Zero-sets of near-symplectic forms

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by  Tim Perutz

ZERO-SETS OF NEAR-SYMPLECTIC FORMS
TIM PERUTZ
Abstract. Wegiveelementaryproofsoftwo‘folklore’assertionsabout
7 near-symplectic forms on four-manifolds: that any such form can be
0 modified, by an evolutionary process taking place inside a finite set of
0 balls,soastohaveanychosenpositivenumberofzero-circles; andthat,
2
on a closed manifold, the number of zero-circles for which the splitting
n of the normal bundleis trivial has the same parity as 1+b1+b+2.
a
J

5
1
1. Statement of results

]
G In this paper we establish some properties of near-symplectic forms, as
S defined in [1]:
.
h Definition 1.1. A two-form ω on an oriented four-manifold X is called
t
a near-positive if at each point x ∈ X, either (i) (ω ∧ ω)(x) > 0, or (ii)
m ω(x) = 0 and the intrinsic gradient (∇ω)(x): T X → Λ2T∗X has rank 3 as
x x
[ a linear map. It is near-symplectic if, in addition, dω = 0.

2
To clarify: at a point where ω(x) = 0, there is an intrinsic gradient
v
(∇ ω)(x) ∈ Λ2T∗X in the direction v, for any v ∈ T X, because x is a zero
0 v x x
2 of a smooth section of a vector bundle. If ω ∧ω ≥ 0 in a neighbourhood
3 of x then (∇ ω)∧(∇ ω) ≥ 0 for all v ∈ T X. The wedge-square quadratic
v v x
1 form Λ2T∗X ⊗Λ2T∗X → Λ4T∗X → R4, which is defined up to a positive
0 x x x
6 scalar, has signature (3,3), so to say that (∇ω)(x) has rank 3 is to say that
0 its image is a maximal positive-definite subspace for the wedge-square form.
/
h
Lemma 1.2. The zero-set Z = {x ∈ X : ω = 0} ⊂ X of a near-positive
t ω x
a form ω ∈Ω2 is a smooth 1-dimensional submanifold.
m X
: Proof. Take z ∈ Z . Working over a small ball B ∋ z, choose a three-
v ω
plane subbundleE ⊂ Λ2T∗B such that E is complementary to im(∇ω)(z).
i z
X Project ω| to a section ω of (Λ2T∗B)/E. Then (∇ω)(z) is surjective,
B
r so, shrinking B if necessary, ω vanishes transversely along a 1-submanifold
a
Z ⊂ B. Moreover, T Z = ker(∇ω)(x) for x ∈ Z, so ω is constant along Z;
x
hence Z = Z ∩B. (cid:3)
ω
A smooth path of two-forms {ωt}t∈[0,1] can be regarded as a section ω• of
the pullback of the vector bundle Λ2T∗X to [0,1]×X.
x
Definition 1.3. Define a near-symplectic cobordism on X to be a path
{ωt}t∈[0,1] of closed two-forms such that for all (x,t) ∈ X × [0,1], either
(i) ωt(x)∧ωt(x) > 0, or (ii) ωt(x) = 0, and (∇ω•)(t,x) has rank 3. Here
(∇ω•)(t,x) is the intrinsic gradient on [0,1]×X.
Date: Revised version: January 8 2007.
1
When ω and ω are near-symplectic, the zero-set Z = {t}×Z
0 1 t∈[0,1] ωt
of the cobordism ω• is a smooth surface in [0,1]×X, realisSing a cobordism
between Z and Z .
ω0 ω1
In this article we prove the following:
Theorem 1.4. Suppose that the connected, oriented four-manifold X car-
ries a near-symplectic form ω whose zero-set has n components, where
0
n ≥ 1. Then
(1) there is a near-symplectic cobordism {ω } , constant on the com-
t t∈[0,1]
plement of a ball in X, such that the zero-set of ω has n+1 com-
1
ponents;
(2) if n ≥ 2, there is a near-symplectic cobordism {ω } , constant
t t∈[0,1]
on the complement of a ball in X, such that the zero-set of ω has
1
n−1 components.
In either case, the zero-surface Z of the cobordism realises an elementary
cobordism between the zero-sets of ω and ω : the projection Z → [0,1] is a
0 1
Morse function with just one critical point, of index 1.
The proof given here is elementary—it relies on stability arguments a` la
Moser. We shall show that given any pair of disjoint arcs embedded in the
zero-set of ω , and a path in X connecting interior points of these arcs,
0
one can find a near-symplectic cobordism whose zero-surface is the result of
attaching a (2-dimensional) 1-handle to the pair of arcs so that the core of
the handle is the chosen path. Thus, given arcs on two distinct components
of the zero-set, one can join them up to make a single component. On the
other hand, taking the arcs to to be on the same component, the surgery
increases by one the number of components.
A statement similar to Theorem 1.4 is mentioned in Taubes’ 1998 ICM
lecture [10], with attribution to unpublishedwork of Karl Luttinger. After I
had written a draft of the present article, Taubes sent me a preprint, which
has now appeared as [12], in which he also shows that the number of zero-
circles can be increased or decreased by a local modification. His approach
also uses 1-handle addition, but aside from that it is different. It does not
proceed through an evolution of the forms, and so does not include the
cobordism statement, butrather through a ‘mutation’ in which one removes
a ball whose intersection with the zero-set is a pair of arcs, then invokes
Eliashberg’s classification of overtwisted contact structures to argue that it
can be glued back so that the ends of the arcs match up differently. Taubes
also proves by means of a ‘birth’ model that any near-symplectic form is
near-symplectically cobordant to one with one more zero-circle.
As asimpler variant, onecan deducethefollowing weaker statement from
a result of Calabi:
Proposition 1.5. Let ω be a near-symplectic form on an oriented four-
manifold whose zero-set has n ≥ 0 components. Then there is a near-
symplectic form whose zero-set has n+2 components and which coincides
with ω outside an embedded ball.
Proof. We consider near-symplectic forms ω on R4 = R3×R which arise
f,g
from functions f ∈ C∞(R3) and metrics g on R3 which coincide with the
2
Euclidean metric g outside a compact set:
0
ω = df ∧dt+∗ df.
f,g g
The two-form ω is closed if and only if df is g-coclosed. If f is Morse then
f,g
ω is near-positive, and its zero-circles are in bijection with the critical
f,g
points. For example, when f (x) = x we recover standard symplectic R4:
0 3
ωf0,g0 = ωR4.
In [2], Calabi gives a recognition criterion for a closed one-form to be
coclosed with respect to some metric. As an example, he shows that one
can start with the function f = x and modify it to a function f which
0 3 1
coincides with f outside a closed ball of chosen radius r, such that df is
1 1
g -coclosed for some g standard outside the r-ball.
1 1
ByDarboux’stheorem,thereexistsr > 0suchthattheopenballD4(0;2r)
ofradius2r,withitssymplecticformω ,embedsintoournear-symplectic
f0,g0
manifold. We can then plug in ω in place of ω , which has the effect
f1,g1 f0,g0
of increasing by two the number of zero-circles. (cid:3)
Example 1.6. When X is closed, connected, and not negative-definite,
there are near-symplectic forms with arbitrary positive numbers of zero-
circles. The existence of a near-symplectic form in this situation is well-
known, and can be proved either using Hodge theory as in [8, 6]—an argu-
ment which originated in the 1980s—or via handle decompositions and the
flexibility of overtwisted contact structures [3].
Remark 1.7. Let ω , ω be two near-symplectic forms on a closed manifold
0 1
X. Assuming either that b+(X) > 1, or that they represent the same coho-
2
mology class, these forms are near-symplectically cobordant (in Ck, k < ∞)
by the work of Honda [6, Thm. 2.24]. When ω and ω are symplectic, the
0 1
zero-set of a cobordism projects to an orientable surface in X whose homol-
ogy class is presumably determined by c (X,ω )−c (X,ω ). Supposing this
1 0 1 1
vanishes, one can ask whether there are constraints on the topology of the
surface, or on the function on it defined by the t-coordinate of the cobordism.
For example, when a three-manifold M fibres over S1, there is a standard
construction of a symplectic form on S1 ×M. Given two possibly inequiv-
alent fibrations on M, representing the same class in H1(M;Z) and having
the same Euler class in H2(M;Z), it is not clear whether the associated
symplectic forms need be deformation-equivalent. One could try first to use
Morse theory on M to construct a near-symplectic cobordism between them
whose zero-set is a union of product tori, and then to attempt to modify this
cobordism so that its zero-set becomes empty.
I also learned the following topological statement from Taubes’ lecture
[10]; he attributes it to Gompf. Though proofs are no doubt known to a
few experts, no proof has appeared in print (and this one was worked out
independently).
A near-symplectic form ω determines a splitting of the normal bundle
N into two sub-bundles, on which the quadratic form N → R,
Zω/X Zω/X
v 7→ hι(z)∇ ω,vi is respectively positive- or negative-definite; here z is a
v
non-vanishing tangent vector field on Z . One of these bundles has rank
ω
1. A component of Z over which this line bundle is topologically trivial is
ω
called ‘even’; one for which it is non-trivial is called ‘odd’.
3
Theorem1.8. Thenumberofevencirclesinthezero-setofanear-symplectic
form on a closed, connected four-manifold X is congruent modulo 2 to
1−b (X)+b+(X).
1 2
For example, there is an even number of them when X admits an almost
complex structure.
In the proof of Theorem 1.4, two chosen components are joined together
to form a single circle. The new circle has even parity if and only if the
parities of the two old circles were different.
The zero-set Z is always nullhomologous: it is dual to the Euler class of
ω
Λ+, which is zero [4] (its components might not be nullhomologous, how-
g
ever). Here is a brief indication of the proof of Theorem 1.8 in the special
casewherethezero-circles arecontained indisjointballsB . Onthecomple-
i
ment of the balls, the form distinguishes an almost complex structure J, up
to homotopy. This has a first Chern class, which extends uniquely to a class
c ∈ H2(X;Z). The number 1(c2 −2e(X)−3σ(X)), which reduces modulo
4
2 to 1−b +b+, is the obstruction to extending J over the punctures. Each
1 2
odd circle makes an even contribution to this obstruction, and vice versa.
The general case can be reduced to this one by carrying out surgery on
all the circles so that they bound disjointly embedded discs.
Remark 1.9. Given a near-symplectic form on X with m even and n odd
′ ′
components, there is another with m even components and n odd compo-
′ ′ ′
nents provided that m − m is even and m + n > 0. To see this, first
reduce the total number of its zero-circles to 0 or 1. Pairs of even circles
can then be introduced as in the proposition above. By Theorem 1.4, a pair
of even circles can be fused to give a single circle, which by Theorem 1.8 is
necessarily odd.
The remaining question is to determine when the last remaining zero-
circle (if it is odd) can be removed so as to obtain a symplectic form.
Acknowledgements. Much of the work presented here appeared in the first
chapter of my doctoral thesis (Imperial College London, 2005), and I am
indebted to my Ph.D. supervisor, Simon Donaldson, for useful discussions
aboutnear-symplectic geometry. Thanksalso toCarlosSimpsonforsending
me a copy of the unpublished manuscript [9]; it contains an example which
we build on here.
2. Local near-symplectic geometry
A two-form is near-symplectic if and only if it is self-dual and harmonic
for some conformal structure and vanishes transversely as a section of the
bundle Λ+ of self-dual forms. This follows from the following lemma (see
e.g. [1]):
Lemma 2.1. (a) Let X be a 4-manifold with chosen conformal structure.
Let ω ∈ Ω+ be a transverse section of Λ+. Then ω is near-positive.
X
(b) Let ω be a near-positive form on an oriented 4-manifold X. The
following sets are in canonical bijection:
(1) conformal structures making ω self-dual;
4
(2) positive-definite three-plane sub-bundles Λ+ ⊂ Λ2 having ω as sec-
X
tion;
(3) almost complex structures J on X \Z such that g (·,·) := ω(·,J·)
ω J
defines a metric whose conformal class extends to one on X.
∞
(c) These sets are non-empty, and contractible in the C topologies.
The following self-dual form Θ on Euclidean R4 is the prototypical near-
symplectic form: let (t,x ,x ,x ) denote standard, positively-oriented coor-
1 2 3
dinates. Write
β = dt∧dx +dx ∧dx ,
1 1 2 3
(1) β = dt∧dx +dx ∧dx ,
2 2 3 1
β = dt∧dx +dx ∧dx .
3 3 1 2
Define Θ ∈ Ω2 by
R4
(2) Θ= x β +x β −2x β ,
1 1 2 2 3 3
so that Θ2 = 2(x2 +x2 +4x2)dt∧dx ∧dx ∧dx . The form Θ is near-
1 2 3 1 2 3
symplectic with zero-set Z = {x = x = x = 0}.
Θ 1 2 3
Twoexamplesofnear-symplecticformswithcompactzero-setarisethrough
symmetries of Θ. These symmetries are
Θ(t,x) = Θ(t−1,x) = Θ(t,σx),
where σ(x ,x ,x ) = (x ,−x ,−x ). By the first of these, Θ descends to
1 2 3 1 2 3
a form Θ on S1 ×R3. By the second, it descends to a form Θ on the
ev odd
mapping torus T(σ) = (R × R3)/(t − 1,x) ∼ (t,σx). Its zero-set is the
zero-section of the mapping torus.
2.1. Birth model (Luttinger and Simpson). The following example is
from [9] (apart from very slight variations in the formulae): There is a path
of closed forms {ωt}t∈[−1/2,1/2] on the unit ball in R4, whose zero-set Zωt is a
circle for t > 0, a point {p} at t = 0, and empty when t < 0. They are near-
R4×R
symplectic when t 6= 0. At t = 0, (∇ω0)(p) has rank 2, but (∇ ω•)(p,0)
has rank 3. The path ω• is a near-symplectic cobordism.
We write down such a path as follows. Using positively oriented coordi-
nates (x ,x ,x ,x ), and writing dx for dx ∧dx , define
1 2 3 4 ij i j
(3) ζ = x (dx +dx )−x (dx +dx ),
2 12 34 4 14 23
1
(4) η = (x2+x2−t)(dx +dx )+x (x dx +x dx ).
t 2 1 3 13 42 4 1 12 3 32
Fix any ǫ ≤ 1/2. Set ω = ζ +ǫη . Then, on the open unit ball D4 ⊂
t t
R4, the path {ω } has the properties just described. To see this,
t t∈[−1/2,1/2]
observe that ω is a C∞(R4)-linear combination of the three forms
t
dx +dx , dx +dx −ǫ(x dx +x dx ), dx +dx .
12 34 14 23 1 12 3 32 13 42
Asimpledeterminantcomputationshowsthatthesespanamaximalpositive-
definite subbundle of the two-forms over D4 when ǫ ≤ 1/2. Hence ω has
t
positive square except at those points where the three coefficients (namely
x , x and(x2+x2−t))arezero. At thosepoints, ∂ ω and∂ ω arelinearly
2 4 1 3 2 t 4 t
independent, so ∇ω has rank ≥ 2. By including also (x ∂ +x ∂ )ω one
t 1 1 3 3 t
5
finds that ∇ω has rank 3 along the zero-set, except when t = 0 and x = 0.
t
In that case, the t-derivative is non-zero.
2.2. Surgery model. We vary the birth model by replacing the family of
′
forms η by a new family η :
t t
1
(5) η′ = (x2−x2−t)(dx +dx )+x (x dx −x dx ).
t 2 1 3 13 42 4 1 12 3 32
′
Taking t ∈ [−1/2,1/2] and ǫ ≤ 1/2, the forms ζ +ǫη are near-symplectic
t
on D4 (except when t = 0, where the rank condition fails at the origin).
However, the zero-set is now {(x ,0,x ,0) : x2 − x2 − t = 0}. Thus as t
1 3 1 3
passesthroughzero, thezero-setundergoesasurgery(itchangesbyaddition
′
of a one-handle). When t = 0, ∇(ζ+ǫη )(p) has rank 2 at the double point
0
p of the zero-set. This model is again a near-symplectic cobordism.
2.3. Geometry near the zero-circles. Taubes [11] has established the
following picture of the differential geometry of a near-symplectic form ω
near its zero-set Z = Z . Fix a metric g with ∗ ω = ω, and identify the
ω g
⊥
intrinsic normal bundle N with the metric complement (TZ) .
Z/X
(a)∇ωdefinesavectorbundleisomorphismN → Λ+| . Inparticular,
Z/X Z
there is a universal procedure for orienting Z (as for the zero-manifold of
any transverse section of an oriented vector bundle).
(b) Let z (resp. ζ) denote the unit-length oriented vector field (resp.
one-form) on Z. The map
Λ+| → (N )∗, η 7→ ι(z)η
Z Z/X
is a bundle isomorphism; its inverse is α 7→ (ζ ∧α)+∗(ζ ∧α).
(c) We have a sequence
N −−∇−−ω→ Λ+| −−ι−(z−)→ (N )∗ −m−−et−r→ic N ,
Z/X Z Z/X Z/X
whose composite we denote by S ∈ End(N ). The endomorphism S
ω,g Z/X ω,g
is a self-adjoint, trace-free automorphism.
(d) It follows that, at each point of Z, S has a basis of eigenvectors,
ω,g
and that two of its eigenvalues are positive and one negative (this statement
pins down our convention for the universal orientation rule). The positive
and negative eigenspaces trace out eigenbundles L+ and L− (here L− has
g g g
rank 1 and is obviously locally trivial; L+ is its orthogonal complement).
g
Thus N = L+⊕L−.
Z/X g g
The auxiliary choice of metric is actually irrelevant. Whilst there are
many conformal classes [g] which make ω self-dual, they are all the same
alongZ,becauseforz ∈ Z,theysatisfyim(∇ω)(z) = Λ+ . Thesub-bundle
[g],z
⊥
TZ ⊂TX| is therefore intrinsic to ω.
Z
We condense these points into a metric-free statement:
Proposition 2.2. A near-symplectic form ω determines
(1) a canonical orientation for the 1-manifold Z ;
ω
(2) a canonical embedding of the intrinsic normal bundle N as a
Zω/X
sub-bundle of TX| complementary to TZ;
Z
(3) a conformal class of quadratic forms S on N , of signature
ω Zω/X
(2,1);
6
(4) a vector bundle splitting N = L+⊕L−, such that S is positive-
Zω/X ω
definite on L+ and negative-definite on L−.
Atrivialthree-planebundleoverS1 admitspreciselytwotopologicalsplit-
tings, distinguished by the orientability of the summands. Thus a basic in-
variant of a near-symplectic form ω is its ‘parity function’ ε : π (Z ) →
ω 0 ω
Z/2, which sends a component-circle Γ of Z to hw (L−),[Γ]i.
ω 1
Thisfunctionisequivariantunderorienteddiffeomorphisms,andinvariant
under isotopies of near-symplectic forms ω (note that the topology of the
t
zero-set cannot change if all the ω are near-symplectic).
t
Each of the model forms Θ and Θ defined afer (2) vanishes along a
ev odd
circle, and their parities are those suggested by the notation.
3. Moser-Honda deformation arguments
The extent to which the Moser deformation argument from symplectic
geometryappliesinthenear-symplecticsettingwasworkedoutbyKoHonda
[5]. He showed that any near-symplectic form can be deformed, through
near-symplectic forms with fixed zero-set, so that near a chosen zero-circle
itembeds into Θ or Θ .1 We useHonda’s method—acarefulapplication
ev odd
of Moser’s argument on the complement for the zero-set—to prove a slight
variant of this result:
Lemma 3.1. Let ω ∈ Ω2 be near-symplectic, and let A be a closed arc
X
embedded in its zero-set Z . Let U be a neighbourhood of A. There exists
ω
a smooth family {ω } of near-symplectic forms, with constant zero-set,
s s∈[0,1]
such that ω (p)= ω(p) if s = 0 or p ∈X\U, and where ω is standard near
s 1
A in the following sense: there is an orientation-preserving open embedding
i: [−1,1]×D3(0;r) → X, for some r > 0, mapping [−1,1]×{0} to A, such
∗
that i ω = Θ.
1
Here D3(0;r) denotes the open ball of radius r centred at 0 ∈R3.
Proof. 1. There is no loss of generality in assuming that X = (−2,2) ×
D3(0;r); that the zero-set of ω is Z = (−2,2)×{0}; that A = [−1,1]×{0};
and that U = (−1−ǫ,1+ǫ)×D3(0;r′), for some r′ ≤ r. Writing t for the
coordinate on (−2,2), we may supposethat ∂ is a positively oriented vector
t
field on Z for the orientation determined by ω.
Take ametric g for whichω is self-dual. We can thenfindanorthonormal
frame (e ,e ,e ) for N such that L+ = span(e ,e ) and L− = span(e ).
1 2 3 Z/X g 1 2 g 3
Themetricandthechoiceofe giverisetonormalcoordinates(t,x ,x ,x )
i 1 2 3
near Z, identifying a small neighbourhood with (−ǫ,1+ǫ)×D3(0;r′′). We
then have three basic two-forms β , as in (1). In the basis (e ,e ,e ), the
i 1 2 3
linear map S (t) (see Section 2.3) is represented by a trace-free symmetric
ω,g
matrix S(t) = S+(t)⊕S−(t), where S+(t) is 2×2 and positive-definite, and
−
S (t) < 0. Regarding x and β as column vectors, we have an expansion
1Onecandowithoutthepreliminary deformation at thepriceofaloss ofsmoothness.
Given two near-symplectic forms ω0, ω1, with zero-circles Z0, Z1 of the same parity, one
has ψ∗ω1=ω0 near Z0 for some map ψ: nd(Z0)→nd(Z1); however, ψ is in general only
Lipschitz continuousalong Z0 (it is C∞ elsewhere).
7
near Z:
ω(t,x) = x·S(t)β +O(|x|2)
= (x x )S+(t) β1 +x S−(t)β +O(|x|2).
1 2 (cid:18) β2 (cid:19) 3 3
The advantage of paying attention to the eigenbundle splitting is that one
can then apply convexity arguments.
Define θ = (1−s)ω+s(x β +x β −2x β ). On a small neighbourhood
s 1 1 2 2 3 3
of Z, this is a family of near-symplectic forms, with common zero-set Z.
2. On the total space of a vector bundlewith a chosen metric one has the
following explicit Poincar´e lemma. Let p: V → M bethebundleprojection,
i : M → V the zero-section. Let h : V → V be the fibrewise dilation
t
x7→ tx, andR theEuler vector field, that is, the vertical vector field defined
on any fibreV by R(y)= y ∂ , wherethe y are orthonormal Euclidean
x j∂yj j
coordinates on V . The maPp
x
1
H: Ω∗ → Ω∗−1; α7→ h∗(ι(R)α)t−1dt
V V Z t
0
∗ ∗
satisfies α−p i α = d(Hα)+Hdα. Notice that if α vanishes along i(M)
then H(α) vanishes to second order along i(M).
Applying this to a neighbourhood of the zero-section in N , we find
Z/X
that η := H(θ ) vanishes to second order along Z, and satisfies dη = θ .
s s s s
′
3. Let U ⊂ U be a neighbourhood of A on which θ is defined and near-
s
′
symplectic. Take a non-negative cutoff function χ, supported in U and
identically 1 near A. We choose χ to have the shape χ(t,x) = χ (t)χ (|x|)
1 2
forfunctionsχ ,χ ∈ C∞(R). Introducethevector fieldsv onX\Z defined
1 2 s
by ι(v )θ +χη = 0.
s s s
The family {vs}s∈[0,1] generates a flow {φs}s∈[0,1] on X \ Z, supported
inside U. The reason is that, along Z, ∇θ˙ (u) is non-zero for all 0 6= u ∈
s
N , while∇(χη )(u) = 0, sothat|v (x)| ≤ k|x|near Z,forsomeconstant
Z/X s s
k. Atrajectoryx ,definedonsomeinterval[0,s′],satisfies d (log|x |) ≥ −k;
s ds s
integrating over [0,s′], we obtain |xs′| ≥ e−ks′|x0|. This shows that the
trajectory stays inside X \Z, and therefore that φ is defined.
s
′ ′ ∗ ′
Now define ω by setting ω = φ ω on X \Z and ω (z) = 0 for z ∈ Z.
s s s s
′ ′
Outside U we have ω = ω. Moser’s argument shows that ω = θ in some
s s s
neighbourhood of A.
′
The path ω gives an isotopy with all the required properties, except
s
′
one: ω fails to be smooth at certain points of Z—the points where χ (t) is
s 1
non-constant. At those points, its coefficients might merely be Lipschitz.
′
4. The ‘problem-points’ of ωs lie strictly inside a pair of regions R− =
I−×D¯3(0;r1) ⊂ U\A and R+ = I+×D¯3(0;r2)⊂ U\A, for closed intervals
I− ⊂ (−1−ǫ,−1) and I+ ⊂ (1,1+ǫ). The map φs embeds R− ∪R+ into
X, and acts as the identity along Z. We can extend these embeddings φ
s
to a path of homeomorphisms s 7→ ψ ∈ Homeo(X), such that (i) ψ = id;
s 0
(ii) supp(ψ ) ⊂ U; (iii) ψ | = φ | ; (iv) the only points where
s s (R−∪R+) s (R−∪R+)
smoothness fails (smoothness of ψs, or of the path) are inside R+∪R−; and
(v) φ is the identity in a neighbourhood of A.
s
8
Finally, setting ω = ψ−1∗ω′, we obtain a smooth path with the required
s s s
properties. (cid:3)
4. Proof of Theorem 1.4
Let θ be near-symplectic, and Z , Z two components of Z = Z , not
0 1 θ
necessarily distinct. Take distinctpointsp ∈ Z andp ∈ Z , andasmooth
0 0 1 1
path γ: [0,1] → X from p to p , transverse to Z, with γ−1(Z) = {0,1}.
0 1
Theplanisthis: first,weshowhowtomodifyθsothatitisstandardinsome
neighbourhoodN of im(γ). We then exhibit a path of forms on N, constant
near ∂N, which realises a surgery on zero-sets; these can be plugged into θ.
Thedelicacyliesinfindingavariantofourearlier‘surgerymodel’whichis
aconstantpathofformsoutsideasuitableneighbourhoodoftheorigin. This
isawkwardpartlybecausewecannotcontroltheshapeoftheneighbourhood
N: in local coordinates we can take it to be an ellipsoid, but it might be
very thin.
Definition 4.1. A letter H in a near-symplectic manifold (X,θ) consists of
threesmoothembeddingsv ,v ,h: [0,1] → X,suchthatv (1/2) = h(0)and
1 2 1
v (1/2) = h(1) (transverse intersections), with no other intersection points.
2
We require that im(v ) and im(v ) lie in the zero-set Z . The interior of the
1 2 θ
arc im(h) joining them then necessarily consists of symplectic points.
Lemma 4.2. Let H be a letter H in (X ,θ ), for i = 0, 1. Then θ is
i i i 0
′
isotopic, through near-symplectic formswithfixedzero-set, toaformθ , such
0
that there are regular neighbourhoods nd(H ) ⊃ H and a diffeomorphism
i i
∗ ′
φ: nd(H ) → nd(H ) with φ θ = θ .
0 1 1 0
Proof. WemayassumethatweareinR4,andthatH coincideswithH . By
1 2
Lemma 3.1, we may also assume that θ and θ coincide in a neighbourhood
1 2
of the two vertical arcs.
LetP bethe‘horizontal stroke’im(h)intheletter H.Fordefiniteness,say
P = [−1,1]×{(0,0,0)} ⊂ R4. Weclaimthatthereexistlinearlyindependent
vector fields (v = ∂ ,v ,v ,v ) along P such that the matrix (θ (v ,v ))
1 x1 2 3 4 0 i j ij
takes the form
J 0 0 −1
λ(t) , J = , λ(t) ≥ 0;
(cid:18) 0 J (cid:19) (cid:18) 1 0 (cid:19)
′ ′ ′ ′
and also linearly independentvector fields (v = ∂ ,v ,v ,v ) along P such
1 x1 2 3 4
′ ′
that the matrix (θ (v ,v )) takes the same form (for the same function λ).
1 i j
Moreover, we may assume that, near ∂P, where θ and θ coincide, we have
0 1
′ ′ ′ ′
(v ,v ,v ,v ) = (v ,v ,v ,v ).
1 2 3 4 1 2 3 4
To see this, decompose P into three subintervals: P = [0,δ]×{(0,0,0)},
1
P = ([δ,1−δ])×{(0,0,0)}, P = [1−δ,1]×{(0,0,0)}. If δ is small, we can
2 3
find such frames over P ∪P , and we may assume, rescaling as necessary,
1 3
that the function λ is identically equal to 1 near the interior endpoints of
P and P . Over P , we can find symplectic trivialisations of the tangent
1 3 2
bundle, for either form, and since the symplectic group is path-connected,
we can make these trivialisations match those on P and P .
1 3
There is now an open neighbourhood N ⊃ P and a diffeomorphism
1
Φ: N → N , such that Φ(x)= x if x∈ P or x lies close enough to ∂P, and
1 1
9
′
such that, along P, DΦ(v ) = v . Define
i i
θ = (1−s)θ +sΦ∗θ ∈Ω2 , s ∈ [0,1].
s 0 1 N1
There is a smaller open neighbourhood N ⊃ P on which these forms are
2
near-symplectic.
The function s 7→ θ ∈ Ω2 has constant derivative dθ /ds = Φ∗ω −θ .
s N′ s 0 0
We may write dθ /ds = dα, where α ∈ Ω1 vanishes near ∂P. Since θ is a
s N2 s
constant path along P, we may assume further that α vanishes along P.
Take an open set N ⊃ P with N ⊂ N , and acutoff function R4 → [0,1]
3 3 2
suchthatχ(N )= {1}andχ(R4\N )= {0}. Defineafamily ofvector fields
3 2
v on R4, supported on N , by ι(v )θ +χα = 0. These fields are zero near
s 3 s s
the common zero-set of the θ . The family {v } generates a flow s 7→ F ,
s s s
∗
whichacts astheidentity alongP. Thenear-symplecticformF θ coincides
1 0
∗
with θ outside N and with θ = Φ θ in some still smaller neighbourhood
0 2 1 0
of P, say N ⊂ N . The result follows.
4 3
(cid:3)
4.1. Sharpeningthesurgerymodel. Theformsappearinginoursurgery
modelfitintoalargerfamily,whichcontainsdifferentexamplesmoreamenable
to embedding in other manifolds.
For any smooth function f: R2 → R, set
∂f ∂f
ηf = f(x ,x )(dx +dx )+x ( dx + dx );
1 3 13 42 4 12 32
∂x ∂x
1 3
1
ωf = ζ + ηf.
2
Here ζ is as in (3). Note that ηf and ζ are closed. If the condition
∂f ∂f
(6) x2 +x x ≤ x2+x2
(cid:12) 4∂x 2 4∂x (cid:12) 2 4
(cid:12) 3 1(cid:12)
(cid:12) (cid:12)
holds then ωf ∧ωf ≥ (cid:12)(x22 +x24)dx1234. M(cid:12)oreover, (ωf ∧ωf)(x1,0,x3,0) =
2f(x ,x )2dx . Hence ωf satisfies the dichotomy that at any point it
1 3 1234
either has positive square or is zero.
We now work in U = {|x | < 2,|x | < 2,x2+x2 < δ}, where δ ≤1. Put
δ 1 3 2 4
1
(7) f(x ,x )= (x2−1).
1 3 4 3
For this f, condition (6) holds, and ωf is near-symplectic in U . Its zero-set
δ
consists of the two straight line segments, {x = ±1,x = x = 0}.
3 2 4
Lemma4.3. Thereisasmooth familyoffunctionsfδ: [−2,2]×[−2,2] → R,
t
parametrised by δ ∈ (0, 1 ] and t ∈ [−1,1], with the following properties:
10
• fδ = f, the function appearing in (7);
−1
• fδ is independent of t outside the square [−1−2δ,1+2δ]×[−1,1];
t
• when t < 0, the zero-set of fδ consists of two arcs, homotopic to
t
those of f; when t = 0, the zero-set is a letter X; and when t > 0 it
again consists of two arcs, connecting the endpoints the other way.
Moreover, the vanishing is of order exactly 1, except at the singular
point when t = 0 where it is quadratic; and
• fδ is bounded in C1, uniformly in t and δ.
t
10