Branched immersions and braids

0.2 MB·English

by  Marina Ville

7 Branched immersions and braids
0
0
2

n Marina Ville
a
J

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2
ABSTRACT. Branch points p of a real 2-surface Σ in a 4-manifold M

] generalize branch points of complex curves in complex surfaces: for example,
G
they can occur as singularities of minimal surfaces. We investigate such a
D
branch point p when Σ is topologically embedded. It defines a link L(p), the
.
h components of which are closed braids with the same axis up to orientation.
t
a If Σ is closed without boundary, the contribution of p to the degree of the
m
normal bundle of Σ in M can be computed on the link L(p), in terms of the
[
algebraic crossing numbers of its components and of their linking numbers

1 with one another.
v
KEYWORDS:surfaces in4-manifolds, branch points, characteristic num-
6
1 bers, braids, transverse knots, twistors, minimal surfaces
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AMS classification: 53C42
1
0
7
0 1 Introduction
/
h
t
a 1.1 Motivation
m
: Weinvestigate herethegeneralizationtotherealcaseofaconstructionwhich
v
i is well understood in the complex algebraic case. Consider a complex alge-
X
braic curve C in C2 which possesses a branch point p. For a small enough
r
a positive real number ǫ the intersection of C with the sphere S (p) centered
ǫ
at p and of radius ǫ is a link which is called an algebraic link. The link type
does not depend on ǫ. Numerical invariants of this link (e.g. its genus if it is
a knot) can be read on the singularity of C at p and vice versa.
Complex branch points generalize to real branch points; in particular
these arethesingularites ofminimal surfaces inRiemannian manifolds. They
are non generic singularities of real surfaces which lowest order term is simi-
1
lar to the lowest order term of a complex branch point.
In the present paper we consider a closed surface S which is immersed in
a 4-manifold M except at a finite number of branch points.
For such a surface we can define a tangent and a normal bundle, which we
denote TS andNS. We will give a precise definition below; let us just say for
now that TS is about the intrinsic topology of the surface and NS reflects
its extrinsic topology (how it sits inside M).
We make an extra assumption which is always satisfied in the complex
case, but not in the real case, namely that the singularities of S are isolated
(even minimal surfaces can have real codimension 1 singularities).
If p is a singular point of S, we consider the sphere S (p) centered at p
ǫ
and of radius ǫ; the intersection of S and S (p) is a link which we denote Lǫ.
ǫ
If there is a single branched disk going through p, Lǫ is a knot which is
transverse to the standard contact structure on S (p); moreover it is immedi-
ǫ
ately available in the form of a closed braid as defined by Bennequin. If there
are several different disks going through p, there is no obvious contact struc-
ture to which Lǫ is transverse. However we can find an axis w.r.t. which and
up to orientation the components of Lǫ are closed braids. The braid indices
of the components and their algebraic crossing numbers do not depend on ǫ;
and neither do the linking numbers of one component with another.
These quantities have an interpretation in terms of the topology of S. We
can derive from the braid index of Lǫ the contribution of p to the degree of
the tangent bundle TS. This has been known for a long time. In the present
paper we show that the algebraic crossing numbers and linking numbers of
the components of Lǫ give the contribution of p to the degree of the normal
bundle NS.
As an application we can recover the (already known) formula for the
self-linking number of iterated torus knots.
ThestartingpointofthisworkwasaconversationwithAlexanderReznikov
many years ago. He gave me advice, friendship andencouragement. Not long
2
ago he died tragically. This paper is inscribed to his memory.
ACKNOWLEDGEMENTS. The author would like to thank Jim Eells for
helpful advice and conversation, Denis Auroux for providing her with much
needed information about 4-dimensional topology and Marc Soret for intro-
ducing her to braids.
1.2 Preliminaries
1.2.1 Transverse knots - Closed braids
We consider S3 as the unit sphere in the complex plane C2. The complex
structure J on C2 enables us to define a contact structure ξ on S3. Namely, if
q isapointinS3, theplaneξ(q)istheplanetangent toS3 atq andorthogonal
to Jq. Stereographic projection maps this to the standard contact structure
in R3 which we also denote ξ: in other words, up to isomorphism, ξ is the
contact structure for which the contact planes are the kernels of the 1-form
written in cylindrical coordinates as
ρ2dθ+dz.
A transverse knot (in S3 or R3) is a smooth map γ from the circle S1 to
S3 or R3 such that
t S1, γ (t) / ξ(γ(t))
′
∀ ∈ ∈
i.e. the tangent vectors to the knot never belong to the contact planes. In
view of the 1-form given above, this means that if the knot is written in
cylindrical coordinates (θ,ρ,z) it verifies
z (t)
t S1, ′ = (ρ(t))2.
∀ ∈ θ (t) 6 −
′
Two transverse knots aretransversally isotopicif they have thesame knot
type and moreover they are isotopic through transverse knots.
A special case of transverse knots (in R3) is given by what Bennenquin calls
closed braids:
Definition 1 A knot K in R3 is a closed braid if, when written in cylin-
drical coordinates
t (ρ(t),θ(t),z(t))
7→
3
it verifies
t S1,ρ(t) = 0,θ (t) > 0.
′
∀ ∈ 6
In this definition the z-axis is called the axis of the braid.
Bennequin ([Be 1]) proved that every transverse knot is transversally isotopic
to a closed braid.
There is a similar definition for closed braids in the 3-sphere; in this case
the axis of the braid is an oriented great circle. For further use we state
Lemma 1 Let L be a link in S3. Let P be a plane in R4 and let (ǫ ,ǫ ) be a
1 2
(not necessarily orthonormal) basis of the orthogonal complement P .
⊥
We denote the orthogonal projection of L to P by
⊥
x (t)ǫ +x (t)ǫ .
1 1 2 2
The following two assertions are equivalent
1) up to orientaton the great circle Γ in P is a braid axis for L
2) the projection of L to P verifies
⊥
x2(t)+x2(t) = 0, x (t)x (t) x (t)x (t) = 0.
1 2 6 1 ′2 − 2 ′1 6
We conclude this section by recalling a few invariants for a closed braid K.
For more details see for example [B-W]. The braid index n(K) of K is the
linking number of K with the oriented z-axis. A generic projection of K
onto the (ρ,θ) plane has only transverse double crossing points; it is called a
closed braid projection. We assign a sign to such an intersection point in the
following manner: consider a basis (u ,u ) of R2 where the u (resp. u ) is
1 2 1 2
the vector tangent to the strand of K which is on top (resp. on the bottom).
If (u ,u ) basis is a positive (resp. negative) basis the point will be counted
1 2
positively (resp. negatively).
The algebraic crossing number e(K) of K is the signed number crossing
points of a closed braid projection of K.
The quantity
sl(K) = n(K) e(K)
−
iscalledtheself-linkingnumberofthetransverseknot. Bennequinintroduced
it in [Be 1] and proved that it is an invariant of transverse isotopy.
4
1.2.2 Branched immersions
We recall a few (standard) definitions. For more details on branched immer-
sions we refer the reader to [G-O-R].
Definition 2 A branched disc in a smooth manifold M is a map from a
disc D centered at 0 in C to M which is an immersion except at 0 and which
writes in a neighbourhood of 0
f1(z) = Re(zN)+o ( z N)
1
| |
f2(z) = Im(zN)+o ( z N)
1
| |
fk(z) = o ( z N) for k > 2
1
| |
where z is a local isothermic coordinate on D around 0, the fi(z)’s are the
coordinates of f(z) in some well-chosen chart on M around f(0) and N is
an integer, N 2.
≥
The notation o ( z N) means that the function is an o( z N) and its first
1
| | | |
derivatives are o( z N 1)’s.
−
| |
The integer m = N 1 is called the branching order of f at 0.
−
Definition 3 A map f : Σ M from a Riemann surface Σ to a smooth
−→
manifold M is a branched immersion if it is an immersion except at a
discrete set of points called branch points in a neighbourhood of which f is
parametrized by branch discs.
Note that there can be several branched discs going through the same branch
point.
If f is a branched immersion as in Def. 3 one can check that the map
from Σ to the Grassmannian of oriented 2-planes G+(M) given by
2
x f (T Σ)
x
7→ ∗
(where T Σ is the tangent plane to Σ at x and f is the derivative of f) ex-
x
∗
tends continuously across the branch points. This yields a bundle Tf above
Σ, called the image tangent bundle and, by taking orthogonal complements a
normal bundle Nf. Note that if M is 4-dimensional and oriented, then Nf
is an oriented 2-plane bundle.
5
1.2.3 Degrees of the tangent and normal bundles
NOTATION. If L is a complex line bundle (or equivalently an oriented real
2-plane bundle) on an oriented closed Riemann surface Σ without boundary,
we will denote its degree by c (L): that is, we use the same notation for the
1
first Chern class and for its representative in the second integral cohomology
group of Σ.
We consider here an oriented closed surface without boundary Σ and a
map f from Σ to an oriented 4-manifold M. The classical Riemann-Hurwitz
formula ([G-H]) generalizes to
Proposition 1 ([Gau]). Suppose f is an immersion with branch points
p ,...,p of respective branching orders m ,...,m . Then the degree of the
1 k 1 k
image tangent bundle Tf is given by
k
c (Tf) = χ(Σ)+ m
1 i
Xi=1
where χ(Σ) denotes the Euler caracteristic of Σ.
Hence we derive the degree of Tf by taking the formula for the degree of
the tangent bundle of an immersed surface (which is χ(Σ)) and we add a
correction term for each branch point, namely its branching order.
One of the purposes of this paper will be to achieve a similar formula for
the degree of the normal bundle if M is 4-dimensional; that is, to estimate
the contribution of a branch point to the degree of the normal bundle of a
branched immersion. To this effect we now recall the classical formula for
the normal bundle in the non branched case:
Proposition 2 . Let Σ be a closed oriented Riemann surface without bound-
ary, let M be an oriented smooth 4-manifold and let f : Σ M be an
−→
immersion with only transverse double points. Then the degree of the normal
bundle Nf is given by
[f(Σ)].[f(Σ)] 2D
f
−
where [f(Σ)].[f(Σ)] denotes the self-intersection number of the 2-homology
class of f(Σ) and D is the signed number of double points of f.
f
6
2 The result
Theorem 1 We considerΣ ,...Σ compactconnected oriented Riemannsur-
1 n
faces without boundary and M an oriented smooth 4-manifold. For each
i = 1,...,n we let
f : Σ M
i i
−→
be a branched immersion and we denote by Nf the corresponding normal
i
bundle. We put = n Σ and we define a map f : M be imposing
S ∪i=1 i S −→
its restriction to each Σ to be equal to f .
i i
We assume f( ) has isolated sigular points and we let [f( )] be its 2-
S S
homology class in M.
We endow M with a Riemannian metric and denote by S (p) the sphere
ǫ
centered at a point p in M and of radius ǫ.
The link S (p) f( ) is a disjoint union of closed braids Γǫ,...,Γǫ which have
ǫ ∩ S 1 s
the same axis up to orientation. Denoting by e(Γǫ) the algebraic crossing
i
number of Γǫ and by lk(Γǫ,Γǫ) the linking number of Γǫ and Γǫ) for i = j,
i i j i j 6
we put
s
E(p) = e(Γǫ)+2 lk(Γǫ,Γǫ)
i i j
Xi=1 1 Xi<j s
≤ ≤
Let p ,...,p be the singular points of f( ). We have
1 k
S
n k
degree(Nf ) = [f( )].[f( )] E(p )
i m
S S −
Xi=1 mX=1
where [f( )].[f( ]) denotes the self-intersection number of f( ).
S S S
REMARK. Since we mention the linking number of the Γǫ’s, we need to
i
specify their orientation. We denote by B (p) the ball in of radius ǫ. Then
ǫ
S
each Γǫ is the image via f of ∂Uǫ where Uǫ is contained in capf 1(B (p)).
i i i S − ǫ
The 2-dimensional surface Uǫ inherits the orientation of : this, in turn,
i S
yields an orientation for ∂Uǫ and thus for Γǫ.
i i
7
2.1 The case of a single branched disk
2.1.1 The knot of the singularity
We first prove the theorem in the case where there is no more than one
branched disc going through each singular point. We consider a branched
disc f : D R4 as in Def. 2. We denote by e ,i = 1,...,4 the values of ∂
−→ i ∂xi
at the origin. Possibly after replacing e by e we assume the basis ( ∂ ) to
4 − 4 ∂xi
be positive w.r.t. the orientation of R4. We define a scalar product g on R4
0
by requiring the basis (e ,e ,e ,e ) of R4 to be orthonormal. We also define
1 2 3 4
a complex structure J on R4 by setting
0
J (e ) = e , J (e ) = e .
0 1 2 0 3 4
Finally if ǫ is a small positive number, we let S be the sphere (for the
ǫ
norm g ) centered at f(0) and of radius ǫ. The complex structure J yields
0 0
isomorphic contact structures on the S ’s (we will denote all these contact
ǫ
structures by ξ).
Proposition 3 Let f : D R4 be a branched disc as in Def. 2. Assume
−→
moreover that f is a topological embedding in a neighborhood of the origin.
There exists a number R, 0 < R < 1 such that for a small enough positive
number ǫ, the curve defined by
Kǫ = S f(D(0,R))
ǫ
∩
is a closed braid with braid index N. For different ǫ’s the Kǫ’s have the same
transverse knot type.
It follows that the Kǫ’s have the same Bennequin self-linking number; since
their braid index is the same, they also have the same algebraic crossing
number which we will denote by e(K). This quantity appears in
Lemma 2 Let Σ be an oriented closed surface without boundary, let M be
an oriented 4-manifold and let f : Σ M be an immersion which has
−→
transverse double points and also branch points p ,...,p .
1 k
Assume that there is only one branched disc going through each p and that
i
the p ’s are isolated singularities of f(Σ).
i
8
For each i we denote by K the closed braid defined by the branch point
pi
p as explained above. Then the degree of the normal bundle Nf is given by
i
k
[f(Σ)].[f(Σ)] 2D + e(K )
− f pi
Xi=1
(for the notations, see Prop.2).
Note the formal similarity with Prop. 1.
2.2 The braids; proof of Prop. 3
We give a proof of Prop. 3, closely inspired by [Mi]. We denote by . the
k k
norm on R4 defined by g . Going back to Def. 2, we derive
0
Lemma 3 There exists a number R, 0 < R < 1 such that the following is
true:
1) The only critical point of z f(z) in D(0,R) is 0
7→ k k
2) f(D(0,R) 0 ) is a smooth submanifold of R4.
−{ }
Thus, for ǫ > 0 small enough Kǫ is smooth. The plane tangent to Kǫ at a
point f(z) of Kǫ is the intersection of the tangent space to S at f(z) and the
ǫ
tangent plane to the surface f(D) at f(z). This last plane, which we denote
T f(D) is generated by the vectors ∂f, ∂f. Close to the origin, f looks very
f(z) ∂x ∂y
much like a holomorphic function, that is, we derive from the expression of
f in Def. 1,
∂f ∂f ∂f
= zN 1 +o( zN 1 ), = J +o( zN 1).
− − 0 − |
k∂xk | | | ∂y ∂x |
It follows that the Kǫ’s are all transverse knots.
The reader can also see that the Kǫ’s are closed braids and that their axis
is the great circle in the plane generated by e ,e .
3 4
To prove that the Kǫ’s are transversally isotopic we will now recall a con-
struction from [Mi]. Milnor constructs a vector field X in a small ball B in
R4 centered at the origin with the following properties:
1) X is everywhere tangent to f(D)
2) for every point p in B distinct from the origin, we have
< X(p),p >> 0.
9
Thus going along the integral curves of X will give an isotopy between Kǫ′
and Kǫ for 0 < ǫ < ǫ.
′
We need to point out that Milnor’s construction is about holomorphic
functions, that is, he assumes the map f in Def. 2 to be holomorphic. How-
ever for this specific part of his book (i.e. the construction of the vector field)
holomorphicity is not necessary and the only thing needed for the construc-
tion to work is Lemma 1 above. This concludes the proof.
We end this section by lemmas which we will use only in 3 below.
§
Lemma 4 For ǫ small enough, there is a function r into R+ such that Kǫ
ǫ
is parametrized by f(r (t)eit).
ǫ
PROOF. Apply the implicit function theorem to the function
(r,θ) < f(reiθ),f(reiθ) > .
7→
We derive
Lemma 5 We let T f be the image tangent plane to f at 0. Let Q be a
0
2-plane in R4 which verifies
(T f) Q = 0 (1)
0 ⊥
∩ { }
(where P denotes the orthogonal complement of P in R4). For ǫ > 0, we
⊥
put Qǫ = Q S . For ǫ small enough, Qǫ is a braid axis (up to orientation)
ǫ
∩
for the knot Kǫ in S .
ǫ
PROOF. We let π : R4 Q be the orthogonal projection and we put
Q
−→
π (e ) = ǫ ,π (e ) = ǫ . Note that ǫ and ǫ are independent vectors. For
Q 1 1 Q 2 2 1 2
z = reit in D, we have
π (f(z)) = rN cos(Nt)ǫ +rN sin(Nt)ǫ +v(z)
Q 1 2
where v(z) = o ( z N). Since the ǫ ’s are independant, there exists a real
1 i
| |
number η > 0 such that
r R,t [0,2π], rN cos(Nt)ǫ +rN sin(Nt)ǫ ηrN (2).
1 2
∀ ∈ ∈ k k ≥
The projection of Γǫ to Q writes as
r (t)N cos(Nt)ǫ +r (t)N sin(Nt)ǫ +v(r (t)).
ǫ 1 ǫ 2 ǫ
We derive Lemma 5 using Lemma 1 2) and (2).
10