Covering manifolds

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Covering manifolds

In this appendix, we summarize some properties of covering spaces in the context of smooth manifolds.

A.1 Topological coverings

Recall that a (topological)covering of a spaceX is another spaceX˜ with a continuous mapπ: ˜X →Xsuch thatXis a union of evenly covered open set, where a connected open subsetUofXis calledevenly coveredif

(A.1.1) π⁻¹U =∪i∈IU˜_i

is a disjoint union of open setsU˜_i of X˜, each of which is mapped home- omorphically onto U under π. In particular, the fibers of π are discrete subsets ofX˜. It also follows from the definition thatX˜ has the Hausdorff property if X does. Further it is usual, as we shall do, to require that X andX˜ be connected, and then the index setI can be taken the same for all evenly covered open sets.

A.1.2 Examples (a)π :R→S¹,π(t) =e^itis a covering.

(b)π:S¹ →S¹,π(z) =zⁿis a covering for any nonzero integern.

(c)π : (0,3π) → S¹,π(t) =e^it is a local homemeomorphism which is not a covering, since1∈S¹does not admit evenly covered neighborhoods.

A.2 Fundamental groups

Covering spaces are closely tied with fundamental groups. Thefundamental groupπ₁(X, x₀)of a topological spaceXwith basepointx₀is defined as follows. As a set, it consists of the homotopy classes of continuous loops based atx₀. The concatenation of such loops is compatible with the equivalence relation given by homotopy, so it induces a group operation onπ₁(X, x₀) making it into a group. IfX is arcwise connected, the isomorphism class

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of the fundamental group is independent of the choice of basepoint (indeed for x₀, x₁ ∈ X and c a continuous path from x₀ to x₁, conjugation by c⁻¹ induces an isomorphism from π₁(X, x₀) and π₁(X, x₁)) and thus is sometimes denoted byπ₁(X). Finally, a continuous map f : X → Y between topological spaces with f(x₀) = y₀ induces a homomorphism f_# : π₁(X, x₀) → π₁(Y, y₀) so that the assignment(X, x₀) → π₁(X, x₀) is functorial. Of course the fundamental group is trivial if and only if the space is simply-connected.

Being locally Euclidean, a smooth manifold is locally arcwise connected and locally simply-connected. A connected spaceX with such local con- nectivity properties admits a simply-connected covering space, which is unique up to isomorphism; an isomorphism between coveringsπ1 : ˜X1 → Xandπ₂ : ˜X₂ → Xis a homeomorphismf : ˜X₁ → X˜₂ such thatπ₂◦f = π₁. More generally, there exists a bijective correspondance between isomorphism classes of coverings π : ( ˜X,x˜₀) → (X, x0) and subgroups of π₁(X, x₀) given by( ˜X,x˜₀) 7→ π_#(π₁( ˜X,x˜₀)); moreover, a change of basepoint inX˜ corresponds to passing to a conjugate subgroupπ₁(X, x₀). A.3 Smooth coverings

Supposeπ : ˜M → M is a covering whereM is a smooth manifold. Then there is a natural structure of smooth manifold onM˜ such that the projec- tionπ is smooth. In fact, for every chart (U, π) of M where U is evenly covered as in (A.1.1), take a chart( ˜U_i, ϕ◦π|_U_˜

i)forM˜. This gives an atlas of M, which is smooth because for another chart˜ (V, ψ) ofM,V evenly covered by∪_i∈IV˜_iandU˜_i∩V˜_j 6=∅for somei,j∈I, we have that the transition map

(ψ◦π|_V_˜

j)(ϕ◦π|_U_˜

i)⁻¹ =ψ◦ϕ⁻¹

is smooth. We already know thatM˜ is a Hausdorff space. It is possible to choose a countable basis of connected open sets forM which are evenly covered. The connected components of the preimages underπ of the el- ements of this basis form a basis of connected open sets for M, which is˜ countable as long as the index setI is countable, but this follows from the countability of the fundamental groupπ₁(M)¹. Now, around any point in M˜, π admits a local representation as the identity, so it is a local diffeomorphism. Note that we have indeed proved more: M can be covered by evenly covered neighborhoodsU such that the restriction of π to a connected component ofπ⁻¹U is a diffeomorphism ontoU. This is the definition of asmooth covering. Note that a topologic covering whose covering map is smooth need not be a smooth covering (e.g.π:R→R,π(x) =x³).

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Next, we can formulate basic results in covering theory for a smooth covering π : ˜M → M of a smooth manifold M. Fix basepointsp˜ ∈ M˜, p∈M such thatπ(˜p) =p. We say that a mapf :N →M admits a lifting if there exists a mapf˜:N →M˜ such thatπ◦f˜=f.

A.3.1 Theorem (Lifting criterion) Letq ∈f⁻¹(p). A smooth mapf :N →M admits a smooth liftingf˜:N →M˜ withf˜(q) = ˜pif and only iff_#(π1(N, q))⊂ π_#(π1( ˜M ,p)). In that case, if˜ N is connected, the lifting is unique.

Takingf :N →M to be the universal covering ofM in Theorem A.3.1 shows that the universal covering ofMcovers any other covering ofMand hence justifies its name.

A.4 Deck transformations

For a topological covering π : ˜X → X, a deck transformation or cover- ing transformation is an isomorphismX˜ → X˜, namely, a homeomorphism f : ˜X → X˜ such thatπ ◦f = π. The deck transformations form a group under composition. It follows from uniqueness of liftings that a deck transformation is uniquely determined by its action on one point. In particular, the only deck transformation admitting fixed points is the identity. Since a smooth covering mapπ : ˜M → M is a local diffeomorphism, in this case the equation π◦f = π implies that deck transformations are diffeomorphisms ofM˜.

An action of a (discrete) group on a topological space (resp. smooth manifold) is a homomorphism from the group to the group of homeomor- phisms (resp. diffeomorphisms) of the space (resp. manifold). For a smooth manifoldM, we now recall the canonical action ofπ₁(M, p)on its universal covering M˜ by deck transformations. First we remark that by the lifting criterion, givenq ∈ M andq˜1, q˜2 ∈ π⁻¹(q), there is a unique deck transformation mappingq˜₁ toq˜₂. Now letγ be a continuous loop inM based at p representing an element[γ] ∈ π₁(M, p). By the remark, it suffices to describe the action of[γ]on a pointp˜∈π⁻¹(p), which goes as follows: lift γuniquely to a pathγ˜starting atp˜; then[γ]·p˜is by definition the endpoint ofγ˜, which sits in the fiberπ⁻¹(p). The definiton independs of the choice made, namely, if we changeγto a homotopic curve, we get the same result.

This follows from Theorem A.3.1 applied to the homotopy, as it is defined on a square and a square is simply-connected. Sinceπ : ˜M → M is the universal covering, every deck transformation is obtained in this way from an element ofπ₁(M, p).

As action of a (discrete) groupΓon a topological spaceX is calledfree if no nontrivial element ofΓ has fixed points, and it is calledproperif any two pointsx, y ∈ X admit open neighborhoodsU ∋ x,V ∋ y such that {γ ∈ Γ | γU ∩V 6= ∅}is finite. The action of π1(M, p) on the universal

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coveringM˜ by deck transformations has both properties. In fact, we have already remarked it is free. To check properness, let p˜, q˜ ∈ M˜. If these points lie in the same orbit ofπ₁(M, p) or, equivalently, the same fiber of π, the required neighborhoods are the connected components of π⁻¹(U) containingp˜andq, resp., where˜ U is an evenly covered neighborhood of π(˜p) = π(˜q). On the other hand, if π(˜p) =: p 6= q := π(˜p), we use the Hausdorff property of M to find disjoint evenly covered neighborhoods U ∋ p,V ∋ q and then it is clear that the connected component ofπ⁻¹(U) containingp˜and the connected component ofπ⁻¹(V)containingq˜do the job.

Conversely, we have:

A.4.1 Theorem If the groupΓacts freely and properly on a smooth manifoldM˜, then the quotient spaceM = Γ\M˜ endowed with the quotient topology admits a unique structure of smooth manifold such that the projectionπ : ˜M → M is a smooth covering.

Proof. The action of Γ on M˜ determines a partition into equivalence classes ororbits, namelyp˜∼ q˜if and only ifq˜= γp˜for someγ ∈ Γ. The orbit throughp˜is denotedΓ(˜p). The quotient spaceΓ\M˜ is also calledorbit space.

The quotient topology is defined by the condition thatU ⊂ M is open if and only ifπ⁻¹(U) is open inM˜. In particular, for an open setU˜ ⊂ M˜ we haveπ⁻¹(π( ˜U)) =∪_γ∈Γγ( ˜U), a union of open sets, showing thatπ( ˜U)is open and proving thatπis an open map. In particular,πmaps a countable basis of open sets inM˜ to a countable basis of open sets inM.

The covering property follows from the fact thatΓis proper. In fact, let

˜

p∈ M˜. From the definition of proeprness, we can choose a neighborhood U˜ ∋p˜such that{γ ∈Γ|γU˜ ∩U˜ 6=∅}is finite. Using the Hausdorff property ofM˜ and the freeness ofΓ, we can shrinkU˜ so that this set becomes empty. Now the mapπ identifies all disjoint homeomorphic open setsγU forγ∈Γto a single open setπ(U)inM, which is then evenly covered.

The Hausdorff property ofM also follows from properness of Γ. In- deed, letp, q ∈ M,p 6= q. Choosep˜∈ π⁻¹(p),q˜∈ π⁻¹(q)and neighbor- hoodsU˜ ∋ p,˜ V˜ ∋ q˜such that{γ ∈ Γ | γU˜ ∩V˜ 6= ∅}is finite. Note that

˜

q6∈Γ(˜p), so by the Hausdorff property forM˜, we can shrinkU˜ so that this set becomes empty. Since π is open,U := π( ˜U) andV := π( ˜V) are now disjoint neighborhoods ofpandq, respectively.

Finally, we construct a smooth atlas forM. Letp ∈ M and choose an evenly covered neighborhoodU ∋ p. Write π⁻¹U = ∪_i∈IU˜_i as in (A.1.1).

By shrinkingU we can ensure thatU˜_iis the domain of a local chart( ˜U_i,ϕ˜_i) ofM˜. Nowϕ_i := ˜ϕ_i◦(π|U˜i)⁻¹ :U →Rⁿdefines a homeomorphism onto the open setϕ˜_i( ˜U_i) and thus a local chart (U, ϕ_i) of M. The domains of such charts coverM and it remains only to check that the transition maps

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are smooth. So let V be another evenly covered neighborhood ofp with π⁻¹V = ∪_j∈IV˜_j and associated local chart ψ_j := ˜ψ_j ◦(π|_V_˜

j)⁻¹ : U → Rⁿ

where( ˜Vj,ψ˜j)is a local chart ofM. Then˜

(A.4.2) ψ_j◦ϕ⁻_i ¹ = ˜ψ_j◦(π|V˜j)⁻¹◦π◦ϕ˜⁻_i¹ However,(π|_V_˜

j)⁻¹◦πis realized by a unique elementγ ∈Γin a neighborhood ofp˜_i =π|⁻¹_˜

Ui(p). SinceΓacts by diffeomorphisms, this shows that the transtion map (A.4.2) is smooth and finishes the proof.