ODD SPHERE BUNDLES, SYMPLECTIC MANIFOLDS, AND THEIR INTERSECTION THEORY

ODD SPHERE BUNDLES, SYMPLECTIC MANIFOLDS, AND THEIR INTERSECTION THEORY

HIRO LEE TANAKA AND LI-SHENG TSENG

Abstract. Recently, Tsai-Tseng-Yau constructed new invariants of symplectic mani- folds: a sequence ofA8-algebras built of differential forms on the symplectic manifold.

We show that these symplectic A8-algebras have a simple topological interpretation.

Namely, when the cohomology class of the symplectic form is integral, theseA8-algebras are equivalent to the standard de Rham differential graded algebra on certain odd- dimensional sphere bundles over the symplectic manifold. From this equivalence, we deduce for a closed symplectic manifold that Tsai-Tseng-Yau’s symplecticA₈-algebras satisfy the Calabi-Yau property, and importantly, that they can be used to define an intersection theory for coisotropic/isotropic chains. We further demonstrate that these symplecticA8-algebras satisfy several functorial properties and lay the groundwork for addressing Weinstein functoriality and invariance in the smooth category.

Contents

1. Introduction 2

2. Recollections 5

2.1. Form decomposition 5

2.2. The cochain complex F_p^‚ 7

2.3. The A8-algebra structure 8

3. Equivalence of algebras 9

3.1. Definition of C 9

3.2. C is equivalent toF 10

3.3. Proof of Theorem 3.7 15

4. Calabi-Yau structure onF 19

5. Homology and intersection whenω is integral 24

5.1. Recollections on currents 24

5.2. Homology of primitive currents 25

5.3. Intersection theory via lifts to E 27

5.4. Example: Kodaira-Thurston four-fold. 27

6. Functoriality ofF 29

6.1. On objects 29

6.2. On morphisms 31

1

6.3. On orthogonal homotopies 32

6.4. Sheaf property 34

6.5. As sheaf cohomology 35

6.6. Looking forward: Weinstein functoriality 36

7. Appendix: Proof that Conepωq is equivalent to differential forms on a sphere

bundle 37

References 39

1. Introduction

Recently, Tsai-Tseng-Yau [TTY16] introduced a family of algebras of differential forms for any symplectic manifoldM. These algebras—denotedF_ppMq, with parameterptaking integral values from zero to dimM{2—have several notable properties. First, eachF_ppMq is anA₈-algebra. Second, the differential of the algebra consists of both first- and second- order differential operators, while the product interestingly involves derivatives. Finally, their cohomologies,F^pH^˚pMq, are all finite-dimensional—they were named the p-filtered cohomologiesofMfor short [TTY16] . These filtered cohomology rings intrinsically depend on, and can vary with, the symplectic structureω [TY12b, TTY16].

In this paper, we equate the algebras F_p (in the A8-algebra sense) to a standard de Rham differential graded algebra—not on the original symplectic manifold M, but on certain odd-dimensional sphere bundles over M. This perspective turns out to give a simple, topological description of F_p, and makes manifest several useful properties of F_p (including some functorial properties).

We can motivate the existence of such an alternative description for F_p on pM²ⁿ, ωq from a broader point of view. For a symplectic manifoldM, the symplectic structure and its powers constitute a distinguished set of forms:

ω, ω², . . . , ωⁿ(

PΩ^˚pMq.

Special forms also arise in geometry as representatives of characteristics classes of bundles overM. So we consider bundles overM with characteristics classes given by powers ofω.

Explicitly, for the distinguished two-form ω, consider the line bundle L over M with c₁“ p1{2πqω. ThisLis the well-known prequantum line bundle in geometric quantization, and has an associated circle bundle with Euler classe“ω. More generally, for the 2pp`1q- formω<sup>p`1</sup>, we have the associated direct sum vector bundleL^{‘p`1</sup>, and we will focus in particular on the associated closedS<sup>2p`1} bundle over M:

S^{2p`1 //}Ep

pM²ⁿ, ωq (1.1)

with its Euler class given byω^p`1. Given thatEp is derived from the symplectic formω, one can ask how the topology ofEp relates to the symplectic geometry of M²ⁿ:

Topological data of E_p ^oo ?

// Symplectic data ofpM²ⁿ, ωq Specifically,

(1) What symplectic data are encoded in the topology of the odd sphere bundle Ep? (2) Can we use the topology of these sphere bundles to gain new insights into the

known symplectic invariants onpM²ⁿ, ωq?

In this paper, we begin to address such questions by analyzing the cohomology and the differential topology ofEp. First, the real cohomology of the sphere bundle—equivalently, its de Rham cohomology—is well-known to fit into the Gysin sequence:

H^˚pE_pq

yy

H^˚pMq ^{^ω}

p`1 //H^˚pMq

ee(1.2)

where again ω^{p`1</sup> is the Euler class of E<sub>p</sub>. This exact triangle identifies H<sup>˚</sup>pE<sub>p</sub>q non- canonically as the direct sum of kernels and cokernels of the ω<sup>p`1} map between the de Rham cohomologies ofM. Coincidentally, exactly the same triangle diagram also appeared in [TTY16] for the symplecticp-filtered cohomologies of differential forms, F^pH^˚pMq; so we can quickly conclude from (1.2) that

H^˚pEpq –cokerrω^{p`1</sup> :H<sup>˚</sup>pMq ÑH<sup>˚</sup>pMqs ‘kerrω<sup>p`1}:H^˚pMq ÑH^˚pMqs (1.3)

–F^pH^˚pMq.

Of course, to construct an honest sphere bundle Ep, the Euler class e“ω^{p`1</sup> must be an integral class, i.e. an element ofH<sup>2p`2}pM,Zq. In the case the cohomology class is not integral, we should instead considerE_pto be a fibration of rational spheres, in the sense of rational homotopy theory. However, from the perspective of the above two questions, the integral condition will often not be relevant. For instance, the resulting cohomology on Ep still fits into a triangle as in (1.3), just as F^pH^˚pMq does regardless of the integrality ofω.

Beyond the cohomology of Ep, we can also consider the de Rham algebra ofEp. When rωsis integral, a standard application of the ˇCech-de Rham double complex shows that

ΩpE_pq »Conepω^p`1q

where Conepω^p`1q is the mapping cone of the map between de Rham chain complexes:

^ω^p`1 : Ω^‚pMqr´2p´2s ÑΩ^‚pMq. We refer the interested reader to the proof in the Appendix.

The cone algebra Conepω^p`1q is a commutative differential graded algebra (cdga). It is also well-defined for any ω whether integral or not. At first glance, this cone cdga is very different from theA₈-algebra F_p, but both fit into the same long exact sequence of

cohomology groups (hence have isomorphic cohomology). This raises the question whether F_p and Conepω^p`1q represent two distinct symplectic algebras onpM, ωq. We prove that they are in fact equivalent:

Theorem 1.1. For each p, there exists a natural equivalence of A8-algebras F_p»Conepω^p`1q.

Moreover, this equivalence suggests thatF_p should come with a non-degenerate pairing when M is compact—after all, when rωs is integral, ΩpEpq has a non-degenerate pairing by Poincar´e Duality. To that end, we prove:

Theorem 1.2. Each A₈-algebra F_p is Calabi-Yau wheneverM is compact.

In other words, each F_p admits a cyclically invariant pairing, and this theorem holds for all symplectic structures onM.

At this point, two natural questions arise from these theorems, whose answers we begin to address in this paper, and which we plan to deepen in future work.

First: Is there an intersection theory associated to eachF_p? After all, common examples of Calabi-Yau algebras are differential forms on compact, oriented manifolds, where the Calabi-Yau pairing encodes the intersection theory of oriented submanifolds. Since the sequence of Calabi-Yau algebrasF_p depends on the choice ofω, an answer to our question amounts to looking for an intersection theory tailored to symplectic manifolds—and one not involving holomorphic curve theory.

In Section 5, we begin to interpret the intersection pairing when ω is an integral co- homology class and p “ 0. The interpretation is as follows: Consider the contact circle bundleE associated to the line bundle classified byω. Given an isotropicI insideM, one can lift I to a section ˜I of E because ω|_I “0. As it turns out, isotropics naturally pair with coisotropicsC ĂM, but C may have some non-trivial, symplectic boundary. (This is a manifestation of the fact that the Poincar´e dual form of C may not be closed under the de Rham differential.) Regardless, our work suggests that C lifts to a submanifold C˜ ĂE in the circle bundle which no longer has boundary. Then the intersection of ˜I with E—hence the usual intersection pairing on˜ E—is the Calabi-Yau pairing on F₀.

Second: What formal properties do the F_p satisfy? For instance, what kinds of sym- plectic morphisms induce maps between them?

We prove that the algebras F_p enjoy several formal properties in Section 6. For in- stance, Theorem 1.1 implies thatF_p—as an A8-algebra—is only an invariant of the co- homology class ofω, rather than ω itself. Second, since there are natural quotient maps Conepω^p`1q ÑConepω^pq, one obtains a sequence of A₈-algebra maps that go in a reverse direction to the filtration that definesF_p:

(1.4) . . .ÑF_p ÑF_p´1 Ñ. . .ÑF₀.

This sequence of algebra maps suggests that theF_pcan be given a simple interpretation in terms of derived geometry—they encode higher order neighborhoods around the subvariety of ΩpMq determined byω.

Furthermore, we begin to develop the functorial properties of the assignment M ÞÑ Conepω^‚q. For instance, this assignment is functorial with respect to maps M Ñ M¹ that respect the symplectic form. Moreover, the assignmentM ÞÑ F_ppMq forms a sheaf of cdgas on the site of symplectic manifolds with open embeddings. Finally, we begin to see the seeds of Weinstein functoriality: For any Lagrangian submanifold L Ă pM₁ ˆ M2,´ω1‘ω2q, one can define a cdga F_ppLq receiving an algebra map from each of the F_ppM_iq—in particular, each Lagrangian correspondence defines a bimodule for the algebras F_ppMiq. It will be a subject of later work to show that this assignment sends Lagrangian correspondences to tensor products of bimodules.

Acknowledgments. We thank Kevin Costello, Rune Haugseng, Nitu Kitchloo, Si Li, Richard Schoen, Xiang Tang, Chung-Jun Tsai, and Shing-Tung Yau for helpful discussions.

This research is supported in part by National Science Foundation under Award No. DMS- 1400761 of the first author and a Simons Collaboration Grant of the second author. We are also grateful for the hospitality of the Perimeter Institute for Theoretical Physics and Harvard University’s Center for Mathematical Sciences and Applications while this research took place. Research at Perimeter Institute is supported by the Government of Canada through the Department of Innovation, Science and Economic Development and by the Province of Ontario through the Ministry of Research and Innovation.

2. Recollections

Here we recall the constructions and results from [TY12a], [TY12b], [TTY16]. This will allow us to set some notation, and to recall theA₈-operations for the algebras F_p. (Here, pcan be any integer between 0 and n, inclusive.)

2.1. Form decomposition. Fix a 2n-dimensional symplectic manifold pM²ⁿ, ωq. Since ω defines an isomorphism between 1-forms and vector fields, ω itself is identified with a bivector field (the Poisson bivector field), under this isomorphism. There are three basic operations on de Rham forms onM:

(1) L: Ω^kÑΩ^k`2 sends ηÞÑω^η.

(2) Λ : Ω^kÑΩ^k´2 sendsηÞÑΛη, the interior product with the Poisson bivector field.

(3) H: Ω^k ÑΩ^k sends a k-formη topn´kqη.

Together, tL,Λ, Hu generate an sl₂ Lie algebra acting on forms. The highest weight forms under this action are called primitive forms, whose space we shall denote by P^‚. Ifβ_kPP^k is a primitive k-form, then its interior product with the Poisson bivector field vanishes, i.e. Λβ_k “ 0. As usual with sl₂ representations, any k-form η_k P Ω^k can be expressed as a sum of primitive forms wedged with powers ofω:

ηk“ ÿ

k“2j`s

L^jβs“βk`Lβk´2`. . .`L<sup>p</sup>βk´2p`. . .

This Lefschetz decomposition of Ω^‚ can be arranged in a “pyramid” diagram as shown in Figure 1. (This can be compared to the well-knownpp, qq“diamond” of forms on complex manifolds.) Associated with this Lefschetz decomposition, we define four more operations:

Ω⁰ Ω¹ Ω² Ω³ Ω⁴ Ω⁵ Ω⁶ Ω⁷ Ω⁸ Ω⁹ Ω¹⁰

´ ´ ´ ´ ´ ´ ´ ´ ´ ´ ´ ´ ´ ´ ´ ´ ´ ´ ´ ´ ´ ´ ´ ´ ´ ´ ´ ´ ´ ´ ´ ´ ´ P⁵

P⁴ LP⁴

P³ LP³ L²P³ P² LP² L²P² L³P²

P¹ LP¹ L²P¹ L³P¹ L⁴P¹

P⁰ LP⁰ L²P⁰ L³P⁰ L⁴P⁰ L⁵P⁰ Figure 1. The decomposition of forms into a “pyramid” diagram in di- mension 2n“10. Note that there is a natural reflection symmetry action about the Ω^n“5 column. This is the˚r action defined in (2.1). To illustrate filtered forms, we havebold-facedthose elements in F^pΩ for p“2 here.

(4) L^´p: the negative power of L with p ą0. Roughly, it is the “inverse” of L^p and removesω^p from each term of the decomposition. Explicitly,

L^´pη_k“ ÿ

k“2j`s

L^´pL^jβ_s“β_k´2p`Lβ<sub>k´2p´2</sub>`. . .

(5) ˚r: the natural reflection action about the central axis of the pyramid diagram in Figure 1. The operation is explicitly defined on each term of the Lefschetz decomposition by

(2.1) ˚rpL^jβsq “L^n´j´sβs. It is worthwhile to note that in terms of˚r,

L^´p “ ˚rL^p˚r,

that is, L^´p is the ˚r conjugate ofL^p. Also, acting on Ω, we have˚rΩ^k “Ω^2n´k as expected.

(6) Π^p: a projection operator that keeps only terms up to the pth power of ω in the Lefschetz decomposition:

(2.2) Π^pη_k “

p

ÿ

j“0

L^jβs“β_k`. . .`L^pβ_k´2p Alternatively, Π^p can be defined as

Π^p“1´L^{p`1</sup>˝L<sup>´pp`1q}. (7) Π^p˚: the˚r conjugate of Π^p defined by

Π^p˚ “ ˚rΠ^p˚r“1´L^{´pp`1q</sup>˝L<sup>p`1}.

The decomposition by powers ofωallows us to group together forms that contain terms only up to certain powers ofω.

Definition 2.1. Define the space ofp-filtered k-formsto be F^pΩ^k:“Π^pΩ^k

forpP t0,1, . . . , nu.

The filtered F^pΩ^k spaces give us a natural filtration of Ω^k:

P^k:“F⁰Ω^kĂF¹Ω^kĂF²Ω^k Ă. . .ĂFⁿΩ^k“Ω^k.

Note that the 0-filtered forms are precisely primitive forms, and thatFⁿΩ^k “Ω^k. Lemma 2.2. The following are equivalent conditions for p-filtered forms:

(i) αk PF^pΩ^k;

(ii) Λ^{p`1</sup>α<sub>k</sub>“0 (or equivalently, L<sup>´pp`1q}α_k“0q; (iii) L^n`p`1´kα_k “0;

(iv) L^p`1˚rα_k“0.

Remark 2.3. TheF^pΩ^k space is only non-zero for k“0,1, . . . ,pn`pq.See Figure 1.

2.2. The cochain complex F_p^‚. There are natural differential operators pd`, d´q that maps betweenp-filtered forms. We recall the key ingredients here:

(1) On symplectic manifolds, the exterior differential has a natural decomposition d“ B``ω^ B´.

where B` :L<sup>j</sup>P<sup>s</sup> Ñ L<sup>j</sup>P<sup>s`1</sup> and B´ :L^jP^s Ñ L^jP^s´1. Note in particular when j“0,B` and B´ map primitive forms into primitive forms. The pair of operators pB`,B´q square to zero (B<sub>`</sub><sup>2</sup> “ B<sub>´</sub><sup>2</sup> “0), commute withL (rL,B`s “ rL, LB_´s “0), and also graded commute after applyingL: LB`B´“ ´LB´B`.

(2) Onp-filtered forms, we can define two natural linear operators:

d_` “Π^p ˝ d , (2.3)

d_´ “ ˚rd˚r, (2.4)

such that d<sub>`</sub> : F<sup>p</sup>Ω<sup>k</sup> Ñ F<sup>p</sup>Ω<sup>k`1</sup> and d_´ : F^pΩ^k Ñ F^pΩ^k´1. Both operators square to zero (pd_{`</sub>q<sup>2</sup> “ pd<sub>´</sub>q<sup>2</sup> “ 0). In fact, when acting on primitive forms, d<sub>˘</sub>P<sup>k</sup>“ B˘P<sup>k</sup>. Furthermore, (d<sub>`}, d_´) is related to pB`,B´q by

d´“ B´` B`L^´1 pB`B´qd<sub>`</sub>“d_´pB`B´q “0

The differentialspd<sub>`</sub>, d<sub>´</sub>,B`B´qcan be used to write down an elliptic complex forF^pΩ^‚: Theorem 2.4. [Section 3 of [TTY16]] The differential complex

0ÝÑF^pΩ⁰ Ý^dÝÑ^{`</sup> F<sup>p</sup>Ω<sup>1</sup> Ý<sup>d</sup>ÝÑ<sup>`} . . .Ý^dÝÑ^{`</sup> F<sup>p</sup>Ω<sup>n`p} ÝÝÝÑ^{B`B´</sup> F<sup>p</sup>Ω<sup>n`p d}ÝÝÑ^´ F^pΩ^n`p´1Ý^dÝÑ^´ . . .Ý^dÝÑ^´ F^pΩ⁰ ÝÑ0 is elliptic for all pP t0,1, . . . , nu.

As a corollary of the ellipticity of the complex, the associated filtered cohomologies labelled by

F^pH“ tF^pH_{`</sub><sup>0</sup>, . . . , F<sup>p</sup>H<sub>`}^{n`p</sup>, F<sup>p</sup>H<sub>´</sub><sup>n`p}, . . . , F^pH_´⁰u are all finite-dimensional.

The cochain complex F_p is defined following the elliptic complex:

F_p “

´

F^pΩ⁰ Ý^dÝÑ^{`</sup> . . .Ý<sup>d</sup>ÝÑ<sup>`} F^pΩ^{n`p</sup>ÝÝÝÝÑ<sup>´B`B´} F^pΩ^{n`p</sup>Ý<sup>´d</sup>ÝÝÑ<sup>´</sup> F<sup>p</sup>Ω<sup>n`p´1} Ý^´dÝÝÑ^´ . . .Ý^´dÝÝÑ^´ F^pΩ⁰

¯ . (2.5)

As vector spaces F^pΩ^k “F^pΩ^k. The extra bar is only inserted to distinguish the second set ofF^pΩ^‚ in the complex from the first. Also, the additional minus signs in some of the differentials are needed to satisfy theA8 relations described below. At times, we will just denote the above cochain complex asF^‚, with

F^j “

$

’&

’%

F^pΩ^j jďn

F^pΩ^2n`2p`1´j jěn`p`1

0 jă0, j ą2n`2p`1.

The notation here is thatF^pΩ^k has degree 2n`2p`1´k in this complex, while F^pΩ^k has degree k. If ambiguity may arise whether a p-filtered k-form is an element of F^k or F^2n`2p`1´k, we will add a bar and write α_kPF^pΩ^k to denote an element inF^2n`2p`1´k. 2.3. The A₈-algebra structure. We use the sign convention that a series of operations

m^l:F^bl ÑFr2´ls satisfies theA₈ relations if and only if

ÿp´1q^{r`st</sup>m<sup>r`t`1}pid^brbm^sbid^btq “0.

As an example, ifm^l“0 forlě3, then the datapF,pm^lqqdefine an ordinary differential graded algebra (dga).

The m² operation.

(1) Fora₁ PF^pΩ^k1 “F^k1,a₂ PF^pΩ^k2 “F^k2, and k₁`k<sub>2</sub> ďn`p, we set m²pa₁, a₂q “Π^ppa₁^a₂q

where Π^p is the projection to the p-filtered components defined above.

(2) Fora₁ PF^pΩ^k1 “F^k1,a₂ PF^pΩ^k2 “F^k2, and k₁`k<sub>2</sub> ąn`p, define DLpa1, a2q “ ´d L<sup>´pp`1q</sup>pa1^a2q ` pL<sup>´pp`1q</sup>da1q ^a2` p´1q^ja1^ pL^´pp`1qda2q.

Then,

m²pa1, a2q “Π^p˚rpDLpa1, a2qq.

In other words, the m² operation measures the difference between the action of d L^{´pp`1q</sup> and the derivation of L<sup>´pp`1q}don the wedge product of filtered forms.

(It then picks out the p-filtered part of the˚r reflection.) (3) Fora1 PF^pΩ^k1 “F^k1 and a2 PF^pΩ^k2 “F^2n`2p`1´k2, we set

m²pa₁, a₂q “ p´1q^k1 ˚rpa₁^ p˚ra₂qq.

(4) Fora₁ PF^pΩ^k1 “F^2n`2p`1´k1 and a₂PF^pΩ^k2 “F^k2, we set m²pa1, a2q “ ˚rpp˚ra1q ^a2q.

This implies thatm² is graded commutative:

m²pa₁, a₂q “ p´1q^a1a2m²pa₂, a₁q.

(5) And for degree reasons, ifa1PF^pΩ^k1 and a2PF^pΩ^k2, then m²pa1, a2q “0.

The m³ operation. Let a_i PF^ki. We set m³pa1, a2, a3q “

$

’&

’% Π^p˚r

“a1^L<sup>´pp`1q</sup>pa2^a3q k1`k2`k3 ěn`p`2,

´L^´pp`1qpa₁^a₂q ^a₃‰

withk₁, k₂, k₃ďn`p

0 otherwise

The higher m^l operations. We setm^l“0 for all lě4.

From [TTY16], we have the following:

Theorem 2.5(Section 5 of [TTY16]). For any symplectic manifoldM, the operationsm^l endow FpMq with the structure of an A8-algebra.

3. Equivalence of algebras

3.1. Definition of C. Let f : X Ñ Y be a map of cochain complexes. In homological algebra, the natural notion of quotient is thecone, ormapping cone off. It is defined to be the cochain complex

Y ‘Xr1s, dpy‘xq “dy`fpxq ‘ ´dx.

Finally, a commutative differential graded algebra, or cdga, is a commutative algebra in the category of cochain complexes. Explicitly, a cdga is a cochain complex A “ pA, dq together with a map of cochain complexes

m² :AbAÑA

which is associative, and which satisfies graded commutativity:

m²pa, bq “ p´1q^{|a| |b|}m²pb, aq.

Note thatm² is a map of cochain complexes if and only if it satisfies the Leibniz rule:

dm²pa, bq “m²pda, bq ` p´1q^|a|m²pa, dbq.

A map of cdgas is a map of cochain complexesf :AÑB such thatf ˝m² “m²˝fbf. Remark 3.1. Given any commutative ring R, and an element x, one has the map of R-modules

RÝÑ^x R

and one can define the quotientR-moduleR{xR; this also has a commutative ring struc- ture. In what follows, we consider the case of R “ Ω^‚pMq, the de Rham complex of

differential forms on a symplectic manifoldM. Our element x is some power of ω.¹ So, heuristically,C should be thought of as “R{xR.”

Consider the degree 0 map

ω^{p`1</sup> : ΩpMqr´2p´2sÝÝÝÝÑ<sup>^ωp`1} ΩpMq.

Then Conepω^pq has in degreekthe vector space

Ω^kpMq ‘θΩ^k´2p´1pMq

whereθis a formal element of degree 2p`1. This has differential dCpη‘θξq “ pdη`ω^p`1ξq ‘ ´θ dξ.

This cochain complex has another description endowing it with an obvious cdga structure:

Definition 3.2. We define C to be the cdga obtained from ΩpMq by freely attaching a degree 2p`1 variableθsatisfyingdθ´ω<sup>p`1</sup> “0. Note thatω andpare implicit in writing C.

Thinking of the cdga C as anA₈-algebra, we will letm¹_C and m²_C denote its differential and its multiplication, respectively, whilem^k_C “0 for all kě3. Explicitly, we have

m²_C`

pα`θα<sup>1</sup>q b pβ`θβ¹q˘

“ pα`θα<sup>1</sup>q ^ pβ`θβ¹q “α^β‘θpα¹^β` p´1q^αα^β¹q.

3.2. C is equivalent to F. Throughout this section we fix an integer 0 ď p ď n. The dependence of C and F on this choice is implicit. Furthermore, the indices j and k take integral valuesjP t0,1, . . . ,2n`2p`1u and kP t0,1, . . . , n`pu.

We shall write any p-filtered form αPF^pΩ^k as

αk“βk`ωβk´2`. . .`ω^pβk´2p

where eachβ_i PPⁱ is a primitive form of degreei.

Let us first make the observation that C^k “Ω^k‘ θΩ^k´2p´1

C^2n`2p`1´k “Ω^2n`2p`1´k‘ θΩ^2n´k “

´

˚rΩ^k´2p´1

¯

‘ θ

´

˚rΩ^k

¯

making use of the relation ˚rΩ^kpMq “ Ω^2n´kpMq for any integer k. The above implies C^k–C^2n`2p`1´k as expected. Following this, we can write for anyC_kPC^k

Ck“ pαk`ω<sup>p`1</sup>βk´2p´2`. . .q `θ2p`1pαk´2p´1`ω<sup>p`1</sup>βk´4p´3`. . .q, (3.1)

and for anyC2n`2p`1´kPC^2n`2p`1´k

C_2n`2p`1´k“ ˚rpα_k´2p´1`ω<sup>p`1</sup>β_k´4p´3`. . .q `θ2p`1˚rpα<sub>k</sub>`ω<sup>p`1</sup>β<sub>k´2p´2</sub>`. . .q. (3.2)

If we further replace theα_k´2p´1terms in (3.1)-(3.2) above with its Lefschetz decomposed form

α_k´2p´1“β_k´2p´1`ωβ<sub>k´2p´3</sub>`. . .`ω^pβ_k´2p,

1In fact, the discussion holds for any even-degree, closed differential form onM.

. . . ^{d //}C^k

αk

d //. . . ^{d //}C^n`p

αn`p

d //C^n`p`1

´pαn`p`w^pB`βn´p´1q

d //. . . ^{d //}C^2n`2p`1´k

´pαk`w<sup>p</sup>B`βk´2p´1q

d //. . .

. . . ^{d` //</sup>F<sup>p</sup>Ω<sup>k d` //}. . . ^{d` //</sup>F<sup>p</sup>Ω<sup>n`p´B`B´//</sup>F<sup>p</sup>Ω<sup>n`p ´d´ //}. . . ^{´d´ //}F^pΩ^{k ´d´ //}. . . Figure 2. The mapf :C Ñ F. The maps are expressed in terms of the decomposition in (3.1)-(3.2).

. . . ^{d //}C^{k d //}. . . ^{d //}C^{n`p d //</sup>C<sup>n`p`1 d //</sup>. . . <sup>d //</sup>C<sup>2n`2p`1´k d //}. . .

. . . ^{d` //}F^pΩ^k

αk´θB´βk´2p

OO

d` //. . . <sup>d` //</sup>F^pΩ^n`p

ω^pβn´p´θB_´βn´p

OO

´B`B´//F<sup>p</sup>Ω<sup>n`p</sup>

´θβn´p

OO

´d´ //. . . ^{´d´ //}F^pΩ^k

´θ˚rαk

OO

´d´ //. . . Figure 3. The map g:F Ñ C. In the maps above, we use the notation β_k´2p “L^´pα_k, and for the two maps in the middle, we have applied the relationF^pΩ^{n`p</sup>“F<sup>p</sup>Ω<sup>n`p} “L^pP^n´p.

then we find the relation

C^k–C^2n`2p`1´k–

˜ à

0ďjďk´2p´1

P^j

¸

‘ F^pΩ^k Therefore, the maps

C_kÞÑ pβ₀, . . . , β_k´2p´1, α_kq (3.3)

C_2n`2p`1´kÞÑ pβ₀, . . . , β_k´2p´1, α_kq (3.4)

are isomorphisms ofRvector spaces. This gives another representation of elements inC^j. We now define a pair of maps pf, gqthat relates C and F.

Definition 3.3. In terms of (3.1)-(3.2), we define f : C^j Ñ F^j

C_j ÞÑ

#

αj jďn`p ,

´pα_k`Π<sup>p˚</sup>w<sup>p</sup>dα<sub>k´2p´1</sub>q jěn`p`1, j“2n`2p`1´k , where Π<sup>p˚</sup>w<sup>p</sup>dα<sub>k´2p´1</sub> “ω<sup>p</sup>B`β_k´2p´1, and

g: F^j Ñ C^j

α_k ÞÑ

#

α_k´θ<sub>2p`1</sub>L<sup>´pp`1q</sup>dα_k jďn`p, k“j ,

´θ2p`1˚rαk jěn`p`1, k “2n`2p`1´j , whereL<sup>´pp`1q</sup>dα_k“ B´β_k´2p.

A pictorial description of thefandgmaps is given in Figure 2 and Figure 3, respectively.

Alternatively, we can also express the maps in terms of the decomposition in (3.3)-(3.4).

We can writef :C^j ÑF^j as

f :pβ₀, . . . , β_k´2p´1, α_kq ÞÑ

#

α_k jďn`p, k “j

´pαk`w<sup>p</sup>B`βk´2p´1q jěn`p`1, k “2n`2p`1´j (3.5)

andg:F^j ÑC^j as g:αkÞÑ

#

p0, . . . ,´L<sup>´pp`1q</sup>dαk, αkq jďn`p, k “j

p0, . . . ,0,´α_kq jěn`p`1, k“2n`2p`1´j (3.6)

Lemma 3.4. The maps f :CÑF and g:F ÑC are chain maps, i.e.

dFf “f dC, dCg“g dF.

Proof. The lemma is proved by direct calculation. We first compute dFf C_j “

$

’&

’%

d`αj jăn`p

´ω^pB`B´βj´2p j“n`p

d_´α_k´ω^pB`B´β<sub>k´2p´1</sub> j“2n`2p`1´kąn`p, kďn`p (3.7)

To computef dC, we consider first the case jăn`p. We have

dCCj “ pd`αj`ω^{p`1</sup>pB´βj´2p` B`βj´2p´2q `. . .q `ω<sup>p`1}pαj´2p´1`ω<sup>p`1</sup>βj`4p´3`. . .q

´θ_{2p`1</sub>dpα<sub>j´2p´1</sub>`ω<sup>p`1</sup>β<sub>j`4p´3}`. . .q

It then follows thatf dCCj “d`αj. Whenj “n`p, the above equation becomes dCC<sub>n`p</sub> “ω<sup>p`1</sup>pB´β_n´p` B`β_n´p´2`β<sub>n´p´1</sub>q `Opωq. . .q

´θ2p`1dpαn´p´1`ω<sup>p`1</sup>βn`3p´3`. . .q

Notice thatθ2p`1dαn´p´1 “θ2p`1pB`βn´p´1`Opωqq “θ2p`1˚rpω<sup>p</sup>B`βn´p´1`Opω<sup>p`1</sup>qq.

Hence, we obtain

f dCCn`p “ ´r´ω<sup>p</sup>B`βn´p´1`ω<sup>p</sup>B`pB´βn´p`βn´p´1qs “ ´ω<sup>p</sup>B`B´βn´p . Now forjąn`p, let j“2n`2p`1´k where 0ďkďn`p. Then

dCC_j “ω^n`2p`2´kpB´β_k´2p´1` B`β_k´2p´3`β<sub>k´2p´2</sub>`Opωqq

´θ<sub>2p`1</sub>L<sup>n´k`1</sup>rpB´β_k` B`β_k´2q `. . .´ω<sup>p</sup>pB´β<sub>k´2p</sub>` B`β<sub>k´2p´2</sub>q `. . .s This gives the desired result

f dCC_2n`2p`1´k “ ´p´d_´α_k´ω^pB`β<sub>k´2p´2</sub>`ω^pB`pB´β<sub>k´2p´1</sub>`β_k´2p´2q

“d_´α_k´ω^pB`B´β_k´2p´1

The calculation for thegmap is straightforward. It can be easily checked that forα_k PF^k, dCgpα_kq “d<sub>`</sub>α<sub>k</sub>`θ2p`1B`B´β_k´2p “g dFpα_kq,

and forαkPF^2n`2p`1´k,

dCgpα_kq “θ_2p`1˚rd_´α¯_k“g dFpα¯_kq

It is straightforward to check thatf g“idF. Hence, we may wonder whetherg f admits a chain homotopy to the identity onC.

Definition 3.5. We define G:C^j ÑC^j´1 as follows. FixηPΩ^j and ξ PΩ^j´2p´1. Gpη`θ^ξq:“L<sup>p</sup>ξ`θL^´pp`1qη.

In terms of the decomposition (3.3)-(3.4), G: C^j Ñ C^j´1 acts by sending the element pβ₀, . . . , β_k´2p´1, α_kq to

$

’&

’%

pβ₀, . . . , β_k´2p´2, ω^pβ_k´2p´1q j ďn`p, k“j

pβ₀, . . . , β_k´2p´1, α_kq j “n`p`1, k“n`p

pβ0, . . . , βk´2p´1, L^´pαk,0q j ąn`p`1, k“2n`2p`1´j where by conventionβ´1“0.

Making use ofG, we have the following result:

Lemma 3.6. The inclusion g : F Ñ C and the maps f and G exhibit F as a strong deformation retract of C:

f g“idF, idC´g f “dCG`GdC.

In particular, bothf and g are quasi-isomorphisms and H^˚pFq –H^˚pCq.

Proof of Lemma 3.6. As mentioned, the proof of the first relationf g“idF follows directly from the definition of the maps. For the second relation, consider first for C^j“k with 0ďkďn`p. For the lefthand side, we have

pgfqC_k“α_k´θ2p`1B´β_k´2p

pidC´gfqC_k“ pω<sup>p`1</sup>β<sub>k´2p´2</sub>`. . .q `θ<sub>2p`1</sub>pB´β_k´2p`α<sub>k´2p´1</sub>`ω<sup>p`1</sup>β<sub>k´4p´3</sub>`. . .q For the righthand side, we have

GC_k“ω^ppα_k´2p´1`ω<sup>p`1</sup>β_k´4p´3`. . .q `θ<sub>2p`1</sub>pβ<sub>k´2p´2</sub>`ωβ_k´2p´4`. . .q dCGCk“ω<sup>p</sup>dpαk´2p´1`ω^{p`1</sup>βk´4p´3`. . .q `ω<sup>p`1}pβk´2p´2`ωβk´2p´4`. . .q

´θ2p`1dpβk´2p´2`. . .q GdCC_k“ ´ω^pdpα_k´2p´1`ω<sup>p`1</sup>β<sub>k`4p´3</sub>`. . .q

`θ2p`1

“B´βk´2p`dpβk´2p´2`. . .q ` pαk´2p´1`ω<sup>p`1</sup>βk`4p´3`. . .q‰ which results in

pdCG`GdCqCk“ω<sup>p`1</sup>pβk´2p´2`ωβk´2p´4`. . .q

`θ2p`1pB´β_k´2p`α<sub>k´2p´1</sub>`ω<sup>p`1</sup>β<sub>k`4p´3</sub>`. . .q which matches exactly with the lefthand side.

We now check the case for Cj“2n`2p`1´k with 0ďkďn`p. For the lefthand side, we have

gf C_2n`2p`1´k “θ2p`1˚rpα<sub>k</sub>`ω^pB`β_k´2p´1q

pidC´gfqC_2n`2p`1´k “ω^n`2p`1´kpα_k´2p´1`ω<sup>p`1</sup>β_k´4p´3`. . .q

`θ<sub>2p`1</sub>˚rp´ω^pB`β<sub>k´2p´1</sub>`ω<sup>p`1</sup>β<sub>k´2p´2</sub>`. . .q For the righthand side, we have

GC_2n`2p`1´k“ω<sup>n´k`p</sup>pα<sub>k</sub>`ω<sup>p`1</sup>β<sub>k´2p´2</sub>`. . .q

`θ<sub>2p`1</sub>ω<sup>n´p`k</sup>pα<sub>k´2p´1</sub>`ω<sup>p`1</sup>β<sub>k´4p´3</sub>`. . .q dCG C_2n`2p`1´k“ω<sup>n´k`p</sup>dpα<sub>k</sub>`ω<sup>p`1</sup>β<sub>k´2p´2</sub>`. . .q

`ω<sup>n´k`2p`1</sup>pα<sub>k´2p´1</sub>`ω<sup>p`1</sup>β<sub>k´4p´3</sub>`. . .q

´θ<sub>2p`1</sub>ω<sup>n`p´k</sup>dpα_k´2p´1`ω<sup>p`1</sup>β_k´4p´3`. . .q dCC<sub>2n`2p`1´k</sub>“ω<sup>n`2p`1´k</sup>“

dpα_k´2p´1`ω<sup>p`1</sup>β_k´4p´3`. . .q ´ B<sub>`</sub>β_k´2p´1‰

`ω<sup>n´k`p`1</sup>pω<sup>p`1</sup>β_k´2p´2`. . .q

´θ<sub>2p`1</sub>ω<sup>n´k</sup>dpα<sub>k</sub>`ω<sup>p`1</sup>β<sub>k´2p´2</sub>`. . .q

GdCC_2n`2p`1´k“ ´ω<sup>n´k`p</sup>dpα<sub>k</sub>`ω<sup>p`1</sup>β<sub>k´2p´2</sub>`. . .q `θ<sub>2p`1</sub>ω^n´k`p`1pβ_k´2p´2`. . .q

`θ<sub>2p`1</sub>ω^n`p´k“

dpα_k´2p´1`ω<sup>p`1</sup>β_k´4p´3`. . .q ´ B<sub>`</sub>β_k´2p´1‰

Note that in the righthand side of the third equation, we have used the property that L<sup>p`1</sup>˚rα<sub>k</sub>“0 and also added the term´ω<sup>n`2p`1´k</sup>B`β_k´2p´1 since it is identically zero.

The above equations give the result

dCG`GdC “ω<sup>n´k`2p`1</sup>pα<sub>k´2p´1</sub>`ω<sup>p`1</sup>β<sub>k´4p´3</sub>`. . .q

`θ2p`1ω^n´k“

´ω^pB`β<sub>k´2p´1</sub>` pω<sup>p`1</sup>β<sub>k´2p´2</sub>`. . .q‰

which is exactly the lefthand side.

Though f and g are chain maps, they are clearly not compatible with the product structure of the algebrasC and F. For instance, it is not difficult to see that for generic a1, a2 PF, g m²_Fpa1b a2q ‰ m²_Cpga1b ga2q. So we treat the cdga C as an A8-algebra.

Recall that a sequence of mapsg^l:F^blÑCr1´lswithl“1,2, . . . is called an A₈ map if the following equations are satisfied (see for example Section 3.4 of [Kel01]):

ÿ

r`s`t“n

p´1q^r`stg^lp1^brbm^s_F b1^btq “ ÿ

i1`...`iq“n

p´1q^um^q_Cpgⁱ¹ b. . . g^iqq, ně1 (3.8)

wherel“r`1`tand the sign on the right-hand side is given by

u“ pq´1qpi1´1q ` pq´2qpi2´1q `. . .`2piq´2´1q ` piq´1´1q.

Theorem 3.7. The sequence of maps g^l :F^blÑCr1´lsgiven by (1) g¹ “g,

(2) g² “ ´θ<sub>2p`1</sub>L<sup>´pp`1q</sup>m²_Cpg¹bg¹q,

(3) g^l “0 for alllě3.

defines anA₈-algebra map F ÑC.

We will give the proof of this theorem in the next subsection. Since the map g is an quasi-isomorphism, Theorem 3.7 gives as an immediate corollary Theorem 1.1, that is, F_p »Conepω^p`1q »ΩpE_pq asA₈-algebras.

To simplify notation in sections following, we will also refer the family of A₈ maps as gas well.

3.3. Proof of Theorem 3.7. For the proof, we will make use of a few useful formulas:

Lemma 3.8. Let η, ξ be forms on M with |ξ| “ |η| ´ p2p`1q. The following hold:

(1)

tm¹_C, θL<sup>´pp`1q</sup>upη`θξq “ω^{p`1</sup>L<sup>´pp`1q}η´θ

´

rd, L^{´pp`1q</sup>sη´L<sup>´pp`1q}ω^p`1ξ

¯ . (2)

rd, L^´pp`1qsη“

#

´L<sup>pp`1q</sup>dΠ<sup>p</sup>η |η| ďn`p Π^p˚dL<sup>´pp`1q</sup>η |η| ąn`p (3) Theg² operation on anya1, a2 PF can be expressed as

g²pa₁, a₂q “

#

´θL<sup>´pp`1q</sup>pa<sub>1</sub>^a<sub>2</sub>q |a<sub>1</sub>|,|a<sub>2</sub>| ăn`p

0 otherwise

Proof of Lemma 3.8. (1): t,uis the graded commutator; sincemC andθL^{´pp`1q</sup>have odd degree, this meanstm<sup>1</sup><sub>C</sub>, θL<sup>´pp`1q}u “m¹_CθL<sup>´pp`1q</sup>`θL^´pp`1qm¹_C. We have

m¹_CpθL<sup>´pp`1q</sup>qpη`θξq “m¹_CpθL^{´pp`1q</sup>ηq “ωL<sup>´pp`1q}η´θdL^´pp`1qη and

θL<sup>´pp`1q</sup>m<sup>1</sup><sub>C</sub>pη`θξq “θL<sup>´pp`1q</sup>pdη`ωξ´θdξq “θpL<sup>´pp`1q</sup>dη`L^´pp`1qωξq.

The result follows.

(2): Decomposing

η“α_k`ω<sup>p`1</sup>β_k´2p´2`. . . we have that

dL<sup>´pp`1q</sup>η“dpβk´2p´2`. . .q “ B`βk´2p´2`ωpB´βk´2p´2` B`βk´2p´4q `. . . while

L^{´pp`1q</sup>dη“L<sup>´pp`1q}p. . .`ω<sup>p`1</sup>pB´β_k´2p` B`β_k´2p´2q `. . .q so we have

rd, L^{´pp`1q</sup>sη “ ´B´β<sub>k´2p</sub> which is equal to´L<sup>´pp`1q}dΠ^pη by definition of Π^p.

When |η| ąn`p, let|η| “n`p`r, withr ą0. Then,

pdL^{´pp`1q</sup>´L<sup>´pp`1q}dqη“ pdL^{´pp`1q</sup>´L<sup>´pp`1q}dqpω<sup>p`r</sup>β<sub>n´p´r</sub>`ω^p`r`1β_n´p´r´2`. . .q

“dpω^r´1βn´p´r`ω<sup>r</sup>βn´p´r´2`. . .q

´L<sup>´pp`1q</sup>pω<sup>p`r`1</sup>pB´β<sub>n´p´r</sub>` B`β<sub>n´p´r´2</sub>q `. . .q

“ω^r´1B`βn´p´r

which is what we claimed.

(3): By definition, we have that g² “ ´θL^{´pp`1q</sup>m<sup>2</sup><sub>C</sub>pg<sup>1</sup>bg<sup>1</sup>q. Note that if any term of m<sup>2</sup><sub>C</sub>pg<sup>1</sup>bg<sup>1</sup>q has a factor of θin it, the operation θL<sup>´pp`1q} renders it zero since θ^θ“0 inC. On the other hand,g¹aforaPF has a term without aθonly when|a| ăn`p.

Proof of Theorem 3.7. We need to show that the sequence of maps g^l : F^bl Ñ Cr1´ls satisfy theA8 map condition of (3.8)

ÿ

r`s`t“n

p´1q^r`stg^lp1^brbm^s_F b1^btq “ ÿ

i1`...`iq“n

p´1q^um^q_Cpgⁱ¹ b. . . g^iqq, ně1 wherel“r`1`tand the sign on the right-hand side is given by

u“ pq´1qpi₁´1q ` pq´2qpi<sub>2</sub>´1q `. . .`2pi<sub>q´2</sub>´1q ` pi_q´1´1q.

Noting thatm^k_C “0 for kě3 and m^k_F “0 for kě4, and further, that g^k“0 for kě3, (3.8) becomes the following four equations:

m¹_Cg¹ “g¹m¹_F (n=1)

m¹_Cg²`g<sup>2</sup>pm<sup>1</sup><sub>F</sub> b1`1bm¹_Fq `m²_Cpg¹bg¹q “g¹m²_F (n=2)

g²p1bm²_F´m²_F b1q `m²_Cpg¹bg²´g²bg¹q “g¹m³_F (n=3)

g²p1bm³_F `m<sup>3</sup><sub>F</sub> b1q `m²_Cpg²bg²q “0. (n=4)

The n“1 equation. We verified g“g¹ is a chain map in the proof of Lemma 3.4.

The n“2 equation. We first simplify the expression involving g² terms:

m¹_Cg²`g<sup>2</sup>pm<sup>1</sup><sub>F</sub> b1`1bm¹_Fq (3.9)

“m¹_Cp´θL<sup>´pp`1q</sup>m<sup>2</sup><sub>C</sub>pg<sup>1</sup>bg<sup>1</sup>qq ` p´θL<sup>´pp`1q</sup>m<sup>2</sup><sub>C</sub>pg<sup>1</sup>bg<sup>1</sup>qqpm<sup>1</sup><sub>F</sub> b1`1bm¹_Fq

“ ´m¹_CpθL<sup>´pp`1q</sup>qm<sup>2</sup><sub>C</sub>pg<sup>1</sup>bg<sup>1</sup>q ` p´θL<sup>´pp`1q</sup>m<sup>2</sup><sub>C</sub>qpm<sup>1</sup><sub>C</sub>b1`1bm¹_Cqpg¹bg¹q

“ ´m¹_CpθL^{´pp`1q</sup>qm<sup>2</sup><sub>C</sub>pg<sup>1</sup>bg<sup>1</sup>q ´ pθL<sup>´pp`1q}m¹_Cm²_Cqpg¹bg¹q

“ ´

´

m¹_CpθL<sup>´pp`1q</sup>q `θL^´pp`1qm¹_C

¯

m²_Cpg¹bg¹q

“ ´tm¹_C, θL^´pp`1qum²_Cpg¹bg¹q .

We have used that g¹ is a chain map, and that the operations m¹_C and m²_C satisfy the Leibniz rule. So the left-hand side of then“2 equation becomes

m¹_Cg²`g<sup>2</sup>pm<sup>1</sup><sub>F</sub> b1`1bm¹_Fq `m²_Cpg¹bg¹q “

´

´tm¹_C, θL<sup>´pp`1q</sup>u `1

¯

m²_Cpg¹bg¹q . (3.10)

Let a1, a2 be elements of F. Let us write

m²_Cpg¹bg¹qpa1ba2q “η`θ ξ.

Then, by Lemma 3.8(1), we find

`m<sup>1</sup><sub>C</sub>g<sup>2</sup>`g²pm¹_F b1`1bm<sup>1</sup><sub>F</sub>q `m²_Cpg¹bg¹q˘

pa₁ba₂q

“

´

´tm¹_C, θL<sup>´pp`1q</sup>u `1

¯

pη`θξq

“ p1´ω^{p`1</sup>L<sup>´pp`1q}qη`θ

´

rd, L<sup>´pp`1q</sup>sη` p1´L^{´pp`1q</sup>ω<sup>p`1}qξ

¯

“Π^pη`θ

´

rd, L<sup>´pp`1q</sup>sη`Π^p˚ξ

¯ . (3.11)

The expression forη and ξ are found by calculating m²_Cpg¹bg¹qpa1ba2q explicitly. Re- calling the definitions ofg¹ in Definition 3.3, we find that m²_Cpg¹bg¹qpa1ba2q equals

$

’’

’&

’’

’%

a1^a2´θpL<sup>´pp`1q</sup>da1^a2` p´1q^a1a1^L<sup>´pp`1q</sup>da2q |a1|,|a2| ďn`p

´θpp´1q^a1a₁^ ˚ra₂q |a₁| ďn`p,|a<sub>2</sub>| ąn`p

´θp˚ra1q ^a2 |a1| ąn`p,|a2| ďn`p

0 |a1|,|a2| ąn`p.

We can now read off the expression forη and ξ and substitute them into (3.11). In the case of|a1|,|a2| ďn, we have

η “a₁^a₂ , ξ“ ´L^{´pp`1q</sup>da<sub>1</sub>^a<sub>2</sub>´ p´1q<sup>a1</sup>a<sub>1</sub>^L<sup>´pp`1q}da₂ . (3.12)

Using Lemma 3.8(2), we find in this case that Π^pη`θ

´

rd, L<sup>´pp`1q</sup>sη`Π^p˚ξ

¯

“

$

’&

’%

Π^ppa1^a2q ´L<sup>´pp`1q</sup>dΠ<sup>p</sup>pa1^a2q |a1| ` |a2| ďn`p θΠ^p˚“

dL<sup>´pp`1q</sup>pa1^a2q |a1| ` |a2| ąn`p

´pL^{´pp`1q</sup>da<sub>1</sub>q ^a<sub>2</sub>´ p´1q<sup>a1</sup>a<sub>1</sub>^L<sup>´pp`1q}da₂‰

The other three cases in (3.11) are straightforward. All in all, we find that the left-hand side of the n “ 2 condition matches exactly the right-hand side (using the definition of m²_F in Section 2.3) given by

g¹m²_Fpa1ba2q

“

$

’’

’’

’’

’’

&

’’

’’

’’

’’

%

Π^ppa1^a2q ´θL<sup>´pp`1q</sup>dΠ<sup>p</sup>pa1^a2q |a1| ` |a2| ďn`p

´θΠ^p˚p´dL<sup>´pp`1q</sup>pa<sub>1</sub>^a<sub>2</sub>q |a<sub>1</sub>| ` |a₂| ąn`p

`L<sup>´pp`1q</sup>da1^a2` p´1q<sup>a1</sup>a1^L<sup>´pp`1q</sup>da2q with |a1|,|a2| ďn`p

´θΠ^p˚pp´1q^a1a1^ ˚ra2q |a1| ďn`p,|a2| ąn`p

´θΠ^p˚p˚ra₁^a₂q |a₁| ąn`p,|a<sub>2</sub>| ďn`p

0 |a₁|,|a₂| ąn`p