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RING STRUCTURE OF THE FLOER COHOMOLOGY OF Σ×S 1
Topology Vol. 38, No. 3, pp. 517—528, 1999 1999 Elsevier Science Ltd All rights reserved. Printed in Great Britain 0040-9383/99 $ — see front matter
PII: S0040-9383(98)00028-7
RING STRUCTURE OF THE FLOER COHOMOLOGY OF &;2 VICENTE MUN OZR (Received 31 October 1997; in revised form 12 May 1998) We give a presentation for the Floer cohomology ring HF*(&;2), where & is a Riemann surface of genus g*1, which coincides with the conjectural presentation for the quantum cohomology ring of the moduli space of odd degree rank two stable vector bundles on & with fixed determinant. We study the spectrum of the action of H (&) * on HF*(&;2) and prove a physical assumption made in [1]. 1999 Elsevier Science Ltd. All rights reserved.
1. INTRODUCTION
Let &"& be a Riemann surface of genus g*1 and let N denote the space of flat E E SO(3)-connections with nontrivial second Stiefel—Whitney class w modulo the gauge transformations that can be lifted to Sº(2). This is a smooth symplectic manifold of dimension 6g!6 and it is the space considered in [6, 4, Section 3]. Alternatively, we can consider & as a smooth complex curve of genus g and N as the moduli space of odd degree E rank two stable vector bundles on & with fixed determinant, which is a smooth complex variety of complex dimension 3g!3. The symplectic deformation class of N only depends E on the genus g and not on the particular complex structure on &. We note that, following the conventions of [8], the moduli space of anti-self-dual instantons on &;"/ with charge i"0 is again isomorphic to N . E We consider the following rings associated with the Riemann surface (we will always use "-coefficients): E QH*(N ) is the quantum cohomology of N (see [13]). This is well-defined since N is E E E a positive symplectic manifold. As vector spaces, QH*(N )"H*(N ), but the multiE E plicative structure is different. The minimal Chern number of N is 2, so QH*(N ) is E E 9/49-graded (the grading comes from reducing mod 4 the 9-grading of H*(N )). The E ring structure of QH*(N ), called quantum multiplication, is a deformation of the E usual cup product for H*(N ). It is associative and graded commutative. We remark E that we do not introduce Novikov rings to define QH*(N ) as in [13, Section 8] E (otherwise said, as H(N )"9, we should introduce an extra variable q of degree 4, E then we equate q"1). E HF (N ) is the symplectic Floer homology of N (with the symplectomorphism * E E
"id). The symplectic manifold N is connected, simply connected and n (N )"9 E E (see [4, introduction]), so the groups HF (N ) are well-defined [5]. They are * E 9/49-graded. HF (N ) is endowed with the pair of pants product [12], which is an * E associative and graded commutative ring structure. The symplectic Floer cohomology of N , HF* (N ), is defined as the dual of the symplectic Floer homology. There E E is a Poincare´ duality [12, remark 2.4] and a pairing 1,2.
R Supported by a grant from Ministerio de Educacio´n y Cultura of Spain.
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E HF (&;2) is the instanton Floer homology of the three manifold ½"&;2 for * the SO(3)-bundle with second Stiefel—Whitney class w "PD[2]3H(½; 9/29). This is defined in [6] and is 9/49-graded. We introduce a multiplication on HF (½) * using a suitable four-dimensional cobordism [14, section 5]. Let X be the four manifold given as a pair of pants times &, which is a cobordism between ½\½ and ½. This gives a map HF (½) HF (½)PHF (½), which is an associative and graded * * * commutative ring structure on HF (½). Again, the instanton Floer cohomology * of ½, HF*(½), is the dual of HF (½). There is a natural isomorphism HF*(½): * HF (!½), where !½ denotes ½ with reversed orientation. As ½"&;2 admits an * orientation reversing self-diffeomorphism, we can identify HF*(!½):HF*(½), so HF*(½):HF (½). This is called the Poincare´ duality isomorphism in Floer * (co)homology. The natural pairing HF*(½) HF (½)P" produces then a pairing * 1,2: HF*(½) HF*(½)P" (see [2], [3]). We will denote HF*"HF*(&;2). E THEOREM 1. ¹here are natural isomorphisms of vector spaces QH*(N ):HF* (N ):HF*(&;2). E E Moreover the first isomorphism respects the ring structures.
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Proof. The second isomorphism is due to Dostoglou and Salamon [4, Theorem 10.1]. It is the particular case where one considers "id : &P&, in which the mapping torus of is &;2 and the SO(3)-bundle has w "PD[2]. The first isomorphism is a standard result obtained by Floer [5]. In [12, Theorem 5.1] it is proved that the first isomorphism intertwines the products. ) CONJECTURE 2. ¹he second isomorphism in (1) is a ring isomorphism. D. Salamon has informed the author that the adiabatic limit techniques of [4] can be extended to give a proof of Conjecture 2. He gave a lecture at Maryland in Spring 1994 on this issue. Many implications of this result to four-dimensional topology were given by Donaldson [3]. The author followed this program in [9] for small genus. Later he obtained very nice results on the behaviour of Donaldson invariants under the operation of connected sum along a Riemann surface [10, 11], exploiting only the isomorphism (1) as vector spaces. Using physical methods, Vafa et al. [1] find a set of generators and relations for QH*(N ). There are two main assumptions in their argument. The first one is Conjecture 2. E The second one is that the spectrum of the action of H (&) on HF*(&;2) can be read off * from the Donaldson invariants of &;3. Later, Siebert and Tian [15] claimed to have found a mathematical proof for the presentation of QH*(N ) given in [1] but they could not yet finish their program. In this E paper, we prove that the set of generators and relations given in [1] is a presentation for HF*(&;2) (Theorem 16), following a method inspired in Siebert and Tian [15]. This together with completion of the work [15] will produce a proof of Conjecture 2 (although in a rather indirect way). We also prove the physical assumption on the spectrum of HF*(&;2) in [1] (cf. Proposition 20). We leave the implications of Theorem 16 to Donaldson invariants of four-manifolds (mostly in the case b>"1) for future work.
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2. RING STRUCTURE OF H*(N ) E
Let us recall the known description of the homology of N [7, 17]. Let UP&;N E E be the universal bundle and consider the Ku¨nneth decomposition as in [7] c (End U)"2a[&]#4t!b C with t" c c , where +c , 2 , c , is a symplectic basis of H (&;9) with c c "[&] for G G E G G>E 1)i)g, and +cC, is the dual basis of H(&). In terms of the map k: H (&)PH\*(N ), G * E given by k(a)"! p (gu) /a (here guP&;N is the associated universal SO(3)-bundle, E and p (gu)3H(&;N ) its first Pontrjagin class), we have E a"2k(&)3H
c "k(c )3H, 1)i)2g G G b"!4k(x)3H
where x3H (&) is the class of the point, and HG"HG(N ). These elements generate H*(N ) E E as a ring [7, 18]. So there is a basis + f , S for H*(N ) with elements of the form E Q QZ f "aLbKc 2c GP Q G for a finite set S of multi-indices of the form s"(n, m; i , 2 , i ), n, m*0, r*0, P 1)i (2(i )2g. The mapping class group Diff (&) acts on H*(N ), with the action P E factoring through the action of Sp (2g, 9) on +c ,. The invariant part, H* (N ), is generated G ' E by a, b and c"!2 E c c . Then G G G>E H* (N )""[a, b, c]/I , (2) ' E E where I is the ideal of relations satisfied by a, b and c. Here deg(a)"2, deg(b)"4, E deg(c)"6. Actually, a basis for H* (N ) is given by the monomials aLbKcP, n#m#r(g ' E (see [17]). For 0)k)g, the primitive component of "IH is "I H"ker(cE\I> : "IHP"E\I>H). Then the Sp (2g, 9)-decomposition of H*(N ) is [7] E E H*(N )"@"I H "[a, b, c]/I . E E\I I PROPOSITION 3 (Siebert and Tian [17]). For g"1, let q"a, q"b, q"c. Define recursively, for g*1,
q "aq#gq E> E E 2g q "bq# q. E> E g#1 E q "cq E> E ¹hen I "(q, q, q ), for all g*1. E E E E Proof. This is the form of the relations given in [17, Proposition 3.2].
)
3. RING STRUCTURE OF HF* E
With the aid of the basis + f , S for H*(N ) we are going to construct a basis Q QZ E for HF*"HF*(&;2) to understand its ring structure. We need to use the gluing E
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properties of the Floer homology of a three manifold. Put ½"&;2 and let w "PD[2]3H(½; 9/29). Let us recall [11] that for a 4-manifold X we define (X)"Sym*(H (X) H (X)) d*H (X). Let us state the result that we shall use. PROPOSITION 4 [2, 3, 9]. For any smooth oriented four-manifold X with boundary *X"½, any w3H(X; 9) with w" "PD[2] in H(½; 9/29), and any z3 (X), we have 7 defined a relative invariant U(X, z)3HF (½). ¹hese relative invariants enjoy the following * gluing property, suppose X"X 6 X is a closed four-manifold split into two open four 7 manifolds X with *X "½, *X "!½, and w3H(X; 9) satisfying w" "PD[2] in G 7 H(½; 9/29). Put w "w" . ¹hen for z 3 (X ), i"1, 2, we have G 6G G G DU (z z )"1 U (X , z ), U (XM , z )2 6
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where DU "DU#DU> (DU is the Donaldson invariant of X for w, see also [9, 10]). 6 6 6 6 ¼hen b>"1, the invariants are calculated for a long neck, i.e. we refer to the invariants defined by [&]. Consider the manifold A"&;D, with boundary ½"&;2, and let *"pt;DLA be the horizontal slice with **"2. Put w"PD[*]3H(A; 9). Clearly (A)" (&)" Sym*(H (&) H (&)) d*H (&). For every s3S, f "aLbKc 2c , define Q G GP z "&LxKc 2c 3 (&) Q G GP e " U(A, z )3HF*(½)"HF* Q Q E (here we identify Floer homology and Floer cohomology through Poincare´ duality). Then +e , S is a basis for HF*. This is a consequence of [11, Lemma 21]. The product E Q QZ HF* HF*PHF* is given by U(A, z ) U(A, z )" U(A, z z ). Then U(A, 1) defines the E E E Q QY Q QY neutral element of the product. As a consequence, the following elements are generators of HF*, E
a"2 U(A, &)3HF E t " U(A, c )3HF, 0)i)2g. G G E b"!4 U(A, x)3HF E
Note that there is an obvious epimorphism of rings
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(&) HF*. E
THEOREM 5. Denote by * the product induced in H*(N ) by the product in HF* under the E E K isomorphism H*(N )P HF* given by f >e , s3S. ¹hen * is a deformation of the cupE E Q Q product graded modulo 4, i.e. for f 3HG(N ), f 3HH(N ), it is f * f " * ' ( f , f ), where E E P P ' 3HG>H\P(N ) and ' "f 6f . P E Proof. First, for s, s 3S, 1e , e 2"DU (z z )"0, Q QY ;"/ Q QY unless deg( f )#deg( f )"6g!6#4r, r*0, as these are the only possible dimensions Q QY for the moduli spaces of anti-self-dual connections on &;"/. Moreover, when deg( f )#deg( f )"6g!6, the moduli space is N , so 1e , e 2"!1 f f ,[N ]2" Q QY E Q QY Q QY E !1 f , f 2 (the minus sign is due to the different convention orientation for Donaldson Q QY invariants).
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Now let f , f be basic elements of degrees i and j respectively. Put f f " c f and Q QY Q QY R R de f * f " d f . This means that e e " d e . Write e e " g , where g " K K D R R Q QY R R Q QY R R Q QY K Kthat are the homogeneous parts. Put g' " d f . Let M be the maximum m such K DK R R g O0. Then there is f of degree 6g!6!M such that 1g' , f 2O0. Since K QYY + QYY 0O!1g' , f 2"1g , e 2"1e e , e 2"DU (z z z ), + Q Q QY Q ;"/ Q QY Q + Q it is deg( f )#deg( f )#deg( f )*6g!6, i.e. M)i#j. Now for m"i#j, any f of Q QY Q Q degree 6g!6!m, it is 1g' , f 2"!DU ;"/ (z z z )"1 f f f , [N ]2"1 f f , f 2. So K Q Q QY Q Q QY Q E Q QY Q g' "f f . G>H Q QY Finally, 1e e , e 2"D U (z z z )"0, whenever deg( f )#deg( f )#deg( f )I Q QY QYY ;"/ Q QY Q Q QY Q 6g!6 (mod 4), so g' "0 unless M,i#j (mod 4). ) K Remark 6. The isomorphism in Theorem 5 is only an isomorphism of H*(N ) and HF* E E as vector spaces (dependent on the chosen basis + f , S ). In general, this is not a ring Q QZ isomorphism (see Example 22), so we do not expect the isomorphism in (1) to coincide with that of Theorem 5. There is again an action of Diff (&) on HF* factoring through an action of Sp (2g, 9) E on +t ,. The invariant part (HF* ) is generated by a, b and c"!2 E U(A, c c ). The G E ' G G>E G epimorphism "[a, b, c] (HF* ) , z> U(A, z), allows us to write E ' HF*(&;2) ""[a, b, c]/J , ' E
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where J is the ideal of relations of a, b and c. Now deg(a)"2, deg(b)"4, deg(c)"6, but E J is not a homogeneous ideal. E LEMMA 7. Suppose cJ LJ , for all g*1. ¹hen we have the Sp (2g, 9)-decomposition E E> E HF*(&;2)"@ "I H "[a, b, c]/J . E\I I Proof. The isomorphisms in Theorem 1 respect the Sp (2g, 9)-action and hence induce isomorphisms on the invariant parts. Then dim(HF* ) "dim H* (N ), for all g*1. Now E ' ' E the lemma is a consequence of the argument in the proof of [7, Proposition 2.2] and the discussion preceding it. )
4. A PRESENTATION FOR (HF*) E '
Theorem 5 and the arguments in [17, Section 2] imply that we can deform the relations of H* (NE ) to get a presentation for (HF* E )'. More explicitly, ' LEMMA 8. It is (HF* E )'""[a, b, c]/(RE , RE , RE ), where RGE"qGE#lower order terms of degrees deg qGE!4r, r'0, as polynomials in "[a, b, c] (qGE are defined in Proposition 3). Proof. Suppose first that g*2. Granted Theorem 5, [17, Theorem 2.2] implies that JE"(RE , RE , RE ), where RGE is qGE expressed in terms of a, b and c and the multiplication of K HF*E . Now we note that under the isomorphism H*(NE )P HF*E of Theorem 5, a>a, b>b, c>c (it can always be arranged so that these elements are in the basis, as g*2, see [16, Proposition 4.2]). So RGE is equal to qGE plus lower-order terms. The case g"1 is computed directly in Lemma 11. )
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As pointed out in [15], the key result to find the generators of JE is the following inclusion of ideals. LEMMA 9. JE>LJE, for all g*1. Proof. We shall prove this through an excision argument for Donaldson invariants. Let &E be a Riemann surface of genus g and consider &E>LAE"&E;DLS"&E;"/ where &E> is given as &E with a trivial handle of genus 1 added internally. This means that we take a point p3&E and a small 4-ball B in S centered at p such that B5&E is a 2-ball in &E. Then we substitute B5&E with an embedded 2-torus in B with a small ball removed, whose boundary coincides with *(B5&E )L*B. This produces &E>. Then the map H* (&E> )P H* (&E ) induces (&E> )P (&E ) which sends (a, b, c) >(a, b, c). Let AE>"&E>;DLAE be a (small) tubular neighbourhood of &E> in S. Now we put S for the complement of the interior of AE> in S, so that S"S6 E>;2AE>. Recall that we have also the decomposition S"AE 6 E;2AE. Now let z3JE>. Then U(&E>;D, z)"0. So for any zQ3 (&E ), s3S, DU (z zQ )"1 U(&E>;D, z), U(S, zQ )2"0. 1 Therefore DU (z zQ )"1 U(&E;D, z), U(&E;D, zQ )2"0, for any s3S. So U(&E; 1 D, z)"0 and z3JE. THEOREM 10. ¹here are numbers cE>, dE>3" such that, for all g*1,
R "aR#gR E> E E 2g R "(b#c ) R# R. E> E> E g#1 E R "cR#d R E> E E> E
Proof. We follow almost literally the argument of Siebert and Tian [15, Proposition 3.2]. As R 3J LJ is a relation on degree 2g#2, it is a linear combination of aR and E> E> E E R. Looking at the leading terms (Proposition 3), we have R "aR#gR. AnalogE E> E E ously, R is a combination of aR, bR, aR and R. Only the term R has degree less E> E E E E E than 2g#4, so R "bR#((2g)/(g#1)) R#c R , for an unknown coefficient c . E> E E E> E E> In the same fashion, R is cR plus a linear combination of R and aR. Adding a suitable E> E E E multiple of R (which is always allowed without loss of generality), we have R " E> E> cq#d R. ) E E> E LEMMA 11. ¹he starting relations ( for g"1) are R"a, R"b!8 and R"c. Proof. HF* is of dimension 1, i.e. HF*"" (see [3, 9]). Let S be the K3 surface and fix an elliptic fibration for S, whose fibre is &"3. The Donaldson invariants are, for w3H(S; 9) with w ) &,1 (mod 2) (see [9]), DU (eR")"!e\/R" . 1 Then DU (1)"!1 and DU (&B)"0, for d'0. Also from [11, Remark 4], DU (x)"2. 1 1 1 Let S be the complement of an open tubular neighbourhood of & in S. Then U(S, 1)
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generates HF* and U(S, &)"0, U(S, x)"!2 U(S, 1) and U(S, c c )"0, i.e. a"0, b!8"0 and c"0 in HF* (recall (4)). ) PROPOSITION 12. For g*2, there exists a non-zero vector v3HF* such that E
av"
g even 4(g!1) (!1 v, g odd
4(g!1) v,
bv"(!1)E\8 v cv"0. Proof. We shall construct such a vector as the relative invariants of an open fourmanifold X with boundary *X"½"&;2, where the closed four-manifold X"X6 A is of simple type with b>'1 and b "0. For concreteness, let X be the 7 manifold C from [11, Definition 25]. We recall its construction. Let S denote the elliptic E E surface of geometric genus p "g!1 and with no multiple fibres. It contains a section E p which is a rational curve of self-intersection !g. Let F be the elliptic fibre. Then p#gF can be represented by an embedded Riemann surface & of genus g and self-intersection g. Blow-up S at g points in & to get B with an embedded Riemann surface & of genus g and E E E self-intersection zero. Then put X"C "B C B (the double of B along & ). By [11, E E E E E E Proposition 27], X is of simple type and #U (e?)"DU ((1#x/2) e?)"!2E\e/? e) ?# 6 6 (!1)E2E\e/? e\) ?, where K3H(X; 9) satisfies K)& "2g!2 (w3H(C ; 9) is a parE E ticular element, which we do not need to specify here). Let us suppose from now on that g is even, the other case being similar. By [11, Proposition 3], DU (e?)"!2E\e/? e) ?#(!1)E2E\e/? e\) ?. 6 Consider X the complement of a small open tubular neighbourhood of & in X, so that E X"X6 A. Then we set v" U(X, g!2)3HF*(&;2)"HF*. Let us prove that 7 E this is the required element. For any z "&LxKc 2c , it is [11, Remark 4], Q G GP
0, 1v, e 2"DU ((g!2) z )" Q 6 Q !2E\(2g!2)L>2K,
r'0 r"0.
Then 1av, e 2"1 U(X, 2&(g!2)), U(A, z )2"DU ((g!2) 2& z )"(4g!4) Q Q 6 Q 1v, e 2, for all s3S. Then av"(4g!4)v. Analogously, cv"0 and bv"!8v. ) Q Notation 13. We set R"1, R"0 and R"0. THEOREM 14. For all g*1, c "(!1)E8 and d "0. E E Proof. The result is true for g"1 by Lemma 11 and Notation 13. Suppose it is true for 1)r)g, and let us prove it for g#1. By Proposition 12, there exists v3HF* with E> bv"(!1)E8 v, cv"0 and av"4g v if g is odd and av"4g(!1 v if g is even. In first place, cv"0 implies Rv"0, for 1)r)g. In second place, bv"(!1)E8 v P implies Rv"(b#(!1)E8) R v"(!1)E16R v E E\ E\ R v"(b#(!1)E\8) R v"0 E\ E\
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R v"(!1)E16R v E\ E\ R v"0, E\ $ In the third place, Rv"aR v#(g!1)R v"aR v, R v"aR v, 2 Also E E\ E\ E\ E\ E\ R v"aR v#(g!2)R v"(a#(g!2)(!1)E16) R v. E\ E\ E\ E\ So finally,
(a#(!1)E16(g!2)2(a#(!1)E16)1)v, R v" E\ (a#(!1)E16(g!2))2(a#(!1)E16)2)av,
g odd g even.
As a conclusion R v"jv, with jO0, and E\ Rv"aR v E\ E Rv"(!1)E16R v E E\ Rv"0. E As v3HF* , we have R v"0, R v"0 and R v"0. Evaluate the equations from E> E> E> E> Theorem 10 on v to get c "(!1)E>8 and d "0. ) E> E> COROLLARY 15. ¼e have cJ LJ LJ , for all g*1. E E> E Proof. The second inclusion is Lemma 9. For the first inclusion, note that cR"R 3J by the third equation in Theorem 10. Then multiplying the first two E E> E> equations in Theorem 10 we get that cR, cR3J . ) E E E> Using this corollary in Lemma 7, we have finally proved that THEOREM 16. ¹he Floer cohomology of &;2, for &"& a Riemann surface of genus g, E has a presentation E HF*(&;2)"@ "I H "[a, b, c]/J . E\I I where J "(R , R , R ) and RG are defined recursively by setting R"1, R"0, R"0 and P P P P P putting for all r*0
R "aR#rR P> P P
2r R "(b#(!1)P>8) R# R P> P r#1 P R "cR. P> P
Remark 17. The presentation obtained for HF* is the conjectural presentation for E QH*(N ) (see [15]). E COROLLARY 18. ker(c : (HF* ) P(HF* ) )"J /J L"[a, b, c]/J "(HF* ) . E ' E ' E\ E E E ' Proof. By Corollary 15, c factors as A "[a, b, c]/J . "[a, b, c]/J "[a, b, c]/J ) E E\ E
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The second map is a monomorphism since a?b@cA, a#b#c(g!1, form a basis for "[a, b, c]/J , and their image under c are linearly independent in "[a, b, c]/J . The E\ E corollary follows. ) For any F3"[a, b, c] define the expectation value by 1F2 "1F, 12 *, where 13HF* E E &$E is the unit element. Therefore 1F , F 2 *"1F F 2 . &$E E COROLLARY 19. For any F3"[a, b, c], 1cF2 "2g1F2 . E E\ Proof. By Corollary 15, the above formula holds for any F3J , as both sides are zero. E\ So it is enough to check it for a set of elements generating HF* , i.e. for F "a?b@cA, E\ ?@A a#b#c(g!1. If (a, b, c) O(0, 0, g!2) , it is 1F 2 "0 and 1cF 2 "0 by degree ?@A E\ ?@A E reasons. Now 1cE\2 "!1cE\, [N ]2, hence the corollary follows from E\ E\ 1cE\, [N ]2"2g1cE\, [N ]2 (see [18]). ) E E\ 5. LOCAL RING DECOMPOSITION OF HF*(&;2)
In [1] it is asserted that the only eigenvalues of the action of k(&), k(x) and k(c ) on G HF*(&;2) are the ones given by looking at the Donaldson invariants of the manifold X"&;3 i.e. if we denote by ¼LHF* the image U(X, (&)), where X"X6 A, E 7 then a, b and c act on ¼ and their eigenvalues are all the eigenvalues of their action on HF*. E The following result is a proof of this physical assertion. PROPOSITION 20. ¹he eigenvalues of (a, b, c) in (HF*) are (0, 8, 0), ($4, !8, 0), E ' ($8(!1, 8, 0), 2 , ($4(g!1)(!1E, (!1)E\8, 0). Proof. Put »"(HF*) . As cJ LJ , one has cE3J , i.e. cE"0 in », so the only E ' E\ E E eigenvalue of c is zero. To compute the eigenvalues of a, b we can restrict to »/c» (if p is a polynomial with p(a)"0 in »/c», then p(a) is a multiple of c in » and p(a)E"0 in »). Now the ideal of relations of » can be written as J "(f , f , f ), where f "1, f "0, E E E> E> \ f "0 and f "af #r(b#(!1)P8)f #2r(r!1)cf , for all r*0. This form of \ P> P P\ P\ the relations follows from rewriting as R"f E E 1 R" (f !af ) E E g E> 1 R" (f !af !(g#1)(b#(!1)E>8)f )). E 2g(g#1) E> E> E Therefore »/c»"" [a, b]/(fM , fM ) E E> where fM "1, fM "0, fM "afM #r(b#(!1)P8) fM , for r*0. From r(b# \ P> P P\ (!1)P8)fM "fM !afM we infer that (b#(!1)P8)fM 3(fM , fM ). Continuing in this P\ P> P P\ P P> way, (b#(!1)E8) (b#(!1)E\8)2(b!8)3(fM , fM ) E E>
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which implies that the only eigenvalues of b in »/c», and hence in », are $8. Let us study the eigenvalues of a for b"8, c"0. Again we only need to study »/(c, b!8) »""[a]/(fK , fK ), where now fK "1, fK "0, fK "afK #r(8#(!1)P8) fK , for E E> \ P> P P\ r*0. Then
fK "(a#(r!2)16)2(a#216)a, r even P fª "(a#(r!1)16)2(a#216)a, r odd P
from where the eigenvalues of a will be 0, $8(!1, $16(!1, 2 , $8[E\](!1. We leave the other case to the reader. ) Remark 21. As mentioned in [1], by the very definition of c"!2 U(A, c c ), it is G G>E cE>"0 in HF*, so the only eigenvalue of c is zero. E Proposition 20 says that (HF*) can be decomposed as a sum of local artinians rings E ' E\ (HF*) " @ R E ' E G G\E\
(6)
where R is a local artinian ring with maximal ideal m"(a!4i, b!(!1)G8, c) if i is odd, E G m"(a!4i(!1, b!(!1)G8, c) if i is even. Also HF* is decomposed as E E E\I\ E\ E\ G \ HF*"@ @ "I H R " @ @ "IH RE\I G. E\I G E I G\E\I\ G\E\ I
(7)
We recall from Lemma 11 that HF*""[a, b, c]/(a, b!8, c). Let us see the next cases. Example 22. For g"2, J "(a#b!8, a(b#8)#c, ac). In (HF*) , c"!a(b#8) ' and ca"0 yield a(b#8)"0. Now a"!(b!8) so (b!8) (b#8)"0 and a(a!16)"0. Also cJ LJ implies ca"c"c(b!8)"0. Finally (c#16a) (a!16)"!16c#16a(a!16)"!16(c#a(b#8))"0. All together proves "[a, b, c] "[a, b, c] "[a, b, c] (HF*) " . ' (a!4, b#8, c) (a, b!8, c#16a) (a#4, b#8, c) We want to remark that HF*P l QH*(N ) (see [12, Example 5.3] for a presentation of the latter ring). The isomorphism sends a>h , b>!4(h !1), c>4(h !h ), where h , h , h are the generators of QH, QH, QH, respectively. This was conjectured in [9, Conjec ture 1.22]. Example 23. For g"3, J "(a(a#b!8)#4(ab#8a#c), (b!8) (a#b!8)# ac, c(a#b!8)). Put »"(HF*) . Then ' »/c»""[a, b]/(a(a#b!8)#4(ab#8a), (b!8) (a#b!8)). In »/c», the first relation yields !5a(b!8)"a#64a and the second a(b!8) (a#b!8)"0. This implies a(a!16) (a#64)"0. Also (b!8)a(a!16)"0. Using (b!8)a"!(b!8), we get a(b!8) (b#8)"0. Therefore, in », a(a!16) (a#64) and a(b!8) (b#8) are multiples of c. As cJ LJ , we have ca(a!16)"0 and ca(b#8)"0 by Example 22. So a(a!16)(a#64)"0,
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a(a!16) (b!8) (b#8)"0 and a(b!8) (b#8)"0. It can be checked now that "[a, b, c] "[a, b, c] (HF*) " ' (a!8(!1, b!8, c) ((a!4), c#3(b#8), c!12(a!4))
"[a, b, c] "[a, b, c] (a, a(b!8), (b!8)! a, c#16a) ((a#4),c!3(b#8), c!12(a#4)) "[a, b, c]
. (a#8(!1, b!8, c)
6. CONJECTURE
We state the following conjecture, that first occurred to Paul Seidel and the author in mid-1996. CONJECTURE 24. ¹he decomposition in equation (7) is HF*:H*(s&) H*(s&) 2 H*(sE\&) E H*(sE\&) H*(sE\&) 2 H*(s&) where sG& is the ith symmetric product of &. Here H*(sG&) is isomorphic to the eigenspace of eigenvalues ($4(g!1!i) (!1E\G, (!1)E\\G8, 0). ¹he isomorphism respects only 9/29grading and is Diff (&)-equivariant. Simple computations establish that the dimensions of both vector spaces appearing in Conjecture 24 are the same, i.e. g2E. The Euler characteristic are both vanishing. Moreover the dimensions of the invariant parts coincide (E>). Examples 22 and 23 agree with the conjecture. A deeper reason for the above conjecture is the fact that HF* is the space for a gluing E theory of Donaldson invariants associated to the three manifold ½"&;2. The gluing theory of Seiberg—Witten invariants should be based on the Seiberg—Witten—Floer homology groups of &;2, which are indexed by a line bundle ¸ (the determinant line bundle of the spinA-structure on ½). The only possibilities are c (¸)"$2(g!1!i)PD [2], 0)i)g!1 (see [9, Section 6]). It is believed that the Seiberg-Witten-Floer groups for ¸ are isomorphic to H*(sG&). Acknowledgements—I am very grateful to Bernd Siebert and Gang Tian for providing me with a copy of [15] which was very enlightening. In particular, the proof of Theorem 10 is due entirely to them. Thanks to the organization of the CIME Course on Quantum Cohomology held in Cetraro (Italy, 1997) for inviting me. Also discussions with Paul Seidel were useful. Finally, thanks to the referee for useful remarks.
REFERENCES 1. Bershadsky, M., Johansen, A., Sadov, V. and Vafa, C., Topological reduction of 4D SYM to 2D p-models. Nuclear Physics, 1995, B448, 166. 2. Donaldson, S. K., On the work of Andreas Floer. Jahresber. Deutsch. Math. »erein,1993, 95, 103—120. 3. Donaldson, S. K., Floer homology and algebraic geometry. »ector Bundles in Algebraic Geometry, London Mathematical Society Lecture Notes Series, vol. 208, Cambridge University Press, Cambridge, 1995, pp. 119—138. 4. Dostoglou, S. and Salamon, D., Self-dual instantons and holomorphic curves. Annals of Mathematics, 1994, 139, 581—640.
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5. Floer, A., Symplectic fixed points and holomorphic spheres. Communications in Mathematical Physics, 1989, 120, 575—611. 6. Floer, A., Instanton homology and Dehn surgery. ¹he Floer Memorial »olume. Progress in Mathematics, 1994, 133, 77—97. 7. King, A. D. and Newstead, P. E., On the cohomology ring of the moduli space of rank 2 vector bundles on a curve. ¹opology, 1998, 37, 407—418. 8. Kronheimer, P. B. and Mrowka, T. S., Embedded surfaces and the structure of Donaldson’s polynomial invariants. Journal of Differential Geometry, 1995, 41, 573—734. 9. Mun oz, V., Gauge theory and complex manifolds. Oxford D. Phil. thesis, 1996. 10. Mun oz, V., Donaldson invariants for connected sums along Riemann surfaces of genus 2, dg-ga/9702004. 11. Mun oz, V., Gluing formulae for Donaldson invariants for connected sums along surfaces. Asian Journal of Mathematics, 1997, 1, 785—800. 12. Piunikhin, S., Salamon, D. and Schwarz, M., Symplectic Floer—Donaldson theory and quantum cohomology. In Contact and Symplectic Geometry, vol. 8, ed. C. B. Thomas, Publ. Newton Institute, Cambridge University Press, Cambridge, 1996, pp. 171—200. 13. Ruan Y. and Tian, G., A mathematical theory of quantum cohomology. Journal of Differential Geometry, 1995, 42, 259—367. 14. Salamon, D., Lagrangian intersections, 3-manifolds with boundary and the Atiyah—Floer conjecture. Proceedings of the International Congress of Mathematicians, vol. 1, Birkha¨user, Basel, 1994, pp. 526—536. 15. Siebert, B., An update on (small) quantum cohomology. Preprint, 1997. 16. Siebert, B. and Tian, G., On quantum cohomology rings of Fano manifolds and a formula of Vafa and Intriligator. Asian Journal of Mathematics, 1997, 1, 679—695. 17. Siebert, B. and Tian, G., Recursive relations for the cohomology ring of moduli spaces of stable bundles. Proceedings of 3rd Go( kova Geometry—¹opology Conference, 1994. 18. Thaddeus, M., Conformal field theory and the cohomology of the moduli space of stable bundles. Journal of Differential Geometry, 1992, 35, 131—150.
Departamento de A¨ lgebra, Geometria y Topologı& a Facultad de Ciencias Universidad de Ma& laga 29071 Ma& laga, Spain
PII: S0040-9383(98)00028-7
RING STRUCTURE OF THE FLOER COHOMOLOGY OF &;2 VICENTE MUN OZR (Received 31 October 1997; in revised form 12 May 1998) We give a presentation for the Floer cohomology ring HF*(&;2), where & is a Riemann surface of genus g*1, which coincides with the conjectural presentation for the quantum cohomology ring of the moduli space of odd degree rank two stable vector bundles on & with fixed determinant. We study the spectrum of the action of H (&) * on HF*(&;2) and prove a physical assumption made in [1]. 1999 Elsevier Science Ltd. All rights reserved.
1. INTRODUCTION
Let &"& be a Riemann surface of genus g*1 and let N denote the space of flat E E SO(3)-connections with nontrivial second Stiefel—Whitney class w modulo the gauge transformations that can be lifted to Sº(2). This is a smooth symplectic manifold of dimension 6g!6 and it is the space considered in [6, 4, Section 3]. Alternatively, we can consider & as a smooth complex curve of genus g and N as the moduli space of odd degree E rank two stable vector bundles on & with fixed determinant, which is a smooth complex variety of complex dimension 3g!3. The symplectic deformation class of N only depends E on the genus g and not on the particular complex structure on &. We note that, following the conventions of [8], the moduli space of anti-self-dual instantons on &;"/ with charge i"0 is again isomorphic to N . E We consider the following rings associated with the Riemann surface (we will always use "-coefficients): E QH*(N ) is the quantum cohomology of N (see [13]). This is well-defined since N is E E E a positive symplectic manifold. As vector spaces, QH*(N )"H*(N ), but the multiE E plicative structure is different. The minimal Chern number of N is 2, so QH*(N ) is E E 9/49-graded (the grading comes from reducing mod 4 the 9-grading of H*(N )). The E ring structure of QH*(N ), called quantum multiplication, is a deformation of the E usual cup product for H*(N ). It is associative and graded commutative. We remark E that we do not introduce Novikov rings to define QH*(N ) as in [13, Section 8] E (otherwise said, as H(N )"9, we should introduce an extra variable q of degree 4, E then we equate q"1). E HF (N ) is the symplectic Floer homology of N (with the symplectomorphism * E E
"id). The symplectic manifold N is connected, simply connected and n (N )"9 E E (see [4, introduction]), so the groups HF (N ) are well-defined [5]. They are * E 9/49-graded. HF (N ) is endowed with the pair of pants product [12], which is an * E associative and graded commutative ring structure. The symplectic Floer cohomology of N , HF* (N ), is defined as the dual of the symplectic Floer homology. There E E is a Poincare´ duality [12, remark 2.4] and a pairing 1,2.
R Supported by a grant from Ministerio de Educacio´n y Cultura of Spain.
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V. Mun oz
E HF (&;2) is the instanton Floer homology of the three manifold ½"&;2 for * the SO(3)-bundle with second Stiefel—Whitney class w "PD[2]3H(½; 9/29). This is defined in [6] and is 9/49-graded. We introduce a multiplication on HF (½) * using a suitable four-dimensional cobordism [14, section 5]. Let X be the four manifold given as a pair of pants times &, which is a cobordism between ½\½ and ½. This gives a map HF (½) HF (½)PHF (½), which is an associative and graded * * * commutative ring structure on HF (½). Again, the instanton Floer cohomology * of ½, HF*(½), is the dual of HF (½). There is a natural isomorphism HF*(½): * HF (!½), where !½ denotes ½ with reversed orientation. As ½"&;2 admits an * orientation reversing self-diffeomorphism, we can identify HF*(!½):HF*(½), so HF*(½):HF (½). This is called the Poincare´ duality isomorphism in Floer * (co)homology. The natural pairing HF*(½) HF (½)P" produces then a pairing * 1,2: HF*(½) HF*(½)P" (see [2], [3]). We will denote HF*"HF*(&;2). E THEOREM 1. ¹here are natural isomorphisms of vector spaces QH*(N ):HF* (N ):HF*(&;2). E E Moreover the first isomorphism respects the ring structures.
(1)
Proof. The second isomorphism is due to Dostoglou and Salamon [4, Theorem 10.1]. It is the particular case where one considers "id : &P&, in which the mapping torus of is &;2 and the SO(3)-bundle has w "PD[2]. The first isomorphism is a standard result obtained by Floer [5]. In [12, Theorem 5.1] it is proved that the first isomorphism intertwines the products. ) CONJECTURE 2. ¹he second isomorphism in (1) is a ring isomorphism. D. Salamon has informed the author that the adiabatic limit techniques of [4] can be extended to give a proof of Conjecture 2. He gave a lecture at Maryland in Spring 1994 on this issue. Many implications of this result to four-dimensional topology were given by Donaldson [3]. The author followed this program in [9] for small genus. Later he obtained very nice results on the behaviour of Donaldson invariants under the operation of connected sum along a Riemann surface [10, 11], exploiting only the isomorphism (1) as vector spaces. Using physical methods, Vafa et al. [1] find a set of generators and relations for QH*(N ). There are two main assumptions in their argument. The first one is Conjecture 2. E The second one is that the spectrum of the action of H (&) on HF*(&;2) can be read off * from the Donaldson invariants of &;3. Later, Siebert and Tian [15] claimed to have found a mathematical proof for the presentation of QH*(N ) given in [1] but they could not yet finish their program. In this E paper, we prove that the set of generators and relations given in [1] is a presentation for HF*(&;2) (Theorem 16), following a method inspired in Siebert and Tian [15]. This together with completion of the work [15] will produce a proof of Conjecture 2 (although in a rather indirect way). We also prove the physical assumption on the spectrum of HF*(&;2) in [1] (cf. Proposition 20). We leave the implications of Theorem 16 to Donaldson invariants of four-manifolds (mostly in the case b>"1) for future work.
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2. RING STRUCTURE OF H*(N ) E
Let us recall the known description of the homology of N [7, 17]. Let UP&;N E E be the universal bundle and consider the Ku¨nneth decomposition as in [7] c (End U)"2a[&]#4t!b C with t" c c , where +c , 2 , c , is a symplectic basis of H (&;9) with c c "[&] for G G E G G>E 1)i)g, and +cC, is the dual basis of H(&). In terms of the map k: H (&)PH\*(N ), G * E given by k(a)"! p (gu) /a (here guP&;N is the associated universal SO(3)-bundle, E and p (gu)3H(&;N ) its first Pontrjagin class), we have E a"2k(&)3H
c "k(c )3H, 1)i)2g G G b"!4k(x)3H
where x3H (&) is the class of the point, and HG"HG(N ). These elements generate H*(N ) E E as a ring [7, 18]. So there is a basis + f , S for H*(N ) with elements of the form E Q QZ f "aLbKc 2c GP Q G for a finite set S of multi-indices of the form s"(n, m; i , 2 , i ), n, m*0, r*0, P 1)i (2(i )2g. The mapping class group Diff (&) acts on H*(N ), with the action P E factoring through the action of Sp (2g, 9) on +c ,. The invariant part, H* (N ), is generated G ' E by a, b and c"!2 E c c . Then G G G>E H* (N )""[a, b, c]/I , (2) ' E E where I is the ideal of relations satisfied by a, b and c. Here deg(a)"2, deg(b)"4, E deg(c)"6. Actually, a basis for H* (N ) is given by the monomials aLbKcP, n#m#r(g ' E (see [17]). For 0)k)g, the primitive component of "IH is "I H"ker(cE\I> : "IHP"E\I>H). Then the Sp (2g, 9)-decomposition of H*(N ) is [7] E E H*(N )"@"I H "[a, b, c]/I . E E\I I PROPOSITION 3 (Siebert and Tian [17]). For g"1, let q"a, q"b, q"c. Define recursively, for g*1,
q "aq#gq E> E E 2g q "bq# q. E> E g#1 E q "cq E> E ¹hen I "(q, q, q ), for all g*1. E E E E Proof. This is the form of the relations given in [17, Proposition 3.2].
)
3. RING STRUCTURE OF HF* E
With the aid of the basis + f , S for H*(N ) we are going to construct a basis Q QZ E for HF*"HF*(&;2) to understand its ring structure. We need to use the gluing E
V. Mun oz
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properties of the Floer homology of a three manifold. Put ½"&;2 and let w "PD[2]3H(½; 9/29). Let us recall [11] that for a 4-manifold X we define (X)"Sym*(H (X) H (X)) d*H (X). Let us state the result that we shall use. PROPOSITION 4 [2, 3, 9]. For any smooth oriented four-manifold X with boundary *X"½, any w3H(X; 9) with w" "PD[2] in H(½; 9/29), and any z3 (X), we have 7 defined a relative invariant U(X, z)3HF (½). ¹hese relative invariants enjoy the following * gluing property, suppose X"X 6 X is a closed four-manifold split into two open four 7 manifolds X with *X "½, *X "!½, and w3H(X; 9) satisfying w" "PD[2] in G 7 H(½; 9/29). Put w "w" . ¹hen for z 3 (X ), i"1, 2, we have G 6G G G DU (z z )"1 U (X , z ), U (XM , z )2 6
(3)
where DU "DU#DU> (DU is the Donaldson invariant of X for w, see also [9, 10]). 6 6 6 6 ¼hen b>"1, the invariants are calculated for a long neck, i.e. we refer to the invariants defined by [&]. Consider the manifold A"&;D, with boundary ½"&;2, and let *"pt;DLA be the horizontal slice with **"2. Put w"PD[*]3H(A; 9). Clearly (A)" (&)" Sym*(H (&) H (&)) d*H (&). For every s3S, f "aLbKc 2c , define Q G GP z "&LxKc 2c 3 (&) Q G GP e " U(A, z )3HF*(½)"HF* Q Q E (here we identify Floer homology and Floer cohomology through Poincare´ duality). Then +e , S is a basis for HF*. This is a consequence of [11, Lemma 21]. The product E Q QZ HF* HF*PHF* is given by U(A, z ) U(A, z )" U(A, z z ). Then U(A, 1) defines the E E E Q QY Q QY neutral element of the product. As a consequence, the following elements are generators of HF*, E
a"2 U(A, &)3HF E t " U(A, c )3HF, 0)i)2g. G G E b"!4 U(A, x)3HF E
Note that there is an obvious epimorphism of rings
(4)
(&) HF*. E
THEOREM 5. Denote by * the product induced in H*(N ) by the product in HF* under the E E K isomorphism H*(N )P HF* given by f >e , s3S. ¹hen * is a deformation of the cupE E Q Q product graded modulo 4, i.e. for f 3HG(N ), f 3HH(N ), it is f * f " * ' ( f , f ), where E E P P ' 3HG>H\P(N ) and ' "f 6f . P E Proof. First, for s, s 3S, 1e , e 2"DU (z z )"0, Q QY ;"/ Q QY unless deg( f )#deg( f )"6g!6#4r, r*0, as these are the only possible dimensions Q QY for the moduli spaces of anti-self-dual connections on &;"/. Moreover, when deg( f )#deg( f )"6g!6, the moduli space is N , so 1e , e 2"!1 f f ,[N ]2" Q QY E Q QY Q QY E !1 f , f 2 (the minus sign is due to the different convention orientation for Donaldson Q QY invariants).
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Now let f , f be basic elements of degrees i and j respectively. Put f f " c f and Q QY Q QY R R de f * f " d f . This means that e e " d e . Write e e " g , where g " K K D R R Q QY R R Q QY R R Q QY K Kthat are the homogeneous parts. Put g' " d f . Let M be the maximum m such K DK R R g O0. Then there is f of degree 6g!6!M such that 1g' , f 2O0. Since K QYY + QYY 0O!1g' , f 2"1g , e 2"1e e , e 2"DU (z z z ), + Q Q QY Q ;"/ Q QY Q + Q it is deg( f )#deg( f )#deg( f )*6g!6, i.e. M)i#j. Now for m"i#j, any f of Q QY Q Q degree 6g!6!m, it is 1g' , f 2"!DU ;"/ (z z z )"1 f f f , [N ]2"1 f f , f 2. So K Q Q QY Q Q QY Q E Q QY Q g' "f f . G>H Q QY Finally, 1e e , e 2"D U (z z z )"0, whenever deg( f )#deg( f )#deg( f )I Q QY QYY ;"/ Q QY Q Q QY Q 6g!6 (mod 4), so g' "0 unless M,i#j (mod 4). ) K Remark 6. The isomorphism in Theorem 5 is only an isomorphism of H*(N ) and HF* E E as vector spaces (dependent on the chosen basis + f , S ). In general, this is not a ring Q QZ isomorphism (see Example 22), so we do not expect the isomorphism in (1) to coincide with that of Theorem 5. There is again an action of Diff (&) on HF* factoring through an action of Sp (2g, 9) E on +t ,. The invariant part (HF* ) is generated by a, b and c"!2 E U(A, c c ). The G E ' G G>E G epimorphism "[a, b, c] (HF* ) , z> U(A, z), allows us to write E ' HF*(&;2) ""[a, b, c]/J , ' E
(5)
where J is the ideal of relations of a, b and c. Now deg(a)"2, deg(b)"4, deg(c)"6, but E J is not a homogeneous ideal. E LEMMA 7. Suppose cJ LJ , for all g*1. ¹hen we have the Sp (2g, 9)-decomposition E E> E HF*(&;2)"@ "I H "[a, b, c]/J . E\I I Proof. The isomorphisms in Theorem 1 respect the Sp (2g, 9)-action and hence induce isomorphisms on the invariant parts. Then dim(HF* ) "dim H* (N ), for all g*1. Now E ' ' E the lemma is a consequence of the argument in the proof of [7, Proposition 2.2] and the discussion preceding it. )
4. A PRESENTATION FOR (HF*) E '
Theorem 5 and the arguments in [17, Section 2] imply that we can deform the relations of H* (NE ) to get a presentation for (HF* E )'. More explicitly, ' LEMMA 8. It is (HF* E )'""[a, b, c]/(RE , RE , RE ), where RGE"qGE#lower order terms of degrees deg qGE!4r, r'0, as polynomials in "[a, b, c] (qGE are defined in Proposition 3). Proof. Suppose first that g*2. Granted Theorem 5, [17, Theorem 2.2] implies that JE"(RE , RE , RE ), where RGE is qGE expressed in terms of a, b and c and the multiplication of K HF*E . Now we note that under the isomorphism H*(NE )P HF*E of Theorem 5, a>a, b>b, c>c (it can always be arranged so that these elements are in the basis, as g*2, see [16, Proposition 4.2]). So RGE is equal to qGE plus lower-order terms. The case g"1 is computed directly in Lemma 11. )
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As pointed out in [15], the key result to find the generators of JE is the following inclusion of ideals. LEMMA 9. JE>LJE, for all g*1. Proof. We shall prove this through an excision argument for Donaldson invariants. Let &E be a Riemann surface of genus g and consider &E>LAE"&E;DLS"&E;"/ where &E> is given as &E with a trivial handle of genus 1 added internally. This means that we take a point p3&E and a small 4-ball B in S centered at p such that B5&E is a 2-ball in &E. Then we substitute B5&E with an embedded 2-torus in B with a small ball removed, whose boundary coincides with *(B5&E )L*B. This produces &E>. Then the map H* (&E> )P H* (&E ) induces (&E> )P (&E ) which sends (a, b, c) >(a, b, c). Let AE>"&E>;DLAE be a (small) tubular neighbourhood of &E> in S. Now we put S for the complement of the interior of AE> in S, so that S"S6 E>;2AE>. Recall that we have also the decomposition S"AE 6 E;2AE. Now let z3JE>. Then U(&E>;D, z)"0. So for any zQ3 (&E ), s3S, DU (z zQ )"1 U(&E>;D, z), U(S, zQ )2"0. 1 Therefore DU (z zQ )"1 U(&E;D, z), U(&E;D, zQ )2"0, for any s3S. So U(&E; 1 D, z)"0 and z3JE. THEOREM 10. ¹here are numbers cE>, dE>3" such that, for all g*1,
R "aR#gR E> E E 2g R "(b#c ) R# R. E> E> E g#1 E R "cR#d R E> E E> E
Proof. We follow almost literally the argument of Siebert and Tian [15, Proposition 3.2]. As R 3J LJ is a relation on degree 2g#2, it is a linear combination of aR and E> E> E E R. Looking at the leading terms (Proposition 3), we have R "aR#gR. AnalogE E> E E ously, R is a combination of aR, bR, aR and R. Only the term R has degree less E> E E E E E than 2g#4, so R "bR#((2g)/(g#1)) R#c R , for an unknown coefficient c . E> E E E> E E> In the same fashion, R is cR plus a linear combination of R and aR. Adding a suitable E> E E E multiple of R (which is always allowed without loss of generality), we have R " E> E> cq#d R. ) E E> E LEMMA 11. ¹he starting relations ( for g"1) are R"a, R"b!8 and R"c. Proof. HF* is of dimension 1, i.e. HF*"" (see [3, 9]). Let S be the K3 surface and fix an elliptic fibration for S, whose fibre is &"3. The Donaldson invariants are, for w3H(S; 9) with w ) &,1 (mod 2) (see [9]), DU (eR")"!e\/R" . 1 Then DU (1)"!1 and DU (&B)"0, for d'0. Also from [11, Remark 4], DU (x)"2. 1 1 1 Let S be the complement of an open tubular neighbourhood of & in S. Then U(S, 1)
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generates HF* and U(S, &)"0, U(S, x)"!2 U(S, 1) and U(S, c c )"0, i.e. a"0, b!8"0 and c"0 in HF* (recall (4)). ) PROPOSITION 12. For g*2, there exists a non-zero vector v3HF* such that E
av"
g even 4(g!1) (!1 v, g odd
4(g!1) v,
bv"(!1)E\8 v cv"0. Proof. We shall construct such a vector as the relative invariants of an open fourmanifold X with boundary *X"½"&;2, where the closed four-manifold X"X6 A is of simple type with b>'1 and b "0. For concreteness, let X be the 7 manifold C from [11, Definition 25]. We recall its construction. Let S denote the elliptic E E surface of geometric genus p "g!1 and with no multiple fibres. It contains a section E p which is a rational curve of self-intersection !g. Let F be the elliptic fibre. Then p#gF can be represented by an embedded Riemann surface & of genus g and self-intersection g. Blow-up S at g points in & to get B with an embedded Riemann surface & of genus g and E E E self-intersection zero. Then put X"C "B C B (the double of B along & ). By [11, E E E E E E Proposition 27], X is of simple type and #U (e?)"DU ((1#x/2) e?)"!2E\e/? e) ?# 6 6 (!1)E2E\e/? e\) ?, where K3H(X; 9) satisfies K)& "2g!2 (w3H(C ; 9) is a parE E ticular element, which we do not need to specify here). Let us suppose from now on that g is even, the other case being similar. By [11, Proposition 3], DU (e?)"!2E\e/? e) ?#(!1)E2E\e/? e\) ?. 6 Consider X the complement of a small open tubular neighbourhood of & in X, so that E X"X6 A. Then we set v" U(X, g!2)3HF*(&;2)"HF*. Let us prove that 7 E this is the required element. For any z "&LxKc 2c , it is [11, Remark 4], Q G GP
0, 1v, e 2"DU ((g!2) z )" Q 6 Q !2E\(2g!2)L>2K,
r'0 r"0.
Then 1av, e 2"1 U(X, 2&(g!2)), U(A, z )2"DU ((g!2) 2& z )"(4g!4) Q Q 6 Q 1v, e 2, for all s3S. Then av"(4g!4)v. Analogously, cv"0 and bv"!8v. ) Q Notation 13. We set R"1, R"0 and R"0. THEOREM 14. For all g*1, c "(!1)E8 and d "0. E E Proof. The result is true for g"1 by Lemma 11 and Notation 13. Suppose it is true for 1)r)g, and let us prove it for g#1. By Proposition 12, there exists v3HF* with E> bv"(!1)E8 v, cv"0 and av"4g v if g is odd and av"4g(!1 v if g is even. In first place, cv"0 implies Rv"0, for 1)r)g. In second place, bv"(!1)E8 v P implies Rv"(b#(!1)E8) R v"(!1)E16R v E E\ E\ R v"(b#(!1)E\8) R v"0 E\ E\
V. Mun oz
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R v"(!1)E16R v E\ E\ R v"0, E\ $ In the third place, Rv"aR v#(g!1)R v"aR v, R v"aR v, 2 Also E E\ E\ E\ E\ E\ R v"aR v#(g!2)R v"(a#(g!2)(!1)E16) R v. E\ E\ E\ E\ So finally,
(a#(!1)E16(g!2)2(a#(!1)E16)1)v, R v" E\ (a#(!1)E16(g!2))2(a#(!1)E16)2)av,
g odd g even.
As a conclusion R v"jv, with jO0, and E\ Rv"aR v E\ E Rv"(!1)E16R v E E\ Rv"0. E As v3HF* , we have R v"0, R v"0 and R v"0. Evaluate the equations from E> E> E> E> Theorem 10 on v to get c "(!1)E>8 and d "0. ) E> E> COROLLARY 15. ¼e have cJ LJ LJ , for all g*1. E E> E Proof. The second inclusion is Lemma 9. For the first inclusion, note that cR"R 3J by the third equation in Theorem 10. Then multiplying the first two E E> E> equations in Theorem 10 we get that cR, cR3J . ) E E E> Using this corollary in Lemma 7, we have finally proved that THEOREM 16. ¹he Floer cohomology of &;2, for &"& a Riemann surface of genus g, E has a presentation E HF*(&;2)"@ "I H "[a, b, c]/J . E\I I where J "(R , R , R ) and RG are defined recursively by setting R"1, R"0, R"0 and P P P P P putting for all r*0
R "aR#rR P> P P
2r R "(b#(!1)P>8) R# R P> P r#1 P R "cR. P> P
Remark 17. The presentation obtained for HF* is the conjectural presentation for E QH*(N ) (see [15]). E COROLLARY 18. ker(c : (HF* ) P(HF* ) )"J /J L"[a, b, c]/J "(HF* ) . E ' E ' E\ E E E ' Proof. By Corollary 15, c factors as A "[a, b, c]/J . "[a, b, c]/J "[a, b, c]/J ) E E\ E
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The second map is a monomorphism since a?b@cA, a#b#c(g!1, form a basis for "[a, b, c]/J , and their image under c are linearly independent in "[a, b, c]/J . The E\ E corollary follows. ) For any F3"[a, b, c] define the expectation value by 1F2 "1F, 12 *, where 13HF* E E &$E is the unit element. Therefore 1F , F 2 *"1F F 2 . &$E E COROLLARY 19. For any F3"[a, b, c], 1cF2 "2g1F2 . E E\ Proof. By Corollary 15, the above formula holds for any F3J , as both sides are zero. E\ So it is enough to check it for a set of elements generating HF* , i.e. for F "a?b@cA, E\ ?@A a#b#c(g!1. If (a, b, c) O(0, 0, g!2) , it is 1F 2 "0 and 1cF 2 "0 by degree ?@A E\ ?@A E reasons. Now 1cE\2 "!1cE\, [N ]2, hence the corollary follows from E\ E\ 1cE\, [N ]2"2g1cE\, [N ]2 (see [18]). ) E E\ 5. LOCAL RING DECOMPOSITION OF HF*(&;2)
In [1] it is asserted that the only eigenvalues of the action of k(&), k(x) and k(c ) on G HF*(&;2) are the ones given by looking at the Donaldson invariants of the manifold X"&;3 i.e. if we denote by ¼LHF* the image U(X, (&)), where X"X6 A, E 7 then a, b and c act on ¼ and their eigenvalues are all the eigenvalues of their action on HF*. E The following result is a proof of this physical assertion. PROPOSITION 20. ¹he eigenvalues of (a, b, c) in (HF*) are (0, 8, 0), ($4, !8, 0), E ' ($8(!1, 8, 0), 2 , ($4(g!1)(!1E, (!1)E\8, 0). Proof. Put »"(HF*) . As cJ LJ , one has cE3J , i.e. cE"0 in », so the only E ' E\ E E eigenvalue of c is zero. To compute the eigenvalues of a, b we can restrict to »/c» (if p is a polynomial with p(a)"0 in »/c», then p(a) is a multiple of c in » and p(a)E"0 in »). Now the ideal of relations of » can be written as J "(f , f , f ), where f "1, f "0, E E E> E> \ f "0 and f "af #r(b#(!1)P8)f #2r(r!1)cf , for all r*0. This form of \ P> P P\ P\ the relations follows from rewriting as R"f E E 1 R" (f !af ) E E g E> 1 R" (f !af !(g#1)(b#(!1)E>8)f )). E 2g(g#1) E> E> E Therefore »/c»"" [a, b]/(fM , fM ) E E> where fM "1, fM "0, fM "afM #r(b#(!1)P8) fM , for r*0. From r(b# \ P> P P\ (!1)P8)fM "fM !afM we infer that (b#(!1)P8)fM 3(fM , fM ). Continuing in this P\ P> P P\ P P> way, (b#(!1)E8) (b#(!1)E\8)2(b!8)3(fM , fM ) E E>
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526
which implies that the only eigenvalues of b in »/c», and hence in », are $8. Let us study the eigenvalues of a for b"8, c"0. Again we only need to study »/(c, b!8) »""[a]/(fK , fK ), where now fK "1, fK "0, fK "afK #r(8#(!1)P8) fK , for E E> \ P> P P\ r*0. Then
fK "(a#(r!2)16)2(a#216)a, r even P fª "(a#(r!1)16)2(a#216)a, r odd P
from where the eigenvalues of a will be 0, $8(!1, $16(!1, 2 , $8[E\](!1. We leave the other case to the reader. ) Remark 21. As mentioned in [1], by the very definition of c"!2 U(A, c c ), it is G G>E cE>"0 in HF*, so the only eigenvalue of c is zero. E Proposition 20 says that (HF*) can be decomposed as a sum of local artinians rings E ' E\ (HF*) " @ R E ' E G G\E\
(6)
where R is a local artinian ring with maximal ideal m"(a!4i, b!(!1)G8, c) if i is odd, E G m"(a!4i(!1, b!(!1)G8, c) if i is even. Also HF* is decomposed as E E E\I\ E\ E\ G \ HF*"@ @ "I H R " @ @ "IH RE\I G. E\I G E I G\E\I\ G\E\ I
(7)
We recall from Lemma 11 that HF*""[a, b, c]/(a, b!8, c). Let us see the next cases. Example 22. For g"2, J "(a#b!8, a(b#8)#c, ac). In (HF*) , c"!a(b#8) ' and ca"0 yield a(b#8)"0. Now a"!(b!8) so (b!8) (b#8)"0 and a(a!16)"0. Also cJ LJ implies ca"c"c(b!8)"0. Finally (c#16a) (a!16)"!16c#16a(a!16)"!16(c#a(b#8))"0. All together proves "[a, b, c] "[a, b, c] "[a, b, c] (HF*) " . ' (a!4, b#8, c) (a, b!8, c#16a) (a#4, b#8, c) We want to remark that HF*P l QH*(N ) (see [12, Example 5.3] for a presentation of the latter ring). The isomorphism sends a>h , b>!4(h !1), c>4(h !h ), where h , h , h are the generators of QH, QH, QH, respectively. This was conjectured in [9, Conjec ture 1.22]. Example 23. For g"3, J "(a(a#b!8)#4(ab#8a#c), (b!8) (a#b!8)# ac, c(a#b!8)). Put »"(HF*) . Then ' »/c»""[a, b]/(a(a#b!8)#4(ab#8a), (b!8) (a#b!8)). In »/c», the first relation yields !5a(b!8)"a#64a and the second a(b!8) (a#b!8)"0. This implies a(a!16) (a#64)"0. Also (b!8)a(a!16)"0. Using (b!8)a"!(b!8), we get a(b!8) (b#8)"0. Therefore, in », a(a!16) (a#64) and a(b!8) (b#8) are multiples of c. As cJ LJ , we have ca(a!16)"0 and ca(b#8)"0 by Example 22. So a(a!16)(a#64)"0,
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a(a!16) (b!8) (b#8)"0 and a(b!8) (b#8)"0. It can be checked now that "[a, b, c] "[a, b, c] (HF*) " ' (a!8(!1, b!8, c) ((a!4), c#3(b#8), c!12(a!4))
"[a, b, c] "[a, b, c] (a, a(b!8), (b!8)! a, c#16a) ((a#4),c!3(b#8), c!12(a#4)) "[a, b, c]
. (a#8(!1, b!8, c)
6. CONJECTURE
We state the following conjecture, that first occurred to Paul Seidel and the author in mid-1996. CONJECTURE 24. ¹he decomposition in equation (7) is HF*:H*(s&) H*(s&) 2 H*(sE\&) E H*(sE\&) H*(sE\&) 2 H*(s&) where sG& is the ith symmetric product of &. Here H*(sG&) is isomorphic to the eigenspace of eigenvalues ($4(g!1!i) (!1E\G, (!1)E\\G8, 0). ¹he isomorphism respects only 9/29grading and is Diff (&)-equivariant. Simple computations establish that the dimensions of both vector spaces appearing in Conjecture 24 are the same, i.e. g2E. The Euler characteristic are both vanishing. Moreover the dimensions of the invariant parts coincide (E>). Examples 22 and 23 agree with the conjecture. A deeper reason for the above conjecture is the fact that HF* is the space for a gluing E theory of Donaldson invariants associated to the three manifold ½"&;2. The gluing theory of Seiberg—Witten invariants should be based on the Seiberg—Witten—Floer homology groups of &;2, which are indexed by a line bundle ¸ (the determinant line bundle of the spinA-structure on ½). The only possibilities are c (¸)"$2(g!1!i)PD [2], 0)i)g!1 (see [9, Section 6]). It is believed that the Seiberg-Witten-Floer groups for ¸ are isomorphic to H*(sG&). Acknowledgements—I am very grateful to Bernd Siebert and Gang Tian for providing me with a copy of [15] which was very enlightening. In particular, the proof of Theorem 10 is due entirely to them. Thanks to the organization of the CIME Course on Quantum Cohomology held in Cetraro (Italy, 1997) for inviting me. Also discussions with Paul Seidel were useful. Finally, thanks to the referee for useful remarks.
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5. Floer, A., Symplectic fixed points and holomorphic spheres. Communications in Mathematical Physics, 1989, 120, 575—611. 6. Floer, A., Instanton homology and Dehn surgery. ¹he Floer Memorial »olume. Progress in Mathematics, 1994, 133, 77—97. 7. King, A. D. and Newstead, P. E., On the cohomology ring of the moduli space of rank 2 vector bundles on a curve. ¹opology, 1998, 37, 407—418. 8. Kronheimer, P. B. and Mrowka, T. S., Embedded surfaces and the structure of Donaldson’s polynomial invariants. Journal of Differential Geometry, 1995, 41, 573—734. 9. Mun oz, V., Gauge theory and complex manifolds. Oxford D. Phil. thesis, 1996. 10. Mun oz, V., Donaldson invariants for connected sums along Riemann surfaces of genus 2, dg-ga/9702004. 11. Mun oz, V., Gluing formulae for Donaldson invariants for connected sums along surfaces. Asian Journal of Mathematics, 1997, 1, 785—800. 12. Piunikhin, S., Salamon, D. and Schwarz, M., Symplectic Floer—Donaldson theory and quantum cohomology. In Contact and Symplectic Geometry, vol. 8, ed. C. B. Thomas, Publ. Newton Institute, Cambridge University Press, Cambridge, 1996, pp. 171—200. 13. Ruan Y. and Tian, G., A mathematical theory of quantum cohomology. Journal of Differential Geometry, 1995, 42, 259—367. 14. Salamon, D., Lagrangian intersections, 3-manifolds with boundary and the Atiyah—Floer conjecture. Proceedings of the International Congress of Mathematicians, vol. 1, Birkha¨user, Basel, 1994, pp. 526—536. 15. Siebert, B., An update on (small) quantum cohomology. Preprint, 1997. 16. Siebert, B. and Tian, G., On quantum cohomology rings of Fano manifolds and a formula of Vafa and Intriligator. Asian Journal of Mathematics, 1997, 1, 679—695. 17. Siebert, B. and Tian, G., Recursive relations for the cohomology ring of moduli spaces of stable bundles. Proceedings of 3rd Go( kova Geometry—¹opology Conference, 1994. 18. Thaddeus, M., Conformal field theory and the cohomology of the moduli space of stable bundles. Journal of Differential Geometry, 1992, 35, 131—150.
Departamento de A¨ lgebra, Geometria y Topologı& a Facultad de Ciencias Universidad de Ma& laga 29071 Ma& laga, Spain