GOMPF'S CONSTRUCTION
Abstract. In this lecture I will present a surgery similar to connected sum
in topology, which works symplectically, and use it to show that any nitely
generated group can be obtained as fundamental group of some compact symplectic manifold.
Suppose we have
i1 , i2 : N → (M1 , ω1 ) , (M2 , ω2 ) compact symplectic submanM1 , M2 respectively. Suppose also both have triv-
ifolds of codimension 2 in
ial normal bundles.
Then from Weinstein's symplectic neighbourhood theorem,
U1 , U2 of
i1 (N ), i2 (N ) in M1 , M2 which are symplectomorphic to the trivial disc bundle
N × B , where B is an Euclidean disc of radius with standard symplectic form
dx ∧ dy . We now remove the origin from every B to get an annulus A . A admits
proof of which in our case we give below, we can nd neighbourhoods
self-symplectomorphism reversing the boundary components (in polar coordinates
√
2
dx ∧ dy = rdr ∧ dθ = d r2 ∧ dθ, so (r, θ) → ( 2 − r2 , −θ) will do it), hence we
can glue the manifolds M1 , M2 along U1 , U2 by means of this symplectomorphism
so that the symplectic forms match on the stitch. We will denote the resulting
symplectic manifold
Remark.
M1 #M2 .
The construction can be generalized to include nontrivial normal bundles
with some condition on normal Euler classes of
N
in
M1 , M 2 ,
but we will use only
the case of trivial normal bundles.
Let N ⊂ (M, ω) be a compact symplectic submanifold with trivial normal
bundle. Then we can nd a neighbourhood U (N ) and a symplectomorphism f :
(N × B , ω × ωstd ) → (U, ω)from trivial disc bundle to U which sends N × {0}to N
in the obvious way.
Lemma.
Proof.
N ⊂ M is symplectic, we can specify its normal bundle by orthogonalω , and since the bundle is trivial, we can x a dieomorphism
f : N × B → U sending N × 0 to N s.t. f∗ sends the bers x × B to bers
Tx N ω of the normal bundle of N , but f is not necessarily a symplectomorphism.
∗
On N × B we have now two forms: ω1 = ω × ωstd and ω0 = f ω , which agree
on N. Let ωt = ω0 + t (ω1 − ω0 ). We will use now Moser's method to get an iso∗
topy φt from φ0 = Id to φ1 s.t. φt ωt = ω0 and φt |N = Id. Once it is done, we
can compose f with φ1 to get the required symplectomorphism. Let Xt denote the
∗
vector eld generating the ow φt . Dierentiating φt ωt = ω0 we get (ωt is closed)
dω
dω
φ∗t dtt + dιXt ωt =0. After we show that dtt = ω1 − ω0 = dη for some 1-form η
vanishing on N , we will be able to nd Xt s.t. ιXt ωt = −η (since every ωt is non
degenerate on N × B ), and integrate it on N × B (possibly for smaller ) with
initial condition φ0 = Id to give the required ow φt .
To construct η , let τ = ω1 − ω0 and let πs : N × B → N × B be berwise
d ∗
∗
∗
∗
∗
multiplication by s. Then
∂ τ = dπs ι ∂ τ and τ = π1 τ − π0 t =
r
ds πs τ = πs Lr ∂r
R1 d ∗
R 1 ∗∂r
R1 ∗
π τ ds = d 0 πs ιr ∂ τ ds, so we can take η = 0 πs ιr ∂ τ ds (η vanishes on N
0 ds s
∂r
∂r
since τ does).
Since
ity with respect to
Now we can use this construction to prove that any nitely generated group
G
is a fundamental group of some compact 4-dimensional symplectic manifold (and
actually construct such a manifold).
1
2
GOMPF'S CONSTRUCTION
Proof.
G = hg1 , . . . , gk | r1 , . . . , rl i and let F be a compact Rieα1 , . . . , αk , β1 , . . . , βk be a standard basis of oriented
simple curves for the rst homology of F . Let γ1 , . . . , γl be smoothly immersed
circles representing the words r1 , . . . , rl (substitute every gi in r1 , . . . , rl by αi ).
Denote for i = 1 . . . k γl+i = βi . Then G = π1 (F ) / hγ1 , . . . , γl+k i (for γi attached
Fix a presentation
mann surface of genus
k.
Let
to a common point).
Suppose
γi
ρ
is a closed one-form on
F
that restricts to a volume form on every
(in this picture it may not exist, but we will show how to change the picture to
construct such a one-form in the next paragraph). Let
1
β =x×S
2
on F × T .
T2
be a torus,
α = S1 × x,
2
T and let ω be the product symplectic structure
Ti = γi × β ⊂ F × T2 are Lagrangian immersed tori. Let θ denote
2
∗
∗
the pullback of volume form on β under projection from T . Then η = π1 ρ × π2 θ
is closed 2-form that restricts to a symplectic form on each Ti . For small enough
t, ω̃ = ω + tη is a symplectic form on F × T2 , and With respect to it the tori Ti
are symplectic immersed manifolds. We can slightly perturb γi in the 3-manifold
F × α to make all Ti be disjointly embedded and disjoint from z × T2 (z ∈ F
1
some point) and if the perturbation is suciently C small, the tori will remain
symplectic w.r.t. ω̃ . Since the normal bundles of perturbed γi in F × α are trivial,
2
so are the normal bundles of Ti in F × T . Let W denote a complex surface with
N ⊂ W a symplectic torus with trivial normal bundle such that W r N is simply
2
connected (such a surface can be obtained from CP by blowing up at 9 points of
intersection of two non-degenerate cubics and N can be chosen as a generic ber of
1
the elliptic bration over CP ). The symplectic summation of F with W along
2
2
a torus i : T → F has the eect of killing i∗ π1 T
in π1 (F ). So to produce a
symplectic manifold with fundamental group G we take the symplectic sum of F
2
with k + l + 1 copies of W along tori Ti and z × T .
Now to construct a closed 1-form ρ as above, we rst modify the picture a little
bit: we can think that all γi 's are transverse, so that their union forms a graph Γ
1
2
in the torus. Let γ = y × S be a circle in T oriented parallel to β and let D be
a small disc disjoint from α and β and intersecting γ in an arc. For every edge
e of the graph Γ choose a small disc that intersects the interior of the edge in an
2
arc and attach (in the sense of symplectic sum) a copy of T identifying this disc
with D so that γ will be summed with e (so that the orientations of γ and e also
match). Call γ1 , . . . , γm the previous collection of γi 's with copies of γ attached,
together with copies of α and β for each attached torus. Clearly we still have
G = π1 (F ) / hγ1 , . . . , γm i, but now we also know that each edge of the graph Γ
formed by γi 's has a segment which lies in a copy of α,β or γ .
2
Let ρ0 be a closed 1-form on T that vanishes near D and has positive integrals
over α, β and γ r D (we can get such form by collapsing a neighbourhood of D to a
2
point, projecting to the diagonal of T so that the projections from α,β and γ have
be oriented circles in
Then
degree 1, and pulling back the volume form from the diagonal circle). Put such a
T2 r D
F . We get a closed one-form
Γ
. Thus we can nd a volume
R
R
form θi on each γi so that
θ = e ρ∗ for each edge e in γi . Hence θi = ρ∗ + df
e i
for some function f on e and we can choose f to vanish at the vertices of Γ. So to
∗
get the required ρ, extend f to a smooth function on F and set ρ = ρ + df .
form on each copy of
ρ
∗
and extend it by zero to
that has a positive integral over every edge
e
of
References
[1] D. McDu, D. Salamon, Introduction to symplectic topology (chapter 3 for the proof of
Weinstein's symplectic neighbourhood theorem, chapter 7 for discussion of Gompf 's construction).
[2] R.E. Gompf, A new construction for symplectic manifolds (for construction in case of nontrivial normal bundles and for more applications).