FRAMED E2STRUCTURES IN FLOER THEORY MOHAMMED ABOUZAID YOEL GROMAN UMUT VAROLGUNES Abstract. We resolve the long-standing problem of constructing the action

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FRAMED E2STRUCTURES IN FLOER THEORY

MOHAMMED ABOUZAID, YOEL GROMAN, UMUT VAROLGUNES

Abstract. We resolve the long-standing problem of constructing the action

of the operad of framed (stable) genus-0 curves on Hamiltonian Floer theory;

this operad is equivalent to the framed E2operad. We formulate the construc-

tion in the following general context: we associate to each compact subset of a

closed symplectic manifold a new chain-level model for symplectic cohomology

with support, which we show carries an action of a model for the chains on

the moduli space of framed genus 0 curves. This construction turns out to be

strictly functorial with respect to inclusions of subsets, and the action of the

symplectomorphism group. In the general context, we appeal to virtual funda-

mental chain methods to construct the operations over ﬁelds of characteristic

0, and we give a separate account, over arbitrary rings, in the special settings

where Floer’s classical transversality approach can be applied. We perform

all constructions over the Novikov ring, so that the algebraic structures we

produce are compatible with the quantitative information that is contained in

Floer theory. Over ﬁelds of characteristic 0, our construction can be combined

with results in the theory of operads to produce explicit operations encoding

the structure of a homotopy BV algebra. In an appendix, we explain how to

extend the results of the paper from the class of closed symplectic manifolds

to geometrically bounded ones.

Contents

1. Introduction 1

2. The Hamiltonian Indexing Multicategory 18

3. The Floer functor 31

4. The Floer algebra of a compact subset 39

5. Comparison of the two models 49

Appendix A. Trees and Riemann surfaces 63

Appendix B. Categorical and algebraic background 68

Appendix C. Dissipative cubes 74

References 82

1. Introduction

1.1. Hamiltonian Floer theory. In his study of the Arnol’d conjecture [Flo89],

Floer associated a cohomology group to each non-degenerate Hamiltonian H:M×

S1→Ron a closed symplectic manifold, based on the gradient ﬂow of the ac-

tion functional on the free loop space of M. Such gradient ﬂow lines correspond

to cylinders satisfying a deformation of the holomorphic curve equation. The fact

that one can study analogous equations on more general Riemann surfaces was

arXiv:2210.11027v3 [math.SG] 11 May 2024

2 MOHAMMED ABOUZAID, YOEL GROMAN, UMUT VAROLGUNES

ﬁrst observed by Donaldson, leading to the construction of an associative prod-

uct on Floer cohomology associated to pairs of pants, which was later shown by

Piunikhin–Salamon–Schwartz [PSS96] to yield a ring that is isomorphic to the quan-

tum cohomology ring for those manifolds satisfying the property that the class of

the symplectic form is a positive multiple of the ﬁrst Chern class. Separately, and in

the technically diﬀerent context of exact symplectic manifolds with contact bound-

ary, Viterbo [Vit99] introduced a circle action in Floer theory, which takes the form

of a degree −1 operator

(1.1) ∆: SH∗(M)→SH∗−1(M),

on a variant of Hamiltonian Floer cohomology, called symplectic cohomology, which

goes back to Hofer and Floer’s work [FH94] on the symplectic topology of open

subsets of Cn.

In this paper, we consider a version of Floer cohomology [Sei12, Gro23, Var21,

Ven18], which we call symplectic cohomology with prescribed support, in a change

from the previous terminology, which vastly generalises both of these frameworks,

but we shall temporarily suppress this point.

The pair of pants product makes sense in the context of symplectic cohomology

as well and, together with the operator ∆ introduced by Viterbo, is known to satisfy

the relation

(1.2) ∆(xyz) = ∆(xy)z+ (−1)|x|x∆(yz)+(−1)(|x|+1)|y|z∆(xy)

−∆(x)yz −(−1)|x|x∆(y)z−(−1)|x|+|y|xy∆(z),

which implies that symplectic cohomology forms a Batalin-Vilkovisky algebra; this

is a folklore result, whose proof appears, for example, in [Abo15, §2.5].

The geometry giving rise to Equation (1.2) can be expressed in terms of the

homology of the moduli space fMR

0,4of framed stable genus 0curves, where the

notion of framing corresponds to a choice of tangent ray at each marked point; the

left hand side corresponds to the class in the ﬁrst homology of fMR

0,4associated

to rotating the tangent ray at a speciﬁc marked point which is distinguished as

output, and the right hand side expresses this class as a sum of the classes associ-

ated to rotating each input and those associated to breaking the domain into two

components (each with three marked points) glued along the node, and rotating

the tangent ray on one side of the node.

This geometric picture suggests that the correct chain-level structure that gives

rise to the Batalin-Vilkovisky structure on symplectic cohomology is that of an

algebra over the operad formed by the moduli spaces fMR

0,k+1 (the case of k= 1 is

exceptional, and we set it to be equal to the circle S1). The operad structure arises

from the concatenation of Riemann surfaces with marked points to nodal Riemann

surfaces, which induces a map of chain complexes

(1.3) C∗(fMR

0,k1+1)⊗ · · · ⊗ C∗(fMR

0,kn+1)⊗C∗(fMR

0,n+1)→C∗(fMR

0,Pki+1),

whereas the algebra structure on symplectic cochains is a collection of operations

(1.4) SC∗(M)⊗ · · · ⊗ SC∗(M)

| {z }

⊗C∗(fMR

0,k+1)→SC∗(M),

FRAMED E2STRUCTURES IN FLOER THEORY 3

for some model C∗(fMR

0,k+1) of the singular chains on fMR

0,k+1, and some chain

complex SC∗(M) whose homology is symplectic cohomology, satisfying the fol-

lowing properties (we suggest [Lei04] as a reference for operads and algebras over

operads):

(1) Invariance under the action of the symmetric group on k-letters, acting by

permuting the ﬁrst kmarked point of elements of fMR

0,k+1, and the copies

of SC∗(M) in the domain of (1.4).

(2) Compatibility with the operations associated to concatenation of the moduli

spaces of framed curves (discussed in Appendix A.2), in the sense that the

map

(1.5)

SC∗(M)⊗Pn

i=1 ki⊗C∗(fMR

0,k1+1)⊗ · · · ⊗ C∗(fMR

0,kn+1)⊗C∗(fMR

0,n+1)

SC∗(M)⊗Pn

i=1 ki⊗C∗(fMR

0,Pn

i=1 ki+1)

obtained either by ﬁrst separately applying the maps in Equation (1.4) for

k=ki, then applying it for k=n, agrees with the map obtained by ﬁrst

applying Equation (1.3), followed by Equation (1.4) for the sum k=Pki.

The main result of the paper establishes the existence of such a structure, which

we moreover show to satisfy the following fundamental properties:

•Independence of auxiliary choices.

•Strict functoriality under restriction maps.

•Compatibility with quantitative structures.

We shall give a precise formulation of our main result in Theorem 1.4 below, which

we precede by a necessary overview of the notion of support for symplectic coho-

mology. Afterwards, we shall discuss our strategy for the proof of these results, as

well as the relationship between the operad formed by the moduli spaces fMR

0,k+1,

and the framed E2operad mentioned in the title, which is more extensively studied

in the literature.

Remark 1.1.It will be apparent from our methods that one can construct a model

for symplectic cochains, satisfying the above list of properties, and carrying an

action of the operad formed by the chains of the union of the moduli spaces of framed

Riemann surfaces fMR

g,k+1 of arbitrary genus (more precisely, one has to shift the

chains by a function of the genus to account for the degree of the corresponding

operations). We leave the details of such an extension to the reader largely because

it is orthogonal to the interesting operations in higher genus, which require one to

consider gluing at multiple points (the operadic structure only allows operations

with one output, which corresponds to gluing at one point). We expect that such an

extension would require a more signiﬁcant use of methods of homotopical algebra

than the present paper.

Remark 1.2.There is a natural analogy between the chain structures we are con-

structing, and those which appear in Lagrangian Floer theory, leading to the ques-

tion of why one cannot construct the operations in Equation (1.4) by a procedure

which follows the existing steps in that context. To explain the essential diﬃculty,

4 MOHAMMED ABOUZAID, YOEL GROMAN, UMUT VAROLGUNES

recall that, notwithstanding the technicalities in resolving questions of anomaly

and obstruction which are required to deﬁne the Floer cohomology groups of a

Lagrangian L, it is by now well-established [FOOO09, Sei08b] that one obtains an

A∞structure on the Lagrangian Floer chain complex, which can be written as a

consistent collection of operations

(1.6) CF∗(L)⊗ · · · ⊗ CF∗(L)⊗C∗(R0,k+1)→CF∗(L),

where R0,k+1 is the moduli space of stable discs with k+ 1 marked points.

The fundamental diﬀerence between Equations (1.4) and (1.6) is that the moduli

spaces R0,k+1 have a particularly simple topology: they can be realised as poly-

topes (the Stasheﬀ associahedra), and the operadic structure maps which control

the consistency of the operations are given by inclusions of products of these poly-

topes as boundary faces. This leads to the algebraic structure being controlled by

relatively simple combinatorics: there is a collection of operation indexed by the

natural numbers, one for each moduli space R0,k+1, satisfying the familiar A∞re-

lation, which is the way that Equation (1.6) is implemented in the literature, with

the model for C∗(R0,k+1) given by the cellular chains of the polytope structure.

Because the geometry of the moduli spaces fMR

0,k+1 is much more complicated

such an approach is not adequate for Equation (1.6). This is already apparent for

the case of operations with only one input, which we recall are given by a copy of S1,

with composition map S1×S1→S1given by the usual multiplication. Evidently,

this map is not given by a cellular inclusion. In fact, in the standard model for

symplectic cochains, the chain level structure corresponding to the circle action is

given [Sei08a] by a sequence of operations ∆i, indexed by the natural numbers,

satisfying d∆n=P∆i∆n−i.

1.2. Support conditions for symplectic cohomology. A standard construc-

tion associates to each compact subset Kof a space Mthe indicator (characteris-

tic) function HKwhich is 0 on Kand is inﬁnite away from it. This construction

is functorial with respect to inclusions in the sense that, whenever Kis a subset of

K′, we have a pointwise inequality

(1.7) HK′≤HK.

When Mis a symplectic manifold (which we now assume to be closed for sim-

plicity), symplectic cohomology with support Kcan be thought of as a lift of the

above construction to cohomology groups: since Floer cohomology is not deﬁned for

discontinuous functions (nor those which take inﬁnite value), one considers instead

a sequence of (non-degenerate) Hamiltonians Hiconverging to HK, to which one

associates the Floer cochain groups CF∗(Hi, Ji) for an auxilliary sequence of almost

complex complex structures Ji. It is crucial at this stage to be careful with the

choice of coeﬃcients: a modern interpretation of Floer’s invariance result [Flo89]

is that the isomorphism type of the Floer cohomology groups does not depend on

the choice of Hamiltonian when the coeﬃcient ring is the Novikov ﬁeld1, whose

elements are series P∞

i=0 aiTλiwith ailying in a chosen ground ring k, and λi

real numbers with the property that limiλi= +∞. In order to retain dynamical

information about the functions Hi, one works instead with the smaller Novikov

1Because we allow the ground ring kto be an arbitrary ring, rather than a ﬁeld, using the

term Novikov ﬁeld in this context is an abuse of terminology. Similarly, the Novikov ring is not

strictly speaking a valuation ring, even though we shall use the term.

FRAMED E2STRUCTURES IN FLOER THEORY 5

ring whose elements consists of series for which the exponents λiare non-negative.

The category of modules inherits a natural completion functor associated to the

T-adic ﬁltration, which is deﬁned as the inverse limit of the tensor product with

quotients of the Novikov ring by powers of T, which is essential to the following:

Deﬁnition 1.3. The symplectic cohomology SH∗

M(K)with support a compact

subset Kof a closed symplectic manifold is the homology of the completion of the

(homotopy) direct limit of the chain Floer complexes of a monotone sequence of

Hamiltonians converging to the indicator functor HK:

(1.8) SC∗

M(K)≡\

hocolim

iCF∗(Hi, Ji).

In the above deﬁnition, the fact that we assume the sequence {Hi}of Hamilto-

nians to be monotone is essential in realizing the maps in the direct limit as maps

of Floer complexes over the Novikov ring, which is what allows us to deﬁne the

symplectic cochains supported on Kby completion. Geometrically, this is a con-

sequence of the fact that monotone continuation maps always have non-negative

(topological) energy, which is a property that fails for general continuation maps.

As we shall discuss in Section 1.5 below, the speciﬁc model for the homotopy

colimit used in [Var21] is the mapping telescope, which is a complete chain complex

receiving a map from each Floer group CF∗(Hi, Ji) in the chosen sequence, together

with a homotopy in each triangle

(1.9)

CF∗(Hi, Ji) CF∗(Hi+1, Ji+1)

SC∗

M(K),

where the horizontal map is the continuation map. In fact the underlying homotopy

type of the mapping telescope can be characterised by a universal property asso-

ciated to this data. This more abstract point of view will be useful to understand

our construction.

Symplectic cohomology supported on Kis independent of the choice of approx-

imating Hamiltonians, recovers ordinary homology when M=K, and is functorial

under inclusions in the sense that there is a restriction map

(1.10) SH∗

M(K′)→SH∗

M(K)

whenever Kis a subset of K′, which is strictly compatible for triple inclusions. In

addition, it satisﬁes a remarkable Mayer-Vietoris property, for a class of coverings

which include those that are arise from a covering of the base of an coisotropic

ﬁbration. This property is crucial for recent applications both to symplectic topol-

ogy [DGPZ21] and to mirror symmetry [GV23]. The last reference includes an

extension of the deﬁnition of symplectic cohomology with support given by com-

pact subsets to the case where the ambient symplectic manifold Mis geometrically

bounded, incorporating all the classes of open symplectic manifolds for which Floer

theory is expected to be deﬁned.

1.3. Statement of results. We now state the main result of this paper, which is

proved in Section 4:

Theorem 1.4. Associated to any compact subset Kof a closed symplectic mani-

fold is a complete torsion free chain complex SC∗

M,fMR

(K)over the Novikov ring,

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FRAMEDE2STRUCTURESINFLOERTHEORYMOHAMMEDABOUZAID,YOELGROMAN,UMUTVAROLGUNESAbstract.Weresolvethelong-standingproblemofconstructingtheactionoftheoperadofframed(stable)genus-0curvesonHamiltonianFloertheory;thisoperadisequivalenttotheframedE2operad.Weformulatetheconstruc-tioninthefollowinggeneralcontext:...

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FRAMED E2STRUCTURES IN FLOER THEORY MOHAMMED ABOUZAID YOEL GROMAN UMUT VAROLGUNES Abstract. We resolve the long-standing problem of constructing the action

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