presumed that, because the chicken had wings, it would be able to fly.^[18] He picked it up and threw it out over the crowd, expecting it to fly away. The chicken instead plummeted into the first few rows occupied by disabled people in wheelchairs, who reportedly proceeded to tear the bird to pieces.^[19]

The next day the incident made the front page of national newspapers, and Zappa phoned Cooper and asked if the story, which reported that he had bitten off the chicken's head and drunk its blood on stage, was true. Cooper denied the rumor, whereupon Zappa told him, "Well, whatever you do, don't tell anyone you didn't do it",^[20] obviously recognizing that such publicity would be priceless for the band.^[21]

The band have later claimed that this period was highly influenced by Pink Floyd, and especially the album Piper at the Gates of Dawn. Glen Buxton could listen to Syd Barrett's guitar for hours at a time.^[22]

1970s[link]

Despite the publicity from the Chicken Incident, the band's second album, Easy Action, released in June 1970, met with the same fate as its predecessor. At around this time the band, fed up with Californians' indifference to their act, relocated to Cooper's birthplace, Detroit, where their bizarre stage act was much better received by the crowds of the Midwest states who were accustomed to the similar hard rock styles of local bands such as The Stooges and The MC5. Despite this, Cooper still managed to receive a cream pie in the face when performing at the Cincinnati Pop Festival. Detroit would remain their steady home base until 1972. "L.A. just didn’t get it," Cooper stated. "They were all on the wrong drug for us. They were on acid and we were basically drinking beer. We fit much more in Detroit than we did anywhere else."^[23]

Alice Cooper appeared at the Woodstock-esque, Strawberry Fields Festival near Toronto, Ontario in August 1970. The band's mix of glam and increasingly violent stage theatrics stood out in stark contrast to the bearded, denim-clad hippie bands of the time.^[24] As Cooper himself stated: "We were into fun, sex, death and money when everybody was into peace and love. We wanted to see what was next. It turned out we were next, and we drove a stake through the heart of the Love Generation".^[25]

In autumn 1970 the Alice Cooper group teamed with producer Bob Ezrin for the recording of their third album Love It to Death. This was the final album in their Straight Records contract and the band's last chance to create a hit. That first success came with the single "I'm Eighteen", released in November 1970, which reached number 21 on the Billboard Hot 100 in early 1971. Not long after the album's release in January 1971 Warner Bros. Records purchased Alice Cooper's contract from Straight and re-issued the album, giving the group a higher level of promotion.^{[citation needed]}

Love It to Death proved to be their breakthrough album, reaching number 35 on the U.S. Billboard 200 album charts. It would be the first of eleven^[26] Alice Cooper group and solo albums produced by Ezrin, who is widely seen as being instrumental in helping to create and develop the band's definitive sound.^[27]

The group's 1971 tour featured a stage show involving mock fights and gothic torture modes being imposed on Cooper climaxing with a staged execution by electric chair, with the band sporting tight, sequined, and color-contrasting glam rock-style costumes made by prominent rock fashion designer Cindy Dunaway (sister of band member Neal Smith, and wife of band member Dennis Dunaway). Cooper's androgynous stage role had developed to present a villainous side, portraying a potential threat to modern society. The success of the band's single and album, and their tour of 1971, which included their first tour of Europe (audience members reportedly included Elton John and a pre-Ziggy David Bowie), provided enough encouragement for Warner Bros. to offer the band a new multi-album contract.

Their follow-up album Killer, released in late 1971, continued the commercial success of Love It to Death and included further single success with "Under My Wheels", "Be My Lover" in early 1972, and "Halo of Flies" which became a Top 10 hit in the Netherlands in 1972. Thematically, Killer expanded on the villainous side of Cooper's androgynous stage role, with its music becoming the soundtrack to the group's morality-based stage show, which by then featured a boa constrictor hugging Cooper on-stage, the murderous axe chopping of bloodied baby dolls, and execution by hanging at the gallows. Back then, the real criticism was aimed at questioning the artists' sexual ambiguity, rather than the stage gore.^{[citation needed]} In January 1972, Cooper was again asked about his peculiar name, and told talk show hostess Dinah Shore that he took the name from a "Mayberry RFD" character.^{[citation needed]}

The summer of 1972 saw the release of the single "School's Out". It went Top 10 in the US, was a No. 1 single in the UK, and remains a staple on classic rock radio to this day. The album School's Out reached No. 2 on the US charts and sold over a million copies. The band now relocated to their new mansion in Greenwich, Connecticut.^[28] With Cooper's on-stage androgynous persona completely replaced with brattiness and machismo, the band solidified their success with subsequent tours in the US and Europe, and won over devoted fans in droves while at the same time horrifying parents and outraging the social establishment.^{[citation needed]} In the United Kingdom, Mary Whitehouse, a well-known campaigner for values of morality and decency, succeeded in having the BBC ban the video for "School's Out"^{[citation needed]} and Member of Parliament Leo Abse petitioned Home Secretary Reginald Maudling to have the group banned altogether from performing in the country.^[29]

In February 1973 Billion Dollar Babies was released worldwide and became the band's most commercially successful album, reaching No. 1 in both the US and UK. "Elected", a late-1972 Top 10 UK hit from the album, which inspired one of the first MTV-style story-line promo videos ever made for a song (three years before Queen's promotional video for "Bohemian Rhapsody"), was followed by two more UK Top 10 singles, "Hello Hooray" and "No More Mr. Nice Guy", the latter of which was the last UK single from the album; it reached No. 25 in the US.^{[citation needed]} The title track, featuring guest vocals by Donovan, was also a US hit single. Around this time Glen Buxton left Alice Cooper briefly due to his waning health.

The group in 1973.

With a string of successful concept albums and several hit singles, the band continued their gruelling schedule and toured the US once again. Continued attempts by politicians and pressure groups to ban their shocking act only served to fuel the myth of Alice Cooper further and generate even greater public interest.^{[citation needed]} Their 1973 US tour broke box office records previously set by The Rolling Stones and raised rock theatrics to new heights; the multi-level stage show by then featured numerous special effects, including Billion Dollar Bills, decapitated baby dolls and mannequins, a dental psychosis scene complete with dancing teeth, and the ultimate execution prop and highlight of the show: the guillotine. The guillotine and other stage effects were designed for the band by magician James Randi, who appeared on stage during some of the shows as executioner. The Alice Cooper group had now reached its peak and it was among the most visible and successful acts in the industry. Beneath the surface, however, the repetitive schedule of recording and touring had begun to take its toll on the band, and Cooper, who was under the constant pressure of getting into character for that night's show, was consistently sighted nursing a can of beer.

Muscle of Love, released at the end of 1973, was to be the last studio album from the classic line-up, and marked Alice Cooper's last UK Top 20 single of the 1970s with "Teenage Lament '74". An unsolicited theme song was recorded for the James Bond movie The Man with the Golden Gun, but a different song of the same name by Lulu was chosen instead. By 1974, the Muscle of Love album had not matched the top-charting success of its predecessor, and the band began to have constant disagreements. For various reasons the band decided to take a hiatus.

During this time Cooper relocated back to Los Angeles and started appearing regularly on TV shows such as Hollywood Squares, and Warner Bros. released the Alice Cooper's Greatest Hits compilation album. It featured classic style artwork and reached the US Top 10, performing better than Muscle of Love. However, the band's 1974 feature film Good to See You Again, Alice Cooper (consisting mainly of 1973 concert footage with 'comedic' sketches woven throughout to a faint storyline), released on a minor theatrical run mostly to drive-in theaters, saw little box office success.

In 1975 Alice released his first solo album, Welcome to My Nightmare. To avoid legal complications over ownership of the group name, Alice Cooper had by then become the singer's new legal name. The success of the solo album marked the final break with the original members of the band with Cooper collaborating with their producer Bob Ezrin, who recruited Lou Reed's backing band, including guitarists Dick Wagner and Steve Hunter, to play on the album. Spearheaded by the US Top 20 hit ballad, "Only Women Bleed", the album was released by Atlantic Records in March of that year and became a Top 10 hit for Cooper. It was a concept album that was based on the nightmare of a child named Steven, featuring narration by classic horror movie film star Vincent Price (who several years after Welcome to My Nightmare guested on Michael Jackson's "Thriller"), and serving as the soundtrack to Cooper's new stage show, which now included more theatrics than ever (including an 8-foot-tall (2.4 m) furry Cyclops which Cooper decapitates and kills).

Cooper (second from right) laughs at Mae West's words after the opening of her last movie in 1978.

By this time, however, alcohol was clearly affecting Cooper's performances. During the Welcome to My Nightmare tour in Vancouver, and only a few songs into the routine, Cooper tripped over a foot-light, staggered a few paces, lost his bearings, and plunged head-first off the stage and onto the concrete floor of the Pacific Coliseum. Some fans, thinking it was all part of the act, reached through the barriers to pull at his blood-matted hair before bouncers could pull him away for help. He was taken to a local hospital, where medical staff stitched his head wound and provided him with a skullcap. Cooper returned to the venue a couple of hours later and tried to perform two more songs, but within minutes he had to call it a night.

Accompanying the album and stage show was the TV special The Nightmare, starring Cooper and Vincent Price, which aired on US prime-time TV in April 1975. The Nightmare (which was later released on home video in 1983 and gained a Grammy Awards nomination for Best Long Form Music Video) was regarded as another groundbreaking moment in rock history. Adding to all that, a concert film, also called Welcome to My Nightmare produced, directed and choreographed by the West Side Story cast member David Winters and filmed live at London's Wembley Arena in September 1975, was released to theaters in 1976.^[30][31] Though it failed at the box office, it later became a midnight movie favorite and a cult classic.

Such was the immense success of Cooper's solo project that he decided to continue alone as a solo artist, and the original band became officially defunct. It was also during this time that Cooper co-founded the legendary drinking club The Hollywood Vampires.

Cooper in 1978.

Following the 1976 US No. 12 ballad hit "I Never Cry";^[32] two albums, Alice Cooper Goes to Hell and Lace and Whiskey; and the 1977 US No. 9 ballad hit "You and Me", it became clear from many performances during his 1977 US tour that Cooper was in dire need of help with his alcoholism (at his alcoholic peak it was rumored that Cooper was consuming up to two cases of Budweiser and a bottle of whiskey a day). Following the tour, Cooper had himself hospitalized in a New York sanitarium for treatment, during which time the live album The Alice Cooper Show was released.

In 1978 a sobered Cooper used his experience in the sanitarium as the inspiration for the semi-autobiographical album From the Inside, which he co-wrote with Bernie Taupin. The release spawned another US Top 20 hit "How You Gonna See Me Now", yet another ballad. The subsequent tour's stage show was based inside an asylum, and was filmed for Cooper's first home video release, The Strange Case of Alice Cooper, in 1979. Around this time, Cooper performed "Welcome to My Nightmare", "You and Me" and "School's Out" on The Muppet Show (episode # 307) on March 28, 1978 (he played one of the devil's henchmen trying to dupe Kermit the Frog and Gonzo into selling their souls). He also appeared in an against-typecasting role as a piano-playing disco bellboy in Mae West's final film, Sextette, and as a villain in the film Sgt. Pepper's Lonely Hearts Club Band. Cooper also led celebrities in raising money to remodel the famous Hollywood Sign in California. Cooper himself contributed over $27,000 to the project, buying an O in the sign in memory of friend and comedian Groucho Marx.

1980s[link]

Cooper's albums from the beginning of the 1980s, Flush the Fashion, Special Forces, Zipper Catches Skin and DaDa, were not as commercially successful as his past releases. Flush the Fashion, produced by Queen producer Roy Thomas Baker, had a thick, edgy New Wave musical sound that baffled even long-time fans, though it still yielded the US Top 40 hit "(We're All) Clones". The album Special Forces featured a more aggressive but consistent form of New Wave style, and included a new version of "Generation Landslide". The following album, Zipper Catches Skin was a more power pop-oriented recording, with lots of quirky high-energy guitar-driven songs. While those three albums engaged the experimental New Wave sound with energetic results, 1983 marked the return collaboration of producer Bob Ezrin and guitarist Dick Wagner with the haunting epic DaDa, the final album in his Warner Bros. contract.

In 1983 after the recording of DaDa, Cooper was re-hospitalized for alcoholism. In a deathly state of health he moved back to Phoenix to save his marriage from collapse, and so that he could receive the support of family and friends. Cooper was finally clean and sober by the time DaDa and The Nightmare home video (of his 1975 TV Special) were released in the fall of that year; however, both releases performed below expectations. Even with The Nightmare scoring a nomination for 1984's Grammy Award for Best Long Form Music Video (he lost to Duran Duran), it was not enough for Warner Bros. to keep Cooper on their books, so in 1984 Cooper became a "free agent" for the first time in his career.

After over a year on hiatus, during which time he spent being a full-time father, perfecting his golf swing every day on the golf course, and finding time to star in the Spanish B-grade horror movie production Monster Dog, Cooper sought to pick up the pieces of his musical career. In 1985 he met and began writing songs with guitarist Kane Roberts. Cooper was subsequently signed to MCA Records, and appeared as guest vocalist on Twisted Sister's song "Be Chrool To Your Scuel". A video was made for the song, featuring Cooper donning his black snake-eyes make-up for the first time since 1979. But any publicity it may have generated toward Cooper's return to the music scene was cut short as the video was promptly banned because of its graphically gory make-up (by Tom Savini), and because of the innumerable zombies in the video and their insatiable appetite for gorging on human flesh.

In 1986 Alice Cooper officially returned to the music industry with the album Constrictor. The album spawned the hits "He's Back (The Man Behind the Mask)" (the theme song for the movie Friday the 13th Part VI: Jason Lives; in the video of the song Cooper was given a cameo role as a deranged psychiatrist) and the fan favorite "Teenage Frankenstein". The Constrictor album was a catalyst for Cooper to make (for the first time since the 1982 Special Forces tour) a triumphant return to the road, on a tour appropriately entitled The Nightmare Returns. The Detroit leg of this tour, which took place at the end of October 1986during Halloween, was captured on film as The Nightmare Returns, and is viewed by some as being the definitive Alice Cooper concert film.^[33] The concert, which received rave reviews in the rock music press,^[34] was also described by Rolling Stone magazine as bringing "Cooper's violent, twisted onstage fantasies to a new generation". The Constrictor album was followed by Raise Your Fist and Yell in 1987, which had an even rougher sound than its predecessor, as well as the Cooper classic "Freedom". The subsequent tour of Raise Your Fist and Yell, which was heavily inspired by the slasher horror movies of the time such as the Friday the 13th series and Nightmare on Elm Street, served up a shocking spectacle similar to its predecessor, and courted the kind of controversy, especially in Europe, that recalled the public outrage caused by Cooper's public performances in America in the early 1970s.

In Britain Labour M.P. David Blunkett called for the show to be banned, saying "I'm horrified by his behaviour – it goes beyond the bounds of entertainment" (even though Blunkett has been blind from birth).^[35] The controversy spilled over into the German segment of the tour, with the German government actually succeeding in having some of the gorier segments of the performance removed.^[36] It was also during the London leg of the tour that Cooper met with a near fatal accident during the hanging execution sequence at the end of the show.^[37]

Constrictor and Raise Your Fist and Yell were recorded with lead guitarist Kane Roberts and bassist Kip Winger, both of whom would leave the band by the end of 1988 (although Kane Roberts played guitar on "Bed of Nails" on 1989's album Trash).

In 1987 Cooper made a brief appearance as a vagrant in the horror movie Prince of Darkness, directed by John Carpenter. His role had no lines and consisted of generally menacing the protagonists before eventually impaling one of them with a bicycle frame. Cooper also appeared at WrestleMania III, escorting wrestler Jake 'The Snake' Roberts to the ring. After the match was over, Cooper got involved and threw Jake's snake Damien at The Honky Tonk Man's manager Jimmy Hart. Jake considered the involvement of Cooper to be an honor, as he had idolized Cooper in his youth and was still a huge fan.

In 1988, Cooper's contract with MCA Records expired and he signed with Epic Records. Then in 1989 his career finally experienced a legitimate revival with the Desmond Child produced and Grammy-nominated album Trash, which spawned a hit single "Poison", which reached No. 2 in the UK and No. 7 in the US, and a worldwide arena tour.

1990s[link]

1991 saw the release of Cooper's 19th studio album, Hey Stoopid, again featuring several of rock music’s glitterati guesting on the record. Released as glam metal's popularity was on the wane, and just before the explosion of grunge, it failed to have the same commercial impact as its predecessor. The same year also saw the release of the video Alice Cooper: Prime Cuts which chronicled his entire career using in depth interviews with Cooper himself, Bob Ezrin, and Shep Gordon. One critic has noted that Prime Cuts demonstrates how Cooper had used (in contrast to similar artists who succeeded him) themes of satire and moralisation to such good effect throughout his career.^[38] It was in the Prime Cuts video that Bob Ezrin delivered his own summation of the Alice Cooper persona: "He is the psycho killer in all of us. He's the axe murderer, he's the spoiled child, he's the abuser, he's the abused; he's the perpetrator, he's the victim, he's the gun slinger, and he's the guy lying dead in the middle of the street".^[39]

By the early 1990s Cooper had become a genuine cultural icon, guesting on records by the most successful bands of the time, such as the Guns N' Roses album Use Your Illusion I, on which he shared vocal duties with Axl Rose on the track "The Garden"; making a brief appearance as the abusive stepfather of Freddy Krueger in the Nightmare On Elm Street film Freddy's Dead: The Final Nightmare (1991).

Cooper made a famous cameo appearance in the 1992 hit comedy film Wayne's World. Cooper and his band first appear onstage performing "Feed My Frankenstein" from Hey Stoopid. Afterwards at a backstage party, the movie's main characters Wayne and Garth discover that when offstage Cooper is a calm, articulate intellectual when he and his band discuss the history of Milwaukee in surprising depth. In a now famous scene, Wayne and Garth respond to an invitation to hang out with Cooper by kneeling and bowing reverently before him while chanting "We're not worthy! We're not worthy!"

In 1994 Cooper released The Last Temptation, his first concept album since DaDa. The album deals with issues of faith, temptation, alienation and the frustrations of modern life, and has been described as "a young man's struggle to see the truth through the distractions of the 'Sideshow' of the modern world".^[40] Concurrent with the release of The Last Temptation was a three-part comic book series written by Neil Gaiman, fleshing out the album's story. This was to be Cooper’s last album with Epic Records, and his last studio release for six years, though during this period the live album A Fistful of Alice^[41] was released, and in 1997 he lent his voice to the first track of Insane Clown Posse's The Great Milenko. In 1999, the four-disc box set The Life and Crimes of Alice Cooper appeared, which contained an authorized biography of Cooper, Alcohol and Razor Blades, Poison and Needles: The Glorious Wretched Excess of Alice Cooper, All-American, written by Creem magazine editor Jeffrey Morgan.

During his absence from the recording studio, Cooper toured extensively every year throughout the latter part of the 1990s, including, in 1996, South America, which he had not visited since 1974. Also in 1996, Cooper sang the role of Herod on the London cast recording of the musical Jesus Christ Superstar.^[42]

He also made an appearance on an episode of That 70s Show, at the end of which he and two other (minor) guest characters parody Dungeons & Dragons.

2000s[link]

Cooper in 2004 on a film set in L.A.

The first decade of the 21st century saw a sustained period of activity from Alice Cooper. In the decade that he turned sixty, he toured extensively and released (after a significant break) a steady stream of studio albums to favorable critical acclaim. During this period Cooper was also recognized and awarded in various ways: he received a Rock Immortal award at the 2007 Scream Awards;^[43] was given a star on the Hollywood Walk of Fame in 2003;^[44] he received (in May 2004) an honorary doctoral degree from Grand Canyon University;^[45] was given (in May 2006) the key to the city of Alice, North Dakota;^[46] he won the living legend award at the 2006 Classic Rock Roll of Honour event;^[47] and he won the 2007 Mojo music magazine Hero Award.

The lengthy break between studio albums ended in 2000 with Brutal Planet, which was a return to horror-lined heavy metal, with industrial rock, and with subject matter thematically inspired by the brutality of the modern world, set in a dystopian post-apocalyptic future, and also inspired by a number of news stories that had recently appeared on the CNN news channel.^[48] The album was produced by Bob Marlett, with longtime Cooper production collaborator Bob Ezrin returning as executive producer. The accompanying world tour, which included Cooper's first concert in Russia, was a resounding success, introducing Alice Cooper to a new audience and producing the live home video, Brutally Live, in 2001. During one memorable episode in Brutally Live, Britney Spears (being played by Alice Cooper's real life daughter, Calico), and representing "everything that my audience hates – the softening of rock and roll...the sweetness of it" is executed by Cooper.

Brutal Planet was succeeded by the sonically similar and widely acclaimed sequel Dragontown, which saw Bob Ezrin back at the helm as producer. The album has been described as leading the listener down "a nightmarish path into the mind of rock's original conceptual storyteller"^[49] and by Cooper himself as being "the worst town on Brutal Planet". Like The Last Temptation, both Brutal Planet and Dragontown are albums which explore Cooper's personal faith perspective (born again Christianity). It is often cited in the music media that Dragontown forms the third chapter in a trilogy begun with The Last Temptation;^[50] however, Cooper has indicated that this in fact is not the case.^[51]

Cooper again adopted a leaner, cleaner sound for his critically acclaimed^[52] 2003 release The Eyes of Alice Cooper. Recognizing that many contemporary bands were having great success with his former sounds and styles, Cooper worked with a somewhat younger group of road and studio musicians who were very familiar with his oeuvre of old. However, instead of rehashing the old sounds, they updated them, often with surprisingly effective results. The resulting Bare Bones tour adopted a less-orchestrated performance style that had fewer theatrical flourishes and a greater emphasis on musicality. The success of this tour helped support the growing recognition that the classic Cooper songs were exceptionally clever, tuneful and unique.

Cooper's radio show, Nights with Alice Cooper, began airing on January 26, 2004 in several US cities. The program showcases classic rock, Cooper's personal stories about his life as a rock icon and interviews with prominent rock artists. The show is broadcast on nearly 100 stations in the US and Canada,^[53] and has also been broadcast all over the world. In 2005 Alice Cooper was inducted into the Michigan Rock and Roll Legends online Hall of Fame.^[54] Two of the hit recordings of the original Alice Cooper band have been voted Legendary Michigan Songs: "Eighteen" in 2008, and "School's Out" in 2009.^[55]

A continuation of the songwriting approach adopted on The Eyes of Alice Cooper was again adopted by Cooper for his 24th studio album, Dirty Diamonds, released in 2005. Dirty Diamonds became Cooper's highest charting album since 1994's The Last Temptation.^[56] The Dirty Diamonds tour launched in America in August 2005 after several European concerts, including a performance at the Montreux Jazz Festival in Switzerland on July 12. Cooper and his band, including Kiss drummer Eric Singer, were filmed for a DVD released as Alice Cooper: Live at Montreux 2005. One critic, in a review of the Montreux release, commented that Cooper was to be applauded for "still mining pretty much the same territory of teenage angst and rebellion" as he had done more than thirty years previously.^[57]

In December 2006 the original Alice Cooper band reunited to perform six classic Alice Cooper songs at Cooper's annual charity event in Phoenix, entitled "Christmas Pudding".^[58]

On July 1, 2007 Cooper performed a duet with Marilyn Manson at the B'Estival event in Bucharest, Romania.^[59] The performance represented a reconciliation between the two artists; Cooper had previously taken issue with Manson over his overtly anti-Christian on-stage antics and had sarcastically made reference to the originality of Manson's choosing a female name and dressing in women's clothing.^[48] Cooper and Manson have been the subject of an academic paper on the significance of adolescent antiheroes.^[60]

In January 2008 he was one of the guest singers on the new Avantasia album The Scarecrow, singing the 7th track, "The Toy Master". In July 2008, after lengthy delays, Cooper released Along Came a Spider, his 25th studio album. It was Cooper's highest charting album since 1991's Hey Stoopid, reaching No. 53 in the US and No. 31 in the UK. The album, visiting similar territory explored in 1987's Raise Your Fist and Yell, deals with the nefarious antics of a deranged serial killer named "Spider" who is on a quest to use the limbs of his victims to create a human spider. The album generally received positive reviews from music critics, though Rolling Stone magazine opined that the music on the record sorely missed Bob Ezrin's production values.^[61] The resulting Theatre of Death tour of the album (during which Cooper is executed on four separate occasions) was described in a long November 2009 article about Cooper in The Times as "epic" and featuring "enough fake blood to remake Saving Private Ryan".

2010s[link]

In January 2010, it was announced that Alice would be touring with Rob Zombie on the "Gruesome Twosome" tour.^[62] In May 2010, Cooper made an appearance during the beginning of the season finale of the reality-show American Idol, in which he sang "School's Out".^[63]

With his daughter and former band member Dick Wagner, Cooper scored the music for the indie horror flick Silas Gore.^[64]

During 2010, Cooper began working on a new album, dubbed Welcome 2 My Nightmare, a sequel to the original Welcome to My Nightmare.^[65] In a Radio Metal interview, he said that "We'll put some of the original people on it and add some new people [...] I'm very happy with working with Bob (Ezrin) again."^[65]

On December 15, 2010, it was announced Cooper and his former band would be inducted into the Rock and Roll Hall of Fame. The official Rock and Roll Hall of Fame induction ceremony took place March 14, 2011 where Cooper was inducted by fellow horror-rocker Rob Zombie. Original members Bruce, Cooper, Dunaway, and Smith all made brief acceptance speeches and performed "I'm Eighteen" and "School's Out" live together, with Steve Hunter filling in for the late Glen Buxton. Alice showed up for the event wearing a (presumably fake) blood-splattered shirt and had a live giant albino boa snake wrapped around his neck.^[5][66] Cooper told Rolling Stone magazine that he was "elated" by the news and that the nomination had been made for the original band, as "We all did go to the same high school together, and we were all on the track team, and it was pretty cool that guys that knew each other before the band ended up going that far".^[67]

On March 10, 2011, Jackson Browne, David Crosby, Graham Nash, Alice Cooper, Jennifer Warnes and others performed at a benefit concert in Tucson, Arizona benefiting The Fund for Civility, Respect and Understanding, a foundation that raises awareness about and provides medical prevention and treatment services to people with mental disorders.^[68] In June 2011 Cooper took his place in the Reasonably Priced Car at the BBC auto show Top Gear.^[69]

Cooper supported Iron Maiden on tour from June to July 21.^[70]

Style and influences[link]

During an interview for the program Entertainment USA in 1986 Cooper stunned interviewer Jonathan King by stating that The Yardbirds were his favorite band of all time.^[71] Perhaps King should not have been so taken aback, as Cooper had as far back as 1969 said that it was music from the mid-sixties, and particularly from British bands The Beatles, The Who, and The Rolling Stones, as well as The Yardbirds, that had the greatest influence on him.^[72] Cooper would later pay homage to The Who by singing "I'm A Boy" for A Celebration: The Music of Pete Townshend and The Who in 1994 at Carnegie Hall in New York, and performing a cover of "My Generation" on the Brutal Planet tour of 2000.

During an interview with Ozzy Osbourne from radio program Nights with Alice Cooper on May 22, 2007, Cooper again affirmed his debt of gratitude to these bands, and to The Beatles in particular. During their discussion, Cooper and Osbourne bemoaned the often inferior quality of songwriting coming from contemporary rock artists. Cooper stated that in his opinion the cause of the problem was that certain modern bands "had forgotten to listen to The Beatles".

On the 25th anniversary DVD of Cabaret, Liza Minnelli stated that Cooper had told her that his whole career was based on the movie Cabaret.^{[citation needed]}

Evidence of Cooper's eclectic tastes in classic and contemporary rock music can be seen in the track listings of his radio show; in addition, when he appeared on the BBC Radio 2 program Tracks of My Years in September 2007, he listed his favorite tracks of all time as being: "19th Nervous Breakdown" (1966) by The Rolling Stones; "Turning Japanese" (1980) by The Vapors; "My Sharona" (1979) by The Knack; "Beds Are Burning" (1987) by Midnight Oil; "My Generation" (1965) by The Who; "Welcome to the Jungle" (1987) by Guns N' Roses; "Rebel Rebel" (1974) by David Bowie; "Over Under Sideways Down" (1966) by The Yardbirds; "Are You Gonna Be My Girl" (2003) by Jet; and "A Hard Day's Night" (1964) by The Beatles,^[73] and when he appeared on Desert Island Discs in 2010 he chose the songs "Happenings Ten Years Time Ago" by The Yardbirds; "I Get Around" by The Beach Boys; "I'm a Boy" by The Who; "Timer" by Laura Nyro; "21st Century Schizoid Man" by King Crimson; "Been Caught Stealing" by Jane's Addiction; "Work Song" by The Paul Butterfield Blues Band and "Ballad of a Thin Man" by Bob Dylan.^[74]

Rob Zombie, former front man of White Zombie, claims his first "metal moment" was seeing Alice Cooper on Don Kirshner's Rock Concert.^[75] Zombie has also claimed to have been heavily influenced by Cooper's costumes.^[76]

In a 1978 interview with Rolling Stone, Bob Dylan stated, "I think Alice Cooper is an overlooked songwriter".^[77]

In the foreword to Alice Cooper's CD retrospective box set The Life and Crimes of Alice Cooper, John Lydon of The Sex Pistols pronounced Killer as the greatest rock album of all time, and in 2002 Lydon presented his own tribute program to Cooper on BBC radio. Lydon told the BBC that "I know the words to every Alice Cooper song. The fact is, if you can call what I have a musical career, it all started with me miming to I'm Eighteen on a jukebox".^[78][79]

The Flaming Lips are longtime Alice Cooper fans and used the bass line from "Levity Ball" (an early song from the 1969 release Pretties for You) for their song "The Ceiling Is Bending". They also covered "Sun Arise" for an Alice Cooper tribute album. (Cooper's version, which closes the album Love It to Death, was itself a cover of a Rolf Harris song.)

In 1999 Cleopatra Records released Humanary Stew: A Tribute to Alice Cooper featuring a number of contributions from rock and metal all-star collaborations, including Dave Mustaine, Roger Daltrey, Ronnie James Dio, Slash, Bruce Dickinson, and Steve Jones.^[80] The album was notable for the fact that it was possible to assemble a different supergroup for each cover version on the record, which gave an indication of the depth of esteem in which Cooper is held by other eminent musicians within the music industry.

A song by alternative rock group They Might Be Giants from their 1994 album John Henry entitled "Why Must I Be Sad?" mentions 13 Cooper songs, and has been described as being "from the perspective of a kid who hears all of his unspoken sadness given voice in the music of Alice Cooper; Alice says everything the kid has been wishing he could say about his alienated, frustrated, teenage world".^[81]

Such unlikely non-musician fans of Cooper included Groucho Marx and Mae West, who both reportedly saw the early shows as a form of vaudeville revue,^[82] and artist Salvador Dalí, who on attending a show in 1973 described it as being surreal, and made a hologram, First Cylindric Chromo-Hologram Portrait of Alice Cooper's Brain.^[83][84]

Personal life[link]

Cooper, a huge fan of The Simpsons, was asked to contribute a storyline for the September 2004 edition of Bongo Comics's Bart Simpson's Treehouse of Horror, a special Monsters of Rock issue that also included stories plotted by Gene Simmons, Rob Zombie and Pat Boone.^[85] Cooper's story featured Homer Simpson being a Jason Voorhees, Friday the 13th style killer and Alice and the citizens of Springfield are being stalked by Homer.

On June 20, 2005, ahead of his June–July 2005 tour, Cooper had a wide-ranging interview with interviewer of celebrities Andrew Denton for the Australian ABC Television's Enough Rope. Cooper discussed various issues during a revealing and frank talk, including the horrors of acute alcoholism and his subsequent cure, being a Christian, and his social and work relationship with his family.^[86] During the interview, Cooper remarked "I look at Mick Jagger and he's on an 18-month tour and he's six [sic] years older than me, so I figure, when he retires, I have six more years. I will not let him beat me when it comes to longevity."^[87]

The actual ownership of the Alice Cooper name is an example of the value of a single copyright or trademark. Since "Alice Cooper" was originally the name of the band, and not the lead singer (cf. Uriah Heep, Jethro Tull, Amy Meredith, etc.), and it was actually owned by the band as whole, Cooper paid, and continues to pay, a yearly royalty to his original bandmates for the right to use the name commercially.

Cooper is a fan of the NHL Phoenix Coyotes.^[88] On February 18, 2012 the Coyotes gave away his bobblehead in a promotion for the first 10,000 fans for a game with the Dallas Stars.^[89]

Relationships and family[link]

In the period when the Alice Cooper group was signed to Frank Zappa's Straight label, Miss Christine of the GTOs became Cooper's girlfriend. Miss Christine (real name: Christine Frka), who had actually recommended Zappa to the group, died on November 5, 1972 of an overdose.^[90]

Another long-time girlfriend of Cooper's was Cindy Lang, with whom he lived for several years. They separated in 1975. Lang sued Cooper for palimony, and they eventually settled out of court in the early 1980s.^[91][92]

After his separation from Lang, Cooper was briefly linked with actress Raquel Welch.^[93] Cooper then reportedly left Welch, however, to marry, on March 20, 1976, ballerina instructor/choreographer Sheryl Goddard, who performed in the Alice Cooper show from 1975 to 1982. In November 1983, at the height of Cooper's alcoholism, Sheryl filed for divorce, but by mid-1984, she and Cooper had reconciled.^[94] The couple has remained together since. In a 2002 television interview, Cooper claimed that he had "never cheated" on his wife in all the time they had been together. In the same interview, he also claimed that the secret to a lasting and successful relationship is to continue going out on dates with your partner.^[95] The couple have three children: elder daughter Calico Cooper (born 1981), an actress and singer who has been performing in the Alice Cooper show since 2000; son Dash (b. 1985), a student at Arizona State University, and also plays in a band called Runaway Phoenix; and younger daughter Sonora Rose (b. 1993).

Drug recovery[link]

In 1986, Megadeth was asked to open for Cooper for dates on his US Constrictor tour. After noticing the hardcore abuse of alcohol and other drugs in the band, Cooper personally approached the band members to try to help them control their abuse, and he has stayed close to front man Dave Mustaine, who considers Cooper to be his "godfather".^[96] Since overcoming his own addiction to alcohol in the mid-1980s, Cooper has continued to help and counsel other rock musicians with addiction problems. "I've made myself very available to friends of mine – they're people who would call me late at night and say, 'Between you and me, I've got a problem.'"^[97] In recognition of the work he has done in helping other addicts in the recovery process, Cooper received in 2008 the Stevie Ray Vaughan Award at the fourth annual MusiCares MAP Fund benefit concert in Los Angeles.^[98]

Religion and politics[link]

Although he originally tended to shy away from speaking publicly about his religious beliefs, Cooper has in recent years been vocal about his faith as a born-again Christian.^[99]

Throughout his career, Cooper's philosophy regarding politics is that politics should not be mixed with rock music. He has usually kept his political views to himself, and in 2010 said "I am extremely non-political. I go out of my way to be non-political. I'm probably the biggest moderate you know. When John Lennon and Harry Nilsson used to argue politics, I was sitting right in the middle of them, and I was the guy who was going 'I don't care.' When my parents would start talking politics, I would go in my room and put on The Rolling Stones or The Who on as long as I could to avoid politics. And I still feel that way".^[67] On occasion he has spoken out against musicians who promote or opine on politics; for example, in the run-up to the 2004 presidential election, he told the Canadian Press that the then crop of rock stars campaigning for and touring on behalf of Democratic candidate John Kerry were committing "treason against rock n' roll". He also added that upon seeing the list of musicians who supported Kerry, "if I wasn't already a Bush supporter, I would have immediately switched. Linda Ronstadt? Don Henley? Geez, that's a good reason right there to vote for Bush."^[100]

Love of golf[link]

Cooper has on several occasions credited golf as having played a major role in helping him to overcome his addiction to alcohol,^[101] and has even gone as far to say that when he took up golf, it was a case of replacing one addiction with another.^[102][103] The importance that the game has had in his life is also reflected in the title to his 2007 autobiography, Alice Cooper, Golf Monster.^[104] Cooper, who has participated in a number of Pro-Am competitions,^[105] plays the game six days a week, off a handicap of two.^[74] Since 1997 he has hosted an annual golf competition, the Alice Cooper Celebrity AM Golf Tournament, all proceeds from which go to his charity, the Solid Rock Foundation. In 2005 while playing with manager Shep Gordon on The Champion Course in Palm Beach Gardens, Florida, Cooper recorded an impressive round of two-over par 74 on the world-class course. Cooper has also appeared in commercials for Callaway Golf equipment, was a guest of veteran British player and broadcaster Peter Alliss on A Golfer's Travels.^[106] He wrote the foreword to the Gary McCord book "Ryder Cup" and participated in the second All*Star Cup in South Wales.^[107] In an interview with VH1, friend and fellow golfer Pat Boone said that Cooper was "'this close' to being a pro".

Discography[link]

See: Alice Cooper discography

Awards and nominations[link]

Grammy Awards

Filmography[link]

See: Alice Cooper filmography

List of Alice Cooper band personnel[link]

Current

• Alice Cooper – vocals, guitar, harmonica (1963–present)
• Chuck Garric – bass guitar, vocals (2002–present)
• Ryan Roxie – guitar, vocals (1996–2006, 2012-presemt)
• Tommy Henriksen – guitar, vocals (2011–present)
• Orianthi – guitar, vocals (2011–present)
• Glen Sobel – drums, percussion (2011–present)

Past

Notes[link]