Download Quotient Spaces and Quotient Maps Definition. If X is a topological

Quotient Spaces and Quotient Maps
Definition. If X is a topological space, Y is a set, and π : X → Y is any surjective
map, the quotient topology on Y determined by π is defined by declaring a subset
U ⊂ Y is open ⇐⇒ π −1 (U ) is open in X.
Definition. If X and Y are topological spaces, a map π : X → Y is called a
quotient map if it is surjective and continuous and Y has the quotient topology
determined by π.
• The fiber of π over a point y ∈ Y is the set π −1 (y).
Definition. If π : X → Y is a map, a subset U ⊂ X is said to be saturated with
respect to π if U is the entire inverse image of its image
U = π −1 (π(U )).
• A subset of X is saturated iff it is the union of fibers.
Lemma 1 (Properties of Quotient Maps). Let π : X → Y be a quotient map.
(a) Characteristic Property: For any topological space B, a map f : Y → B is
continuous iff f ◦ π : X → B is continuous.
(b) The quotient topology is the unique topology on Y for which the characteristic
property holds.
Lemma 2 (Passing to the Quotient). Suppose π : X → Y is a quotient map,
B is a topological space, and f : X → B is a continuous map that is constant on
the fibers of π (i.e., π(p) = π(q) implies f (p) = f (q)).
Then there exists a unique continuous map fe : Y → B such that f = fe ◦ π.
X


πy
fe
X

f
y
Y −−−−→ B
Lemma 3 (Uniqueness of Quotient Spaces).
If π1 : X → Y1 and π2 : X → Y2 are quotient maps that are constant on each
othere’s fibers (i.e. π1 (p) = π1 (q) iff π2 (p) = π2 (q)),
then there exists a unique homeomorphism ϕ : Y1 → Y2 such that ϕ ◦ π1 = π2 .
X


π1 y
ϕ
X

π
y 2
Y1 −−−−→ Y2 .
Definition. Let X be a topological space and ∼ is an equivalence relation on X.
Let X/ ∼ denote the set of equivalence classes in X, and let
π : X → X/ ∼
be the natural projection sending each point to its equivalence class. Endowed with
the quotient topology determined by π, the space X/ ∼ is called the quotient
space of X determined by the equivalence relation.
Typeset by AMS-TEX
1
2
Example (Real Projective Spaces).
Definition. The n-dimensional real projective space, denoted by RPn (or sometimes just Pn ), is defined as the set of 1-dimensional linear subspace of Rn+1 .
• We give it the quotient topology determined by the natural map
π : Rn+1 \ {0} → RPn
sending each point x ∈ Rn+1 \ {0} to the subspace spanned by x.
— ∀x ∈ Rn+1 \ {0}, denote [x] = π(x) ∈ RPn .
• For each i = 1, · · · , n, let
ei ⊂ Rn+1 \ {0} = {(x1 , · · · , xn+1 ), xi 6= 0)},
U
n
e
e
and
let Ui = π(Ui ) ⊂ RP . Since Ui is a saturated open set, Ui is open and
ei → Ui is a quotient map.
π : U
Ũi
— Define a map ϕi : Ui → Rn by
1
n+1
ϕi [x , · · · , x
]=
xi−1 xi+1
xn+1
x1
.
,··· , i , i ,··· , i
xi
x
x
x
— This map is well-defined, because its value is unchanged by multiplying x by a
nonzero constant.
— Because ϕi ◦ π is continuous, ϕi is continuous by the characteristic property of
quotient maps.
In fact, ϕi is a homeomorphism, because its inverse is given by
1
n
1
i−1
, 1, ui , · · · , un ].
ϕ−1
i (u , · · · , u ) = [u , · · · , u
Because the sets Ui cover RPn , this shows that RPn is locally Euclidean of
dimension n.
RPn is Hausdorff and second countable.
RPn is compact, since the restriction of π to Sn is surjective.
• The coordinate charts (Ui , ϕi ) constructed in that example are all smoothly
compatiable; indeed, for i > j, it is straightforward to compute that
1
uj−1
ui−1 1 ui
un
u
1
n
ϕj ◦ ϕ−1
(u
,
·
·
·
,
u
)
=
, · · · j , , · · · , j , j , j , · · · , j ),
i
j
u
u
u
u u
u
which is a diffeomorphism from ϕi (Ui ∩ Uj ) to ϕj (Ui ∩ Uj ).
Example (the Complex Projective Space)
Definition. Complex projective n-space, denoted by CPn , is defined to be the
set of 1-dimensional complex-linear subspaces of Cn+1 , with the quotient topology
inherited from the natural projection
π : Cn+1 \ {0} → CP n .
• CPn+1 is a smooth 2n-manifold.
• The quotient map π : Cn+1 \ {0} → CP n is smooth.
• CP 1 is diffeomorphic to S2 .
3
Submersions
Definition. A smooth map F : M → N of constant rank is called a submersion
if F∗ is surjective at each point (or equivalently, if rank F =dim N ).
Constant-Rank Level Set Theorem. Let M and N be smooth manifolds, and
let Φ : M → N be a smooth map with constant rank to k. Each level set of Φ is a
closed embedded submanifold of codimension k in M .
Proof. Let c ∈ N be arbitrary, and let S denote the level set Φ−1 (c) ⊂ M .
Clearly, S is closed in M by continuity.
— To show that is an embedded submanifold, we need to show that ∀p ∈ S, there
is a slice chart for S in M near p.
— From the rank theorem, there are smooth charts (U, ϕ) centered at p and (V, ψ)
centered at c = Φ(p) in which Φ has a coordinate representation of the form
Φ(x1 , · · · , xk , xk+1 , · · · , xm ) = (x1 , · · · , xk , 0, · · · , 0),
and therefore S ∩ U is the slice {(x1 , · · · , xm ) ∈ U : x1 = · · · xk = 0}.
Submersion Level Set Theorem. If Φ : M → N is a submersion, then each
level set of Φ is a closed embedded submanifold whose codimension is equal to the
dimension of N .
Proof. A submersion has constant rank equal to the dimension of N .
f → M is any continuous map, a section of π is a continuous
Definition. If π : M
f
M
f such that π ◦ σ = IdM : π ↓↑ σ
map σ : M → M
M.
f defined on some open set
• A local section is a continuous map σ : U → M
U ⊂ M and satisfying the analogous relation π ◦ σ = IdU .
Definition. A (smooth) k-dimensional vector bundle is a pair of smooth
manifolds E (the total space) and M (the base), together with a surjective
(smooth) map π : E → M (the projection), satisfying the following conditions:
(1) Each set Ep = π −1 (p) (called the fiber of E over p) is endowed with the
structure of a vector space.
(2) For each p ∈ M , ∃a nbhd U of p and a homeomorphism (diffeomorphism)
ϕ : π −1 (U ) → U × Rk , called a (smooth) local trivilization of E such
that the following diagram commutes.
ϕ
π −1 (U ) −−−−→ U × Rk



 π1
πy
y
U
U
(where π1 is the projection onto the first factor).
(3) The restriction of ϕ to each fiber ϕ : Ep → {p}×Rk is a linear isomorphism.
4
Proposition 4 (Properties of Submersion). Let π : M → N be a submersion.
Then the following are true.
(a) π is an open map.
(b) Every point of M is in the image of a smooth local section of π.
(b*) π is locally a trivial fibration.
(c) If π is surjective, it is a quotient map.
Proof. Given p ∈ M , let q = π(p) ∈ N . Because a submersion has constant
rank, we can choose smooth coordinates (x1 , · · · , xm ) centered at p and (y 1 , · · · , y k )
centered at q in which π has the coordinte representation
π(x1 , · · · , xk , xk+1 , · · · , xm ) = (x1 , · · · , xk ).
(b) If ε is sufficiently small positive number, the coordinate cube
Cε = {x : |xi | < ε, for i = 1, · · · , m}
is a nbhd of p whose image under π is the cube
Cε0 = {y : |y i | < ε, for i = 1, · · · , k}
The map σ : Cε0 → Cε whose coordinate representation is
σ(x1 , · · · , xk ) = (x1 , · · · , xk , 0, · · · , 0)
is a smooth local section of π satisfying σ(q) = p.
(a) Suppose W is any open subset of M and q = π(W ).
– For any p ∈ W with π(p) = q, W contains an open coordinate cube Cε centered
at p as above, and thus π(W ) contains an open coordinate cube centered at π(p).
– This proves that π(W ) is open, so (a) holds.
(c) Since a surjective open map is a quotient map, (c) follows from (a). • The general philosophy of the proofs of the next three propositions is this:
To “push” a smooth objects (such as a smooth map) down via a submersion,
pull it back via local sections.
Proposition 5. Suppose M , N and P are smooth manifolds, π : M → N is a
surjectice submersion, and F : N → P is any map.
Then F is smooth iff F ◦ π is smooth:
M


πy
M


yF ◦π
N −−−−→ P.
F
5
Proof. (⇒)If F is smooth, then F ◦ π is smooth by composition.
(⇐) Suppose that F ◦ π is smooth and let q ∈ N is arbitrary.
∀p ∈ π −1 (q), Proposition 4(b) guarantees the existence of a nbhd U of q
M
=
σ ↑↓ π
smooth local section σ : U → M of π such that σ(q) = p.
F
→
U
Then π ◦ σ = IdU implies
F = F ◦ IdU = F ◦ (π ◦ σ) = (F ◦ π) ◦ σ
U
U
and a
M
↓
P .
U
which is a composition of smooth maps. This shows that F is smooth in a nbhd of
each point, so it is smooth. • The next proposition gives a very general sufficient condition under which a
smooth map can be “pushed down” by a submersion.
Proposition 6 (Passing Smoothly to the quotient). Suppose π : M → N is
a surjective submersion.
If F : M → P is a smooth map that is constant on the fiber of π,
then there is a unique smooth map Fe : N → P such that Fe ◦ π = F :
M


πy
M


yF
N −−−−→ P.
e
F
Proof. Clearly, if Fe exists, it will have to satisfy Fe(q) = F (p) whenever p ∈ π −1 (q).
— We use this to define Fe : Given q ∈ N , let Fe (q) = F (p), where p ∈ M is any
point in the fiber over q. (Such a point exists because we are assuming that π is
surjective).
— This is well-defined because F is constant on the fiber of π,
and it satisfies Fe ◦ π = F by construction.
— Thus Fe is smooth by Proposition 5. • Our third proposition can be interpreted as a uniqueness result for smooth manifold defined as quotients of other smooth manifolds by submersion.
Proposition 7 (Uniqueness of Smooth Quotients). Suppose π1 : M → N1
and π2 : M → N2 are surjective submersions that are constant on each other’s
fibers. Then ∃ a unique diffeomorphism F : N1 → N2 such that F ◦ π1 = π2 :
M


π1 y
M

 π2
y
N1 −−−−→ N2 .
F
6
Quotients of Manifolds by Group Actions
• Suppose a Lie group G acts on a manifold M (on the left, say).
— Define a relation on M by setting p ∼ q if there exists g ∈ G such that g · p = q.
— This is an equivalence relation, whose equivalence classes are exactly the orbits
of G in M .
Definition. The set of orbits is denoted by M/G; with the quotient topology it
is called the orbit space of the action.
• It is of great importance to determine conditions under which the orbit space
is a smooth manifiolds.
Example. Consider the action of Rk on Rk × Rn by translation in the Rk factor:
θv (x, y) = (v + x, y).
The orbits are the affine subspaces parallel to Rk ,
and the orbit space (Rk × Rn )/Rk is homeomorphic to Rn .
The quotient map π : Rk × Rn → Rn is a smooth submersion.
Lemma 7. For any continuous action of a Lie group G on a manifold M , the
quotient map π : M → M/G is open.
S
Proof. For any open set U ⊂ M , π −1 (π(U )) = g∈G θg (U ).
Since θg is a homeomorphism, θg (U ) is open ∀g ∈ U , and therefore π −1 (π(U )) is
open in M . Because π is a quotient map, this implies that π(U ) is open in M/G. Theorem 8 (Quotient Manifold Theorem). Suppose a Lie group G acts
smoothly, freely, and properly on a smooth manifold M . Then the orbit space
M/G is a topological manifold of dimension equal to dimM −dimG, and has a
unique smooth structure with the property that the quotient map π : M → M/G
is a smooth submersion.
Proof. (I) Claim: uniqueness of the smooth structure.
— Suppose that M/G has two different smooth structures such that π : M → M/G
is a smooth submersion.
— Let (M/G)1 and (M/G)2 denote M/G with the first and second smooth structures, respectively. By Proposition 5, the identity map is smooth from (M/G)1
to (M/G)2 :
M
M



π
πy
y
(M/G)1 −−−−→ (M/G)2
Id
The same argument shows that it is also smooth in the opposite direction, so
the two smooth structures are identical.
(II) Prove that M/G is a topological manifold.
— Assume for definiteness that G acts on the left, and let θ : G × M → M denote
the action and Θ : G × M → M × M the proper map
Θ(g, p) = (g · p, p).
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(i) Claim: M/G is second countable.
If {Ui } is a countable basis for the topology of M ,
then {π(Ui )} is a countable collection of open subsets of M/G
(because π is an open map),
and it is easy to check that it is a basis for the topology of M/G.
(ii) Claim: M/G is Hausdorff.
— Define the orbit relation O ⊂ M × M by
O = Θ(G × M ) = {(g · p, p) ∈ M × M : p ∈ M, g ∈ G}.
(It is called the orbit relation because (g, p) ∈ O iff p and q are in the same
G-orbit.)
— Since proper continuous maps are closed, O is a closed subset of M × M .
— If π(p) 6= π(q), then p and q lie in distinct orbits, so (p.q) ∈
/ O.
– If U × V is a product nbhd of (p, q) in M × M that is disjoint from O, then π(U )
and π(V ) are disjoint open subsets of M/G containing π(p) and π(q) respectively.
(iii) Claim: M/G is locally Euclidean.
(iii.a) Claim: The G-orbits are embedded submfds of M diffeomorphic to G.
— For any p ∈ M , define the orbit map θ(p) : G → M by
θ(p) (g) = g · p.
This is a smooth map whose image is exactly the G orbits of p.
— Claim: θ(p) is a smooth embedding.
(a.1) θp is an injective immersion.
(a.1.1) θ(p) is injective.
Indeed, if θ(p) (g 0 ) = θ(p) (g), then g 0 · p = g · p, which inplies (g −1 g 0 ) · p = p.
Since we are assuming that G acts freely on M , this can happen only if
g −1 g 0 = e, which means that g = g 0 .
(a.1.2) θ(p) has constant rank.
– Indeed, observe that
θ(p) (g 0 g) = g 0 · θ(p) (g),
θ (p)
G −−−−→


Lg0 y
M

 (p) 0
yθ (g )
(θ (p) )∗
Tg G −−−−→ Tθ(p) (g) M



 (p) 0
(Lg0 )∗ y
y(θ (g ))∗
G −−−−→ M
θ (p)
Tg0 g G −−−−→ Tθp (g0 g) M,
(θ (p) )∗
so θ(p) is eqievariant w.r.t. the left translation on G and the given action on
M . Since G acts transitively on a G-orbit, the equivariant rank theorem implies
that θ(p) has constant rank.
(a.2) θ(p) is a proper map.
Indeed, if K ⊂ M is compact, then (θ(p) )−1 (K) is closed in G by continuity,
and since it is contained in
GK∪{p} = {g ∈ G : ∃p ∈ K such that g · p ∈ K ∪ {p}},
it is compact since G acts properly on M .
8
(iii.b) Let k=dim G and n=dim M −dim G.
Definition. Smooth chart (U, ϕ) on M , with coordinate functions
(x, y) = (x1 , · · · , xk , y 1 , · · · , y n )
is said to be adapted to the G-action if
(1) ϕ(U ) is a product open set U1 × U2 ⊂ Rk × Rn , and
(2) each orbit intersects U either in the empty set or in a single slice of the form
{y 1 = c1 , · · · , y n = cn }.
Claim: ∀p ∈ M , ∃an adapted coordinate chart centered at p.
(a) To prove this, we begin by choosing any slice chart (W, ϕ0 ) centered at p for the
orbit G · p in M . Writing the coordinate functions of ϕ0 as
(u1 , · · · , uk , v 1 , · · · , v n ),
so that (G · p) ∩ W is the slice {v 1 = · · · = v n = 0}.
— Let S be the submfd of W defined by u1 = · · · = uk = 0.
(This is the slice “perpndicular” to the orbit in these coordinates.)
Thus Tp M decomposes as the following direct sum:
Tp M = Tp (G · p) ⊕ Tp S,
where Tp (G · p) is the span of (∂/∂ui ) and Tp S is the span of (∂/∂v i ).
— Let ψ : G × S → M denote the restriction of the action θ to G × S ⊂ G × M .
Claim: ψ∗ : T(e,p) (G × S) → Tp M is surjective, (and then for dimensional
reason, it is bijective.) For this, it suffices to
claim: Tp (G · p) ⊂ ψ∗ (T(e,p) (G × S)) and Tp S ⊂ ψ∗ (T(e,p) (G × S)).
(1) Let ip : G → G × S be the smooth embedding given by ip (g) = (g, p).
The orbit map θ(p) : G → M is equal to the composition
ip
ψ
G −−−−→ G × S −−−−→ M.
Since θ(p) is a smooth embedding whose image is the orbit G · p, it follows
(p)
that θ∗ (Te G) is equal to the subapace Tp (G · p) ⊂ Tp M , and thus the
image of ψ∗ : T(e,p) (G × S) → Tp M contains Tp (G · p).
(2) Similarly, if je : S → G × S is the smooth embedding je (q) = (e, q), then
the inclusion ı : S ,→ M is equal to the composition
je
ψ
S −−−−→ G × S −−−−→ M.
Therefore, the image of ψ∗ also includes Tp S ⊂ Tp M .
— By this and the inverse function theorem, there exist a nbhd X × Y of (e, p)
in G × S and a nbhd U of p in M such that ψ : X × Y → M is a diffeomorphism.
9
(b) We need to claim: Y ⊂ S can be chosen small enough that each G-orbit
intersects Y in at most a single point.
— Suppose this is not the case.
Then if {Yi } is a countable nbhd basis for Y at p (e.g., a sequence os coordinate
balls whose diameters decrease to 0), for each i there exists distinct points pi ,
p0i ∈ Yi that are in the same orbit, namely
gi · pi = p0i for some gi ∈ G.
– Since {Yi } is a nbhd basis, both sequences {pi } and {p0i = gi · pi } converge to p.
– Since G acts properly on M , we may pass to a subsequence and assume that
gi → g ∈ G.
⇒ g · p = limi→∞ gi · pi = limi→∞ p0i = p, by continuity.
⇒ g = e, since G acts freely.
⇒ gi ∈ X, when i large enough.
— But this contradicts the fact that ψ = θ
is injective, because
X×Y
θgi (pi ) = p0i = θe (p0i ).
and we are assuming pi 6= p0i .
(c) Choose diffeomorphisms α : Bk → X and β : Bn → Y (where Bk and Bn are the
open unit balls in Rk and Rn , respectively). Define γ : Bk × Bn → U by
γ(x, y) = θα(x) (β(y)).
— Because γ is equal to the composition of diffeomorphisms
α×β
ψ
Bk × Bn −−−−→ X × Y −−−−→ U,
γ is a diffeomorphism.
— The map ϕ = γ −1 is therefore a smooth coordinate map on U .
Claim: ϕ is adapted to the G-action.
(1) is obvious from the construction.
(2) Observe that each y =constant slice is contained in a single orbit, because
it is of the form
θ(X × {p0 }) ⊂ θ(G × {p0 }) = G · p0 ,
where p0 ∈ X is the point whose y-coordinate is the given constant.
Since an orbit can inersect Y at most once, and each y =constant slice has
a point in Y , each orbit intersects U in either precisely one or no slice.
10
(iii.c) To finish the proof that M/G is locally Euclidean, let q = π(p) be an arbitrary
point of M/G, and let (U, ϕ) be an adapted coordinate chart for M centered at
p, with ϕ(U ) = U1 × U2 ⊂ Rk × Rn .
– Let V = π(U ), which is an open subset of M/G because π is an open map.
— With the coordinate functions of ϕ denoted by (x1 , · · · , xk , y 1 , · · · , y n ) as before,
let Y ⊂ U be the slice {x1 = · · · = xk = 0}.
– Note that π : Y → V is bijective by the definition of an adapted chart.
– Moreover, if W ia any open subset of Y , then
π(W ) = π({(x, y) : (0, y) ∈ W })
is open in M/G, and thus π Y is a homeomorphism.
−1
— Let σ = (π Y ) : V → Y ⊂ U , which is a local section of π.
Define a map η : V → U2 by sending the equivalence class of points (x, y) to y;
this is well-defined by the definition of an adapted chart.
— Formally, η = π2 ◦ ϕ ◦ σ, where π2 : U1 × U2 → U2 ⊂ Rn is the projection onto
the second factor.
σ
Y
V
π



η
ϕy
y
U1 × U2 −−−−→ U2 .
π2
– Because σ is a homeomorphism from V to Y and π2 ◦ ϕ is a homeomorphism
from Y to U2 , it follows that η is a homeomorphism.
This completes the proof that M/G is a topological n-manifold.
(III) Claim: M/G has a smooth structure such that π is a submersion.
— We use the atlas consisting of all charts (V, η) as constructed in the preceeding
paragraph.
— W.r.t. any such chart from M/G and the corresponding adapted chart for M , π
has the coordinate representation π(x, y) = y, which is certainly a submersion.
Thus we only need claim: any two such charts for M/G are smoothly
compatible.
e , ϕ)
— Let (U, ϕ) and (U
e be two adapted charts for M ,
and let (V, η) and (Ve , ηe) be the corresponding charts for M/G.
(III.1) First consider the case in which the two adapted charts are both centered at the
same point p ∈ M .
— Write the adapted coordinates as (x, y) and (e
x, ye).
– The fact that the coordinates are adapted to the G-action means that any two
points with the same y-coordinate are in the same orbit, and therefore also have
the same ye coordinates.
– This means that the transition map between these coordinates can be written
(e
x, ye) = (A(x, y), B(y)), where A and B are smooth maps defined on some nbhd
of the origin.
– The transition map ηe ◦ η −1 is ye = B(y), which is clearly smooth.
e , ϕ)
(III.2) In the general case, suppose (U, ϕ) and (U
e are adapted charts for M , and
e
p ∈ U , pe ∈ U are points such that π(p) = π(e
p) = q.
11
— Modifying both charts by adding constant vectors, we can assume that they are
centered at p and pe, respectively.
— Since p and pe are in the same orbit, ∃a group element g such that g · p = pe.
— Because θg is a diffeomorphism taking orbits to orbits, it follows that ϕ
e0 = ϕ
e ◦ θg
is another adapted chart centered at p.
– Moreover, σ
e0 = θg−1 ◦ σ
e is the local section corresponding to ϕ
e0 , and therefore
e0 ◦ σ
e 0 = π2 ◦ ϕ
e ◦ θg ◦ θg−1 ◦ σ
e = π2 ◦ ϕ
e◦σ
e = ηe.
ηe0 = π2 ◦ ϕ
Thus we are back in the situation of the preceeding paragraph, and the two
charts are smoothly compatible.