ON STRICT CATEGORY WEIGHT
AND THE ARNOLD CONJECTURE
Yuli B. Rudyak
Abstract. In [R2] and [RO] the Arnold conjecture for symplectic manifolds*
* (M; !)
with ss2(M) = 0 was proved. This proof used surgery and cobordism theory.*
* Here
we give a purely cohomological proof of this result.
Introduction
Given a smooth (=C1 ) manifold M, we set CritM := min {critf} where f runs
over all smooth functions M ! R.
Let (M; !) be a symplectic manifold. Given a smooth function f : M ! R, let
sgrad f denote the symplectic gradient of f, i.e., the vector field defined as *
*follows:
!(sgrad f; ) = -df()
for every vector field .
In [A] Arnold proposed the following remarkable conjecture. Let (M; !) be a
closed symplectic manifold, and let H : M x R ! R be a smooth function such that
H(m; t) = H(m; t + 1) for every m 2 M; t 2 R. We define Ht : M ! R by setting
Ht(m) := H(m; t). Consider the time-dependent differential equation
_x= sgradHt(x(t)): (*)
The Arnold conjecture claims that the number of 1-periodic solutions of (*) is *
*at
least CritM.
This conjecture admits another interpretation. The equation (*) yields a fam*
*ily
't : M ! M; t 2 R, where, for every x 2 M, 't(x) is the integral curve of (*). A
Hamiltonian symplectomorphism is a diffeomorphism OE : M ! M which has the
form OE = '1 for some function H : M x R ! R as above. So, the Arnold conjecture
can be refolmulated as follows:
Fix OE CritM
for every Hamiltonian symplectomorphism OE, where Fix OE denotes the number of
fixed points of OE.
This conjecture was proved for many special cases, see [MS] and [HZ] for a s*
*urvey.
Here we notice the following result of Floer [Fl] and Hofer [H]: the number Fix*
* OE
can be estimated from below by the cup-lenght of M. So, here we have a weak
form of the Arnold conjecture.
1
2 YULI B. RUDYAK
In [R2] and [RO] the Arnold conjecture was proved for every closed connected
symplectic manifold (M; !) with !|ss2(M) = 0 = c1|ss2(M). In greater detail, in
[R2] the conjecture was proved under the additional condition cat M = dim M,
and it was proved in [RO] that the condition !|ss2(M)= 0 implies the condition
cat M = dim M. Because of the last result, it turns out to be that CritM = 2n +*
* 1
provided !|ss2(M)= 0, and actually we have the inequality Fix OE 2n + 1.
The proof of the Arnold conjecture in [R] uses surgery and cobordism theory.
Here we give another proof of the Arnold conjecture (under the same restriction
!|ss2(M)= 0 = c1|ss2(M)). This proof uses the ordinary cohomology H* only; prob-
ably, it is more convenient for people which work in the area of dynamical syst*
*ems
and are not very familiar with extraordinary cohomology. The main line of the p*
*roof
follows Rudyak-Oprea [RO], but here we use the strict category weight instead t*
*he
category weight.
Remarks. 1. Hofer and Zehnder [HZ, p.250] mentioned that the Theorem 2.2
below is true without the restriction c1|ss2(M) = 0. Because of this, the Arno*
*ld
conjecture turns out to be valid for all closed connected symplectic manifolds *
*with
!|ss2(M)= 0.
2. Actually, in Theorem 2.2 below the number Fix OE is the number of 1-perio*
*dic
solutions of the equation (*), while Rest is the number of contractble 1-perio*
*dic
solutions of this equation. So, here (as well as in [R2] and [RO]) it is proved*
* that
the number of contractible 1-periodic solutions of (*) is at least 2n + 1.
The paper is organized as follows. In x1 we discuss strict category weight,*
* in
x2 we use Floer's results in order to reduce the Arnold conjecture to a certain
topological problem, in x3 we prove main results, in Appendix we discuss an ana*
*log
of the Arnold conjecture for locally Hamiltonian symplectomorphisms.
The cohomology group Hn (X; G) is always defined to be the Alexander-Spanier
cohomology group with coefficient group G, see [M] or [S] for the details.
We reserve the term "map" for continuous functions.
"Connected" always means path connected.
x1. Strict category weight
1.1. Definition ([LS], [Fox], [F], [BG]). Given a map ' : A ! X, we define the
Lusternik-Schnirelmann category cat' of ' to be the minimal number k with the
following property: A can be covered by open sets A1; : :;:Ak+1 such that '|Ai
is null-homotopic for every i. Furthrmore, we define the Lusternik-Schnirelmann
category catX of a space X by setting catX := cat1X .
1.2. Proposition ([BG]). (i) For every diagram A -'!Y -f! X we have catf'
min {cat '; catf}. In particular, catf min {cat X; catY }.
(ii) If ' ' : A ! X then cat' = cat .
(iii) If h : Y ! X is a homotopy equivalence then cat ' = cat h' for every
' : A ! X.
Given a connected pointed space X, let " : SX ! X be the map adjoint to
1X , here X is the loop space of X and S denotes the suspension, see e.g. [Sw].
ARNOLD CONJECTURE 3
1.3. Theorem ([Sv, Theorems 3, 190 and 21]). Let ' : A ! X be a map of
connected Hausdorff paracompact spaces. Then cat ' < 2 iff there is a map :
A ! SX such that " = '.
1.4. Definition ([R1]). Let X be a Hausdorff paracompact space, and let
u 2 Hq(X; G) be an arbitrary element. We define the strict category weight of
u (denoted by swgt u) by setting
fi
swgt u = sup{k fi'*u = 0 for every map ' : A ! X with cat' < k}
where A runs over all Hausdorff paracompact spaces.
We use the term "strict category weight", since the term "category weight" is
already used (introduced) by Fadell-Hussein [FH]. Concerning the relation betwe*
*en
category weight and strict category weight, see [R1].
We remark that swgt u = 1 if u = 0.
1.5. Theorem. Let X and Y be two Hausdorff paracompact spaces. Then for
every u 2 H*(X) the following hold:
(i) for every map f : Y ! X we have catf swgt u provided f* u 6= 0. Further-
more, if X is connected then swgt u 1 whenever u 2 eH*(X);
(ii) for every map f : Y ! X we have swgt f* u swgt u;
(iii) for every u; v 2 H*(X) we have swgt (uv) swgt u + swgt v.
Proof. (i) This follows from the definition of swgt.
(ii) This follows from 1.2(i).
(iii) Let swgt u = k; swgt v = l with k; l < 1. Given f : A ! X with catf <
k + l, we prove that f* (uv) = 0. Indeed, catf < k + l, and so A = A1 [ . .[.Ak*
*+l
where each Ai is open in A and f|Ai is null-homotopic. Set B = A1 [ . .[.Ak and
C = Ak+1 [ . .[.Ak+l. Then catf|B < k and catf|C < l. Hence f* u|B = 0 =
f* v|C, and so f* (uv)|(B [ C) = 0. i.e., f* (uv) = 0.
The case of infinite category weight is leaved to the reader.
x2. Floer's reduction and related things.
2.1. Recollection. A flow on a topological space X is a family = {'t}; t 2 R
where each 't : X ! X is a self-homeomorphism and 's't = 's+t for every s; t 2 R
(notice that this implies '0 = 1X ).
A flow is called continuous if the function X x R ! X; (x; t) 7! 't(x) is co*
*ntin-
uous.
A point x 2 X is called a rest point of if 't(x) = x for every t 2 R. We de*
*note
by Rest the number of rest points of .
A continuous flow = {'t} is called gradient-like if there exists a continuo*
*us
(Lyapunov) function F : X ! R with the following property: for every x 2 X we
have F ('t(x)) < F ('s(x)) whenever t > s and x is not a rest point of .
The following Theorem can be found in [Fl, Th. 7], cf. also [HZ].
4 YULI B. RUDYAK
2.2. Theorem. Let (M; !) be a closed connected symplectic manifold such that
!|ss2(M)= 0 = c1|ss2(M), and let OE : M ! M be a Hamiltonian symplectomorphism.
Then there exists a map f : X ! M with the following properties:
(i) X is a compact metric space;
(ii) X possesses a continuous gradient-like flow such that Rest FixOE;
(iii) The homomorphism f* : Hn (M; G) ! Hn (X; G) is a monomorphism for
every coefficient group G.
The following theorem (of Lusternik-Schnirelmann type) is proved in [R2].
2.3. Theorem. Let be a continuous gradient-like flow on a compact metric
space X, let Y be a Hausdorff space which can be covered by open and contracti*
*ble
in Y subspaces, and let f : X ! Y be a map. Then
Rest 1 + catf:
We need also the following well-known fact which follows from [LS] and [T].
2.4. Theorem. For every closed smooth manifold M we have
1 + catM CritM 1 + dim M:
x3. Proof of the Arnold conjecture
3.1. Theorem (cf. [FH], [RO], [St]). Let ss be a discrete group. Then for e*
*very
u 2 Hk (K(ss; 1); G) with k > 1 we have swgt u 2.
(Actually, Strom [St] proved that swgt u k. Moreover, it is easy to see th*
*at
swgt u k provided u 6= 0, and so swgt u = k if u 6= 0.)
Proof. Because of 1.3, it suffices to prove that "*u = 0 where " is a map from
1.3 and "*; H*(K(ss; 1); G) ! H*(SK(ss; 1); G) is the induced homomorphism.
But K(ss; 1) is homotopy equivalent to the discrete space ss, and so SK(ss; 1) *
*is
homotopy equivalent to a wedge of circles. Hence, Hi(K(ss; 1(; G) = 0 for k > *
*1,
and thus "*u = 0.
3.2. Theorem (cf. [RO]). Let Y be a connected finite CW -space, and let y 2
H2(Y ; G) be such that y|ss2(Y ) = 0. Then swgt y 2.
Proof. Let ss = ss1(Y ), and let g : Y ! K(ss; 1) be a map which induces an
isomorphism of fundamental groups. First, we prove that
y 2 Im {g* : H2(K(ss; 1); R) ! H2(Y ; R)}:
Indeed, since Y is a finite CW -space, its singular cohomology coincides with *
*H*,
and so we have the universal coefficient sequence
0 ! Ext(H1(Y ); R) -! H2(Y ; R) -l!Hom (H2(Y ); R) ! 0:
On the other hand, there is a Hopf exact sequence
ss2(Y ) ! H2(Y ) ! H2(K(ss; 1)) ! 0;
ARNOLD CONJECTURE 5
and so we have the following commutative diagram with exact rows and colomn:
Ext(H1(K); R) - ---! H2(K); R) ----! Hom (H2(K); R)) ----! 0
? ? ?
g0?y~= ?yg* ?yg00
Ext(H1(Y ); R) - ---! H2(Y ; R) ---l-! Hom (H2(Y ); R) ----! 0
??
y
Hom (ss2(Y ); R)
where K denotes K(ss; 1). Now, since y|ss2(Y ) = 0, we conclude that l(y) 2 Im *
*g00.
Since g0 is an isomorphism, an easy diagram hunting shows that y 2 Im g*.
Thus, by 3.1 and 1.5(ii), swgt y 2.
3.3. Corollary. Let Y be a connected finite CW -space, let R be a commutative
ring, let y 2 H2(Y ; R) be such that y|ss2(Y ) = 0, and let X be a Hausdorff pa*
*ra-
compact space. If f : X ! Y is a map with f* (yn ) 6= 0, then catf 2n.
Proof. By 1.5,
catf swgt yn n swgty 2n:
3.4. Corollary ([RO]). Let (M2n ; !) be a closed connected symplectic manifold
with !|ss2(M)= 0. Then catM = 2n and CritM = 2n + 1.
Proof. Since !n 6= 0, we conclude that, by 3.3, catM = cat1M 2n. So, catM =
2n since catM dim M. The second equality follows from 2.4.
3.5. Corollary. Let (M2n ; !) be a closed symplectic manifold with !|ss2(M) =
0 = c1|ss2(M), and let OE : M ! M be a Hamiltonian symplectomorphism. Then
Fix OE 2n + 1. In particular, the Arnold conjecture holds for (M; !).
Proof. Let f : X ! M and be a map as in 2.2. Since every closed connected
smooth manifold is a finite polyhedron, and since !n yields a non-trivial cohom*
*ology
class in H*(X; R), we conclude that, by 3.3, catf 2n. Now, by 2.2 and 2.3,
Fix OE Rest 1 + catf 2n + 1:
Thus, because of 3.4, the Arnold conjecture holds for (M; !).
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6 YULI B. RUDYAK
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Universit"at-GH Siegen, FB6/ Mathematik, 57068 Siegen, Germany
E-mail address: rudyak@mathematik.uni-siegen.de