Download General Topology of Ramified Coverings

General Topology of Ramified Coverings
Lucio Guerra
Rapporto Tecnico n. 02-1993 (seconda versione)
Dipartimento di Matematica - Università di Perugia
Contents
Introduction
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
2
Background
Quasi-coverings, Spreads
Restrictions
Ramified coverings
Ramification
Locally compact spaces
Complete spreads
Completion
Classification
Group actions
Complex analytic spaces
3
3
4
6
8
10
12
13
16
17
20
Bibliography
23
1
Introduction
The concept of a ramified covering arises in the theory of Riemann
surfaces, and is suddenly realized to belong indeed to the topology
of spaces. It occurs in complex geometry in several important situations such as, for instance, the usual representations of singularities of
complex analytic spaces as ramified coverings of complex polydiscs. A
purely topological formulation is found in [Fox], where certain restrictions are imposed on the covering and base spaces that in particular
avoid some kind of singularities, namely those with several branches.
A treatment of finite coverings only is in [E].
We propose in this paper a general setting, including all the above
mentioned examples, and the usual analytic coverings of complex spaces,
and a general class of orbit maps. It is based on a definition only consisting of the very characteristic properties, leaving out some special
requirements, such as (global) connectedness and completeness, which
serve to guarantee some special features of the theory. We explain that
the full content of Fox’s definition is in a classification theorem, involving his beautiful process of “completion”, modeled on the normalization
of complex spaces.
An effort has been made in stating results in their full generality
and detail, providing satisfactory motivations of axioms, introducing
relative properties of maps rather than conditions on the spaces. In
particular, we allow 0-dimensional fibres, and do not confine to locally
compact spaces nor to proper maps; in the locally compact case the
theory is simplified, mainly because ramified coverings are then open
maps.
2
1. Background
1.1. Let f : X → Y be a continuous map. We assume that X, Y
are locally connected spaces. This implies unique decomposition into
sheets over sufficiently small evenly covered open sets. A property of
the map f suitable to replace local connectedness of the spaces would
be a prescribed “slicing” [H; p. 97]. This is the only non-relative axiom
that we introduce, for the sake of a reasonably simple exposition.
The only separation axiom needed somewhere is that f is a Hausdorff map, i.e. the diagonal x1 = x2 is closed within the relative product
X ×Y X, the subspace f (x1 ) = f (x2 ) of X 2 . This is something more
than requiring the fibres to be Hausdorff spaces.
A special attention will be payed to proper maps. Recall that f is
said to be proper iff it is closed and has compact fibres; when X, Y are
locally compact spaces, an equivalent requirement is that f −1 (K) is
compact for every compact K ⊂ Y . For the basic properties of proper
maps we refer to [B; ch. I, §10]. A finite map is a proper map with
finite fibres.
1.2. Let A be a dense subspace of a locally connected topological
space X. We say that A is locally connected in X if there is a basis of
open subsets U in X such that every A∩U is connected or, equivalently,
if A ∩ U is connected for every connected open U in X [Fox; p. 247,
lemma]. A necessary condition is that A is a locally connected space,
in the absolute sense. The complement A0 = X − A is often said to be
non-separating in X, as U − A0 is connected for every connected U .
2. Quasi-coverings, Spreads
Consider a continuous map f : X → Y .
2.1. If there is some neighborhood U of x ∈ X which is mapped
by f homeomorphically onto a neighborhood f (U ) of y = f (x), we say
that x is a smooth or regular point of f . Then x is an isolated point in
the fibre f −1 (y).
A non-smooth point x ∈ X is also called a critical point, and its image f (x) is called a critical value of f (the smooth/critical terminology
comes from differential topology).
We denote by X0 the open subset of smooth points of f . If X0 = X,
then f is a local homeomorphism.
2.2. If there is some open neighborhood V of y ∈ Y such that
f −1 (V ) can be written as a disjoint union of open subsets U of X, each
mapped homeomorphically onto V by f , then V is said to be evenly
3
covered by f , and y is said to be an ordinary value of f . The fibre
f −1 (y) is then a discrete subspace in X.
We denote by Y0 the open set of ordinary values of f . Then
f −1 (Y0 ) ⊂ X0 .
If Y0 = Y , we call f an ordinary covering. We are not requiring
neither Y0 connected, implying fibres with one and the same cardinality,
nor f −1 (Y0 ) connected, which is necessary for possibly having universal
coverings.
We say that f is a quasi-covering iff f −1 (Y0 ) is dense in X. Then
Y0 is dense in f (X). A quasi-covering f has an associated ordinary
covering f0 : f −1 (Y0 ) → Y0 .
2.3. Following [Fox], we say that a continuous map f is a spread if
it has the following property:
for every x ∈ X and every neighborhood N of x there is some open
neighborhood V of y = f (x) such that f −1 (V ) = U ∪˙ U 0 , with U, U 0
disjoint opens and x ∈ U ⊂ N .
In other words, f is a spread iff the closed-open subsets of the inverse
images of the open sets in Y form a basis for the open sets in X or
equivalently, since X, Y are locally connected spaces, iff the connected
components of the inverse images of the connected open sets in Y form
a (smaller) basis for the open sets in X.
A spread f has 0-dimensional (⇒ totally disconnected) fibres f −1 (y), y ∈
f (X). An ordinary covering is a spread.
The concept of a spread may be considered as a suitable relative
version of 0-dimensionality for a map. We will see (6.3) that a spread
of locally compact spaces is nothing more than a continuous map with
0-dimensional fibres.
2.4. A finite (Hausdorff ) map f is a spread.
Proof. If y ∈ Y , x ∈ f −1 (y), and N is a neighborhood of x, since
f (y) is a finite set, we can find disjoint opens U, U 0 with x ∈ U ⊂ N
and f −1 (y) ⊂ U ∪˙ U 0 ; since f is a closed map, there is some open
neighborhood V of y in Y with f −1 (V ) ⊂ U ∪˙ U 0 .
−1
3. Restrictions
3.1. Let f : X → Y be a quasi-covering. For some open V ⊂ Y ,
let U be closed-open in f −1 (V ) and let f¯ : U → V be the restriction
of f . Then f¯ still is a quasi-covering, whose ordinary locus contains
V 0 := f (U ) ∩ Y0 .
4
Proof. The set V 0 = f (U ) ∩ Y0 = f (U ∩ f −1 (Y0 )) is a non-empty
open in V . We have to show that the set V0 of ordinary values of f¯
contains V 0 . Since f¯−1 (V 0 ) = U ∩ f −1 (Y0 ) is dense in U , it follows that
f¯ is a quasi-covering.
Let y ∈ V ∩ Y0 . There is some connected open W with y ∈ W ⊂
V ∩ Y0 and f −1 (W ) = ∪˙ i∈I Ai , Ai open and mapped homeomorphically
onto W by f . As U is closed-open in f −1 (V ), every Ai ∩U is closed-open
in Ai hence, as Ai is connected, either one has Ai ⊂ U or Ai ∩ U = ∅.
Thus y ∈ V 0 iff some Ai ⊂ U ; in this case W ⊂ V 0 and we can write
f¯−1 (W ) = f −1 (W ) ∩ U = ∪˙ i∈J Ai , where J = {i ∈ I : Ai ⊂ U }. This
gives V0 ⊃ V 0 .
3.2. In the setting of 3.1, the following also holds.
(a) U dominates f (U ) ∩ V = V 0 ∩ V ;
(b) V 0 is closed-open in V ∩ Y0 ;
(c) if C is a connected component of U ∩ f −1 (Y0 ), then f (C) is a
connected component of V ∩ Y0 , contained in V 0 ;
(d) U dominates V ⇔ V 0 = V ∩Y0 ⇔ every connected component
of V ∩ Y0 is the image of some connected component of U ∩ f −1 (Y0 );
(e) if Y0 is locally connected in Y , then V 0 = V ∩ Y0 and U dominates V ;
(f) if f is an open map and V := f (U ), then V 0 = V ∩ Y0 holds by
definition, with no further hypothesis on Y0 .
Proof. (a) U ∩ f −1 (Y0 ) dense in U ⇒ V 0 = f (U ∩ f −1 (Y0 )) dense
in f (U ) ∩ V . (b) In the proof of 3.1 we also have that y ∈ V − V 0 iff
every Ai ∩ U = ∅, and in this case W ∩ V 0 = ∅. (c) C is closed-open
in U ∩ f −1 (Y0 ), because f −1 (Y0 ) is locally connected, therefore f (C) is
closed-open in V ∩ Y0 , by (b), and connected. (d) The first equivalence
follows from (a), the second from (c). (e) Without loss of generality
we may assume V connected, so V ∩ Y0 is connected (1.2), and the
assertion then follows from (b). (f) is obvious.
3.3. The proofs of 3.1 and 3.2 only depend on f being ordinary on
f (Y0 ) dense in X, and not on the maximality of Y0 as the whole set
of ordinary values of f ; therefore 3.2 still holds after replacing Y0 with
any dense open subset Y1 of Y0 .
−1
3.4. Let f : X → Y be a proper (Hausdorff ) map, let U be open in
f (V ), with V open in Y , and let f¯ : U → V be the restriction of f .
Then f¯ is still proper if and only if U is closed-open in f −1 (V ).
Proof. Assume U closed-open in f −1 (V ). Every fibre f¯−1 (y) =
f −1 (y) ∩ U is then closed in the compact f −1 (y), hence it is compact
−1
5
too. If F is closed in U , it is closed in f −1 (V ), i.e. F = F ∩ f −1 (V );
also f (F ) = f (F ) as f is a closed map; it follows that f (F ) = f (F ∩
f −1 (V )) = f (F ) ∩ V = f (F ) ∩ V and f (F ) is closed in V .
Conversely, assume that f¯ is proper. We need show that every
x ∈ U ∩ f −1 (V ) is in U . Suppose, on the contrary, that x 6∈ U . Let
y = f (x). Since f is Hausdorff and U ∩ f −1 (y) is compact, there are
disjoint opens A containing x and B containing U ∩ f −1 (y). Then F :=
U − B is closed in U . We have y 6∈ f (F ) and x ∈ F ⇒ y ∈ f (F ). Thus
f (F ) is not closed in V and f¯ is not a closed map, a contradiction. In particular, a necessary and sufficient condition for a proper map
being “locally proper” is being a spread.
3.5. Let f : X → Y be a (proper) quasi-covering and a spread. The
collection of U closed-open in some f −1 (V ), V open in Y , is a basis
for the open sets in X, such that every restriction f |U : U → V is a
(proper) quasi-covering and a spread.
Proof. Immediate from 3.1 and 3.4.
4. Ramified coverings
4.1. We say that a (continuous, Hausdorff) map f : X → Y is a
ramified covering iff it is:
(1) a quasi-covering;
(2) a spread;
(3) locally dominant: the collection of opens U in X such that f (U )
is dense in some open V in Y is a basis for the open sets in X.
For simplicity, we also assume that f is dominant, i.e. f (X) = Y .
4.2. An equivalent formulation of properties (2) and (3) together
is the requirement that the collection of U ⊂ X such that U is closedopen in some f −1 (V ), V open in Y , and f (U ) is dense in V , form a
basis for the open sets in X. Because of property (1), that f (U ) is
dense in V means (for such a basic U ) that f (U ) ∩ Y0 = V ∩ Y0 or,
equivalently, that every connected component of V ∩ Y0 is the image of
some connected component of U ∩ f −1 (Y0 ) (3.2.d).
It is useful to remark here that the property of being a dominant
map is in general not inherited by the local restrictions, and cannot
therefore be put in a general definition which has to be of a local
character, unless of course the property of being locally dominant also
appears, though in studying a single given map one can always reduce
to a dominant one.
6
In a quasi-covering spread, the property of being locally dominant
avoids “identifications” like, for instance, in the quotient map of X =
R2 onto the quotient space Y obtained by identifying two points x, x0 ∈
X into a single point y ∈ Y . This is a quasi-covering spread, and not
locally dominant both at x and x0 , for every open neighborhood V of
y is the connected sum of open neighborhoods U of x, U 0 of x0 , with x
and x0 identified, so U, U 0 dominate each a “half” of V .
4.3. The definition in [Fox] of a branched covering is more restrictive. It is a quasi-covering and a spread. Property (3) is replaced
with:
(30 ) both f −1 (Y0 ) and Y0 are locally connected in X, Y respectively.
This is not a relative property of the map f but imposes restrictions
on the spaces X, Y as well. For instance, it is not satisfied for the usual
representations of singularities of complex analytic spaces as “ramified
coverings” of complex polydiscs.
If Y0 is locally connected in Y , then property (3) holds (3.2.e),
so definition 4.1 includes the ramified coverings in the sense of Fox.
They are subject to some more requirements that, in the aim to a
comprehensive setting, we prefer to leave out of the definition. Namely,
they are (global) connectedness of f −1 (Y0 ) and Y0 (a short comment
in 2.2), and a property of “completeness”, that will be considered later
on in §§7, 8. The full content of completeness plus property (30 ) will
be exploited in 9.5.
4.4. Let f : X → Y be a (proper) covering. The collection of U
closed-open in some f −1 (V ), V open in Y , with f (U ) dense in V , is a
basis for the open sets in X, such that every restriction f |U : U → V
is a (proper) covering.
Proof. Immediate, using 4.2 and 3.4.
4.5. A finite map which is a covering is called a finite covering.
(a) A proper covering f is surjective and open.
Proof. Closed and dominant imply surjective. There is a basis
of opens U in X such that f |U is a proper covering dominating some
open V in Y (4.4), so f (U ) = V by the same reason.
(b) In order for a finite surjective map f being a covering it is
necessary and sufficient that it is:
(1) a quasi-covering;
(300 ) an open map.
7
Proof. Necessity is from (a). Sufficiency is because an open map
is locally dominant (3.2.f) and a finite map is a spread (2.4).
It is now clear that definition 4.1 also includes the finite branched
coverings of [E].
5. Ramification
5.1. Let f : X → Y be a ramified covering. Define:
d(f ) := sup card f −1 (y).
y∈Y0
Note that card f −1 (y) is constant for y ∈ Y0 , if Y0 is connected. In this
case, the number d(f ) may be called the covering degree of f .
For every x ∈ X there is a basis of neighborhoods U of x such that
every restriction f |U is a ramified covering of some open neighborhood
of f (x) (4.4). Define:
d(x; f ) := inf d(f |U ).
U
The number d(x; f ) counts how many points in the fibre f −1 (y) tend
to x as y tends to f (x). It may be called the ramification index or
multiplicity, or the local degree of f at x.
Clearly U 0 ⊂ U ⇒ d(f |U 0 ) ≤ d(f |U ). It follows that the basic
neighborhoods U of x such that the restriction f |U is a covering of
smallest degree d(f |U ) = d(x; f ) still form a neighborhood basis at x.
If a restriction of f to some open neighborhood U of x is still a
covering, then:
d(x; f |U ) = d(x; f ).
Thus the local degree is a local concept.
5.2. The function x 7→ d(x; f ) is upper semicontinuous.
Proof. If α is a cardinal number, and d(x; f ) < α, there is some
neighborhood U of x with d(f |U ) < α, whence d(x0 ; f ) ≤ d(f |U ) < α
for x0 ∈ U , and the set {x ∈ X : d(x; f ) < α} is therefore open in
X.
A point x ∈ X where d(x; f ) > 1 is called a ramification point of f
and its image f (x) is called a branch point or value of f (the branchramification terminology is used in algebraic geometry). The closed
set R of ramification points is called the ramification locus of f , and
B := f (R) is called the branch locus of f . Clearly R ⊂ X − X0 and
we will see (6.6) that R = X − X0 for locally compact X. If d(x; f ) is
finite for every x ∈ X we say that f is finitely ramified.
8
5.3. Let f be a covering with d(f ) finite. For every y ∈ Y :
card f −1 (y) ≤ d(f ).
Proof. Let y ∈ Y . Let S = {xi } be a finite subset of f −1 (y). Take
disjoint neighborhoods Ni of xi . Since f is a spread, there is an open
neighborhood V of y such that each Ni contains some neighborhood
Ai of xi with Ai closed-open in f −1 (V ).
Since f is a covering, shrinking V if necessary, we may assume that
each Ai contains some neighborhood Ui of xi with Ui closed-open in
f −1 (V ), f (Ui ) dense in V , and such that fi := f |Ui : Ui → V has
˙ i )∪U
˙ 0 for some U 0 .
degree d(fi ) = d(xi ; f ). Then f −1 (V ) = (∪U
Since f is locally dominant, then f (Ui ) ∩ Y0 = V ∩ Y0 (3.2.d) and
every fi is ordinary over V ∩ Y0 (3.1). Pick any y0 ∈ V ∩ Y0 . Then
card fi −1 (y0 ) ≥ 1 and
X
d(f ) ≥ card f −1 (y0 ) ≥
card fi −1 (y0 ) ≥ card S.
i
Hence d(f ) ≥ card f
−1
(y) since d(f ) is finite.
5.4. A finitely ramified covering f is discrete.
Proof. Every x ∈ X has a neighborhood U such that f |U is a
covering of finite degree of some neighborhood of y = f (x). Because of
5.3, the fibre f −1 (y) ∩ U of f |U is finite, so x is an isolated point in its
fibre f −1 (y).
5.5. A proper covering f with Y0 connected is finite and has finite
degree.
Proof. The ordinary fibres are compact and discrete, hence finite,
all having one and the same cardinality as Y0 is connected, so d(f ) is
finite. Every fibre is then finite because of 5.3.
5.6. (ramification formula) Let f be a finite covering with Y0 connected and locally connected in Y . For every y ∈ Y :
X
d(x; f ) = d(f ).
x∈f −1 (y)
Cf. [M; 3.25, p. 53].
Proof. Take up again the proof of 5.3, with S = f −1 (y). Since
f is proper, there is an open neighborhood V 0 of y with V 0 ⊂ V and
˙ i . In other words, shrinking V if necessary, we may
f −1 (V 0 ) ⊂ ∪U
P
−1
˙ i . Then card f −1 (y0 ) = i card fi −1 (y0 ).
assume f (V ) = ∪U
9
We may also take V connected so that V ∩ Y0 is connected too,
since Y0 is locally connected in Y . Then d(f ) = card f −1 (y0 ) as Y0 is
connected, and d(xi ; f ) = d(fi ) = card fi −1 (y0 ) as V ∩ Y0 is connected.
5.7. Assume that f is finitely ramified. Assume furthermore that
Y is a first-countable space and that Y0 is path-connected. It follows
that for every y ∈ Y there is some path u : [0, 1] → Y with u(1) = y
and u(t) ∈ Y0 for t 6= 1 (hence Y is path-connected too). For every
y ∈Y:
card f −1 (y) ≤ d(f ).
This follows from the possibility of lifting such a path u.
Proof. Let y0 = u(0). For every x0 ∈ f −1 (y0 ) there is a unique
lifting u0 : [0, 1) → X of u|[0, 1) with u0 (0) = x0 . If u0 extends to a
path ū : [0, 1] → X, then ū(1) ∈ f −1 (y) and ū is a lifting of u. We are
going to prove that for every x ∈ f −1 (y) there is at least one such ū
with ū(1) = x. It follows that card f −1 (y) ≤ card f −1 (y0 ) ≤ d(f ).
Let x ∈ f −1 (y). For every neighborhood U of x closed-open in
some f −1 (V ) there is 0 ≤ θ < 1 such that u(t) ∈ V ∩ Y0 for θ ≤ t < 1.
Let W be the connected component of V ∩ Y0 containing u[θ, 1). The
function card U ∩ f −1 (y) takes on W a constant value δ(U, u). Define
δ(x, u; f ) := inf U δ(U, u). Clearly δ(x, u; f ) ≤ d(x; f ). Those U as
before such that δ(U, u) takes the smallest value δ(x, u; f ) still form a
neighborhood basis at x. For every such U there are precisely δ(x, u; f )
liftings u0 : [θ, 1) → U of u|[θ, 1).
Defining u0 (1) = x we obtain a lifting u0 : [θ, 1] → U which is
continuous at 1. Indeed, every neighborhood of x contains some basic
U 0 closed-open in f −1 (V 0 ) with U 0 ⊂ U . There is some θ ≤ < 1 such
that u[, 1] ⊂ V 0 , and there are δ(x, u; f ) liftings of u|[, 1) in U 0 , as
many as there are in U . Hence u0 |[, 1) is one of them, and u0 [, 1] ⊂ U 0 .
Easily u0 can now be extended to a lifting ū : [0, 1] → X of u with
ū(θ) = u0 (θ).
6. Locally compact spaces
6.1. Let X be a locally compact space. Let F ⊂ X be a closed
subspace, C a connected component of F , and assume that C is compact. There is a relatively compact open neighborhood U of C such that
∂U ∩ F = ∅. Cf. [WD; p. 102, lemma].
Proof. There is some relatively compact open neighborhood N of
C. If M := ∂N ∩ F 6= ∅, for every x ∈ M there are disjoint opens
˙ 0 , C ⊂ A, x ∈ A0 . As M is compact, there
A, A0 such that F ⊂ A∪A
10
˙ 0i ,
is a finite collection of pairs of disjoint opens Ai , A0i with F ⊂ Ai ∪A
0
C ⊂ Ai , and such that M ⊂ ∪Ai . Thus U := (∩Ai ) ∩ N is a relatively
compact open neighborhood of C, U 0 := (∪A0i ) ∪ (X − N ) is open and
˙ 0 , hence
contains M , and easily one sees that U ∩ U 0 = ∅ and F ⊂ U ∪U
∂U ∩ F = ∅.
6.2. (criterion of local properness) Let X be a locally compact space
and f : X → Y a continuous map. Let y ∈ f (X), C a connected
component of the fibre f −1 (y), and assume that C is compact. There
are a neighborhood V of y and a relatively compact open neighborhood
U of C such that U is closed-open in f −1 (V ) and the restriction f |U :
U → V is a proper map. Cf. [S; p. 77, Hilfssatz 3].
Proof. Because of 6.1 there is some relatively compact open neighborhood U 0 of C with ∂U 0 ∩ f −1 (y) = ∅. Then f (∂U 0 ) is compact and
does not contain y. Thus V := Y − f (∂U 0 ) is a neighborhood of y
and U := U 0 ∩ f −1 (V ) is a relatively compact open neighborhood of C.
˙
As V ∩ f (∂U 0 ) = ∅ then f −1 (V ) ⊂ U 0 ∪(X
− U 0 ), so U is closed-open
−1
in f (V ). That f |U is a proper mapping follows applying 3.4 to the
restriction f |U 0 : U 0 → Y .
6.3. (criterion for a spread) Let X be a locally compact space, and
f : X → Y a continuous map. If f has totally disconnected fibres then
f is a spread. Cf. [J; p. 332, lemma]; a counterexample with X not
locally compact is in [Fox; p. 255, footnote].
Proof. Let x ∈ X, y = f (x), N a neighborhood of x. Proposition
6.2 applied to the restriction f |N gives us open neighborhoods U of x,
V of y, with U ⊂ N and U closed-open in f −1 (V ).
6.4. (criterion for a ramified covering) Let X be locally compact.
In order for f : X → Y being a ramified covering it is necessary and
sufficient that it is:
(1) a quasi-covering;
(20 ) with totally disconnected fibres;
(300 ) an open map.
Then f (X) is locally compact too.
A covering f with Y0 locally connected in Y is finitely ramified and
discrete.
Proof. Sufficiency is easy: that (20 ) ⇒ (2) is 6.3, that (300 ) ⇒ (3)
is clear (3.2.f). Necessity. From 4.4 and 6.2 it follows that the collection
of U closed-open in f −1 (V ), V open in Y , with f (U ) dense in V and
f |U : U → V a proper covering, is a basis for opens in X. Because of
4.5.a, we have f (U ) = V for every U , so that f is an open map. This
11
proves (300 ). Local compactness of f (X) is then a consequence of a well
known property of proper maps [B; p. 106, cor.].
If Y0 is locally connected in Y , we may assume that V0 is connected
for every basic proper covering U → V , which has therefore a finite
degree by 5.5. Thus f is finitely ramified and so discrete by 5.4.
6.5. (local properness of coverings) Let X be locally compact, and
let f : X → Y be a covering. There is a basis of open sets U in X such
that every restriction f |U : U → f (U ) is a proper covering.
Proof. Seen in the proof of 6.4.
6.6. (criterion of smoothness) Let X be locally compact, and let
f : X → Y be a covering. Let x ∈ X and y = f (x). Then d(x; f ) = 1
if and only if f is a local homeomorphism at x.
Proof. Clearly d(x; f ) = 1 if f is a local homeomorphism at x.
Conversely, let d(x; f ) = 1. Take some open neighborhood U of x such
that f |U is a covering of degree 1. We know from 6.4 that f (U ) is
open in Y and that f |U is an open map. It is bijective by 5.3, hence
an homeomorphism.
7. Complete spreads
Let f : X → Y be a spread. Our background hypothesis that f
is a Hausdorff map is slightly more general than the one in [Fox] that
X, Y are T1 spaces (a spread with T1 fibres is Hausdorff), and makes no
significant change. Without local connectedness the subject is worked
out in [Mi].
7.1. Let y ∈ Y . Consider functions ϕ assigning to every open
neighborhood V of y a connected component ϕV of f −1 (V ) in such a
way that V 0 ⊂ V ⇒ ϕV 0 ⊂ ϕV . We define N (ϕ) := ∩V ϕV . Clearly
N (ϕ) ∩ f −1 (y) contains at most one point, as f is Hausdorff. If Y is
T1 then N (ϕ) = N (ϕ) ∩ f −1 (y).
We say that f is complete at y iff N (ϕ) ∩ f −1 (y) 6= ∅ for every
function ϕ based at y, and f is complete iff it is complete at every
y ∈ Y . An ordinary covering is a complete spread.
7.2. A proper spread f is complete.
Proof. Let y ∈ Y and let ϕ based at y be as in the definition
above. We claim that ϕV ∩ f −1 (y) 6= ∅ for every V . Indeed, if some
ϕV ∩ f −1 (y) = ∅, the complement U := f −1 (V ) − ϕV is then an
open neighborhood of f −1 (y) (ϕV is closed in f −1 (V )) and there is
an open neighborhood V 0 of y with f −1 (V 0 ) ⊂ U , since f is proper,
12
whence ϕV 0 ⊂ U ∩ ϕV so ϕV 0 = ∅, a contradiction. Now the family
of non-empty closed subsets ϕV ∩ f −1 (y) of f −1 (y) clearly has the
finite intersection property (i.e. every finite subfamily has non-empty
intersection) and f −1 (y) is compact, so the whole family has non-empty
intersection.
7.3. (lifting of paths) Let f : X → Y be a complete spread. For
every path u : [0, 1] → Y , every “open” lifting u0 : [0, 1) → X of u|[0, 1)
has a unique extension to a lifting ū : [0, 1] → X of u.
This will be an easy consequence of 8.5 in the next section.
7.4. (ramification formula) Let f : X → Y be a finitely ramified
and complete covering, and assume that Y is first-countable and Y0 is
path-connected and locally connected in Y . For every y ∈ Y :
X
d(x; f ) = d(f ).
x∈f −1 (y)
Cf. [AS; I.21.B, p. 42].
Proof. Take any path u : [0, 1] → Y with u(1) = y and u(t) ∈ Y0
for t 6= 1. The collection of open liftings u0 : [0, 1) → X of u|[0, 1) has
cardinality d(f ). We know from 7.3 that every u0 extends to ū : [0, 1] →
X, so that ū(1) ∈ f −1 (y). We have seen in the proof of 5.7 that for
every x ∈ f −1 (y) there are δ(x, u; f ) paths ū with ū(1) = x. Coming
back to that proof, we may assume that V is connected, so V ∩ Y0 is
connected too as Y0 is locally connected in Y , hence W = V ∩ Y0 and
therefore δ(x, u; f ) = d(x; f ).
7.5. Here is the place to mention some related facts concerning local homeomorphisms. An ordinary covering has the path lifting property and the converse fact, that a local homeomorphism of manifolds
having the path lifting property is an ordinary covering [F; p. 28,
4.19], was sometimes called the “monodromy theorem”. Because of
7.3, completeness also is equivalent to the path lifting property for a
local homeomorphism of manifolds. Another version of completeness
for a local homeomorphism of Riemann surfaces is in [AS; I.14.F, p.
29].
8. Completion
8.1. Same setting as in §7. An extension of a spread f : X → Y
is a spread g : X 0 → Y together with an embedding i : X → X 0 such
that g ◦ i = f . It is clear what isomorphism of extensions will mean.
13
A complete extension f¯ : X → Y with i(X) dense in X and locally
connected in X is called a completion of f .
8.2. (extension theorem) Consider a commutative diagram
X
f¯
.
. . ā
..
i 6 ~
a
X - X0
f?
g
?
b
-Y
- Y0,
where f¯ is a completion of the spread f and g is a complete spread.
There is a (unique) extension ā of a making the whole diagram commute.
The fact that g ◦ a = b ◦ f is expressed saying that the map a
“covers” the map b. So in other words the theorem states that a has
an extension ā that also covers b. The proof given in [Fox; p. 247]
to the following statement, which is in fact a corollary of the former,
actually works.
8.3. (functoriality) In a commutative diagram
ā- X 0
X ...
6
i 6
i0
f¯
a-
X
X0
ḡ
f?
g
?
b
-Y
- Y 0 ,
where f¯, ḡ are completions of f, g, respectively, there is a unique extension ā of a.
8.4. (universal property) For every commutative diagram
X
f¯
.
. . ā
..
i 6 ~
a
X - X0
f?
g
?
-Y = Y ,
14
where g is a complete spread and f¯ is a completion of f , there is a
unique extension ā making the whole diagram commute.
This universal property makes a completion of f (if exists) uniquely
determined up to isomorphisms as the “minimal” complete extension
of f .
8.5. (covering homotopies) Let f : X → Y be a complete spread.
Let Q be a locally connected space and let a : Q × [0, 1] → Y be a
homotopy. Every “open” homotopy a0 : Q × [0, 1) → X covering the
“open” homotopy a|Q × [0, 1) extends to a homotopy ā : Q × [0, 1] → X
that covers a.
The inclusion mapping of Q×[0, 1) into Q×[0, 1] is clearly a spread,
and its completion is the identity mapping of Q × [0, 1] upon itself, so
the statement is a corollary of 8.2.
In the special case where Q is a point, we obtain 7.3.
8.6. (existence theorem) Every spread has a completion.
The proof is in [Fox; §2]. Here we only quote how a canonical
completion of a spread f : X → Y is constructed.
The points of X are the functions ϕ, based at some point y ∈ Y ,
assigning to every open neighborhood V of y a connected component
ϕV of f −1 (V ) in such a way that V 0 ⊂ V ⇒ ϕV 0 ⊂ ϕV . The map
f¯ : X → Y is defined by f¯(ϕ) = y. The inclusion i : X → X sends x
into the function i(x), based at y = f (x), assigning to V the connected
component of f −1 (V ) containing x. A basis for the topology of X
consists of all the sets of the following form: for V open in Y and U
closed-open in f −1 (V ), the set U |V of points ϕ, based at some y ∈ V ,
and such that ϕV ⊂ U .
8.7. The completion f¯ of a quasi-covering spread f is a quasicovering.
Proof. The restriction of f¯ to f¯−1 (Y0 ) is still complete. Also
f −1 (Y0 ) is dense and locally connected in f¯−1 (Y0 ) (X locally connected
in X, U open in X ⇒ U ∩ X locally connected in U ). Thus the restriction of f¯ is a completion of the associated ordinary covering f0 ,
which is already complete. From the uniqueness of completions (8.4)
we conclude that f −1 (Y0 ) = f¯−1 (Y0 ), so f¯ is ordinary over Y0 .
15
In general, the completion process does not preserve coverings, because it does not preserve the property of being locally dominant. However, if Y0 is locally connected in Y , then both the quasi-covering spread
f and its completion f¯ are locally dominant and so coverings by 3.2.e.
9. Classification
9.1. Same setting as in §7. To every complete quasi-covering spread
f : X → Y we associate the pair (f0 , i), where f0 : f −1 (Y0 ) → Y0 is the
associated ordinary covering, and i : Y0 → Y is the inclusion mapping.
In the opposite direction, for every pair (g, j) consisting of an ordinary covering g : S → A and a dense open embedding j : A → Y , the
completion of j ◦ g is a quasi-covering (8.7).
We are going to investigate to what extent these two correspondences are inverse each other.
9.2. Consider a pair (g, j) as above, form the completion f of j ◦ g,
and take the associated pair (f0 , i). This needs not be the same as the
one we started with. What we can assert is the following.
In the diagram
S ,→ X
g↓
↓f
j
A → Y,
arguing as in the proof of 8.7 shows that:
S = f −1 (A) and g = f |f −1 (A),
where we identify A and j(A). This says that f is ordinary over A,
i.e. the locus B of ordinary values of f contains A and the ordinary
covering f0 extends g.
The situation is summarized in the following diagram:
S = f −1 (A) ,→ f −1 (B) ,→ X
g↓
↓ f0
↓f
A
→
B
,→ Y,
where g and f0 are ordinary coverings and where A is locally connected
in B (as S is locally connected in f −1 (B) and f0 is ordinary).
9.3. Conversely, consider a commutative diagram:
S1 = g2 −1 (A1 ) ,→ S2
g1 ↓
↓ g2
A1
→ A2
,→ Y,
16
where g1 and g2 are ordinary coverings, and A1 is dense and locally
connected in A2 . We claim that j1 ◦ g1 and j2 ◦ g2 have isomorphic
completions (j1 , j2 the inclusion mappings of A1 , A2 into Y ).
In fact, if f : X → Y is the completion of j2 ◦ g2 , then S2 is dense
and locally connected in X; also S1 is dense and locally connected in
S2 (as A1 is dense and locally connected in A2 and g2 is ordinary), so
S1 is dense and locally connected in X and f is a completion of j1 ◦ g1 .
9.4. We make the situation symmetric in A1 , A2 , by defining an
equivalence of pairs (g1 , j1 ) ∼ (g2 , j2 ) if and only if A := A1 ∩ A2 is locally connected both in A1 , A2 and there is an isomorphism g1 |g1 −1 (A) ∼
=
g2 |g2 −1 (A) over A (where we are writing embeddings like inclusions).
Easily from 9.3 it follows that (g1 , j1 ) ∼ (g2 , j2 ) implies that j1 ◦ g1
and j2 ◦ g2 have isomorphic completions. Furthermore, what we have
seen in 9.2 is that, for any pair (g, j), the completion f of j ◦ g has an
associated pair (f0 , i) ∼ (g, j).
Recalling the final remark in 8.7, we end with the following classification theorem.
9.5. Sending a pair (g, j), consisting of an ordinary covering g of
Y0 and a dense open embedding j of Y0 into Y , into the completion of
j ◦ g, induces an injection of the set of equivalence classes of pairs (g, j)
under ∼ into the set of isomorphism classes of complete quasi-covering
spreads over Y . A left inverse is induced by the map sending f into
(f0 , i).
Furthermore, this gives a bijection from equivalence classes of pairs
with Y0 locally connected in Y onto isomorphism classes of complete
ramified coverings with both f −1 (Y0 ), Y0 locally connected in X, Y respectively, i.e. satisfying property (30 ) in 4.3.
10. Group actions
10.1. Let G be a subgroup of the group of automorphisms of a
locally connected topological space X. We will denote by Y = X/G
the resulting quotient space and by π : X → Y the canonical quotient
map. It is an open map, so Y is locally connected too.
We assume that the action of G is discontinuous, or discrete, i.e.
that every x ∈ X has a basis of open neighborhoods U (we call them
good neighborhoods) such that:
(1) U ∩ gU = ∅ for every g ∈ G with gx 6= x;
(2) U = gU if gx = x, that is to say U is invariant under the
stabilizer Nx .
17
The quotient map π is then a Hausdorff map and a spread with
discrete fibres (the orbits of G).
An action is necessarily discrete if every x has some open neighborhood U satisfying (1) and if every stabilizer Nx is finite. In particular,
the action of a finite group G on a Hausdorff space X is necessarily
discrete.
10.2. π is a complete spread.
Proof. A neighborhood basis at π(x) ∈ Y is the collection of
V = π(U ) where U is a good connected neighborhood of x ∈ X. Then
π −1 (V ) = ∪˙ g∈G/Nx gU is the decomposition into connected components. A choice function ϕ, based at π(x), is determined by its values
on all basic V . We may assume that ϕV = U for some given V = π(U ).
Then, if U 0 ⊂ U and V 0 = π(U 0 ), one has ϕV 0 ⊂ π −1 (V 0 ) ∩ ϕV =
(∪˙ gU 0 ) ∩ U = U 0 , whence ϕV 0 = U 0 and N (ϕ) = ∩V ϕV = ∩U =
{x}.
10.3. The function card Nx is an upper semicontinuous and Ginvariant function on X.
Proof. Given any cardinal number α, let x be such that card Nx <
α. Let U be a good neighborhood of x. If y ∈ U , and g ∈ G is
such that gy = y, then U ∩ gU 6= ∅ ⇒ g ∈ Nx . Hence Ny is a
subgroup of Nx and card Ny ≤ card Nx < α for every y ∈ U . The set
{x ∈ X : card Nx < α} is thus open in X.
It follows that the set
X0 := {x ∈ X : Nx = 1}
is open in X (possibly empty), and clearly G-invariant. Hence Y0 :=
π(X0 ) is open in Y and Y0 = X0 /G.
A point x ∈ X − X0 is said to be a fixed point of the action, and
the action is called free, or without fixed points, iff X = X0 .
10.4. If X0 6= ∅, then the restriction π0 : X0 → Y0 is an ordinary
covering of degree d(π0 ) = card G. This is usually called a regular
ordinary covering.
Proof. For every x ∈ X0 , every good neighborhood U of x is
necessarily contained within X0 (10.3), so V = π(U ) is an open neighborhood of π(x) which is evenly covered by π, and π −1 (V ) = ∪˙ g∈G gU
is the decomposition into connected components.
Let us say that the group G acts discretely and almost freely on X
iff X0 is dense in X.
18
10.5. If the group G acts discretely and almost freely on X, then
the quotient map π : X → X/G is a complete ramified covering of
degree d(π) = card G and local degrees
d(x; π) = card Nx , x ∈ X.
In particular, X0 and Y0 are exactly the sets of ordinary points, resp.
values, of the covering. We call this a regular ramified covering.
Proof. What has to be proved is the assertion on the local degrees
only. Let U be a good neighborhood of x. It is invariant under Nx and
clearly π(U ) ∼
= U/Nx . Since X0 is dense, then U ∩ X0 6= ∅ and the
restriction U ∩X0 → π(U )∩Y0 is an ordinary covering of degree card Nx
(10.4).
Remark that the quotient map π is a proper covering if and only if
G is a finite group. If π is proper, the orbit of any x ∈ X0 is compact
and discrete, hence finite, so G is finite. Conversely, if G is finite then
π is a closed map.
10.6. Let G act freely on X with quotient (ordinary) covering π :
X → X/G, and let i : X/G → Y be a dense embedding. Let π̄ : X → Y
be the completion of i ◦ π.
The action of G extends to a discrete and almost free action on X
and there is a factorization:
π̄Y
X
0@
π R i0
X/G
for a unique i0 , which is a quasi-covering spread of degree 1. Moreover
i0 is locally dominant, hence a covering, iff so is π̄.
This is a generalization of [E; 4.1, p. 299].
Proof. Use the canonical completion 8.6. A point of X over y ∈ Y
is a choice function ϕ which gives for every open neighborhood V of
y a connected component ϕV of π −1 (i−1 (V )). This set is G-invariant,
hence g(ϕV ) is another connected component, for every g ∈ G, and
(gϕ)V := g(ϕV ) defines a choice function gϕ based at y. In this way
every g ∈ G is seen as an automorphism of X, which is continuous
because g(U |V ) = (gU )|V for every U closed-open in π −1 (i−1 (V )).
Consider a point ϕ ∈ X over y ∈ Y . For every open neighborhood
V of y the restriction π|ϕV : ϕV → i−1 (V ) is an ordinary covering.
Define d(ϕ) := inf V d(π|ϕV ). Those V such that d(π|ϕV ) = d(ϕ), the
smallest value, form a neighborhood basis at y. If V 0 ⊂ V are basic
neighborhoods, then ϕV 0 = ϕV ∩ π −1 (i−1 (V 0 )). Every neighborhood of
19
ϕ of the form ϕV |V for basic V is good for the action of G. If gϕ 6= ϕ
then g(ϕV 0 ) 6= ϕV 0 for some V 0 ⊂ V , whence g(ϕV ) 6= ϕV by the
last formula; then g(ϕV ) ∩ ϕV = ∅ and (ϕV |V ) ∩ g(ϕV |V ) = ∅. So
property (1) is satisfied and property (2) is obvious. The action of G
on X is therefore discrete, and clearly almost free.
Since π̄(ϕ) = π̄(gϕ), π̄ factors through π 0 by means of a unique i0 .
For every V and U closed-open in π −1 (i−1 (V )), U 0 := ∪gU is closedopen in U 00 := π −1 (i−1 (V )), so π 0 −1 (π 0 (U |V )) = U 0 |V is closed-open in
π 0 −1 (i0 −1 (V )) = U 00 |V and π 0 (U |V ) is closed-open in i0 −1 (V ). Hence i0
is a spread. The remaining assertions about i0 are easy.
Using the classification theorem 9.5, the correspondence between
regular ramified and ordinary coverings given by π 7→ π0 (associated
ordinary part) and backwise by π 7→ π 0 (in the factorization of the
completion π̄) can be worked out in detail. More on finite regular
coverings is found in [E; §4].
11. Complex analytic spaces
11.1. (branches) Let X be a reduced complex analytic space. We
denote by X sm the smooth locus of X. Let Y be a nowhere dense
(rare) analytic subset of X, and let A := X − Y . The following is a
basic fact, called “local decomposition” relative to the pair (X, A).
Every point x ∈ X has a connected open neighborhood U ∗ such
that:
- A ∩ U ∗ has a finite number of connected components C, each having
x ∈ C;
- for every connected open neighborhood U of x with U ⊂ U ∗ , each
C ∩ U is (non-empty) connected and so is a connected component of
A ∩ U.
The germ at x of each C is independent of U ∗ and is called a branch
at x of the pair (X, A).
In this situation, the pair (X, A) is unibranch at every x ∈ X iff
A is locally connected in X (Y is non-separating in X) in the sense of
1.2. If X is smooth, any A as before is locally connected in X. The
branches of the pair (X, X sm ) are simply called the branches of X.
11.2. For every x ∈ X and any A = X − Y , there is some neighborhood U ∗ of x such that the branches of X and (X, A) are the germs
of connected components of X sm ∩ U ∗ and A ∩ U ∗ , respectively. For
every connected component C of X sm ∩ U ∗ still C ∩ A is connected
and is therefore contained within a unique connected component C 0 of
A ∩ U ∗ . Sending the germ of C into the germ of C 0 a map is defined
20
from the set of branches of X at x into the set of branches of (X, A)
at x. This is always a surjective map, and is bijective if A ⊂ X sm . Cf.
[M; p. 43].
11.3. (analytic quasi-coverings) Let f : X → Y be an analytic map
of complex analytic spaces, assumed to be dominant.
We call f an analytic quasi-covering iff there is some analytic subset
A ⊂ Y such that:
- the restriction X − f −1 (A) → Y − A is an analytic ordinary covering
in the usual sense (topologically ordinary + locally bianalytic);
- f −1 (A) is nowhere dense in X (⇒ A nowhere dense in Y ).
It follows that f is a quasi-covering in the sense of 2.2, and Y −Y0 ⊂
A is a thin subset in Y .
To some extent, it would be enough to assume A a thin subset in
the definition above. The analytic set A is assumed to be negligible in
[GR; III, B], where the case Y a domain in Cn is considered only.
11.4. Taking the smallest analytic set A as above, we define Y1 :=
Y − A, dense open in Y0 .
Because of 3.3, assertion 3.2.d holds with Y1 ∩Y sm replacing Y0 . We
know from 11.2 that the branches of Y1 ∩ Y sm and of f −1 (Y1 ∩ Y sm ) =
f −1 (Y1 ) ∩ X sm are in 1-1 correspondence with the branches of Y and
X, respectively. It follows that f is locally dominant at x, in the sense
of 4.1, if and only if every branch of Y at y = f (x) is the image of some
branch of X at x.
11.5. (spreads) Since a spread of locally compact spaces is just a
continuous map with 0-dimensional fibres (6.3), and a 0-dimensional
complex space is just a discrete set, then an analytic map is a spread
iff it has discrete fibres.
11.6. A finite analytic map of complex spaces is an analytic quasicovering.
If an analytic map f has a point x isolated in its fibre f −1 (f (x)),
then there are open neighborhoods V of f (x) and U of x in f −1 (V ) such
that the restriction f |U : U → V is finite, hence a finite analytic quasicovering. In particular, f has a “local degree” d(x; f ). Furthermore,
applying 6.4 and 11.4 to f |U , a characterization of openness of f at x
is obtained.
However, a 0-dimensional analytic map can be nowhere locally trivial on Y , even in the topological sense Y0 = ∅. An example is mentioned
in [Fox; p. 250, footnote].
21
11.7. (analytic coverings) An analytic quasi-covering f will be called
an analytic covering when it is a covering as a continuous map. Because
of 11.5 and 6.4, this happens iff f is discrete and open.
11.8. (about ramification) If f is an analytic covering, the set Rk :=
{x ∈ X : d(x; f ) ≥ k} is (closed) analytic for every integer k.
Proof. The relative product Π := X ×Y · · · ×Y X (k times) is the
analytic subset f (xi ) = f (xj ) of X k . Let ∆ be the analytic subset of
X k where some coincidence xi = xj occurs, and let δ be the diagonal
embedding of X into X k . We have: δ(Rk ) = clos(Π − ∆) ∩ δ(X). The same holds for the local degree of a 0-dimensional analytic map
(11.6).
11.9. (normalization) Let X be a reduced complex analytic space.
The structure sheaf OX can be identified with a subsheaf of the sheaf
OX of weakly holomorphic functions on X. The space X is said to be
normal iff there is equality OX = OX . The following is a description
of the topology of the normalization of a complex space. Compare the
one given in [W; p. 258, after 3.C], using 8.6.
If f : X → X is an analytic map of reduced complex spaces such
that:
(i) f −1 (X sm ) → X sm is biholomorphic,
(ii) f −1 (X sm ) is dense in X,
(iii) f is finite,
(iv) X is normal,
then, as a continuous map, f is the completion in the sense of Fox of
the inclusion X sm ,→ X.
sm
Proof. (i) implies that f −1 (X sm ) is contained in X , hence it is
sm
locally connected in X (11.1). A normal X is locally irreducible [W;
sm
p. 251, 1.B(b)], i.e. X is locally connected in X (11.2). It follows
that f −1 (X sm ) is dense and locally connected in X. As a finite map is
a complete spread (2.4 plus 7.2), this means that f is a completion of
the inclusion X sm ,→ X.
The statement above can be read as follows. The underlying continuous map of such an analytic map f is uniquely determined up to
homeomorphisms. The structure sheaf on X is uniquely determined
too. Indeed a normal X has OX = OX and OX = f −1 (OX ) follows
from (i). Therefore, the content of the normalization theorem, that f
in the statement above does exist, is essentially in the assertion that
the ringed space (X, f −1 (OX )) is a complex analytic space.
22
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23