A SHARP LOWER BOUND ON FIXED POINTS OF SURFACE SYMPLECTOMORPHISMS IN EACH MAPPING CLASS

. Given a closed, oriented surface Σ, possibly with boundary, and a mapping class, we obtain sharp lower bounds on the number of ﬁxed points of a surface symplectomorphism (i


Geometry & Topology msp
Volume 27 (2023) A sharp lower bound on fixed points of surface symplectomorphisms in each mapping class 1 Introduction 1.1 Sharp fixed point bounds for surface symplectomorphisms 1.1.1Overview Let .†; !/ be a compact surface of negative Euler characteristic, 1  possibly with boundary, with ! a symplectic form (ie area form).For any mapping class we give sharp lower bounds on the number of fixed points of an area-preserving map in the mapping class, both in the case in which is assumed to have nondegenerate fixed points 2 and in the general case in which degenerate fixed points are allowed.This generalizes the Poincaré-Birkhoff fixed point theorem, which states that areapreserving twist maps of the annulus have at least two fixed points, to arbitrary surfaces and mapping classes.
1.1.3Mapping classes and Thurston's classification Let Diff C .†/ denote the space of orientation-preserving diffeomorphisms of †.For closed surfaces, mapping classes are elements of 0 .Diff C .†//.For surfaces with boundary, we define Diff C @ .†/ to be the space of orientation-preserving diffeomorphisms with no fixed points on the boundary, and use the term mapping classes to refer to elements of 0 .Diff C @ .†//, though this is not standard.By Moser's trick, in dimension two, these are homotopy equivalent to the versions with Diff replaced by Diff Vol D Symp.
We state our results in terms of Thurston's classification of surface diffeomorphisms (see Thurston [20] and Fathi, Laudenbach and Poénaru [4]), which states that given a compact, oriented surface † (with or without boundary) every element of 0 .Diff C .†// is precisely one of the following types: Periodic (finite order) For some representative , we have `D id for some `2 Z >0 .
Pseudo-Anosov Some singular representative preserves two transverse singular measured foliations, expanding the measure on one and contracting the measure on the other.See Cotton-Clay [1] for symplectic smoothings of these singular representatives.
Reducible Some representative fixes setwise a collection of curves C , none of which are nullhomotopic or boundary parallel (and the mapping class is not periodic).
In the reducible case, we call such a curve C above a reducing curve.Cutting along a maximal collection of pairwise nonhomotopic reducing curves C gives a map on each component of † n C , given by the smallest power of which maps that component to itself, which is periodic or pseudo-Anosov.We call these components of † n C reducible components or geometric components when extending this to the case of the entire surface for nonreducible maps.We say a component abuts the reducing curves making up its boundary, as well as any curves making up its actual boundary.

Fixed annuli and standard representatives
To understand fixed points of surface symplectomorphisms, we are interested in components which map to themselves.In addition to the geometric components described above, we must extend the notion of components to allow for a number of fixed annuli between geometric components or near a boundary when there is sufficient twisting along the reducing curve or at a boundary curve.After standard perturbations (see Section 4.1.1),near a reducing curve or boundary curve C , we choose standard representatives to have twisting regions (à la Dehn twists) connecting the components.Such a twisting region may have any number (in Z 0 ) of circles of fixed points, which we think of as fattened to annuli and consider as fixed annuli.See Section 4.1.2for a definition and, for an optional further elucidation of these, see Section 4.1.3relating counting the number of these to a variant of fractional Dehn twist coefficients, related to the concept in Honda, Kazez and Matić [7].
These fixed annuli are important to consider as they can show up in one of two types of components (see Section 1.1.5below) which contribute to excess fixed points over the traditional Nielsen theory lower bound.Indeed, the case of the Poincaré-Birkhoff fixed point theorem can be thought of as a solitary fixed annulus in this context.

Excess over Nielsen lower bound: components of types A and B
In what follows, these fixed annuli are additionally considered components, on which the induced map is considered to be the identity.Additionally, we consider mapping classes on a surface with boundary with fixed annuli reducible in 0 .Diff C @ .†//, even if they are not reducible in 0 .Diff C .†//.
There are two settings, in terms of standard representatives, in which the minimum for area-preserving maps exceeds the topological minimum from Nielsen theory.Both involve components on which the induced map is the identity:

Components of type A We say a component is of type A if
its induced map is the identity, it is genus-zero but nonannular, it does not abut with twist 0 2 Q any pseudo-Anosov components, all its boundaries are nullhomologous or boundary-parallel, and all its boundaries have twist in Q with the same sign or 0.

Components of type B We say a component is of type B if
it is a fixed annulus (as above), and it is nullhomologous or boundary-parallel.
Here twist is in the sense of the fractional Dehn twist coefficient of [7].

Theorems and discussion
Theorem 1.1 Let .†; !/ be a compact, oriented surface, possibly with boundary, with area form !.The minimum number of fixed points of an area-preserving map with nondegenerate fixed points in a mapping class h is given by Here A is the number of components of type A, and B the number of components of type B, of a standard representative of the mapping class h.
The upper bound here is given by construction, which comes from perturbing maps which are nice with respect to the Nielsen-Thurston geometry, which we call standard form maps, with particular symplectic vector fields.The lower bound comes from Floer homology computations for these standard form maps with certain twisted coefficients, based on computations we performed in [1], plus a result discussed below showing how computations with appropriate twisted coefficients give fixed point bounds over entire mapping classes.
In this case, counting nondegenerate fixed points, components of types A and B each contribute an additional two fixed points, analogous to the Poincaré-Birkhoff fixed point theorem, which can be thought of as the case of a solitary fixed annulus.

Theorem 1.2
The minimum number of fixed points of an area-preserving map in a mapping class h is given by where A and B are as in Theorem 1.1.
Geometry & Topology, Volume 27 (2023) To clarify Theorem 1.2, both components contributing to A and components contributing to B contribute two to the minimum number of fixed points, but those contributing to A have nonzero index and thus have already contributed one in the traditional Nielsen bound.For type A components, degeneracy allow collapsing down to two fixed points, but (also with type B components) no further.
Though we have restricted to surfaces of negative Euler characteristic, we note that a twist map on the annulus may be considered to have one component of type B, and the two fixed points guaranteed by the Poincaré-Birkhoff fixed point theorem appear in the 2B term here.
Again the upper bound is given by construction.The lower bound comes from "quantum cup-length" computations for a certain cohomology operation4 on the summand of Floer homology corresponding to the given Nielsen class, plus a compactness argument of Taubes.This cohomology operation is morally given by counting intersections of holomorphic cylinders with cycles in H 1 .S; @S/ for S a reducible component on which the map is the identity whose fixed points are in the Nielsen class.The fact that this is well defined comes from an understanding of the homotopy type of the component of the space of sections .M / of M !S 1 corresponding to the given Nielsen class, plus an algebraic invariance result given in [1] for certain types of cohomology operations on Floer homology.

Fixed point bounds for monotone symplectic manifolds.
Let .X; !/ be a symplectic manifold, and consider the problem of finding fixed point bounds for symplectic mapping classes, which are connected components of Symp.X /.
The symplectic Floer homology of a symplectomorphism is the homology of a chain complex generated by the fixed points of ; see Section 2 for more details.Thus the rank of Floer homology gives a bound on the number of fixed points of , and results on the invariance of Floer homology under deformation of give more general fixed point bounds.A main challenge in obtaining fixed point bounds for symplectic mapping classes is that we are interested in fixed point bounds on connected components of Symp.X / as opposed to on Ham.X /-cosets of Symp.X /.Floer homology is invariant under Hamiltonian perturbations between maps with nondegenerate fixed points, so as long as we can define it on a given Ham.X /-coset, the rank of the Floer homology for any map in the coset is a bound on the number of fixed points for all nondegenerate maps in the coset.
To deal with this, we give a general method for finding fixed point bounds for symplectic mapping classes on monotone symplectic manifolds .X; !/.We say that .X; !/ is monotone if OE! 2 H 2 .X / is a positive multiple of c 1 .X /. 5 The method consists of performing a single Floer homology computation, for a suitable map in the symplectic mapping class, with suitable twisted coefficients.Along the way, we show that Floer homology computations for one suitable map with various twisted coefficients give computations of HF .; ƒ /, the Floer homology of with its natural Novikov coefficients ƒ , for any in the symplectic mapping class.
We identify a subset of each mapping class, which we call weakly monotone, such that Floer homology is defined with any coefficients and is invariant under deformations through such maps.In [1] we have shown that standard form maps on surfaces are in this subset.Let !be the two-form on M induced by ! on † R and let c be the first Chern class of the vertical tangent bundle of M !S 1 .
where T 0 .M / H 2 .M I R/ is generated by tori T with c .T / D 0 such that j T W T !S 1 is a fibration with fiber S 1 , where the map W M !S 1 is the projection.
In Section 3.1 we define a flux map.Monotone maps have zero flux, which allows their Floer homology with untwisted coefficients to be computed.Weakly monotone maps need not have zero flux, but they still have well-defined Floer homology with untwisted coefficients.Further, if we apply well-chosen twisted coefficients, the Floer homology gives a lower bound on fixed points for non-weakly-monotone maps in the same symplectic mapping class.
Consider a flux map Flux./W N h !R.Here N h denotes the image under the map H 2 .M / !ker.
id/ H 1 .†/ of ker.c / H 2 .M /.Let N 0 h be the image under the map from the long exact sequence for the mapping torus fibration H 2 .M / !ker.
Theorem 1.4 Let X be a monotone symplectic manifold , h a symplectic mapping class on X , and 2 h.
(i) Flux./W N h =tors !R is well-defined.
(ii) Flux./j N 0 h =tors D 0 if and only if is weakly monotone.(iii) Let be such that Flux./j N h D 0. Then the rank of HF .I Q.Z=2OEN 0 h =tors// gives a lower bound on the rank HF .I ƒ / for any map in the class h, where ƒ is the Novikov ring over which HF ./ is naturally defined , and thus gives a lower bound on the number of fixed points of a map in the mapping class with nondegenerate fixed points.
Here Q.Z=2OEN 0 h =tors/ is the quotient field of the group ring of N 0 h =tors over Z=2.The Novikov ring ƒ is a ring over which HF ./ can be defined, with generators given by fixed points, even if is not weakly monotone.The main idea is that in some cases the field Q.Z=2OEN h / injects into ƒ , in which case we have a field extension and the ranks of the homology are the same.When we do not have an injection, we can extend ƒ to a larger Novikov ring into which Q.Z=2OEN h / does inject.When homology is computed over the larger Novikov ring, the rank can only decrease, giving the lower bound in Theorem 1.4.Yi-Jen Lee's bifurcation analysis [12; 13] is vital to this argument, giving a way to compare the Floer homology of maps with proportional fluxes.We note that Lee and Taubes [14,Corollary 6.6] have a similar Theorem for periodic Floer homology coming from their isomorphism with Seiberg-Witten Floer homology.Lee also informed us that she was independently aware of the existence of results such as Theorem 1.4.

Organization of the paper
In Section 2 we review Floer homology and Nielsen classes; develop general twisted coefficients for Floer homology; review Novikov rings; and discuss invariance results due to Yi-Jen Lee [12; 13].
In Section 3 we give a general method for finding fixed point bounds on a monotone symplectic manifold using a Floer homology computation for any weakly monotone symplectomorphism with a particular choice of twisted coefficients.
In Section 4 we carry out this method in the case of surface symplectomorphisms to give a lower bound, using computations from [1].We obtain an equal upper bound by explicit constructions.
In Section 5, we give fixed point bounds for surface symplectomorphisms with possibly degenerate fixed points.We use a certain cohomology operation in place of the quantum cap product, which vanishes in the situation of interest, to give cup-length-type bounds.An equal upper bound is again given by explicit constructions.
Acknowledgements This paper is adapted from part of my thesis [2] at UC Berkeley under Michael Hutchings, who gave invaluable advice and support.I'd also like to thank MIT for their hospitality during my last year of graduate school, and Denis Auroux, for his advising and helpful discussions.The reviewers and editors also gave numerous valuable comments and suggestions.Let .M / denote the space of smooth sections of W M !S 1 .Note that a fixed point x 2 † of can be interpreted as a constant section x .Let J denote the space of almost complex structures J on R M which are R-invariant, preserve the vertical tangent bundle of W R M !R S 1 , and for which is .J; j /-holomorphic, given the standard complex structure j on the cylinder R S 1 .
Suppose has nondegenerate fixed points, in the sense that d does not have 1 as an eigenvalue at any fixed point.Let P x;y .M / denote the space of paths from x to y in .M /.Let C 2 0 .P x;y .M //.Given a generic J 2 J , the moduli space M. ; x; y; C / of holomorphic sections R S 1 !R M in the homotopy class C is smooth and compact, of dimension ind.C /; see [5] or [6].
Geometry & Topology, Volume 27 (2023) For a statement of the index formula, see [16], or [1, Section 2.2] for a discussion tailored to the current setting.Other than computations that we will cite below from [ We have the following conditions.Definition 2.3 A map 2 Symp @ .†; !/ is monotone if ! vanishes on the kernel of c .Definition 2.4 A map 2 Symp @ .†; !/ is weakly monotone if ! vanishes on T 0 .M /, where T 0 .M / H 2 .M I R/ is generated by tori T with c .T / D 0 such that j T W T !S 1 is a fibration with fiber S 1 , where the map W M !S 1 is the projection.
Under either of these conditions, the moduli space M. ; x; y; C / is compact.Thus we may define they symplectic Floer homology HF ./ of with coefficients in Z=2 to be the homology of the Z=2-graded where the R-action is by translation in the R-direction in R M , and the Z=2-grading of a fixed point x is given by the sign of det.1 d /.The homology of this chain complex is invariant under deformations through monotone or weakly monotone maps.

Nielsen classes and Reidemeister trace
There is a topological separation of fixed points due to Nielsen: given a fixed point x of , we obtain an element OE x 2 0 ..M //, the free homotopy class of x in the space of sections .M /.We denote this homotopy class by .M / OE x .The chain complex CF ./ defined above, as well as all the variants to be defined below, split into direct summands for each Nielsen class Á 2 0 ..M //: HF .; Á/: We note that in the case of surfaces of negative Euler characteristic, Nielsen classes are well-defined on entire mapping classes (ie there is no monodromy).Thus the rank of HF .; Á/ is a lower bound on the number of fixed points in the Nielsen class Á on, for example, the weakly monotone subset of the mapping class.
A simpler lower bound on the number of nondegenerate fixed points in a Nielsen class Á 2 0 ..M // is given by the absolute value of the index of the Nielsen class ind.Á/ D X xWOE x DÁ ind.x/: We note that ind.Á/ is the Euler characteristic of HF .; Á/.A lower bound for the number of fixed points of a map with nondegenerate fixed points in the same mapping class as is thus given by X jind.Á/j: For maps with possibly degenerate fixed points, the bound from Nielsen theory is the number of essential Nielsen classes; that is, the number of Nielsen classes Á for which ind.Á/ ¤ 0.

Twisted coefficients
We have a groupoid G whose objects are sections of the mapping torus M and whose morphisms from x to y are given by homotopy classes of paths from x to y Geometry & Topology, Volume 27 (2023) in .M /.Given a ring R, we can take the groupoid algebroid RhG i, a category enriched over R-modules with the same objects as G and with Hom RhG i .x;y/ given by the free R-module generated by elements of Hom G .x; y/.Note that we have a homomorphism indW G ! Z given by the index.
If we have a representation from RhG i to an R-module M , we can define Floer homology with coefficients in .orM if is understood/ as the homology of a chain complex over M with generators the fixed points of and differential given by where OEC is the homotopy class of the path in .M / associated to the flow line C , and we abuse notation by identifying y with the section y 2 .M /.This is defined for weakly monotone for arbitrary (because then M 1 .; x; y/ is compact).In Section 2.4 we discuss representations ƒ suitable for each .
We will typically suppress the ring R, which throughout this paper may be assumed to be Z=2.
The standard representation st into the group ring of H 1 ..M // is defined as follows.
For every pair of sections x; y and every index i , we choose a path C i x;y in .M / between them, of index i if possible.We require that C i x;y C j y;z ' C iCj x;y ; C i x;y D C i y;x and C 0 x;x D : Here signifies appending paths, signifies reversal of a path, and signifies the constant path.Then st .OEC /, for C a path from x to y, is defined to be OEC C ind.C / y;x 2 H 1 ..M //.Note that in fact this lies in ker.c / H 1 ..M //.We have made choices, but the resulting Floer homology is well-defined up to a change of basis.
We typically compose this with the map H 1 ..M // !H 2 .M / to get what we call fully twisted coefficients, over Z=2OEH 2 .M /.If we desire to have field coefficients, we may for example take the quotient field of the group ring of H 2 .M /=tors.

Novikov rings
Definition 2.6 For R a ring, G an abelian group and N W G ! R a homomorphism, the Novikov ring Nov R .G; N / is defined to be the ring whose elements are formal sums P g2G a g g, where a g 2 R are such that for each r 2 R, only finitely many of the a g with N .g/< r are nonzero.
There is also the universal Novikov ring: Definition 2.7 For R a ring, the universal Novikov ring ƒ R is defined to be the ring whose elements are formal sums P r 2R a r T r , where a r 2 R are such that, for each s 2 R, only finitely many of the a r with r < s are nonzero.
Note that this is a field if R is a field.We have maps Nov R .G; N / !ƒ R given by g 7 !T N .g/ .If R is a field and N is injective, this is an extension of fields.
We define a representation, the natural Novikov coefficients for , denoted by ƒ , as follows.We have a representation into Nov Z=2 .ker.c /; !/, where the maps c ; !W H 1 ..M // !R are defined as in Section 2.1.The representation is defined in the same manner as the standard representation; we have simply taken a submodule ker.c / H 1 ..M // in which the image must lie, and allowed certain infinite sums.We further compose this with the map to ƒ Z=2 .This all is simply to say we take We may define HF .; ƒ /, the Floer homology of with coefficients in ƒ for any .The point is simply that, while we may not have finiteness for M 1 .x;y/, we do have that there are only finitely many C 2 M 1 .x;y/ with !.C / < r for any given r 2 R, by Gromov compactness, and this is precisely what is required.

Bifurcation analysis of Yi-Jen Lee
Yi-Jen Lee [12; 13] has worked out a general bifurcation argument for what she calls Floer-type theories.Michael Usher [21] has a nice summary of the invariance result this gives (which Lee conjectured in an earlier paper [11, equation 3.2] but did not explicitly state as a theorem in [12; 13]) and its algebraic aspects.We have adapted the statement to our setting involving representations of Z=2hG i.
Geometry & Topology, Volume 27 (2023) We point out that in item (i) we are indeed taking the Floer homology of 1 with coefficients given by the representation ƒ 0 .What this means is that we use !0 to determine the power of T in .OEC /.
3 Bounds on fixed points in symplectic mapping classes

Representations and flux
Throughout this section consider .X; !/ a monotone symplectic manifold.The goal of this section is to give a lower bound on fixed points for symplectomorphisms in a symplectic mapping class h 2 0 .Symp.X; !// in terms of the rank of HF .; / for one suitable choice of a pair .; / with 2 h and a representation of Z=2.G /.In this subsection we define the representations m for which it will be shown that .; m / is such a pair for any monotone , and wm for which it will be shown that .; wm / is such a pair for any weakly monotone .

Monotone case
We have a representation m that works for any monotone .The representation m is defined as follows: We have the representation st into Z=2OEH 1 ..M //.We first compose7 with the map H 1 ..M // !H 2 .M /, noting that the image lies in the kernel of c W H 2 .M / !R. Now we compose with the map to H 1 .X / in the long exact sequence for the mapping torus, a part of which is We denote the image of ker.c / in H 1 .X / by N h (this depends only on the mapping class h).Finally, we mod out by torsion and then take the quotient field, so that our coefficients lie in the field where Q denotes taking the quotient field.Essentially we have taken fully twisted field coefficients, but we have been careful about where the image lies so that we can define flux in this context, which both will be useful in specifying which maps are suitable to work with and will be a useful tool in what is to come.

Andrew Cotton-Clay
We have a characterization of monotone maps in terms of Flux, which we define using the long exact sequence (1) above: Definition-Lemma 3.1 The flux of any symplectomorphism , denoted by and defined as Flux./. / D ! .C /; where C 2 H 2 .M / is such that @.C / D and c .C / D 0, is well-defined.Here @ is the map from the long exact sequence (1).
Proof First, we note that such a C exists because N h is the image under @ (in the long exact sequence) of ker.c /.Next we note that !.C / is well-defined because if we take any other C 0 such that @.C 0 / D and c .C 0 / D 0, the difference C C 0 is in the kernel of c and, being in the kernel of @, must also come from an element of H 2 .X /, which we denote by B. We have that c as well, we conclude that !.C / !.C 0 / D 0. Finally, we remark that this is a homomorphism, and thus any torsion in N h must map to 0 2 R. Lemma 3.2 For a symplectomorphism , we have is monotone ( ) Flux. / D 0: Proof The statement Flux./ D 0 is equivalent to ! vanishing on the kernel of c .This implies that ! is some multiple of c .
Weakly monotone case With .X; !/ monotone, we also give a representation wm that works for any weakly monotone .The representation wm is defined as follows: Let the image of the map H 1 ..M // !H 2 .M / be denoted by T .M /.This is generated by 2-tori in M standardly fibering over S 1 .When composing the standard representation into Z=2OEH 1 ..M / with the map H 1 ..M // !H 2 .M /, the image thus lies not only in ker.c / but in ker.c j T .M / /.In fact it lies moreover in T 0 .M /, generated by 2-tori T in M standardly fibering over S 1 such that c .T / D 0. We denote the image of T 0 .M / under @ in the long exact sequence as id/ H 1 .X /: Geometry & Topology, Volume 27 (2023) As before, we mod out by torsion and then take the quotient field, so our coefficients lie in the field Q.Z=2OEN 0 h =tors/: We restrict Flux./ to N 0 h =tors N h =tors.
Lemma 3.3 For a symplectomorphism , we have is weakly monotone Nielsen class Á ( ) Flux. /j N 0 h =tors D 0: Proof The statement Flux./ N 0 h =tors D 0 is equivalent to ! vanishing on T 0 .M /.Let T 0 .M / Á be generated by 2-tori T in M standardly fibering over S 1 such that c .T / D 0 and which moreover have a section in Nielsen class Á.Then the vanishing of ! on T 0 .M / is equivalent to ! vanishing on T 0 .M / Á for every Á, because every such torus has a section, which lies in some Nielsen class, ie component of .M /.

Corollary 3.4
The space Symp wm h .X; !/ of maps in the mapping class h which are weakly monotone Nielsen class Á is homotopy equivalent under the inclusion to Symp h .X; !/.

Fixed point bounds
Theorem 3.5 Let .X; !/ be a monotone symplectic manifold , h a symplectic mapping class on X , and 2 h.Let be such that Flux./j N 0 h D 0. Then the rank of gives a lower bound on the rank HF .I ƒ / for any map in the class h.
Corollary 3.6 Let X be a monotone symplectic manifold , h a symplectic mapping class on X , and 2 h.Let be such that Flux./j N 0 h D 0. Then the rank of gives a lower bound on the number of fixed points of any map in the mapping class with nondegenerate fixed points.
Remark 3.7 The corresponding versions with such that Flux./ D 0 and coefficients in Q.Z=2OEN h =tors/ also hold, either with the same proofs or as a formal consequence. .Note that such a family exists as we can modify Flux in any direction we desire by flowing by an S 1 -valued Hamiltonian representing, as a map to S 1 , the desired cohomology class of the modification.Then (for a generic family of almost complex structures) the conditions of (i) in Theorem 2.8 are met with f .r/ D 1 r .Summing over Nielsen classes gives the result.Lemma 3.9 With and as above, the rank of HF. ; ƒ / equals the rank of The first is by allowing some infinite sums, and the second is the map discussed in Section 2.4.Field extensions are flat, so the ranks are equal.over Q and as such their group rings over Z=2 and the aforementioned augmentation map are as in Lemma 3.10, where n and m are their respective dimensions, with k D Z=2.Thus, by Lemma 3.10, the rank of the homology of the displayed complexes is less than or equal to the rank of HF. ; Z=2OEN 0 h =tors/, which is the same as the rank of HF. ; Q.Z=2OEN 0 h =tors//, giving the result.
4 Bounds on fixed points for surface symplectomorphisms

Standard form maps and Nielsen classes
In this section we describe standard form maps and their Nielsen classes.See also [1,, which has more background.In this case every fixed point is in the same Nielsen class.This Nielsen class has index given by the Euler characteristic of the surface.
Definition 4.2 For nonidentity periodic mapping classes, a standard form map is an isometry with respect to a hyperbolic structure on the surface with geodesic boundary.
Every fixed point is in a separate Nielsen class and each of the Nielsen classes for which there is a fixed point has index one.See [1, Section 3.2] for further details, including an introduction to pseudo-Anosov maps.For the specified smoothing, see [1, Figure 3 and surrounding].See also the top images in Figure 1 for a rendition in an example.
There is a separate Nielsen class for every smooth fixed point, which is of index one or minus one; for every fixed singularity, which when symplectically smoothed gives p 1 fixed points all of index minus one if the rotation number is zero modulo p, or one fixed point of index one otherwise [1]; and for every fixed boundary component with rotation number zero modulo p, which when symplectically smoothed gives p fixed points all of index minus one.
In the pseudo-Anosov case, note that what we are using as a boundary is a deformation of a punctured singularity in terms of the standard singular representative.
Remark 4.4 From this discussion, we see that for nonidentity periodic and pseudo-Anosov mapping classes, the standard form map is such that all fixed points are nondegenerate and, for every Nielsen class Á, the number of fixed points in Á is jind.Á/j.

Standard form for reducible maps and fixed annuli
In addition to the geometric components described above, we extend the notion of components to allow for a number of fixed annuli between geometric components or near a boundary when there is sufficient twisting along the reducing curve or at a boundary curve.
To obtain a standard form for reducible mapping classes, the standard forms for geometric components, described above, are joined by twisting regions (à la Dehn twists).Each twisting region may have any number (in Z 0 ) of circles of fixed points, which we think of as fattened to annuli and consider as fixed annuli.Formally we have a standard form whose existence follows from Thurston's classification [20; 4]; see also [1,Definition 4.6].
Definition 4.5 A reducible map is in standard form if there is a -and 1 -invariant finite union of disjoint noncontractible (closed) annuli N † such that: (i) For A a component of N and `the smallest positive integer such that `maps A to itself, the map `jA is either a twist map or a flip-twist map.That is, with respect to coordinates .q;p/ 2 OE0; 1 R=Z, we have one of .q;p/ 7 !.q;p f .q//(twist map), .q; p/ 7 !.1 q; p C f .q//(flip-twist map), where f W OE0; 1 !R is a strictly monotonic smooth map.We call the (flip-) twist map positive or negative if f is increasing or decreasing, respectively.Note that these maps are area-preserving.
(ii) Let A and `be as in (i).If `D 1 and j N is a twist map, then Im.f / OE0; 1.
That is, j int.A/ has no fixed points.(If we are to twist multiple times, we separate the twisting region into the parallel fixed annuli separated by regions on which the map is the identity.)We further require that parallel twisting regions twist in the same direction.
(iii) For S a component of †nN and `the smallest integer such that `maps A to itself, the map `jS is area-preserving and is either isotopic to the identity, periodic, or pseudo-Anosov.In these cases, we require the map to be in standard form as above.
We are most interested in the fixed annuli defined in item (ii) and which are parallel and interstitial to the twist regions, as that is where fixed points may occur.

Counting fixed annuli
The fixed annuli described above can be counted in terms of a variant of fractional Dehn twist coefficients, related to those in [7].This subsection is optional, as the arguments proceed from the definition of fixed annuli in the preceding subsection, but may be of interest.We first discuss fractional twist coefficients, and then how to use that to count the number of fixed annuli.
Relative fractional twist coefficient Consider a mapping class g with a reducing (setwise-fixed non-nullhomotopic, non-boundary-parallel) curve C or a setwise-fixed boundary C .Suppose the induced map on the homology of C is the identity (rather than 1).Then for each component abutting C , a geometric (periodic or pseudo-Anosov) representative of that component gives a relative fractional twist coefficient in Q=Z at its boundary.In the periodic case, this is simply the fraction by which it rotates its circle boundary.In the pseudo-Anosov case, the singular measured foliations have p prongs arranged around C and these are permuted, which we interpret as a rational rotation.
Number of fixed annuli: reducing curve case Across such a reducing curve C , relative to the geometric representatives on each side, we obtain a total fractional Dehn twist coefficient x 2 Q at C , which is a lift of the sum of the two relative fractional twist coefficients.Obtain these lifts by isotoping the map to pointwise fix C and using the fractional Dehn twist coefficient [7] from each side.(Such isotopies are related by a free Z-action which adjusts the coefficients on each side in complementary fashion.) We say the mapping class g has a fixed annulus parallel to the reducing curve C for each (necessary) more-than-full twist.That is, we say g has ceil.jxj1/ fixed annuli parallel to C .
Number of fixed annuli: boundary curve case In the case of a setwise-fixed boundary curve C , we are considering the mapping class g, which is an element of 0 of the space of maps in Diff C @ .†/, which fix no points on the boundary.There are then two neighboring mapping classes g 1 ; g 2 in 0 of the space of maps Diff C .†; @ †/ which fix pointwise the boundary.Each of these has its own fractional Dehn twist coefficient [7], which we denote by y 1 ; y 2 2 Q respectively; these necessarily differ by 1.We may think of the total twisting for g at C as being allowed to take any value in the open interval between y 1 and y 2 .
Again, for each (necessary) more-than-full twist at C we say the mapping class g has a fixed annulus parallel to the boundary curve C .That is the minimum over the possible total twisting values, ie over the values in the open interval between y 1 and y 2 , of the number of more-than-full twists.More simply, this is the minimum (over i D 1; 2) of floor.jyi j/.

Fixed points of standard form maps
The fixed points of our standard form reducible maps are as follows: Type Ia The entire component of components S of †nN on which the induced map is the identity, with .S/ < 0.
Type Ib The entire component of components S of †nN on which the induced map is the identity, with .S / D 0. These are annuli and only occur when we have multiple parallel Dehn twists.
Type IIa Fixed points x of periodic components S of †nN with .S/ < 0 which are setwise fixed by .These are each index one.
Type IIb Fixed points x of flip-twist regions.These are each index one.Note that each flip-twist region has two fixed points.
Type III Fixed points x of pseudo-Anosov components S of †nN which are setwise fixed by .These come in 4 types (note that there are no fixed points associated to a rotated puncture): -Type IIIa Fixed points which are not associated with any singularity or boundary component of the pre-smoothed map.These are index one or negative one.
-Type IIIb-p Fixed points which come from an unrotated singular point with p prongs.There are p 1 of these for each such singular point, each of index negative one.
-Type IIIc Fixed points which come from a rotated singular point.These are each of index one.
-Type IIId-p Fixed points which come from an unrotated boundary component with p prongs.There are p for each such boundary component, each of index negative one.
4.1.5Nielsen classes of fixed points of standard form maps We review Nielsen classes of fixed points of standard form reducible maps, as discussed in [1].
In [1], adapting the work of [10] to the area-preserving case, we showed that we have a separate Nielsen class for every component of type Ia or type Ib, for every single fixed point of type IIa, IIb, IIIa, or IIIc, and for every unrotated singular point of the pre-smoothed map for type IIIb (ie the collection of fixed points associated to a single unrotated singular point are all in the same Nielsen class).
Type IIId fixed points associated to the same boundary component are in the same Nielsen class.They may also be in the same Nielsen class as fixed points of the component they abut if that component has fixed points at that boundary also of type IIId or of type Ia (they cannot abut regions of with type Ib fixed points).In the former case this Nielsen class is again separate from all others already specified, and has combined index p q from the p C q index negative one fixed points.In the latter case, we have already stated that this Nielsen class is separate from all others already specified.Thus we have: The combined index of the Nielsen class Á associated to a fixed component of S is Finally, the index of a fixed component of type Ib is zero.

Floer homology with twisted coefficients
We compute the Floer homology HF .; Q.Z=2OEN 0 h =tors// for a standard form reducible map.This splits into a direct sum over Nielsen classes.For Nielsen classes not associated to a fixed component, has fixed points all of the same index and thus there are no flow-lines at all and the differential vanishes.Consider first components S on which the induced map is the identity, possibly abutting components at a boundary with type IIId fixed points of pseudo-Anosovs.We perturb with the Hamiltonian flow of a small Morse-Smale function that patches together with the perturbation of the function on a neighborhood of any boundary components with type IIId fixed points.This is given by a Hamiltonian flow of a Morse-Smale function with p saddle points; see [1] for details.In [1] we showed that the flow-lines we get between fixed points in the Nielsen class Á corresponding to the fixed points in this component are only those which correspond to Morse flow-lines.
We are interested in the rank of the summand of HF .; Q.Z=2OEN 0 h =tors// corresponding to such a Nielsen class Á.The key is to understand the extrema of the Morse-Smale function.If the component has boundaries which rotate in opposite directions, we may choose the Morse-Smale function to have no extrema, and then there are jind.Á/j fixed points in the Nielsen class all of the same index and thus no flow-lines.If the component has boundaries rotating all in the same direction, we may choose a Morse-Smale function with one extremum.There are jind.Á/j C 2 fixed points in the Nielsen class Á.Finally, if there is no boundary, we may choose a Morse-Smale function with two extrema.There are jind.Á/j C 4 fixed points in the Nielsen class Á.In these latter two cases, we must further understand the flow-lines.
We consider the one-boundary-component case first.For the purposes of computing rank, we may assume the extremum is a minimum by duality.Suppose first that we have a type IIId boundary component abutting our fixed component S. In [1] we show that there is precisely one flow-line from each of the p type IIId fixed points to the minimum; see Figure 1.In this case we have a cancellation because we are working with field coefficients and whatever element (even zero) of N 0 h =tors this flow-line corresponds to under the representation wm , it corresponds to a nonzero and thus invertible element of Q.Z=2OEN 0 h =tors/.Thus the rank of the summand corresponding to the Nielsen class Á is jind.Á/j.Suppose next that S does not abut any type IIId boundary components.Denote the minimum by y.Then for every saddle point x we have two flow-lines to y. Lemma 4.7 We have that @x D a x y, where a x is nonzero if and only if the class of the closure of the descending manifold of x in N 0 h =tors H 1 .†/ is nonzero.
Proof See Figure 2. We have two flow-lines C 1 and C 2 .Thus x /y D a y b y: Here, considered as elements of the group ring, the values a; b 2 N 0 h =tors are such that a b in N 0 h =tors is the class of the descending manifold of x.This value is zero if and only if a D b.The result follows.
If a x ¤ 0 for some x, then y is a boundary because our coefficients lie in a field, and so we have a cancellation and the rank is only jind.Á/j.If a x D 0 for all x, this is the statement that the differential vanishes in the summand corresponding to the Nielsen class Á, and so the rank is jind.Á/j C 2. Lemma 4.8 In the above situation, a x D 0 for every saddle point x if and only if S has genus zero and every boundary component of S is nullhomologous in H 1 .†; @ †/.Proof In the case in which S has genus, we can find two descending manifolds which meet algebraically once.It follows that neither of them are nullhomologous.See Figure 3, left.
If S is genus zero, it appears as in Figure 3, right.The homology class of each of the boundary components is given by either (plus or minus) the homology class of one the descending manifolds (if on either end) or the difference of two such.We see all the descending manifolds are nullhomologous if and only if all of the boundary components are.
In the case in which S has no boundary, we see that a x is never zero for saddle points x and thus we cancel the minimum.Dually, we also cancel the maximum, say with any saddle point we haven't used to cancel the minimum.Thus in this case the rank is jind.Á/j.Summing up, we have shown: Proposition 4.9 Consider a Nielsen class Á corresponding to a fixed component S. If S does not abut any type IIId boundary components, every boundary component rotates in the same direction, S has genus zero, and every boundary component of S is nullhomologous in H 1 .†; @ †/, then the rank of the summand of HF .; Q.Z=2OEN 0 h =tors// corresponding to such a Nielsen class Á is jind.Á/j C 2. Otherwise it is jind.Á/j.

Construction of maps using symplectic vector fields
We now construct a map in our mapping class h so that in every Nielsen class Á, the number of fixed points in this Nielsen class equals the rank of the summand of HF .; Q.Z=2OEN 0 h =tors// corresponding to the Nielsen class Á, which we computed in the previous section.
We start with a map st , a standard-form reducible map but which is the identity on any fixed components as opposed to having been perturbed by Hamiltonian flows.We next perturb by Hamiltonian flows on any components for which there was no cancellation in the previous section; that is, components for which the number of fixed points equaled the rank.These are components which have boundary components rotating in different directions as well as components which satisfy all of the criteria in Proposition 4.9.
Next we use a modified Hamiltonian on components S which meet a type IIId boundary, geometrically canceling the extremum with one of the type IIId fixed points.We start with a Hamiltonian function with at most one extremum on such a component, which patches together with the Hamiltonian on a neighborhood of the boundary of the pseudo-Anosov region as in the previous section.Proof We geometrically cancel one of the p type IIId fixed points and the fixed point corresponding to the minimum as in Figure 4. To do this, we consider, as in Figure 5, the perturbed situation but without the central disk excised.We draw a loop , excising the disk it bounds and replacing it with "the rest" of S , ie the portion to the left of in Figure 4.This reduces the situation to one standard case.We rescale the Hamiltonian on S to be small if necessary so there are points on which the Hamiltonian evaluates to a number less than the evaluation of the Hamiltonian at the (local) minimum.Next we consider components S whose boundary components rotate all in the same direction and which meet no type IIId boundaries, but have nonzero genus.In this case, we can use an S 1 -valued Hamiltonian to remove extrema.Lemma 4.11 There exists an S 1 -valued Hamiltonian on such a component whose associated symplectic vector field is parallel to the boundary and whose critical points are all nondegenerate saddle points.
Proof We modify the Hamiltonian away from the boundary as in Figure 6.Now we perturb this whole map by the flow of a small symplectic vector field.Lemma 4.12 There exists a symplectic vector field V transverse to every boundary C of a reducible component for which OEC ¤ 0 in H 1 .†; @ †/.
Proof The plan is to choose a totally irrational flux class.In order to see that we can choose the flow to be transverse to every reducing curve, we choose a handlebody bounding † such that the reducing curves all bound disks.Next we collapse this to the underlying graph and put flows on each edge of the graph whose only rational dependences are given by balancing conditions at the vertices.We then use this as a guide to build the flow on the surface.
We rescale the symplectic vector field so that it is small enough that its time-1 flow does not move any of the fixed points too much and does not create any new fixed points.
For components S whose boundary components rotate all in the same direction, which meet no type IIId boundaries, and are genus zero but which have a non-nullhomologous boundary component, we now modify the symplectic vector field V in a neighborhood of the component, keeping it transverse to each reducing curve.
Lemma 4.13 There exists an S 1 -valued Hamiltonian on such a component whose associated symplectic vector field is transverse to the boundary and whose critical points are all nondegenerate saddle points.
Proof We modify the Hamiltonian away from the boundary as in Figure 7.
We've shown: Proposition 4.14 Given a reducible mapping class h, there exists a map in the mapping class such that in every Nielsen class Á, the number of fixed points in this Nielsen class equals the rank of the summand of HF .; Q.Z=2OEN 0 h =tors// corresponding to the Nielsen class Á.
Combining this with our observations regarding identity, periodic, pseudo-Anosov mapping classes and, for reducible mapping classes, with Proposition 4.9, we obtain (summing over Nielsen classes): Theorem 4.15 The minimum number of fixed points of an area-preserving map with nondegenerate fixed points in a mapping class h is given by P Á jind.Á/j if h is periodic or pseudo-Anosov, P Á jind.Á/j C 2A if h is reducible, where A is the number of genus zero components of the reducible mapping class on which the map is the identity, which do not abut any pseudo-Anosov components, and all of whose boundary components rotate in the same direction and are nullhomologous or homologically boundary parallel.

Degenerate fixed points
If we are allowed degenerate fixed points, each of the Nielsen classes Á for which the bound for nondegenerate fixed points was jind.Á/j can be reduced to a single degenerate fixed point.To see this, note that the only of these situations in which we are not already reduced to a single fixed point are in cases in which the Nielsen class is associated with a p-prong pseudo-Anosov singularity or in which the Nielsen class is associated to a fixed component (possibly meeting type IIId boundaries).In the former case, we simply modify the singular Hamiltonian H sing D r 2 cos.pÂ / D Re.z p /=jzj p 2 to a smooth Hamiltonian which agrees with H sing outside a small ball and inside a yet smaller ball is C r p cos.pÂ / D C Re.z p /, which is smooth at the origin, where it has a (generalized) monkey saddle.In the latter case, all of the fixed points are index negative one, ie are given by saddles, and we can again combine them all into the appropriate generalized monkey saddle, using p D jind.Á/j C 1 (the index of such a degenerate fixed point is 1 p).
Similarly, each of the Nielsen classes Á for which the bound for nondegenerate fixed points was jind.Á/j C 2 can be reduced to two fixed points: each of these corresponds to a genus zero component of the standard form map on which the map is the identity and for which each boundary component rotates in the same direction.Each of these in the nondegenerate case one index one fixed point, some number k of index negative one fixed points, where the component has k C 1 boundary components.All of the index negative one fixed points can be combined in one degenerate fixed point of the same sort as in the previous paragraph.See Figure 8.Our task is now to show that we can do no better.That is, we cannot combine these two fixed points into one.
To show this, we use a cohomology operation.The argument has similarities with arguments that cup lengths give bounds on (even degenerate) fixed points, but even though we have, by [1, Section 2.5], a deformation-invariant module structure over the quantum homology of † (which agrees with H . †/), the module structure is trivial.Every element of H 1 .†/ acts as zero because each of the descending manifolds from Lemma 4.7 is nullhomologous, and thus has zero algebraic intersection with any element of H 1 .†/.
All is not lost, however.If we restrict our attention to one Nielsen class Á, there is a sense in which this descending manifold is homologically essential for this Nielsen class.We choose our map (amongst those in the mapping class) to be a standard form one which is the identity on S and choose our basepoint to be the constant path at some point in S. We claim that the image of this map is 1 .S; .0//.We note that the image consists of elements of 1 .†; .0//represented by loops ˛.s/ based at .0/ which are homotopic through loops based at .0/ (because is the constant path) to .˛.s//.Thus the image contains 1 .S; .0//.
We now claim that any ˛.s/ based at .0/ homotopic to .˛.s// through loops based at .0/ can be homotoped inside S.This would give the result.This follows from [10,Lemma 3.4], with its modification to the standard form maps for the areapreserving case (in which we need to consider multiple parallel Dehn twist regions with fixed annuli in between) given in [1, Lemma 3.8 and Corollary 3.9].These state that any path between two fixed points of a standard form map which is homotopic rel endpoints to applied to itself can be homotoped inside the fixed point set of .The component of Fix./ containing .0/ is simply S .
Finally, we note that the result continues to hold for any other map in the mapping class.Nielsen classes are well-defined on the entire mapping class h by [1,Lemma 4.2], so this statement is sensible.Now we restrict our attention to fixed genus zero components S which do not meet type IIId boundaries, and whose boundary components all rotate in the same direction and are all nullhomologous.
Lemma 5.2 The Floer homology chain complex CF .; Z=2OEH 1 ..M / Á /I Á/ of a map 2 h, restricted to Nielsen class Á summand of such a fixed component S, with coefficients in the representation Z=2OEH 1 ..M / Á /, is well defined on the entire mapping class h and invariant up to quasi-isomorphism.The same holds for CF .; Z=2I Á/, where the coefficients are the trivial representation Z=2.
Proof It is well-defined for any because !W H 1 ..M / Á / !R is the zero map for any (which implies that every is Á-weakly monotone as defined in [1]).This follows because !W H 1 ..M / Á / !R agrees with the map Thus, by Lemma 5.2, @ ˇis a well-defined operation HF .h;Z=2I Á/ on the Á component of the Floer homology for the mapping class h.This operation is what replaces quantum cup products such as in [17] in a cup-length argument.
Lemma 5.3 There is a ˇsuch that @ ˇxi D y.In fact, for any sum P i c i x i with c 1 2 Z=2 not all zero, there exists a ˇsuch that @ ˇPi c i x i D y.
Proof We simply take ˇto be the class in H 1 .†; @ †I Z=2/ of an arc meeting the closure of the descending manifold of the saddle point corresponding to an x i appearing with coefficient one once and all others zero times.See Figure 3 for a picture of the descending manifolds.
Thus the "cup-length" of HF .h;Z=2I Á/ as an H 1 .S/-module is two.We will be taking a limit in which degenerate fixed points are allowed to appear.We have need of a Gromov compactness result appropriate for such a situation.We use Taubes's currents-in-the-target version of Gromov compactness (valid in dimension four) [19,Proposition 3.3] as applied to compact subsets OEa; b M in [8, Lemma 9.9].From this we see that we have C 0 -convergence to the orbit corresponding to a fixed point at each end.
Theorem 5.5 There must be at least two fixed points in the Nielsen class Á corresponding to such a fixed component S even if we allow degenerate fixed points.
Proof Suppose there is just one, necessarily degenerate, fixed point in Nielsen class Á.
Perturb by a small Hamiltonian flow such that all fixed points are nondegenerate.Because HF .h;Z=2I Á/ has rank one in even degree, we have at least one index-one fixed point which survives in homology; choose one such and call it y.By Lemma 5.3, for any index negative one fixed point x which survives in homology, there is a ˇ2 H 1 .S; @S/ such that @ ˇOEx D OEy.Thus there are (at least) two flow-lines from x to y, and there are two such that the difference between their classes, mapped to H 1 .SI Z=2/, is nonzero.

Theorem 3 .Lemma 3 . 8
5 follows from the following three lemmas.Let 2 h be such that Flux./j N 0 h D 0 and let 2 h.Then HF. ; ƒ / Š HF. ; ƒ /.Proof We use (i) from Theorem 2.8.Consider a smooth family r of symplectomorphisms from 0 D to 1 D with Flux.r /j N 0 h D .1 r /Flux./j N 0 h

4. 1 . 1
Standard form for geometric components Definition 4.1 For the identity mapping class, a standard form map is a small perturbation of the identity map by the Hamiltonian flow associated to a Morse function for which the boundary components are locally minima and maxima.

Definition 4 . 3
For a pseudo-Anosov mapping classes, a standard form map is a (specified) symplectic smoothing of the singularities and boundary components of the standard singular representative.Each singularity has a number p 3 of prongs, and each boundary component has a number p 1 of prongs.If a singularity or boundary component is (setwise) fixed, it has some relative fractional twist coefficient in Q=Z with denominator p.

Figure 1 :
Figure 1: Top left: level sets of a Morse-Smale function for a neighborhood of an unperturbed IIId boundary, viewed as a puncture.Top right: level sets of the perturbed Morse-Smale function.Note that the central disk, which rotates under the Hamiltonian flow, is excised to glue with other components.Bottom: the plane represents part of the pseudo-Anosov with type IIId boundary.The remainder of the surface represents a component with type Ia fixed points.The unique flow-line is shown in bold.

C 2 C 1 Figure 2 : 1 a x 1 a x 2 x 2 : : : x n 1 a x n 1 yFigure 3 :
Figure 2: The closure of the descending manifold.

Lemma 4 . 10
On a component S meeting a type IIId boundary, there exists a modification of the aforementioned Hamiltonian perturbation whose critical points are all nondegenerate saddle points.

Figure 4 :
Figure 4: Geometrically canceling the minimum when meeting a type IIId boundary.Left: before.Right: after.

Figure 5 :
Figure 5: The perturbed type IIId boundary with shaded -bounded disk to be excised and replaced with the rest of the component S , and unshaded region in which we are modifying the Hamiltonian.Left: before.Right: after.

Figure 6 :Figure 7 :
Figure 6: On a component which has genus, we can locally cancel the minimum geometrically with an S 1 -valued Hamiltonian.Left: before.Right: after.The top and bottom circles marked with arrows are identified.

Figure 8 :
Figure 8: A three boundary component genus zero component with one minimum and, instead of two nondegenerate index negative one fixed points, one 3-prong monkey saddle.

Proposition 5 . 4
[19, Proposition 3.3; 8, Lemma 9.9] Let u k W R OE0; 1 !R M be a sequence of holomorphic sections with energies bounded by some E 0 .Then we can pass to a subsequence such that (i) the u k converge weakly as currents in R M to a proper pseudoholomorphic map uW R OE0;1 !R M , and(ii)for any compact K R M , lim k!1 h sup x2Im.uk /\K dist.x;Im.u// C sup x2Im.u/\Kdist.x;Im.u k // i D 0: Note that this is a †-bundle over S 1 with projection W M !R=Z D S 1 .
1], we have need only of the change of homology class formula.In preparation, we define the cohomology class c 2 H 2 .M / D H 2 .R M / to be the first Chern class of the vertical tangent bundle to the projection .