choose a countable basis of connected open sets for M which are evenly covered. The connected components of the preimages under π 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). Now, around any point in M˜, π admits a local representation as the identity, so it is a local diffeo-morphism. Note that we have indeed proved more: M can be covered by evenly covered neighborhoods U such that the restriction of π to a con-nected component ofπ−1U is a diffeomorphism ontoU. This is the defini-tion of asmooth covering. Note that a topological covering whose covering map is smooth need not be a smooth covering (e.g.π:R→R,π(x) =x3).
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 transformationis 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 trans-formation 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 diffeomor-phisms 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. diffeomorhomeomor-phisms) 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 transfor-mation mappingq˜1 toq˜2. Now letγ be a continuous loop inM based atp representing an element[γ]∈ π1(M, p), and fix a pointp˜∈ π−1(p). By the
remark, it suffices to describe the action of[γ]onp, which goes as follows:˜ liftγ uniquely to a path˜γ starting atp; then˜ [γ]·p˜is by definition the end-point of˜γ, which sits in the fiberπ−1(p). The definition 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).
An action of a (discrete) groupΓon a topological spaceXis calledfree if no nontrivial element ofΓhas fixed points, and it is calledproper if 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 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 in M˜. 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 properness, 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 consists of the identity only. Now the mapπ identifies all disjoint homeomorphic open setsγUforγ ∈Γto a single open setπ(U)inM, which is then evenly
A.4. DECK TRANSFORMATIONS 129 covered.
The Hausdorff property of M also follows from properness of Γ. In-deed, letp,q ∈ M, p 6= q. Choose p˜ ∈ π−1(p), q˜∈ π−1(q) and neighbor-hoodsU˜ ∋ p,˜ V˜ ∋ q˜such that{γ ∈ Γ |γU˜ ∩V˜ 6=∅}is finite. Note that
˜
q 6∈Γ(˜p), so by the Hausdorff property forM, we can shrink˜ U˜ 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 neighborhood U ∋ 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( ˜Ui,ϕ˜i) ofM˜. Nowϕi := ˜ϕi◦(π|U˜i)−1 :U → Rndefines a homeomorphism onto the open set ϕ˜i( ˜Ui) and thus a local chart (U, ϕi) of M. The domains of such charts coverM and it remains only to check that the transition maps are smooth. So let V be another evenly covered neighborhood of p with π−1V = ∪j∈IV˜j and associated local chartψj := ˜ψj ◦(π|V˜j)−1 : U → Rn where( ˜Vj,ψ˜j)is a local chart ofM. Then˜
(A.4.2) ψj◦ϕ−1i = ˜ψj◦(π|V˜j)−1◦π◦ϕ˜−1i
However,(π|V˜j)−1◦π is realized by a unique elementγ ∈Γin a neighbor-hood ofp˜i =π|−1U˜i(p). SinceΓacts by diffeomorphisms, this shows that the transtion map (A.4.2) is smooth and finishes the proof.
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131
action
transitive, 80 adjoint representation
of Lie algebra, 77 of Lie group, 77 atlas
oriented, 92 smooth, 4 topological, 3 bump function, 19 chain rule, 10 cohomology, 101
homotopy invariance, 102 in top degree, 109
of sphere, 107
complex projective space, 82 contractible, 105
contraction, 56 coordinate system, 3
distinguished, 31 coordinate vector, 8 cotangent bundle, 48 covering
smooth, 126, 127 topological, 125
covering transformation, 127 critical point, 16
critical value, 16
de Rham cohomology, 101 deck transformation, 127 deformation
retract, 105 retraction, 105 degree, 110
diffeomorphism, 4 local, 12 differential, 10, 23 differential form, 48
closed, 100 exact, 100
directional derivative, 9 distribution, 31
integrable, 33 involutive, 34
embedded submanifold, 12 embedding, 14
proper, 14 exponential, 69
of a matrix, 69 exterior
algebra, 45 derivative, 51 product, 45 exteriork-bundle, 48 exterior algebra bundle, 48 foliation, 31
integral, 33
fundamental group, 125 Grassmann manifold, 36, 89 group
complex general linear, 66 complex special linear, 66 general linear, 6, 66 orthogonal, 17, 66 special linear, 66 special unitary, 66 unitary, 66 132
INDEX 133 Heisenberg algebra, 68
homogeneous coordinates, 7 homotopy, 102
equivalent, 104 inverse, 105 type, 104
immersed submanifold, 12 immersion, 1, 12
local form, 13 initial submanifold, 15 integral manifold, 33
maximal, 33 integration, 94
on a Lie group, 118
on a Riemannian manifold, 117 interior multiplication, 56
isotropy group, 80 Jacobi identity, 28 left translation, 67 Lie algebra, 67
representation, 84 Lie bracket, 27
Lie derivative, 29 Lie group, 65
action, 80 discrete, 66
exponential map, 69 homomorphism, 71
one-parameter subgroup, 74 linking number, 112
local chart, 3 adapted, 12 distinguished, 31 local frame, 30 manifold
boundary, 96 homogeneous, 80 integral, 33 orientable, 92 oriented, 92 parallelizable, 40 smooth, 4 topological, 3 with boundary, 95 with corners, 99 measure zero, 90
orientation, 92 induced, 96 paracompactness, 21 parametrization, 1 partition of unity, 19 plaque, 31
Poincaré lemma, 105 projective space
complex, 82 real, 6 proper map, 14 pull-back, 54
real projective space, 6 regular value, 1, 16 retraction, 105
Riemannian metric, 63 right translation, 67 singular value, 16 smooth
curve, 11 function, 3, 4 manifold, 4 map, 4 structure, 4 spheres, 5
that are Lie groups, 106 that are parallelizable, 106 Stiefel manifold, 82
submanifold, 12 codimension, 18 embedded, 2, 12 immersed, 2, 12 initial, 15
properly embedded, 14 submersion, 1, 16
local form, 16
system of local coordinates, 3 tangent
bundle, 22 map, 10 space, 7, 8 vector, 8, 11 tensor, 45
algebra, 45 field, 48
homogeneous, 45