Symplectic homology via generating functions

Geometric And

1016-443X/94/0600718-3151.50+0.20/0

Functional Analysis

r 1994 Birkhguser Verlag, Basel

Vol. 4, No. 6 (1994)

SYMPLECTIC

HOMOLOGY

VIA GENERATING

FUNCTIONS

LISA TRAYNOR

1. I n t r o d u c t i o n Exciting discoveries in the symplectic topology of (R 2n, w0) have been made recently via a technique that combines Hofer's capacity theory and Floer's homology theory into a homology theory with a real filtration ([FH], [FHWy]). The theories of Floer and Hofer naturally fit together since their constructions are both based on studying the action functional on the loop space of a symplectic manifold. In IV], Viterbo made important progress in capacity theory by viewing the action functional on the loop space as an infinite dimensional limit of generating functions. By means of generating functions, the set of lagrangian submanifolds of a cotangent bundle that can be "defined" by functions is enlarged by considering functions on a vector bundle. Due to results of Sikorav and Laudenbach and of Viterbo, a large class of lagrangians admit "unique" generating functions (w Viterbo's capacities for lagrangian submanifolds are critical values of the associated generating function. In this manner, capacities are defined for open subsets of (R2n,w0) via the lagrangians associated to compactly supported symplectomorphisms. Eliashberg outlined how this work of Viterbo could be extended to an alternate construction of symplectic homology for open subsets of (R 2n, w0) (JEll). Details of this construction are given in w167 In essence, symplectic homology groups for a symplectomorphism h, denoted by G(.a'bl(h), are the relative homology groups H.(E b, Ea), E b := {x E E : S(x) ~_ b}, where S : E -* R is the generating function for the lagrangian associated to h, Definition 3.5. Symplectic homology groups for an open set U C R 2~ are constructed via a limit of homology groups associated to symplectomorphisms supported on U. These groups are invariants of U under the action Research at M a t h e m a t i c a l Sciences Research I n s t i t u t e supported by an MSRI postdoctoral fellowship. Research at Stanford supported by an NSF M a t h e m a t i c a l Sciences Postdoctoral Research Fellowship. Research at Centre Emile Borel, I n s t i t u t Henri Poincar~, Paris funded by Ministate des Affaires Etrang~res, France.

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of symplectic diffeomorphisms of R2=. Henceforth, Hofer and Floer's construction will be referred to as F-H homology and this generating function construction as gf-homology. As opposed to the infinite dimensional analysis behind the construction of F-H homology, the analysis for gf-homology is finite dimensional and elementary. The Conley-Zehnder index and pseudo-holomorphic curves present in the F-H homology theory are replaced by the Morse index of the generating function and the natural boundary operator in a finite dimensional manifold. It is important to point out that the F-H theory extends to more general symplectic manifolds, [CiFH], while the following is a construction only for open subsets of (R2~,~0). This technique has potential to be generalized using, for example, ideas of Givental in [G]. In w it is shown that many important functorial properties satisfied by F-H homology for (R 2n, w0) are also satisfied by gf-homology. For example, both homology theories have an isotopy invariance property, Theorem 6.8. These functorial properties are central in applications of symplectic homology. Many of the results in [FHWy] can be reproved via gf-homology. A difference between the homology theories is that gf-homology is based on symplectomorphisms defined by compactly supported hamiltonians while the F-H homology is based on hamiltonians that are quadratic at infinity. Hamiltonians which are quadratic at infinity are well adapted to the study of products while it is natural to study disjoint unions via compactly supported symplectomorphisms, Theorem 6.15. In w the gf-homology groups of an open ellipsoid are calculated. These groups are similar to the F-H homology groups ([FHWy]) in the sense that the groups obtained by both theories detect the actions of closed characteristics in the boundary of the ellipsoid. As an illustration, consider 1 tx22 + y2) < 1} . E(1,2) := {(Xl2 + yl2) + 3' Examples of gf-homology groups with Z2 := Z/2Z coefficients are, for 0 < E 0, Ft := Ct(O) is a graph over the zero section: Ft = FaF~, ~ : W ~ ---* R. Then

ff--iF'(q)

Gt

(

q' Oq ]

To see this, first notice that the family F, can be viewed as a function on extended phase space: F : i~'~ • [0, cx~) ~ R, F(q, t) :-- Ft(q). Consider (q',p') = r 0). In a neighborhood of the trajectory Ct(qo, 0), t e [0, t'], in t3,Ft, it is possible to take new coordinates x = (q,t)

r

= x.

There exists a path a : [0,e] ---* ( T ' R " N { p = 0 } ) N { t = 0}, a(0) --(q0,0) such that (r = o F" ' t '' T h e q u a d r a n g l e o : = s E [0, ~], t E [0, t'], is foliated by curves that are annihilated by the form A + Gdt and thus

O= f d ( , k + G d t ) = ~o~ A + G d t . Oa = ao + fl~ - 6 -/30 where a0 := (a(s), 0), 6 := (a(s), t') for s E [0, c] and /3o := (a(0), t), f~ := (a(r t) for t E [0, tq. Since both A and Gdt vanish on

ot 0 ,

In the limit as e --~ O, the left-hand side equals ~ hand side equals Ge (q,,pr).

t') while the righto

726

LISA TRAYNOR

GAFA

4.6 C o n s t r u c t i o n o f a g e n e r a t i n g f u n c t i o n f o r Fg,a ("Chekanov's formula", cf. [C2]). Let S0(q, ~) be a generating function quadratic at infinity for Fh and let F(p, Q) be a compactly supported generating function for F~ (see (4.4)). T h e n for S( Q; p, q, ~) := So(q,~) + F(p, Q ) + p( Q - q) ,

Es =

{(Q , p, q, r ) : _-~p_=_q _- _Q, _-~q OSo O=Fp, --~OSo= o }

---- ( O , P , q , ~ ) : (q,P) E Fh, ~ p (P Q) = q - O and

Q} ={(q =

OF

OF

- -~p (p, Q), ~--~ (p, Q) p ) : (q,p) c

+p)

:

(q,p)

E

Fa}

} = Y( h) 9

Since F(p, Q) has compact support and So(q, ~) is quadratic at infinity, it is easy to check that outside a compact set in each fiber, S differs from a quadratic only by the lower order pQ term. There t h e n exists a fiber preserving diffeomorphism 4) so that S o 4) is quadratic at infinity. For example, consider 4 ) I ( Q , P , q, ~) = (Q,p, Q - q, ~). T h e n S o 4)1 agrees with a quadratic outside a compact set in each fiber. There exists another fiber preserving diffeomorphism q)2 so that outside a compact set, S o 4)1 o 4)2 does not vary from fiber to fiber, o With the above formula, it is sometimes possible to prove t h a t properties of lagrangians that are graphs over the zero section hold for more general lagrangians. LEMMA 4.7. Let go ~ ~'~0(R2n) 9 For r E D~0(R2~), gl := r

o go o r

Let go, gl be the associated symplectic diffeomorphisms of T* A. H go is sufficiently close to the identity, there exist Fo, F1 : A • A ---, R so that F~o = FdFo, F~I -- FdF1, and an isotopy Ot of ~ • A, t E [0, 1], 00 --- id, 0 F1 = Fo.

Proof: Let L0 := F~0, L1 := F ~ . There exists Ct E 7/~ t E [0,1], such that Co = id, r162 = r162 Ct induces gt E 9/~ 2~) and corresponding Lt C T * ( A x A). It is not hard to check t h a t there exists a hamiltonian isotopy 5t : T * ( A x A) --, T * ( A • ~ ) such that St(L0) = Lt, 6t(z) = z for all z E O ~ • and for z outside some compact set. Assuming

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g is sufficiently close to id, it follows that there exist Ft : / k x / X --+ R such t h a t Lt = Fag,. Next a Moser argument will be applied to construct 0t such t h a t F t o 0t = F0. If Xt is an integrable vector field such that

OFt

dFt (Xt) + 0---[- = O, then X t integrates to Ot such that d ( Ft o Ot) = O, Oo = id. To show the existence of Xt, first notice that if Gt is the compactly supported hamiltonian of the isotopy 5t then, as remarked in 4.5,

Ot (q) = Gt

q,--~q (q)

9

Since 5, fixes all points of O,

O OGt (q, s p ) p d s = at(q,p)=at(q,O)+~o 1 Nat(q, sp)ds=O+~o I -~p

(Xt(q,p),p)

where J(t is a smooth, compactly supported vector field and < , ) denotes the standard inner product. (This is sometimes referred to as Hadamard's lemma.) In particular,

--~q ] =aFt

Oq ] ]

and thus X , ( q ) : = - ) ~ , (q, oF,~ is the desired vector field. Oq I PROPOSITION 4.8. Given a smooth family ht E ~/~ t E [0,1], and r E D~o (R2~), consider h~ := r 1 6 2 -1 . There exists a smooth 1-parameter family of generating functions St : R 2n x R N ~ It for Fh, and a smooth 1-parameter family of diffeomorphisms Ot of R 2'* x [t N so that St o Ot is a generating function for Phi. Both St and St o Ot are asymptotically quadratic at intinity, for all t E [0, 1].

Proof: This will be proved by induction using L e m m a 4.7 and the formula in w Assume there exists a generating function So(q,() for Fho and a diffeomorphism r0 such that So o TO(q, () is a generating function for Fh~. By L e m m a 4.7, this is true when h0 is sufficiently close to the identity. Next assume that gt := ht o ho 1 and g~ := r o 9t o r t E [0, 1] are sufficiently close to the identity, t E [0, 1]. By L e m m a 4.7, there exists a function Ft and a diffeomorphism 0t such that F0 = Pdr,, F0' = Fa(F, o0,). Then by formula (4.6),

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St(Q;p,q,~) := So(q, ~) + Ft(p,Q) + p(Q - q) S~(Q;p,q,~) := St oOt where Ot(Q,p,q,~):= (Ot(p,Q),To(q,~)) are generating functions for Fh,, Fhi, respectively, and the proposition follOWS.

0

5. Homology Groups for an Open Set Homology groups will be associated to open sets by using a partial ordering on 7-/~ that induces a h o m o m o r p h i s m between the corresponding homology groups.

DEFINITION 5.1 (cf.

[FH, 3.3], IV, 4.10]):

Let hi, hj E 7-E~

hi -.< hj ~. ~ hj o h~ 1 is the time-l-flow of Ht(x) >_0. Remark 5.2: hi -~ hj implies hi ~v hj where "~v denotes Viterbo's partial ordering ([V]). Thus it is easy to check that --< is a partial ordering. Note that 7-/~ C 7-/~ 2n) is directed with respect to - 0. T h e result then follows since, as remarked in w O F (p,Q) = Gt p,Q, Op ' aQ ]

Vol.4, 1994

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HOMOLOGY VIA GENERATING

FUNCTIONS

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DEFINITION 5.4: For an open set U C R 2" and c 9 R, define ~ ~ C 7-/~ to be those hamiltonian diffeomorphisms such that the associated generating function does not have c as a critical value. Then 7-l~ := ~~ n ~(U), ~~ := ~ ~ For a,b # 0, ~ ~ # ~ and is directed with respect to -~. If Si < Sj the inclusion E~ C E~, -cx~ < c _< oc, induces a homomorphism

a a,bl(h ) ( I: ~ k(o,bj ( h i ) ,

hi-~ hj ,

o R2n )" h i , h j E~/,,b(

(5.5)

Moreover it is easy to check that given hj, hi, hh E ,~.~0 a,bt(~2n~l, hh -'~ hi -.4 hj,

f~ o f j = fJh, fi i = id. Thus {G(~'b](hi)}t,.eno du) form an inversely directed family of groups. See, for example, [L] for background on inverse limits. DEFINITION 5.6: Let U C R 2~ be an open set. For - ~ < a < b _< ~ ,

a,b~O, G(~'bl(u) : = li2,G(~'b](hi) , =

{(Xl'X2'''')

" xi

hi 9 7-l~ 9

~(,,b] "-~k (hi)

,

xi

:

.fJ(xj)

,

Vj > - i}.

Remark 5. 7: An important ingredient in computing the homology groups is to make a "good choice" of a sequence of hamiltonians hj E ~~ ). More precisely, the homology groups can be calculated by any sequence hi -~ h2 -~ 9 .. such that the associated hamiltonian functions get arbitrarily large. This guarantees that the sequence is unbounded in the sense that for any g E 7-f~ there exists j such that g -~ hj. By careful choices, it will often be easy to understand G(k"'bl(hj) and the homomorphism in (5.5). o

Remark 5.8: of R 2n.

The homology groups are defined for unbounded subsets o

6. Functorial Properties of the H o m o l o g y Groups In calculations and applications of gf-homology, it will be important to have various functorial properties. Most of these definitions and properties have analogs in F-H homology (compare [FH]). Throughout this section, U, If, W C R 2'~ will denote open sets, k will denote an integer, and a, b will be non-zero, -cx~ < a < b < c~. T H E O R E M 6.1 (Symplectic Invariance). Let U C R 2'~, r E :D~o(R2n). Then there is an isomorphism r G(a'H(V(U)) ---* G(a'b](u).

730

LISA T R A Y N O R

GAFA

Proof: If (hi), hi 9 ~~ is an unbounded, ordered sequence, h} := o hi o r 9 7Y~ form an unbounded, ordered sequence. Proposition 4.8 implies ~(-,bl 9"k (hi) is isomorphic to G("'b](h'~k ~ i J, for all i, and that these r

isomorphisms commute with the homomorphisms from (5.5). inverse limits are isomorphic.

Thus the D

T H E O R E M 6.2 (Monotonicity). If U C V then there exists a restriction homomorphism i*: G(k" ,bl(v ) ~ G("'bl (U).

Proof: Given an unbounded, ordered sequence of hi 9 ~ ] b ( U ) , there exists an unbounded, ordered sequence of gi 9 ~~ such that hi -~ gi. The homomorphism from (5.5) then induces a homomorphism between the inverse limits. DEFINITION 6.3 (Induced Homomorphism): If ~b : U ---* V, ~b 9 D~0(R2"), there exists an induced homomorphism r = r oi*: G(k"'bl(v) --+ G(a'bl(u). One of the powerful properties of symplectic homology is that it provides a means for studying symplectic isotopies. To prove the isotopy invariance property, the following notions will be used.

DEFINITION 6.4: For hi, h29176 d(hl, h2):=infshi,sh~ 211Sh~--Sh21l where Sh~, Sh2 : It 2n x R g ~ [t are generating functions for Fh~, Fh2 and ][ ]l denotes the sup norm. By the constructions in w it follows that there exist generating fimctions Sh~, Sh2 that agree outside a compact set of R 2~ x il N and thus d(hl, h2) < oo. DEFINITION 6.5 (Critical Gap): For c, d 9 R, let Xh[c, d] denote the critical values of Sh in [c, d]. Given h 9 7-{~ define the (endpoint) critical gap, g(h,(a,b]) > O, by

g(h,(a,b]) = sup { 7 : Xh[a-- 7, a + 71 = 0 and X h [ b - "),,b+ ~/1 = 0} when b < ~z

g(h,(a, oc]) = sup {'y: Xh[a--'~,a + 7] = 0 } . Given hi,h2 e ~.~0 .,b~[R2n'~,, g(hl,h2; ( a , b ] ) : = min{g(h,,(a, bl),g(h2,(a,b]) }. PROPOSITION

6.6

(Stability). If hi,h2

7-Ia,b\ 0 (R 2'~1, d(hl,h2)

E

9(ha,h2;(a,b]), then C("'b](hl) is isomorphic to


0,

=--c_ i}

where hj form an unbounded, ordered family in 7-l~

and fJ

: G(,a'bl(hj)

--~ G(,a'b](hi) is the homomorphism from (5.5) induced by inclusion. Since for all i, G(,a'bl(hi) is either 0 or Z2, it follows

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LISA TRAYNOR

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t h a t G(~'b](E(cq,..., an)) has at most two elements:

G(,a'b](E((~l,...,~,)) C { ( 0 , 0 , 0 , . . . ) , ( 0 , 0 , . . . , 1 , 1 , . . . ) }

.

f~(d,-c,d,]/~t_

It is possible for, say, ~-'2n ( ~ t ~ l , . . . , an)), dl < e, to be non-zero since from w it is known that for an u n b o u n d e d sequence of hi, l i m c l ( j ) = dl f,( dl -e,dl ][~~- l~^1 , . . . , ~ , ) ) and G(,~l-~'a'](hj) ~ O, for all j >> 1. ~'2n will be isomorphic to 12 if

G(,dl-~,dd(hj) ,__ (~,,~(a,-~,d~lt~,t~j+l) is an isomorphism, for all j >> 1. In general, to understand these induced homomorphisms, it suffices to understand

(h )

(9 )

where g, h are the hamiltonian diffeomorphisms corresponding to a, p E ~ , ! a > p, a'(0) < if(0) < 0. Suppose p I[0,~] -- c. There exists a 1-parameter family of v~ E ~', A E [0, 2], satisfying the following conditions:

to=p,

r2=a,

d ~-~r~>_0;

VA E [0, I],

z15~< 5,

VAE[1,2],

30 cdA ) > ... > el(A) > Coo(A) = 0, for all A E [1,2]. For each j, the corresponding path of critical values A~-~ cj(A), A e [1,2] are continuous, non-decreasing functions bounded above by the jth distinct capacity dj. These critical values will get arbitrarily close to dj for an appropriate choice of a. Suppose a(A), b(A) satisfy cj(A) < a(A) < Cj+l(A), ck(A) < b(A) < Ck+l(A). Critical non-crossings Proposition 6.7 then implies that there is an isomorphism

G(aO)'b(i)](hi) *-- a(a(;~)'b(;~)](h),), A e [1,21 and the result easily follows. This completes the proof of Theorem 7.1. o Remark 7.6: Let D2(1) and Bd(1) be the open 2-disc and 4-ball of radii 1. Notice that the unbounded, open sets D2(1) x R 4 and Bd(1) x R 2 can be thought of as E(1, o,oo) and E(1, 1, co), respectively. Calculations as above show that, for e < 7r, G('-~M(D2(1) • Rd)

{12, 0,

=

9 { •

12, 0,

=

,=6, 7 otherwise ; *=6, 9 otherwise .

8. Homology Groups of Shells Consider two open ellipsoids E ( a l , . . . , an) C E(/31,...,/3n), ai < /3i, as defined in w Then consider the corresponding "shell": the open set that is the interior of the set difference of these ellipsoids, g2~(~,~) := E ( / 3 1 , . . . , ~ ) -

E(al,...,a~)

9

In the terminology of Eliashberg and Gromov ([E1Gr]), these are symplectically non-convex subsets of (R 2n, w0). Many homology groups of these shells can be calculated using arguments similar to those in w For simplification of notation, the homology groups are calculated for B(a, t3), the difference of two 4-dimensional balls: +

+

c R

.

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T H E O R E M 8.1. (a) For k E N , O < e ~ < T r a , O < e z < ~ r f l ,

G(-k~r a-c~ ,-klr a] .

(B(a,~)) = a ( kr fl-E a ,kr fl]

12,

*=-4k,

o,

otherwise

{19,

-4k+3

, = 4k, 4k + 3

otherwise

(B(a,z)) =

o,

G(,-,~,ea] (B(a, fl)) =

Z2@Z2,

*=0

O,

otherwise.

If c ~ { - k r a , kTrfl, 0: k E N} then there exists ec such that a?-"~ (b) F o r k >

=o

v,, w < ~.

1, a < O < b < o % G(ka,b]( ~ ( a , ~ ) ) = _ k(2_(a,b] ( B 4 0 ) )

where B4(fl) denotes the open 4-dimensional ball of radius V~. Proof: Both (a) and (b) can be proved using sequences of h e 7-l~ fl)) corresponding to functions of the form H = p o q where q(xl, Yl, x2, Y2) = y := { p : [0, oo) -~ R: m.-

a + fl 2 '

pllo,~M~,~) = 0 ; p I [(,,,m) > 0

,

P'l(m,~)

< 0 9 ,

961 : a < 61 < 62 < m < 53 < 54 < fl

P"[(~,~)u(,h,~) > 0 , p'It~,,~j - c , ,

P"i(~,,h) < 0

P'l[~,,~,1 - c~,

Cl,C~ r ~-z}.

There exists a sequence pi e ~ with limSi(i) = a, lim54(i) = fl, limci(i) = 0% limc2(i) = - o c so that the associated hi 9 :H~ fl)) form an unbounded sequence. Let

C(p) := { a:~,m,b:~ : p ' ( a ~ ) = jTr , p"(a +) > O, p"(a-~) < O, p'(b~) = - j r ,

p'(b +) > O, p'(b;) < O, j 9

N}.

For each point of C(p) there is a critical submanifold diffeomorphic to S 3. Let A~,M,B t C E , a~,cm, flt 9 R (8.2) denote the critical submanifolds corresponding to {q = a~ }, {q = m}, {q = b~}, respectively, and their corresponding critical values. For a sequence p/ as described above, l i m a + ( / ) = -jTra, limfl+(i) = jTrfl, l i m a f ( i ) = limfl~-(i) = limcm(i) = oo.

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LEMMA 8.3. M, A~, B~ are non-degenerate critical submanifolds with i n d ( M ) = 1, ind(A + ) = -42, i n d ( A ; ) = - 4 2 + 1, ind(B + ) = 42, i n d ( B ; ) = 4 j + 1.

Proof: The calculations in the proof of L e m m a 7.4.1 prove the index statements for A j+, Bj+. The index of A y (respectively Bj-) will be one more t h a n the index of A + (B +) due to the comment about the convexity of p at the end of the proof of L e m m a 7.4.1. Also note that even though p'(m) -= O, since p"(m) ~ O, the calculations in the proof of 7.4.1 show that M is non-degenerate, i n d ( M ) = 1. [] Proof of 8.1(a): In addition to A f , M , B ? , there exist two disjoint critical submanifolds with critical value (} corresponding to {q < a}, {q >/3}. Using arguments and the notation of w for e sufficiently small, 2,

b.(-e,e] =

.=0

0 , otherwise .

Given e~ and e~ as described in the statement, there exists an unbounded sequence of hi E 7-/~ such that 0 is the only critical value of Sh, in (--e~,eZ]. By critical non-crossing (6.7), the statement about the 0dimensional homology group follows. To show, for example, that when 0 < a < ~r/3 < b < 2v/3, G(.a'bl(B(a,/3)) = /2 when * -- 4, 7 and equals 0 for o t h e r . , choose an ordered sequence of hi corresponding to pi E 3v such that for each i, of all the critical values O, af(i),cm(i),/3f(i), only /31+(i) e (a,b]. (Direct calculations show that this is possible.) Bott-Morse theory implies G(a'bl(h~) = 1-2, when 9 = 4, 7 and equals 0 for other *. By critical non-crossings (6.7), the homomorphisms from the orderings are the identity. Thus the statement about G!~ follows. I f c ~ {-k~ra, 0, k~r/3 : k 6 hl} then there exists an ec > 0 and an unbounded sequence of hi so that, for each i, there exist no critical values of Sh~ in [c - ec, c + e~] and thus all homology groups vanish, as claimed, n Note that although the critical submanifolds have distinct indices, in contrast to the calculations in w now a given Betti number might be nonzero in intervals around different critical values. For example, if there exists b+ E C(p), for e sufficiently small, bs(/3~--e,/31+~]=l,

bs(/3+-e,/3+Te]=l.

For this reason, calculating homology groups for all a, b is more difficult than in the case of an ellipsoid. However, homology groups for k _> 1, b < oo, can be easily calculated.

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Proof of(b): Since 0 < b < oo, there existsg E NU{0}, g~r/3 < b < (g+l)Tr/3. Consider p E ~- such that - ( n + 1)Tr < m i n p ~ < -nTr, nTr < m a x p ~ < (n + 1)r, n E N, n >_ g + 1, and the critical values as defined in (8.2) satisfy 8 + < l,