floer homology and arnold conjecture

74 Pages·1999·0.61 MB·English

j. differentialgeometry
49 (1998)1-74
FLOER HOMOLOGY AND ARNOLD CONJECTURE
GANG LIU & GANG TIAN
1. Introduction
Let V be a closed symplectic manifold with a symplectic form !.
This means that ! is a closed non-degenerate two-form. Because of the
non-degeneracy of !, with any time-dependent periodical Hamiltonian
1
function H : V (cid:2)S ! R; we can associate a (cid:18)-dependent vector (cid:12)eld
XH(cid:18) given by:
!(XH(cid:18);(cid:1))= dH(cid:18);
1
where (cid:18) 2 R is the usual angular coordinate of S and H(cid:18) = HjS1(cid:2)f(cid:18)g:
Consider the Hamiltonian equation:
dz
(0.1) = XH(cid:18)(z):
d(cid:18)
Let P(H) be the set of periodic-1 solutions of (0.1). Clearly P(H) is
one to one correspondence to the set of (cid:12)xed points of the time-1 (cid:13)ows
H
(cid:30)1 of V associated to (0.1). For a \generic" choice of H, the graph
H
(cid:0)(cid:30)H1 of (cid:30)1 is transversal to the diagonal 4V in V (cid:2)V: It follows that
P(H) is (cid:12)nite in this case. We refer this as a nondegenerate case. By
the Lefschetz (cid:12)xed point theorem, the algebraic cardinality of P(H) is
just the Euler characteristic (cid:31)(V) of V, which is the alternating sum
of the Betti number bi(V) of V. However, it has been conjectured by
V.I. Arnold in [1] that the geometric cardinality of P(H)should satisfy
a Morse inequality, #P(H) (cid:21) ibi(V): This yields a much stronger
estimate than what is expected by algebraic topology and re(cid:13)ects the
P
remarkable symplectic rigidity (see [2] and [9]). This famous conjecture
Received September 24, 1996, and, in revised form,July 21, 1997.
1
2 gang liu & gang tian
has been a major driving force for the developments of various the-
ory and techniques in symplectic topology and many special cases have
been proved. The (cid:12)rst breakthrough was made by Conley and Zehnder
2n
in 1982, who proved the conjecture for the torus T with the standard
symplectic structure. In the subsequent years, this result was extended
n
by Floer and Sikorav to certain other quotients of R , which include
H 0
all the twodimensional orientable surfaces. When (cid:30)1 is C -close to the
identity, the conjecture was proved to be true in general by Weinstein
(See [21]). However, despite of various interesting results, the subject
did not achieve a uni(cid:12)ed frameworkuntil the adventof Floer homology.
In 1985, Gromov introduced the idea of using J-holomorphic curves in
symplectic topology, which yields many important new results in the
subject. In the very next year, combining the variational method pre-
viously used by Conley and Zehnder with the theory of J-holomorphic
curves,Floerintroducedhiscelebrated Floerhomologytheoryforclosed
monotonic symplectic manifolds, and consequently proved the Arnold
conjecture for this class of symplectic manifolds. Later on, this result
was extended by Hofer and Salamon in [10] to semi-positive case with
a mild extra restriction on the minimal Chern number, which includes
all Calabi-Yau manifolds. Soon after that, this extra condition was
removed by Ono in [17]. However, there were serious obstructions to
extend the Floer homology, hence to prove the Arnold conjecture in
general, because of the appearance of multiply covered J-holomorphic
curves with negative (cid:12)rst Chern class. Unlike the semi-positive case,
the moduli spaces used in the general case to construct Floer chain
complex arenot compactany more. Their natural compactifcation may
contain stratawhose dimension maybe greaterthan that of the moduli
space itself. It was completely unclear whether or not a cohomology
theory of Floer-type could be ever assembled from Hamiltonian sys-
tems because of these wild strata. One has to develop a new method of
counting contributions from those stratain the boundary of the natural
compacti(cid:12)cation of the moduli space, so that a cohomology theory of
Floer-type can be well-de(cid:12)ned and consequently, the Arnold conjecture
can be proved. In fact, a similar di(cid:14)culty also appeared in establish-
ing a mathematical theory of quantum cohomology and GW-invariants
beyond the scope of semi-positive symplectic manifolds.
Recent development in theory of GW-invariants casts new light on
the subject and reveals the possibility to overcome the di(cid:14)culty. In
1995, J. Li and the second author of this paper introduced the method
of constructing virtual moduli cycles in the setting of algebraic geome-
floer homology and arnold conjecture 3
try ( see [12] ). Their idea is to use the global two-termfree resolutions
of the deformation-obstruction complexes. Inspired by this, J. Li and
the second author of this paper constructed virtual moduli cycles and
de(cid:12)ned G-W invariants for general symplectic manifolds in [13], while
we developed in this paper a di(cid:11)erent method of constructing virtual
moduli cycles in our dynamic setting of Hamiltonian system. As a
consequence of this, we extended Floer (co-)homology to all symplectic
manifolds withoutany positivity assumption and provedArnold conjec-
ture in general. One of the main techniques of this paper is the gluing
of J-holomorphic curves for which the transversality may fall. Gluing
ofJ-holomorphic curvesunder transversalityassumptionwasdeveloped
before in [11] and [19]. The method we used in this paper was based
on the work of the (cid:12)rst author in [11]. The method in [19] can also be
adapted here.
To motivate our construction, we (cid:12)rst need to introduce some ideas
and notations prevailed in previous Floer (co)homology theory.
Recall that the question of (cid:12)nding 1-periodic orbits of (0.1) has a
variational formulation.
Let L be the space of contractible loops in V and L be its universal
coveringwithcoveringgroup(cid:25)2(V):Eachelement [z;w]ofLcanberep-
1 2
resented bya C -map w :D ! V with boundary valuee z = wj@D2=S1:
We denote this representation by (z;w): e
However, we will introduce a weaker relation for the de(cid:12)nition of
L, namely, we de(cid:12)ne [z1;w1] (cid:24) [z2;w2] if z1 = z2 and w1 and w2 are
homologous to each other. Under this equivalence relation, we have
L(V) = L(V)=(cid:0), where (cid:0) is the image of (cid:25)2(V) under the Hurewicz
map (cid:25)2(V) ! H2(V): The symplectic action functional aH : L ! R is
de(cid:12)ned bye
(cid:3) e
aH([z;w])= w !+ H(cid:18)(z((cid:18)))d(cid:18):
D2 S1
Z Z
The critical pointsofaH arejustthose[z;w]with z being the1-periodic
solution of (0.1). We will use P~(H) to denote the set of critical points
of aH, which is just the \lifting" of P(H) in L~(V):
Let J be a !-compatible almost complex structure in the sense
that for any x;y 2 TvV;!(Jx;Jy) = !(x;y) and the symmetric bi-
linear form gJ(x;y) = !(x;Jy) is positive on TvV for any v 2 V.
Clearly, gJ is a J-invariant Riemannian metric on V, which, in turn,
2
induces an L -metric on L(V): With respect to this metric, a gradi-
1
ent (cid:13)ow line of aH is just a connecting orbit f : R (cid:2) S ! V with
4 gang liu & gang tian
bounded energy, satisfying the equation @(cid:22)J;Hf = 0 and the limit condi-
1 (cid:6)
tion along the ends of R(cid:2)S , namely, lims!(cid:6)1f(s;(cid:18))= z ((cid:18)), where
@(cid:22)J;Hf 2 (cid:0)(^0;1(f(cid:3)TV)) is given by
@ @f @f
@(cid:22)J;Hf( )= +J(f) +rxH(f;(cid:18))
@s @s @(cid:18)
(cid:6) DT (cid:0) +
and z 2 P(H): We will use M (J;H;z~ ;z~ ) to denote the space
(cid:0) +
of the connecting orbits de(cid:12)ned as above with z~ #f = z~ : Now the
energy E(f) of f is de(cid:12)ned by f
1 @f 2 @f 2
E(f)= (j j +j (cid:0)XH(cid:18)(f)j )dsd(cid:18);
2 S1 R @s @(cid:18)
Z Z
DT (cid:0) +
and any element f 2 M (J;H;z~ ;z~ ) has a (cid:12)xed energy
+ (cid:0)
fE(f)= aH(z~ )(cid:0)aH(z~ ):
Let
DT (cid:0) + DT (cid:0) +
M (J;H;~z ;z~ )= M (J;H;z~ ;z~ )=R
be the moduli space of unparametrized connecting orbits, where R acts
DT (cid:0) + f
on M (J;H;z~ ;z~ ) by s-translations.
DT (cid:0) +
For a \generic" choice of (J;H), M (J;H;z~ ;z~ ) is a smooth
manfifold of dimension (cid:22)(z~+) (cid:0) (cid:22)(z~(cid:0)); where (cid:22) : P~(H) ! Z is the
Conley-Zehnder index. f
With such data one can attempt to develop a Morse theory for aH
to get an estimate on #P(H):The \classical" Floer cohomology is just
a such device constructed for some ideal situations, such as in the case
of semi-positive symplectic manifolds.
(cid:3)
The idea is to construct a chain complex (C (H);(cid:14)J;H); whose ho-
(cid:3) (cid:3) (cid:3)
mology H (C (H);(cid:14)J;H) is isomorphic to H (V); in such a way that
C(cid:3)(H) is generated by the elements of P~(H) as a Q-vector space,
and the coboundary operator (cid:14)J;H is de(cid:12)ned by \counting" the num-
(cid:3)
ber of discrete connecting orbits. More precisely, we de(cid:12)ne C (H) =
k k
(cid:8)kC (H),andanyelement(cid:24) 2 C (H)isaformalsum(cid:24) = (cid:22)(z~)=k(cid:24)z~(cid:1)z~
with (cid:24)z~2 Q, such that for any c > 0;
P
#fz~j(cid:24)z~6= 0; aH(z~)(cid:20) cg< 1:
(cid:3)
In general C (H) is of course in(cid:12)nite dimensional over Q, but it is a
(cid:12)nite dimensional vector space over the Novikov ring ^!, which is a
floer homology and arnold conjecture 5
(cid:12)eld in our case (see the relative de(cid:12)nition in Section 5). In fact the
(cid:3)
dimension of C (H) over ^! is just #P(H):
k k+1
Now (cid:14)J;H :C ! C is de(cid:12)ned by
(cid:14)J;H(x~)= n(x~;y~)y~
(cid:22)(y~)=(cid:22)(x~)+1
X
k DT
foranyx~ 2 C ;wheren(x~;y~)istheoriented number#M (J;H;x~;y~):
If we have
(i) n(x~;y~) is (cid:12)nite when (cid:22)(y~)(cid:0)(cid:22)(x~)= 1;
k k+2
(ii) (cid:22)(y~)=k+1n(x~;y~)(cid:1)n(y~;z~)= 0 for any x~ 2 C and z~2 C ,
P (cid:3)
then (cid:14)J;H is well-de(cid:12)ned and (C (H);(cid:14)J;H)) is a chain complex. The
(cid:3)
\classical" Floer cohomology is just the homology of (C (H);(cid:14)J;H) for
a generic (J;H) when (V;!)is semi-positive so that (i) and (ii) hold.
Note that the left-hand side of (ii) can be interpreted as the (ori-
ented) number of pairs of \broken" connecting orbits between x~ and z~
when (cid:22)(z~)(cid:0)(cid:22)(x~)= 2: In the \ideal" situation, the space
DT DT
[(cid:22)(y~)=k+1M (J;H;x~;y~)(cid:2)M (J;H;y~;z~)
of such \broken" connecting orbits is just the \boundary" of
DT DT
M (J;H;x~;z~): We denote its union with M (J;H;x~;z~) by
M(J;H;x~;z~): One can show that it is compact. In fact, in this ideal
case, this compactmoduli space of\broken"connecting orbitscoincides
withthemoduli spaceofstable(J;H)-mapsconnecting x~andz~(Seethe
de(cid:12)nition in Sec.2). Therefore, we have a \good" compacti(cid:12)cation of
DT
M (J;H;x~;z~) with boundary components of codimension 1. Putting
this in a more algebraic form, we can summarize the \classical" Floer
cohomology (for good cases) in the following statement:
k k+2
(iii) When x~ 2 C , z~2 C , the moduli space M(J;H;x~;z~) of sta-
ble (J;H)-mapsconnecting x~and z~is compactandis aone-dimensional
manifold with boundary. It can be viewed as a relative virtual 1-cycle
with
@M(J;H;x~;z~)= [(cid:22)(y~)=k+1M(J;H;x~;y~)(cid:2)M(J;H;y~;z~):
Clearly, (i) and (ii) follow from (iii).
Nowforageneralclosedsymplecticmanifoldotherthansemi-positive
DT
ones,thenaturalcompacti(cid:12)cationM(J;H;x~;z~)ofM (J;H;x~;z~),the
6 gang liu & gang tian
stable compacti(cid:12)cation, contains not only those \broken" connecting
orbits as above, but also some bubbles of J-holomorphic spheres. A
\boundary" component of the stable compacti(cid:12)cation containing some
multiply covered bubbles with negative (cid:12)rst Chern class may have a
DT
higher dimension than that of M (J;H;x~;z~) itself. Consequently
both (i) and (ii) may fail.
To overcome this di(cid:14)culty, we will construct a virtual moduli
(cid:23)
Q-cycle C(M (J;H;x~;z~)) such that its underlying moduli space
(cid:23)
M (J;H;x~;z~) is compact. Here (cid:23) stands for certain \generic" pertur-
bation of the @(cid:22)J;H-operator. Below is the outline of our construction.
The construction consists of twoparts, local and global one. Firstly,
note that M(J;H;x~;y~) consists of all unparametrized stable (J;H)-
maps connecting x~ and y~, which is contained in the in(cid:12)nite dimensional
p 2
space B(x~;y~) of unparametrized stable Lk-maps, k (cid:0) p > 1 (see the
relevant de(cid:12)nitions on page 17).There is an in(cid:12)nite dimensional \bun-
dle" L ! B(x~;y~) with each (cid:12)ber L[f] of [f] 2 B(x~;y~) consisting of all
p 0;1 (cid:3)
Lk(cid:0)1-sections of the bundle ^ (f TV);f 2 [f]; modulo the equiva-
lence relation induced by reparametrization of the domains. In general,
we do not expect to get any useful smooth structure for B(x~;y~) due to
the non-compactness of reparametrization group. However, there ex-
ists an open set W (cid:26) B(x~;y~) such that M(J;H;x~;y~) (cid:26) W and W is
a strati(cid:12)ed Banach orbifold, called partially smooth orbifold, strati(cid:12)ed
according to the topological types of the domains of the stable maps.
m
In fact , W = [i=1Wi and each Wi = W(fi) is an open neighborhood
in B(x~;y~) of [fi], with [fi]2 M(J;H;x~;y~); such that Wi is uniformized
W
by (cid:25)i : Wi = W(fi;Hi) ! Wi with a (cid:12)nite automorphism group (cid:0)i,
p
where the uniformizer Wi consists of all those stable Lk-maps in the
f f (cid:0)1
neighborhood Wi(fi) = (cid:25) (Wi) of fi which send their marked points
into some particularly cofnstructed family oflocal hypersurfaces Hi: Let
(cid:3)
Li = (cid:25)i(L) !fWi. Then
L W
e (cid:25)i = (f(cid:25)i ;(cid:25)i ): (Li;Wi)! (Li;Wi); i= 1;(cid:1)(cid:1)(cid:1);m;
gives rise to a uniformizing system for the orbifold bundle (LjW;W):
e f
@J;H-operator can be interpreted as a section of Li which is (cid:0)i- equiv-
W
ariant. Let Mi be the zero set of @J;H in Wi: Then Mi = (cid:25)i (Mi) is
just e
f M(J;H;x~;y~)\Wfi: f
Now the wrong dimension of the boundary of the stable compacti(cid:12)-
cation simply means that @J;H is not a transversal section yet in some
floer homology and arnold conjecture 7
Wi though we have chosen a \generic" pair (J;H): In terms of this
orbifold structure, the non-su(cid:14)ciency of perturbing (J;H) to achieve
tfransversality for @J;H is quite easy to understand. It is simply because
that any perturbation in J (cid:2)H ofthe set of pairs (J;H)will yield a (cid:0)i-
equivariant change in Wi, but for a given pair (J;H), which is e(cid:11)ective
in the sense that there exists some u, say, in Wi, such that @J;Hu = 0,
the cokernel Ri(u) of tfhe linearization of @J;H at u may not be gener-
ated by (cid:0)i -invariant sections ofLi: Because offthis, our remedy for this
non-transversality problem becomes quite plain at least locally. What
we need to do is to choose a \geeneric" perturbation (cid:23)i in Ri = R(fi),
which in general may not be generated by (cid:0)i-invariant sections of Li;
and consider the (cid:23)i-perturbed section @J;H +(cid:23)i : Wi ! Li: It directly
follows fromtheconstruction thatthis new section is transversaltozeero
section and the local moduli space f e
(cid:23)i (cid:0)1
Mi = (@J;H +(cid:23)i) (0)
(cid:23)i
and its projection Mi tfo Wi certainly have the right dimension at all
their stratum as expected by the index theorem.
To complete our construction of the virtual moduli Q-cycle
(cid:23)
C(M (J;H;x~;y~)), we need to globalize the above construction. The
main di(cid:14)culty here is how to transform each non-equivariant section (cid:23)i
in Wi into Wj when Wi\Wj is not empty.
Inordertogetsuchatransformation,letWij = Wi\Wj andconsider
f f
(cid:0)1
(cid:25)i :W^ij = (cid:25)i (Wij) ! Wij
and
f
(cid:0)1
(cid:25)j : Wi^j = (cid:25)j (Wij)! Wij:
(cid:0)ij
Wede(cid:12)neWij tobetheirf(cid:12)berproductW^ij(cid:2)WijWi^j overWij;which in
some sense can be thought of as a substitute for \Wi\Wj". In general,
f f f
let N be the nerve of the covering W = fWi;i= 1;(cid:1)(cid:1)(cid:1);mg: For each
f f
WI = Wi1;(cid:1)(cid:1)(cid:1);in = Wi1 \Wi2(cid:1)(cid:1)(cid:1)\Win
with I = (i1;(cid:1)(cid:1)(cid:1) ;in)2 N; let
(cid:0)1
Wi1;(cid:1)(cid:1)(cid:1);^ik;(cid:1)(cid:1)(cid:1);in = (cid:25)k (WI):
Then we have n \(cid:12)nite" morphisms
f
(cid:25)k :Wi1;(cid:1)(cid:1)(cid:1);^ik;(cid:1)(cid:1)(cid:1);in ! WI
f
8 gang liu & gang tian
(cid:0)I
with automorphism group (cid:0)k: As above, we de(cid:12)ne WI as the (cid:12)ber
product of Wi1;(cid:1)(cid:1)(cid:1);^ik;(cid:1)(cid:1)(cid:1);in(1(cid:20) k (cid:20) n) over Wi1;(cid:1)(cid:1)(cid:1);in; where
f
f (cid:0)I = (cid:0)i1 (cid:2)(cid:0)i2(cid:1)(cid:1)(cid:1)(cid:2)(cid:0)in:
(cid:0)I W W W
Obviously (cid:0)I actsonWI and(cid:25)I (cid:1)(cid:27) = (cid:25)I forany(cid:27) 2 (cid:0)I;where (cid:25)I is
(cid:0)I
the natural projection from WI toWI: We have a similar construction
L (cid:0)I f L (cid:0)1
(cid:25)I : LI ! LI for \bundles" LI = ((cid:25) ) (WI): If J (cid:26) I 2 N, there
exists a morphism f
e
I (cid:0)I (cid:0)I (cid:0)J (cid:0)J
(cid:25)J :(LI ;WI ) ! (LJ ;WJ )
such that
e f e f
W I I W I
(i) (cid:25)J (cid:14)(cid:25)J = EJ (cid:14)(cid:25)I ; where EJ :WI ! WJ is the inclusion;
I (cid:0)1
(ii) #(((cid:25)J) (u)) is NI=NJ for a generic u, where NI = j(cid:0)Ij:
Now we can construct an open subset VI (cid:26) WI for each I 2 N to
remove those \extra"overlaps between these WI’s (see detail in Section
4). By replacing WI and all induced construction above by VI’s, we get
a system of morphisms of bundles:
I (cid:0)I (cid:0)I
f(cid:25) :(EI ;VI )! (EI;VI);I 2 Ng:
(cid:0)I (cid:0)I
Note that each WI aned VIe are not (partially) smooth manifolds,
but rather (partially) smooth varieties (see the de(cid:12)nition on page 59).
(cid:0) (cid:0)
We now use abovfe system (eE ;V ) to globalize our local moduli space
(cid:23)i (cid:0)I (cid:0)I
Mi : Since these (EI ;VI ) relate to each other by those \semi-global"
morphisms (cid:25)JI, a global sectieon seof the system (E(cid:0);V(cid:0)) can be de(cid:12)ned
f e e (cid:0)I (cid:0)I J (cid:3)
asacollection ofsections fsI;I 2 Ngof(EI ;VI )such that((cid:25)I) sI =
sJ is valid over smooth points in their overlap. Celeaerly, @J;H gives rise
to a global section of this system, and eaech eleement (cid:23)i 2 R(fi) can be
W
transformedas aglobal section ofthe system by using these (cid:25)I ’s tolift
(cid:0)I (cid:0)I m
it into a collection of sections of (EI ;VI );I 2 N: Let R= (cid:8)i=1R(fi).
We will prove in Section 4 that for a \generic" choice of (cid:23) 2 R, @J;H+(cid:23)
(cid:0)I (cid:0)I
is a transversal global section of (eEI ;eVI ):
Now
(cid:0)1 (cid:0)1
(@J;H +(cid:23)) (0)= f(@eJ;H +e (cid:23)I) (0);I 2 Ng
are certainly compatible with each other. Let
(cid:23) (cid:0)1
MI = (@J;H +(cid:23)I) (0)
f
floer homology and arnold conjecture 9
(cid:23) (cid:23) (cid:23) (cid:23)
and MI = (cid:25)I(MI): Then M = fMI;I 2 Ng is the compact moduli
space induced by (cid:23) underlying the virtual moduli Q-cycle that we are
looking for. Thefresulting relative virtual cycle is \formally" de(cid:12)ned to
be
(cid:23) 1 (cid:23)I
C(M )= MI
NI
I2N
X
f
(see the precise de(cid:12)nition on page 64). The following theorem, which is
proved in Sec.4, serves as a technique base of this paper.
(cid:23)
Theorem 1.1. Theabove C(M (x~;z~))isa rationalcycleinB(x~;z~)
of dimension (cid:22)(z~)(cid:0)(cid:22)(x~)(cid:0)1. Moreover, we have
(cid:23) (cid:23) (cid:23)
@(C(M (x~;z~))= [y~C(M (x~;y~))(cid:2)C(M (y~;z~)):
(cid:23)
Inthecasethat(cid:22)(y~)(cid:0)(cid:22)(x~)= 1,itfollowsfromthisthatM (x~;y~)is
(cid:23)
a(cid:12)niteset,andorientednumber#(C(M (x~;y~)))2 Qiswell-de(cid:12)ned. If
(cid:23)
wede(cid:12)nen(x~;y~)= #(C(M (x~;y~))),itiseasytoseethat(i)and(ii)will
follow from above theorem. With this new interpretation of n(x~;y~), we
now can extend Floer (co-) homology to all closed symplectic manifolds
by the very same formulae as before. By using a parametrized version
of above theorem we can prove that the resulting Floer cohomology
(cid:3)
FH (V;!;J;H;(cid:23)) is independent of the parameter (J;H;(cid:23)): In fact,
with certain suitable modi(cid:12)cation of the above theorem, we can also
de(cid:12)ne both the intrinsic and exterior multiplicative structures in the
Floer cohomology for all closed symplectic manifolds, which were only
de(cid:12)ned for semi-positive case before.
This paper is organized as follows.
In Section 2, we will de(cid:12)ne the moduli space M(J;H;x~;y~) of stable
(J;H)-maps connecting x~ and y~and its ambient space B(x~;y~) of stable
p 2
Lk-maps, k (cid:0) p > 1: We then prove in Lemmas 2.6 and 2.7 that for
each [f]2 M(J;H;~x;y~), there is an open neighborhood W(f) of [f] in
B(x~;y~),which isa(partially)smoothorbifold witha(partially)smooth
uniformizer W(f;H)andautomorphismgroup(cid:0)f;andanorbifold bun-
dle L(f) overW(f)with uniformizer L(f):The @J;H-operatorgives rise
f
to a (cid:0)f-equivariant section of L(f):
In Section 3, we will establish theemain local transversality of @J;H-
section perturbed by a \genereic" section (cid:23) of the (cid:12)nite dimensional
\obstruction" bundle R(f). The main technical part of this section is
the main estimate in Proposition 3.1. As a consequence of the transver-
sality of @J;H +(cid:23), in Lemma 3.9 we will prove that the local perturbed
10 gang liu & gang tian
(cid:23)
moduli space M (f) has the \right" dimension as expected by index
theorem for each of its stratum. Another corollary of the transversality
isthegluingconfstructionofProposition3.2andCorollary3.3whichwill
p
serve as a basis for comparing the strong Lk-topology for M(J;H;x~;y~)
1
de(cid:12)ned in this section with the weak C -topology used before by Floer
and Gromov.
(cid:23)
Section 4 is devoted to globalize above local moduli space M (f).
Wewill give thedetails ofourconstructionoftherelativevirtualmoduli
(cid:23)
Q-cycle C(M (J;H;x~;y~)) sketched in this introduction. f
In Section 5, wewill use the theorywhich we developed in the previ-
(cid:3)
ous sections toextend Floer cohomologyFH (V;!;J;H;(cid:23))toa general
closed symplectic manifold and prove that it is invariant with respect
to the parameter (J;H;(cid:23)). We conclude our proof of Arnold conjecture
(cid:3)
in Theorem 5.3 and Corollary 5.4 showing that FH (V;!;J;H;(cid:23)) is
(cid:3)
isomorphic to H (V)(cid:10)^!:
During the preparation of this paper, we learned that Fukaya and
Ono obtained a di(cid:11)erent proof of the Arnold Conjecture in [8].
The authorsare grateful toreferees fortheir suggestions forimprov-
ing the writing of the paper. In paticular, we are very grateful to one
of the referees, who pointed out that the parametrized moduli space
introduced in the (cid:12)rst version was not needed. Though the proof in
the present version of this paper is the same as the (cid:12)rst version, this
suggestion of the referee makes our presentation much more clear and
simpler.
2. Moduli space of stable maps
Inthissectionwewill de(cid:12)nethemoduli spaceM(J;H;x~;y~)ofstable
p
(J;H)-mapsandits ambientspaceB(x~;y~)ofstable Lk-mapsconnecting
x~ and y~. Near M(J;H;x~;y~), B(x~;y~) has a (partially) smooth orbifold
structure. Locally, this amounts to say that for each stable (J;H)-
map [f], there exists an open neighborhood U(f) of f in B(x~;y~) such
that U(f) is uniformized by a connected (partially) smooth manifold
U(f;H):Overeachuniformizer U(f;H),wewill de(cid:12)ne aBanachbundle
L(f): The @J;H-operator is an equivariant section of the bundle, which
ies smooth on each strata of U(fe;H):
e