# Essential loops of Hamiltonian homeomorphisms

Vincent Humilière  Alexandre Jannaud  Rémi Leclercq

###### Abstract

We initiate the study of the fundamental group of the group of Hamiltonian
homeomorphisms denoted by $\Hambar(M,\omega)$, i.e. the $C^{0}$-closure of the
group of Hamiltonian diffeomorphisms $\Ham(M,\omega)$ in $\mathrm{Homeo}(M)$.
We prove that in some situations, namely complex projective spaces and
rational Hirzebruch surfaces, certain Hamiltonian loops that were known to be
non-trivial in $\pi_{1}\Ham(M,\omega)$ remain non-trivial in
$\pi_{1}\Hambar(M,\omega)$. This yields in some cases, including $\CP^{2}$ and
$S^{2}\times S^{2}$, the injectivity of the map
$\pi_{1}\Ham(M,\omega)\to\pi_{1}\Hambar(M,\omega)$ induced by the inclusion.

Our method relies on results from $C^{0}$ symplectic topology and on
computations of the valuation of Seidel elements and hence of the spectral
norm on $\pi_{1}\Ham(M,\omega)$. Some of these computations were known before,
but we also present new ones which might be of independent interest.

##  1 Introduction

Let $(M,\omega)$ be a closed symplectic manifold. We denote by
$\Ham(M,\omega)$ its group of Hamiltonian diffeomorphisms and by
$\Hambar(M,\omega)$ the closure of $\Ham(M,\omega)$ with respect to the
$C^{0}$-topology in the set of all homeomorphisms of $M$. The elements of
$\Hambar(M,\omega)$ are called _Hamiltonian homeomorphisms_. Their behavior is
quite well understood on surfaces, in particular thanks to Le Calvez’s
foliation techniques established in [LC05], see e.g. Le Calvez’s work on the
subject starting from [LC06a, LC06b]. In higher dimension, partial results
were obtained very recently by Buhovsky, Seyfaddini, and the first author, see
e.g. [BHS18, BHS21].

In this note, we study the natural map

$\iota:\Ham(M,\omega)\to\Hambar(M,\omega)$

starting at the level of fundamental groups, on some symplectic manifolds of
arbitrary dimension. Indeed, while the homotopy type of $\Ham(M,\omega)$ has
been _extensively_ studied, see below for some references which are used here,
absolutely nothing is known about $\Hambar(M,\omega)$ beyond the case of
surfaces where the map $\iota$ is known to be a homotopy equivalence. Before
getting to the heart of the matter, let us point out that the analogous map
$\Symp(M,\omega)\to\overline{\Symp}(M,\omega)$, between the groups of
symplectic diffeomorphisms and homeomorphisms, was studied at the $\pi_{0}$
level by the second author [Jan21, Jan22].

###  1.1 Main results

The upshot of this work is a method which detects non-trivial elements in the
image of $\iota_{*}$, i.e. non-trivial elements in
$\pi_{1}\big{(}\Ham(M,\omega)\big{)}$ which survive in the fundamental group
after taking the $C^{0}$-closure. For our method to work, all symplectic
manifolds will be required to be _rational_ , meaning that their group of
periods $\langle\omega,\pi_{2}(M)\rangle$ is generated by a unique positive
element, which will be denoted by $\Omega$. In other words,
$\langle\omega,\pi_{2}(M)\rangle=\Omega\bb Z$.

The first application of our method concerns Hamiltonian circle actions and
relies on deep work by McDuff and Tolman [MT06]. Recall that a fixed point
component of a circle action is _semifree_ if it admits a neighborhood in
which the stabilizer of every point is either trivial or the whole circle.

Consider a Hamiltonian circle action $\Lambda$ on a compact, rational
symplectic manifold $(M,\omega)$. Assume that its extremal fixed point
components are semifree. Then $\iota_{*}([\Lambda])$ is non-trivial in
$\pi_{1}(\overline{\Ham}(M,\omega))$.

As a direct consequence, we deduce the injectivity of $\iota_{*}$ in two
different specific situations. The map
$\iota_{*}:\pi_{1}\big{(}\Ham(M,\omega)\big{)}\to\pi_{1}\big{(}\Hambar(M,\omega)\big{)}$
is injective when $(M,\omega)$ is the monotone product $S^{2}\times S^{2}$ and
for $\CP^{2}$ endowed with the Fubini–Study symplectic form.

This consequence is straightforward since, in these two cases, all non-trivial
elements of $\pi_{1}(\Ham(M,\omega))$ may be represented by a Hamiltonian
circle action satisfying the conditions of Theorem 1.1. Whenever
$\pi_{1}(\Ham(M,\omega))$ has an element which does not admit such a
representative, our method needs some extra work to produce an essential loop
of Hamiltonian homeomorphisms. Notice in particular that semifreeness is not
preserved under taking non-trivial powers. Therefore, we cannot directly
extract information from Theorem 1.1 about elements of the form $h^{k}$ with
$|k|\neq 1$, even when $h$ can be represented by a circle action with semifree
extremal components.

######  Example 1.1.

The fundamental group of the Hamiltonian diffeomorphism group of $S^{2}\times
S^{2}$ endowed with a non-monotone product symplectic form is generated by two
order-2 elements and the class of a loop $\Lambda$ of infinite order. All
three are ensured to survive in $\pi_{1}\big{(}\Hambar(M,\omega)\big{)}$ by
Theorem 1.1, however our method does not detect the even powers of $\Lambda$
so that, as far as we know, $\iota_{*}([\Lambda])$ might as well be of order
2.

However, we also have results in these more interesting situations, namely for
complex projective spaces of any dimension and for all rational 1-point blow-
ups of $\CP^{2}$. First, recall that Seidel [Sei97] proved that the group
$\pi_{1}(\Ham(\CP^{n},\omega_{\mathrm{FS}}))$ admits a non-trivial element
$h_{n}$ of order $n+1$. For any $n\geq 1$, the element $\iota_{*}(h_{n})$ has
order $n+1$ in the group $\pi_{1}(\Hambar(\CP^{n},\omega_{\mathrm{FS}}))$.
Note that the above corollary can also be obtained by non-symplectic methods,
as was pointed out to us by Randal-William. Indeed, Sasao shows in [Sas74]
that the action of $U(n+1)$ on $\CP^{n}$ induces an isomorphism between
$\Z/(n+1)\Z$ and the fundamental group of the group of degree-1 continuous
maps from $\CP^{n}$ to itself. This immediately implies that the class
$h_{n}$, which is induced by the aforementioned action, is non-trivial in
$\Ham(\CP^{n},\omega_{\mathrm{FS}})$ and in its closure
$\Hambar(\CP^{n},\omega_{\mathrm{FS}})$.

Second, recall that the symplectic $1$-point blow-ups $\bb
F^{\mu}=(\CP^{2}\\#\overline{\CP}^{2},\omega^{\prime}_{\mu})$ of $\CP^{2}$
admit a standard symplectic form $\omega^{\prime}_{\mu}$ parameterized by a
positive real number $\mu$. The fundamental group of their respective
Hamiltonian diffeomorphism groups was computed by Abreu and McDuff [AM00]: it
is generated by a unique Hamiltonian circle action of infinite order. The map
$\iota_{*}$ is injective on all rational $1$-point blow-ups of $\CP^{2}$. In
other words, whenever $\mu\in\bb Q$, the map $\iota_{*}:\pi_{1}\big{(}\Ham(\bb
F^{\mu})\big{)}\to\pi_{1}\big{(}\Hambar(\bb F^{\mu})\big{)}$ is injective.

######  Remark 1.2.

Consider the completion $\Hamhat(M,\omega)$ of $\Ham(M,\omega)$ with respect
to the spectral norm $\gamma$ as in [Hum08, Vit22]. It turns out that all the
above results still hold when $\Hambar(M,\omega)$ is replaced with
$\Hamhat(M,\omega)$. This is thus also true when $\Hambar(M,\omega)$ is
replaced by the completion with respect to Hofer’s metric. Such variations are
obtained via minor modifications of the proof; see Remark 2.1 below for more
details.

###  1.2 The method

Our method is based on the following proposition of independent interest. In
order to state it, we need to briefly recall a few well-established notions
(necessary preliminaries are given in Section 2). The quantum homology of
$(M,\omega)$ is denoted by $QH(M,\omega)$. We let
$\nu:QH(M,\omega)\to\Omega\Z\cup\\{-\infty\\}$ be the quantum valuation map,
and $\mathcal{S}:\pi_{1}(\Ham(M,\omega))\to QH(M,\omega)^{\times}$ be the
Seidel morphism. We consider the map
$\Gamma:\pi_{1}(\Ham(M,\omega))\to\Omega\Z$ defined by $\Gamma(h)=\nu\big{(}\m
S(h)\big{)}+\nu\big{(}\m S(h^{-1})\big{)}$.

######  Proposition 1.

Assume that $(M,\omega)$ is a rational symplectic manifold. Then, the map
$\Gamma:\pi_{1}(\Ham(M,\omega))\to\Omega\Z$ factors through
$\pi_{1}(\overline{\Ham}(M,\omega))$, i.e. there exists a map
$\overline{\Gamma}:\pi_{1}(\overline{\Ham}(M,\omega))\to\Omega\Z$ such that
the following diagram commutes

${\pi_{1}(\Ham(M,\omega))}$${\pi_{1}(\overline{\Ham}(M,\omega))}$${\Omega\Z}$$\scriptstyle{\Gamma}$$\scriptstyle{\iota_{*}}$$\scriptstyle{\overline{\Gamma}}$
(1)

Based on this proposition, the proof of Theorem 1.1, Corollary 1.1, and
Theorem 1.1 boils down to computing $\Gamma$ in order to show that it is non-
zero on elements of $\pi_{1}\big{(}\Ham(M,\omega)\big{)}$. The diagram above
then ensures that their image via $\iota_{*}$ in
$\pi_{1}(\overline{\Ham}(M,\omega))$ cannot be trivial.

######  Remark 1.3.

McDuff already used the map $\Gamma$ to estimate the length of loops of
Hamiltonian diffeomorphisms of 1-point blow-ups of $\CP^{2}$, see [McD02,
Lemma 5.1].

######  Remark 1.4.

All the examples appearing in this work are _strongly semi-positive_
symplectic manifolds, which is the setting of Seidel’s seminal paper [Sei97].
However, the method introduced in this note extends to a much more general
setting, see e.g. [LMP99] and [McD00].

The proof of Proposition 1 has two ingredients. The first one is the action of
the Seidel homomorphism on spectral invariants. Indeed, it is not hard to see
that, when $(M,\omega)$ is rational, $\Gamma$ coincides with the restriction
of the spectral pseudo-norm $\wt\gamma:\wt\Ham(M,\omega)\to\bb R$ to
$\pi_{1}\big{(}\Ham(M,\omega)\big{)}$, see Section 2.4 for details.

The second ingredient is the following result of $C^{0}$-continuity of the
spectral pseudo-norm up to the action of the group of periods
$\langle\omega,\pi_{2}(M)\rangle=\Omega\bb Z$, due to Kawamoto [Kaw22, Theorem
1]. [Kawamoto] Let $(M,\omega)$ be a rational symplectic manifold. For any
$\varepsilon>0$, there exists $\delta>0$ such that for any
$\phi\in\Ham(M,\omega)$, if $d_{C^{0}}(\id,\phi)<\delta$, then for any lift
$\wt\phi\in\wt\Ham(M,\omega)$ of $\phi$ there exists an integer $\ell\in\Z$
such that

$|\widetilde{\gamma}(\wt\phi)-\ell\cdot\Omega|<\varepsilon.$

Let us now see how these two ingredients are combined to give a proof.

######  Proof 1.5 (Proof of Proposition 1).

The map $\wt\gamma:\wt\Ham(M,\omega)\to\bb R$ descends to a map
$\Ham(M,\omega)\to\R/\Omega\Z$. It follows from Kawamoto’s theorem that this
map extends continuously to $\Hambar(M,\omega)$, hence factors through the
inclusion:

$\Ham(M,\omega)\to\Hambar(M,\omega)\to\R/\Omega\Z\,.$

Applying the functor $\pi_{1}$, this yields a factorisation

$\pi_{1}(\Ham(M,\omega))\to\pi_{1}(\overline{\Ham}(M,\omega))\to\Omega\Z$

of the restriction of $\wt\gamma$ to $\pi_{1}(\Ham(M,\omega))$, which is
nothing but $\Gamma$.

###  1.3 Computation of the spectral pseudo-norm

As mentioned above, the proof of our main results boils down to computing the
valuation of the Seidel elements associated to the elements of the fundamental
group of Hamiltonian diffeomorphism groups. These computations are based on
work by Entov and Polterovich [EP03] in the case of complex projective spaces,
and on works by McDuff [McD02], by Ostrover [Ost06], and by Anjos and the
third author [AL18, AL17] in the case of the Hirzebruch surfaces ($S^{2}\times
S^{2}$ and the 1-point blow-ups of $\CP^{2}$).

Since the resulting function $\Gamma$ coincides with the restriction of the
spectral pseudo-norm $\wt\gamma$ to $\pi_{1}(\Ham(M,\omega))$ when the
manifold is rational, we get explicit computations of the latter. We collect
below some phenomena of independent interest concerning $\wt\gamma$.

######  Proposition 2.

Let $\wt\gamma:\pi_{1}\big{(}\Ham(M,\omega)\big{)}\to\bb R$ denote the
restriction of the spectral pseudo-norm.

  1. _(F1)_

Let $(M,\omega)=(S^{2}\times S^{2},\omega_{\mu})$ with $\omega_{\mu}$ the
product symplectic form with area $\mu$ on the first factor and $1$ on the
second.

When $\mu\in\bb Q$ and $\mu\neq 1$, $\wt\gamma$ is _degenerate_ :
$\wt\gamma^{-1}(\\{0\\})=\\{h^{2p}\,|\,p\in\bb Z\\}$ where $h$ is the
generator of infinite order of $\pi_{1}\big{(}\Ham(M,\omega)\big{)}$.

  2. _(F2)_

Let $(M,\omega)$ be any symplectic 1-point blow-up
$(\CP^{2}\\#\overline{\CP}^{2},\omega^{\prime}_{\mu})$ of $\CP^{2}$ or the
$n$-dimensional complex projective space $(\CP^{n},\omega_{\mathrm{FS}})$,
endowed with the standard Fubini–Study symplectic form.

When $\mu\in\bb Q$, $\wt\gamma$ is _non-degenerate_.

We now focus on 1-point blow-ups
$(\CP^{2}\\#\overline{\CP}^{2},\omega^{\prime}_{\mu})$ of $\CP^{2}$. The
symplectic $\omega^{\prime}_{\mu}$ has area $\mu>0$ on the exceptional
divisor, and $1$ on the fiber111Recall that $\CP^{2}\\#\overline{\CP}^{2}$ is
the total space of the only non-trivial Hamiltonian fibration over $\CP^{1}$
with fiber $\CP^{1}$.. Under these conventions, the only monotone such
symplectic manifold is the one for which $\mu=\frac{1}{2}$.

  1. _(F3)_

The spectral norm $\wt\gamma$ is _not bounded_ on “small” rational 1-point
blow-ups of $\CP^{2}$, for which $\mu<\frac{1}{2}$ and $\mu\in\bb Q$.

  2. _(F4)_

On the monotone 1-point blow-up, that is when $\mu=\frac{1}{2}$, we have
$\wt\gamma(k)=2$ for all non-trivial elements $k$ of
$\pi_{1}\big{(}\Ham(\CP^{2}\\#\overline{\CP}^{2},\omega^{\prime}_{\mathrm{mon}})\big{)}$.

  3. _(F5)_

The image of the spectral norm is _bounded_ on “big” rational 1-point blow-ups
of $\CP^{2}$, for which $\mu\geq\frac{1}{2}$ and $\mu\in\bb Q$.

  4. _(F6)_

Let $h^{\prime}$ be the generator of
$\pi_{1}\big{(}\Ham(\CP^{2}\\#\overline{\CP}^{2},\omega^{\prime}_{\mu})\big{)}$
and $p\in\bb Z$, then the function $\mu\mapsto\Gamma(h^{\prime p})$, whose
restriction to $\bb Q$ is $\mu\mapsto\wt\gamma(h^{\prime p})$, is continuous
and piecewise linear on $\bb R$.

######  Remark 1.6.

Concerning (F1) above, it was proved in [AL17] that Seidel’s morphism is
injective for _all_ Hirzebruch surfaces. The computations of Section 3.3 show
that $\nu(\m S)$ is also injective. However, $\nu(\m S)$ does not factor
through $\pi_{1}(\Hambar)$ a priori and $\Gamma$ is _not_ injective.

The fact (F2) for 1-point blow-ups of $\CP^{2}$ was proved in [McD02], by
showing that $\Gamma$ is positive. In the second part of Section 3.3, we
compute the specific values of $\Gamma$ on the fundamental group of the
Hamiltonian diffeomorphism group. These computations yield the facts (F2) to
(F6).

#### Acknowledgments.

We thank Oscar Randal-Williams and Sobhan Seyfaddini for useful conversations.
The first and third authors are partially supported by the ANR grant
21-CE40-0002 (CoSy). The first author is also partially supported by the
Institut Universitaire de France, and the ANR grant ANR-CE40-0014. The second
author is supported by the Deutsche Forschungsgemeinschaft (DFG, German
Research Foundation) under Germany’s Excellence Strategy EXC 2181/1 -
390900948 (the Heidelberg STRUCTURES Excellence Cluster). Some progress on
this topic were made during a stay of the second author at the Institut
Mathématique d’Orsay. We thank the Laboratoire Mathématique d’Orsay for making
that stay possible.

##  2 Preliminaries

###  2.1 Quantum homology

Let $\bb k$ be a field (which will be chosen to be $\Q$ in general, except for
the case of $\CP^{n}$ in Section 3.2 for which $\bb k=\bb C$). The (small)
_quantum homology_ of a strongly semi-positive symplectic manifold
$(M,\omega)$ is the $\Z$-graded algebra defined as
$\QH_{*}(M;\Lambda)=\HH_{*}(M;\bb k)\otimes_{\bb k}\Lambda$ where
$\Lambda=\Lambda^{\mathrm{univ}}[q,q^{-1}]$ has coefficients in the ring of
generalized Laurent series in the degree-0 variable $t$:

$\Lambda^{\mathrm{univ}}=\Big{\\{}\sum_{\kappa\in\R}r_{\\!\kappa}t^{\kappa}\,\Big{|}\,r_{\\!\kappa}\in\bb
k\,\mbox{ s.t. }\forall c\in\R,\;\\#\\{\kappa>c\,|\,r_{\kappa}\neq
0\\}<\infty\Big{\\}}$

and $q$ is a variable of degree 2. The grading of an element of the form
$a\otimes q^{d}t^{\kappa}$ with $a\in\HH_{l}(M;\bb k)$ is simply given by
$\deg(a\otimes q^{d}t^{\kappa})=l+2d$.

The _quantum intersection product_ on $\QH_{*}(M;\Lambda)$ is a deformation of
the usual intersection product on $\HH_{*}(M;\bb k)$ by counts of certain
Gromov–Witten invariants. More precisely, for $a\in\HH_{k}(M;\bb k)$ and
$b\in\HH_{l}(M;\bb k)$,

$a*b=\sum_{B\in H_{2}^{S}(M;\Z)}(a*b)_{B}\otimes q^{-c_{1}(B)}t^{-\omega(B)}$

where the sum runs over all spherical homology classes $B$, i.e. classes $B$
in the image $H_{2}^{S}(M;\Z)$ of the Hurewicz map $\pi_{2}(M)\to
H_{2}(M;\Z)$.

The class $(a*b)_{B}\in\HH_{*}(M;\bb k)$ has degree $k+l-\dim(M)+2c_{1}(B)$
and is defined by requiring its usual intersection product with any class
$c\in\HH_{*}(M;\bb k)$ to be given by the Gromov–Witten invariant

$(a*b)_{B}\cdot c=\mathrm{GW}^{M}_{B,3}(a,b,c)\in\bb k$

which counts the number of spheres in $M$, in the class $B$, which meet cycles
representing $a$, $b$ and $c$. The specific definition of
$\mathrm{GW}^{M}_{B,3}$ is not necessary in this note. Only the following
facts will be of interest:

  1. (i)

As expected, $\deg(a*b)=\deg(a)+\deg(b)-\dim(M)$.

  2. (ii)

The quantum intersection product turns the ring $\QH_{*}(M;\Lambda)$ into a
$\Z$-graded commutative unital algebra.

  3. (iii)

The unit of this algebra is the fundamental class $[M]$ of the symplectic
manifold, seen as an element in $\QH_{2n}(M;\Lambda)$.

#### Quantum valuation

The quantum homology algebra comes with a natural _valuation_
$\nu:\QH_{*}(M;\Lambda)\to\R\cup\\{-\infty\\}$ defined by

$\nu\Big{(}\sum_{\kappa\in\R}a_{\kappa}\otimes
q^{d_{\kappa}}t^{\kappa}\Big{)}=\max\\{\kappa\,|\,a_{\kappa}\neq 0\\}\,.$

Notice that any non-zero class has finite valuation because of the finiteness
condition required in the definition of $\Lambda^{\mathrm{univ}}$. By
convention, the zero class has valuation $-\infty$.

#### The monotone case

Morally speaking, the variables $q$ and $t$ respectively remember the first
Chern number and the symplectic area of classes of spheres in $\pi_{2}(M)$.

In the (positively) monotone case, namely when

$\exists\lambda>0\;\mbox{ such that }\;\omega|_{\pi_{2}(M)}=\lambda\cdot
c_{1}|_{\pi_{2}(M)}\,,$

the morphisms $\omega$ and $c_{1}$ are linearly dependent on $\pi_{2}(M)$.
Hence there is no need to carry around two variables: in this case, $\Lambda$
is usually replaced by $\bb k[[s]$ where $s$ is of degree $2N$, i.e. twice the
_minimal Chern number_ $N$ of $M$ which is the positive generator of $\langle
c_{1},\pi_{2}(M)\rangle=N\bb Z$.

An element of the form $a\otimes s^{j}$ thus corresponds to $a\otimes
q^{jN}t^{j\Omega}$ in the above description, with $\Omega$ the positive
generator of $\langle\omega,\pi_{2}(M)\rangle$, which satisfies by
monotonicity $\Omega=\lambda N$. The definition of the valuation has to be
adapted consequently to $\nu\big{(}\sum_{j\in\Z}a_{j}\otimes
s^{j}\big{)}=\Omega\cdot\max\\{j\,|\,a_{j}\neq 0\\}$.

This is for example the setting of [EP03] whose results are used below.

###  2.2 Floer homology

In order to prove Arnold’s conjecture, Floer developed at the end of the 80’s
[Flo88a, Flo88b, Flo89a, Flo89b] an infinite dimensional Morse–Bott-type
homology for the symplectic action functional. As Gromov–Witten invariants,
this construction is another striking consequence of Gromov’s celebrated work
on pseudo-holomorphic curves [Gro85]. (And as Gromov–Witten invariants, it has
had countably many applications.)

The upshot of Floer’s construction, as far as this note is concerned, is that
with a generic pair $(H,J)$ formed of a time-dependent, non-degenerate
Hamiltonian function $H$ and a $\omega$-compatible almost complex structure
$J$, one can define a $\Z$-graded complex $(\CF_{*}(M\wrt
H),\partial_{(H,J)})$ whose homology $\HF_{*}(M,\omega)=\HH_{*}(\CF(M\wrt
H),\partial_{(H,J)})$ satisfies the following properties.

  1. (i)

There exist canonical _continuation isomorphisms_ between Floer homologies
built from different admissible pairs of Floer data $(H,J)$ and
$(H^{\prime},J^{\prime})$.

  2. (ii)

The _pair-of-pants product_ $\star$ turns Floer homology into a graded
algebra, with unit $[M]\in\HF_{2n}(M,\omega)$, the fundamental class of $M$
seen as a Floer homology class.

  3. (iii)

There exist isomorphisms $\PSS:\QH_{*}(M;\Lambda)\to\HF_{*}(M,\omega)$ of
graded commutative unital algebras.

  4. (iv)

Floer complexes are _filtered_ : any given $\alpha\in\R$ defines a subcomplex
$(\CF_{*}^{\alpha}(M\wrt H),\partial_{(H,J)})$. Its homology does not depend
on the choice of $J$ and is denoted by $\HF_{*}^{\alpha}(M\wrt H)$.

###  2.3 The Seidel morphism

The _Seidel morphism_ was described in [Sei97] in two quite different but
equivalent ways. On one hand, it can be seen as a morphism

$\m
S:\pi_{1}\big{(}\Ham(M,\omega)\big{)}\longrightarrow\QH_{*}(M;\Lambda)^{\times}\,,\qquad
h\longmapsto\m S(h)$

where $\QH_{*}(M;\Lambda)^{\times}$ denotes the multiplicative group of
invertible elements of $\QH_{*}(M;\Lambda)$. The quantum class $S(h)$ is
called the _Seidel element_ associated to $h$. It is defined by counting
pseudo-holomorphic sections of a Hamiltonian fibration over $S^{2}$ with fibre
$M$ obtained from a loop $(\phi^{t})_{t\in[0,1]}\in h$ via the clutching
construction.

On the other hand, it can be seen as a _representation_

$\m
S:\pi_{1}\big{(}\Ham(M,\omega)\big{)}\longrightarrow\mathrm{Aut}(\HF_{*}(M,\omega))\,,\qquad
h\longmapsto\m S_{h}\,.$

The relation between these two viewpoints is straightforward: the automorphism
of Floer homology $\m S_{h}$ is nothing but the pair-of-pants multiplication
by $\m S(h)$ seen as a Floer homology class via the PSS morphism, i.e.

$\forall b\in\HF_{*}(M,\omega)\,,\quad\m S_{h}(b)=\PSS(\m S(h))\FP b\,.$

###  2.4 The spectral norm

The spectral norm is a norm defined on the universal cover of Hamiltonian
diffeomorphism groups. It is based on the theory of _spectral invariants_ ,
introduced by Viterbo [Vit92] via generating functions, and adapted to the
Floer-theoretic setting by Schwarz [Sch00] for symplectically aspherical
manifolds and Oh [Oh05] for monotone manifolds. Since then, they have been
defined in a wide range of situations (and also had countably many deep
consequences).

#### Spectral invariants

Let $a\in\QH_{*}(M;\Lambda)$ be a non-zero quantum homology class. For any
non-degenerate Hamiltonian function $H$ on $M$, the _spectral invariant
associated to $a$ with respect to $H$_ is the real number

$c(a\wrt H)=\inf\\{\alpha\in\R\,|\,\mathrm{PSS}(a)\text{ belongs to the image
of }\iota^{\alpha}\\}$

where $\iota^{\alpha}:HF_{*}^{\alpha}(M\wrt H)\to HF_{*}(M,\omega)$ is the map
induced by the inclusion. These numbers enjoy a list of standard properties.
First, they are “spectral” in the sense that $c(a\wrt H)$ belongs to the set
of critical values of the action functional associated to $H$; this property
which explains their name will not be needed here. They are also Hofer
continuous, namely

$\int_{0}^{1}\\!\min_{M}(H_{t}-K_{t})\,dt\,\leq c(a\wrt H)-c(a\wrt
K)\leq\int_{0}^{1}\\!\max_{M}(H_{t}-K_{t})\,dt$

for any two Hamiltonians $H(t,x)=H_{t}(x)$, $K(t,x)=K_{t}(x)$. In particular,
$c(a\wrt H)$ continuously depends on $H$ and the map $c(a\wrt\,\cdot\,)$
extends continuously to all (possibly degenerate) Hamiltonians.

Recall that a smooth path of Hamiltonian diffeomorphisms $(h^{t})_{t\in[0,1]}$
with $h^{0}=\id$ is generated by a unique mean normalized Hamiltonian $H$,
i.e. satisfying $\int_{M}H_{t}\,\omega^{n}=0$ for all $t$. Thus we may define
spectral invariants of the isotopy $(h^{t})$ by setting
$c(a\wrt(h^{t}))=c(a\wrt H)$. These invariants applied to isotopies have the
nice feature that they are invariant by homotopy with fixed end points.
Therefore, we obtain a well-defined map on the universal cover of
$\Ham(M,\omega)$:

$c(a\wrt\,\cdot\,):\wt\Ham(M,\omega)\to\R\,.$

These maps satisfy the so-called _triangle inequality_ :

$c(a\ast b\wrt\wt h\wt k)\leq c(a\wrt\wt k)+c(b\wrt\wt h)$

for any classes $a$, $b\in QH_{*}(M,\Lambda)$ such that $a\ast b\neq 0$, and
any $\wt h$, $\wt k\in\wt\Ham(M,\omega)$.

We will mainly use the spectral invariants associated to the fundamental
class. Therefore, it will be convenient to introduce the notation
$c_{+}(H)=c([M]\wrt H)$ and $c_{+}(\wt h)=c([M]\wrt\wt h)$. Note that the
triangle inequality for $c_{+}$ takes the form

$c_{+}(\wt h\wt k)\leq c_{+}(\wt h)+c_{+}(\wt k)\,.$ (2)

#### Relation between $c$, $\m S$, and $\nu$

In the language of spectral invariants, the quantum valuation defined in
Section 2.1 behaves as the spectral invariant computed with respect to the
zero Hamiltonian / the identity in $\Hamtilde$. By this, we mean that for any
non-zero quantum homology class $a\in\QH_{*}(M;\Lambda)$

$c(a\wrt\wt\id)=\nu(a)\,.$ (3)

Moreover, because Seidel’s representation provides automorphisms of _filtered_
Floer complexes, the action of any $h\in\pi_{1}(\Ham(M,\omega))$ on
$\wt\Ham(M,\omega)$ (or in other words the difference between lifts of
$\phi\in\Ham(M,\omega)$ to the universal cover) yields, in terms of spectral
invariants:

$\forall a\in\QH_{*}(M;\Lambda)\,,\quad c(a\wrt\wt kh)=c(\m S(h)*a\wrt\wt k)$
(4)

for all $\wt k\in\wt\Ham(M,\omega)$. In the specific situation where $\wt
k=\wt\id$ and $a=[M]$, we get

$c_{+}(h)=c([M]\wrt h)=c(\m S(h)*[M]\wrt\wt\id)=\nu(\m S(h))$ (5)

respectively by definition of $c_{+}$, by (4), and finally by (3) and the fact
that $[M]$ is the unit of the quantum homology algebra. If not litterally
_well-known_ , this equality has been successfully used before, see e.g.
Proposition 4.1 in [EP03], Section 6.3 of [Ost06], and Property (2.4) of
spectral numbers in [McD10].

#### The spectral pseudo-norm

The spectral pseudo-norm is defined as a group pseudo-norm
$\wt\gamma:\wt\Ham(M,\omega)\to\R$ by

$\wt\gamma(\wt h)=c_{+}(\wt h)+c_{+}(\wt h^{-1})=c_{+}(H)+c_{+}(\overline{H})$

with $H$ generating $(\phi_{H}^{t})_{t\in[0,1]}\in\wt h$ and $\overline{H}$
the Hamiltonian function defined by $\overline{H}_{t}(x)=-H_{1-t}(x)$.
Standard computations show that $\overline{H}$ generates the isotopy
$\phi_{H}^{1-t}(\phi_{H}^{1})^{-1}\in\wt h^{-1}$. The triangle inequality for
$\wt\gamma$ as well as its non-negativity follow from (2) and the fact that
$\wt\gamma(\wt\id)=0$.

Notice that (5) yields for any $h\in\pi_{1}(\Ham(M,\omega))$

$\wt\gamma(h)=c_{+}(h)+c_{+}(h^{-1})=\nu(\m S(h))+\nu(\m S(h^{-1}))=\Gamma(h)$
(6)

where $\Gamma$, defined by the last equality, is the central object of
Proposition 1 from the introduction. Notice that this implies that

$\wt\gamma(h)\in\langle\omega,\pi_{2}(M)\rangle,\quad\forall
h\in\pi_{1}(\Ham(M,\omega)).$

The pseudo-norm $\wt\gamma$ induces a genuine (non-degenerate) norm $\gamma$
on the group $\Ham(M,\omega)$, called the _spectral norm_ , by the formula

$\gamma(\phi)=\inf\left\\{\wt\gamma(\wt h)\,\big{|}\,\wt
h\in\wt\Ham(M,\omega),\pi(\wt h)=\phi\right\\}.$

######  Remark 2.1.

Assume $(M,\omega)$ is rational and let $\Omega$ be the positive generator of
$\langle\omega,\pi_{2}(M)\rangle$. It is known that if $\wt h_{1}$, $\wt
h_{2}$ are two lifts to $\wt\Ham(M,\omega)$ of a given
$\phi\in\Ham(M,\omega)$, then the difference $\wt\gamma(\wt
h_{1})-\wt\gamma(\wt h_{2})$ belongs to
$\langle\omega,\pi_{2}(M)\rangle=\Omega\Z$. It follows that for any
$\phi\in\Ham(M,\omega)$ and any lift $\wt h$ of $\phi$ to $\wt\Ham(M,\omega)$,
there exists an integer $\ell\in\Z$ such that

$\wt\gamma(\wt h)-\ell\cdot\Omega=\gamma(\phi).$

Therefore, Kawamoto’s theorem from Section 1.2 holds after replacing the
$C^{0}$ distance $d_{C^{0}}$ with the spectral distance $\gamma$. We then
deduce that Proposition 1 similarly holds with the $\gamma$-completion
$\Hamhat(M,\omega)$ replacing $\Hambar(M,\omega)$. We can then use this
modified version of Proposition 1 to adapt our proofs of Theorems 1.1, 1.1,
1.1 as well as Corollary 1.1 to the case of $\Hamhat(M,\omega)$. This
justifies the part of Remark 1.2 related to $\gamma$.

For the part related to Hofer’s distance (which we denote here by $\delta$),
the Hofer continuity of spectral invariants yields $\gamma\leq\delta$ and thus
a natural map $\Hamhat^{\delta}\\!(M,\omega)\to\Hamhat^{\gamma}\\!(M,\omega)$
between the respective completions, and hence a factorization

$\pi_{1}(\Ham(M,\omega))\to\pi_{1}(\Hamhat^{\delta}\\!(M,\omega))\to\pi_{1}(\Hamhat^{\gamma}\\!(M,\omega))\,.$

Thus, our claims from Remark 1.2 for $\gamma$ yield the same claims for
$\delta$.

##  3 Computations

###  3.1 Proof of Theorem 1.1

This is a simple remark based on the following deep result from McDuff and
Tolman which is part of [MT06, Theorem 1.10].

[McDuff–Tolman] Let $\Lambda$ be a Hamiltonian circle action on a compact
symplectic manifold $(M,\omega)$, generated by the moment map $K:M\to\R$.
Assume $K$ to be normalized and let $K_{\max}=\max_{M}K$. Assume that the
fixed point component $F_{\max}=K^{-1}(K_{\max})$ is semifree. Then there are
classes $a_{B}\in H_{*}(M)$ so that

$\m S(\Lambda)=[F_{\max}]\otimes q^{-m_{\max}}t^{K_{\max}}+\sum a_{B}\otimes
q^{-m_{\max}-c_{1}(B)}t^{K_{\max}-\omega(B)}$ (7)

where the sum runs over all spherical class $B\in\HH^{S}_{2}(M)$ with
$\omega(B)>0$.

Recall that semifreeness means that the action acts semifreely: the stabilizer
of each point is trivial or the whole circle in a neighbourhood of $F_{\max}$.
The integer $m_{\max}$ will not be used below. Let us simply mention that it
is determined by the degree of $\m S(\Lambda)$ and that it corresponds to the
sum of the weights at a point $x\in F_{\max}$. Note that McDuff and Tolman
were also able to specify the structure of the lower order terms in exchange
of additional requirements on $\omega$-compatible almost complex structures.

For our purpose, it is enough to notice that, since $[F_{\max}]$ and the
$a_{B}$’s are honest classes in $\HH_{*}(M)$, the valuation of $\m S(\Lambda)$
is $K_{\max}$. Moreover, $\Lambda^{-1}$ is generated by $-K$ whose maximal
fixed point component is nothing but
$(-K)^{-1}(\max_{M}(-K))=K^{-1}(\min_{M}K)=F_{\min}$. Requiring both extremal
fixed point components to be semifree and applying McDuff–Tolman’s Theorem to
$K$ and $-K$, yield

$\Gamma(\wt\Lambda)=\nu(\m S(\Lambda))+\nu(\m
S(\Lambda^{-1}))=K_{\max}-K_{\min}\,.$

Since $K$ is not constant, $\Gamma(\wt\Lambda)>0$. Under the additional
assumption that $(M,\omega)$ is rational, Proposition 1 ensures that
$\iota_{*}(\wt\Lambda)$ is not trivial in
$\pi_{1}(\overline{\Ham}(M,\omega))$. $\square$

###  3.2 Proof of Theorem 1.1

Recall from [Sei97] that Seidel’s morphism detects an element $h_{n}$ of order
$n+1$ in $\pi_{1}(\Ham(\CP^{n},\omega_{\mathrm{FS}}))$. We claim that $\Gamma$
is positive on all classes detected by Seidel’s morphism. Since
$(\CP^{n},\omega_{\mathrm{FS}})$ is monotone, it is rational and Proposition 1
ensures that $\iota_{*}(h_{n})$ is of order $n+1$ in
$\Hambar(\CP^{n},\omega_{\mathrm{FS}})$.

We now proceed and show that

$\forall\,h\in\pi_{1}(\Ham(\CP^{n},\omega_{\mathrm{FS}})),\quad h\notin\ker(\m
S)\;\Longrightarrow\;\Gamma(h)=\Omega$ (8)

where $\Omega=\omega_{\mathrm{FS}}([\CP^{1}])>0$ is the generator of
$\langle\omega_{\mathrm{FS}},\pi_{2}(\CP^{n})\rangle$. This obviously implies
our claim, hence concluding the proof.

We recall that the quantum homology algebra of $\CP^{n}$ is isomorphic to the
algebra $\mathrm{k}[A]/\\{A^{n+1}=s^{-1}\\}$ with $\mathrm{k}=\bb k[[s]$, see
our comment on the monotone case at the end of Section 2.1. Here, $A$ is the
hyperplane class of degree $2n-2$ and $s$ is of degree $2N=2(n+1)$. Notice
that for any $m\in\Z$, $A^{m}$ has degree $2n-2m$.

Moreover, recall from Proposition 4.2 of [EP03], that all Seidel elements are
monomials of the form $A^{m}s^{\beta}$. Since they belong to
$\QH_{2n}(\CP^{n};\Lambda)$, $\beta=\frac{m}{n+1}$ for degree reasons. Notice
that when $m$ is a multiple of $n+1$, we get $A^{m}s^{\beta}=[\CP^{n}]$ as
expected (see Proposition 4.3 of [EP03]).

We now assume that $m$ is not a multiple of $n+1$. We need to compute
$\nu(A^{m}s^{\beta})+\nu(A^{-m}s^{-\beta})$ so that we may assume that $m\geq
0$. Then

$m=q\cdot(n+1)+r\quad\mbox{ and }\quad-m=-(q+1)\cdot(n+1)+(n+1-r)$

with $q=\lfloor\frac{m}{n+1}\rfloor$, and $r$ and $n+1-r$ are integers in
$(0,n+1)$. Hence,

$A^{m}s^{\frac{m}{n+1}}=(A^{n+1})^{q}\cdot
A^{r}s^{\frac{m}{n+1}}=([\CP^{n}]s^{-1})^{q}\cdot
A^{r}s^{\frac{m}{n+1}}=A^{r}s^{\frac{m}{n+1}-q}$

whose valuation is
$\Omega\cdot(\frac{m}{n+1}-q)=\Omega\cdot(\frac{m}{n+1}-\lfloor\frac{m}{n+1}\rfloor)$
since $A^{r}\in\HH_{*}(\CP^{n};\Q)$. Similarly, by replacing $q$ by $-(q+1)$
and $r$ by $n+1-r$, we get that $A^{-m}s^{\frac{-m}{n+1}}$ has valuation
$\Omega\cdot(\lfloor\frac{m}{n+1}\rfloor+1-\frac{m}{n+1})$, so that
$\nu(A^{m}s^{\beta})+\nu(A^{-m}s^{-\beta})=\Omega$.

Hence, $\Gamma(h)=\Omega$ whenever $h$ has a non-trivial associated Seidel
element which proves (8) and ends the proof of Theorem 1.1. $\square$

###  3.3 About Example 1.1 and the proof of Theorem 1.1

In this section, we compute the Seidel elements associated to all elements of
the fundamental group of the group of Hamiltonian diffeomorphisms of all
Hirzebruch surfaces. This completes partial computations from [McD02],
[Ost06], and [AL17].

Recall that in [AL18], Anjos and the third author computed Seidel elements of
many 4-dimensional toric manifolds starting from the fundamental
aforementioned result from McDuff and Tolman. In [AL17], these computations
were used to prove that Seidel’s morphism detects all the generators of the
fundamental group of the Hamiltonian diffeomorphism group of all Hirzebruch
surfaces, namely all symplectic products of $S^{2}\times S^{2}$ and 1-point
blow-ups of $\CP^{2}$.

We now show that, when we consider $\Gamma$ rather than the Seidel elements
themselves, the situation is more subtle:

  * •

$\Gamma$ vanishes on certain essential loops of Hamiltonian diffeomorphisms of
all products of spheres except the monotone one, this will justify Example 1.1
and Fact (F1) of Proposition 2 from the introduction ;

  * •

$\Gamma$ is positive on all essential loops of Hamiltonian diffeomorphisms of
the 1-point blow-ups of $\CP^{2}$, this will prove Theorem 1.1 and justify
Fact (F2) of Proposition 2.

The computations done in the latter case will also clarify Facts (F3) to (F6)
of Proposition 2.

######  Remark 3.1.

Note that all the quantities $\m S$, $\nu$, $\Gamma$, $\wt\gamma$ depend on
the symplectic forms, themselves being parameterized by a real number $\mu$ in
both cases. However, we omit $\mu$ in the notation for short throughout this
section.

#### Even Hirzebruch surfaces.

Recall that $\F_{2k}^{\mu}$ is identified to $M=S^{2}\times S^{2}$ endowed
with the symplectic form $\omega_{\mu}$ with area $1$ on the first factor and
$\mu$ on the second.

We denote by $u=[S^{2}\times\\{\mathrm{pt}\\}]\otimes q$ and
$v=[\\{\mathrm{pt}\\}\times S^{2}]\otimes q$ the degree 4 quantum homology
classes induced by each component of the product $M=S^{2}\times S^{2}$. The
quantum homology algebra of $(M,\omega_{\mu})$ is

$\QH_{*}(M,\omega_{\mu})\simeq\Lambda^{\mathrm{univ}}[u,v]/\langle
u^{2}=t^{-1},v^{2}=t^{-\mu}\rangle\,.$

The fundamental group of the group of Hamiltonian diffeomorphisms of
$\F_{2k}^{\mu}$ was computed in [AM00].

$\bullet$ When $\mu=1$, the manifold $(M,\omega_{1})$ is monotone hence
rational, and the fundamental group of $\Ham(M,\omega_{1})$ is generated by
two elements of order 2, each generated by one factor of the product. These
generators are Hamiltonian circle actions which satisfy the assumption of
Theorem 1.1 so that they induce non-trivial elements in
$\Hambar(M,\omega_{1})$. Since they are of order 2, there is nothing more to
prove.

$\bullet$ When $\mu>1$ and rational, $(M,\omega_{\mu})$ is rational and
$\pi_{1}(\Ham(M,\omega_{\mu}))$ is generated by the same order-2 circle
actions, together with a third Hamiltonian circle action of infinite order
$\Lambda$ (denoted $\Lambda^{2}_{e_{1}}$ in [AL17]). The Seidel element of the
latter and of its inverse are given by

$\m S(\Lambda)=(u+v)\otimes t^{\frac{1}{2}-\epsilon}\quad\mbox{ and }\quad\m
S(\Lambda)^{-1}=(u-v)\otimes\frac{t^{\frac{1}{2}+\epsilon}}{1-t^{1-\mu}}$

with $\epsilon=\frac{1}{6\mu}$. Hence, for any positive integer $\ell$,

$\m S(\Lambda)^{\ell}=\sum_{k=0}^{\ell}\Big{(}\\!\\!\begin{array}[]{c}\ell\\\
k\end{array}\\!\\!\Big{)}u^{k}v^{\ell-k}t^{(\frac{1}{2}-\epsilon)\ell}\,.$

Notice that, depending on the parity of $k$ and $\ell$ and up to some power of
$t$, $u^{k}v^{\ell-k}$ is $[M]$, $u$, $v$, or $uv$, so that no products of
powers of $u$ and $v$ vanish. Moreover, for all $k$ and $\ell$,

$\nu\big{(}u^{k}v^{\ell-k}t^{(\frac{1}{2}-\epsilon)\ell}\big{)}=-\Big{\lfloor}\frac{k}{2}\Big{\rfloor}-\mu\Big{\lfloor}\frac{\ell-k}{2}\Big{\rfloor}-\Big{(}\frac{1}{2}-\epsilon\Big{)}\ell\,.$

Since $\mu>1$, we get that $\nu(\m
S(\Lambda)^{\ell})=-\lfloor\frac{\ell}{2}\rfloor+(\frac{1}{2}-\epsilon)\ell$.

The computation of $\nu(\m S(\Lambda)^{-\ell})$ is similar. Only notice
additionally that $\frac{1}{1-t^{1-\mu}}=1+t^{1-\mu}+t^{2(1-\mu)}+\ldots$ so
that, since $1-\mu<0$, $\nu\big{(}(\frac{1}{1-t^{1-\mu}})^{\ell}\big{)}=0$.
Hence, we get $\nu(\m
S(\Lambda)^{-\ell})=-\lfloor\frac{\ell}{2}\rfloor+(\frac{1}{2}+\epsilon)\ell$
which yields

$\Gamma(\Lambda^{\ell})=\ell-2\Big{\lfloor}\frac{\ell}{2}\Big{\rfloor}=\left\\{\begin{array}[]{l}0\mbox{
if $\ell$ is even},\\\ 1\mbox{ if $\ell$ is odd}.\end{array}\right.$

Hence, we see that $\Gamma$ detects only odd powers of $\Lambda$, it follows
that $\iota_{*}([\Lambda])\neq 0$ (but it could be of order 2). $\square$

#### Odd Hirzebruch surfaces.

We follow the conventions from [AL17, Section 3.2]: $\bb F_{2k+1}^{\mu}$ is
identified with $\CP^{2}\\#\overline{\CP^{2}}$ endowed with the symplectic
form $\omega^{\prime}_{\mu}$ such that

  * •

the exceptional divisor $B$ of self-intersection $-1$ has area $\mu>0$,

  * •

the fiber $F$ has area $1$,

  * •

the projective line $B+F$ has area $\mu+1$.

As a vector space, its quantum homology is generated by:
$u_{0}=[\mathrm{pt}]\otimes q^{2}$, $u=F\otimes q$, $u_{3}=B\otimes q$, and
$\bb 1$. It is also convenient to denote by $u_{1}=(B+F)\otimes q=u+u_{3}$.
These classes satisfy the following relations222The last one does not appear
as such in [AL17] as it was not needed to get the expression of the quantum
homology algebra. It is necessary here and can easily be computed with the
methods used there. See also [MT06, Remark 5.6] and [Ost06, Section 3.3] in
which it is explicitly computed _under different conventions_ though.:

$u_{1}\,u_{3}=t^{-1}\,,\quad u^{2}=u_{3}\,t^{-\mu}\,,\quad
u^{-1}=u_{0}\,t^{\mu+1}\,.$ (9)

As an algebra, the quantum homology is given as the quotient

$\QH_{*}(\bb F^{\mu}_{2k+1})\simeq\Lambda^{\mathrm{univ}}[u]/\langle
u^{4}t^{2\mu}+u^{3}t^{\mu}-t^{-1}\rangle\,.$

Last (but not least!), recall from [AM00] that the fundamental group of the
group of Hamiltonian diffeomorphisms of $\bb F_{2k+1}^{\mu}$ is generated by a
single class of infinite order. This class is induced by a circle action
$\Lambda$ whose associated Seidel element is $\m
S(\Lambda)=u^{-1}t^{-\varepsilon}$ where
$\varepsilon=\frac{3\mu^{2}+3\mu+1}{3(1+2\mu)}$. Hence we get, for all
integers $p$,

$\Gamma(\Lambda^{p})=\nu(\m S(\Lambda^{p}))+\nu(\m
S(\Lambda^{-p}))=\nu(u^{-p})+\nu(u^{p})\,.$

The following proposition collects the results of the computations of
$\Gamma(\Lambda^{p})$ for all $p$ (greater than some $p_{0}$) for all possible
values of the parameter $\mu>0$.

######  Proposition 3.

  * •

For $0<\mu\leq\frac{1}{2}$,

$\displaystyle\forall p\geq
7,\quad\Gamma(\Lambda^{p})=-2\Big{(}\Big{\lfloor}\frac{p-1}{3}\Big{\rfloor}-1\Big{)}\cdot\mu+\Big{(}\Big{\lfloor}\frac{p-1}{3}\Big{\rfloor}+1\Big{)}\,.$
(10)

  * •

For $\mu>\frac{1}{2}$, the value of $\Gamma(\Lambda^{p})$ depends on the rest
$\ell$ of the Euclidean division of $p$ by $4$, and on the sign of $\mu-1$.
Namely, for all $p\geq 12$, $\Gamma(\Lambda^{p})$ is given by the following
table:

$\displaystyle\begin{array}[t]{c||c|c}\phantom{\Big{\lfloor}}\ell\phantom{\Big{\lfloor}}&\frac{1}{2}<\mu\leq
1&1<\mu\\\ \hline\cr 0&2&2\\\ 1&2&\mu+1\\\ 2&-2\mu+3&1\\\
3&2&\mu+1\end{array}$ (16)

This proposition shows that, for any $\mu>0$, $\Gamma(\Lambda^{p})\neq 0$ for
$p$ big enough333Computing the first few terms shows that this fact actually
holds for all integers $p$. The formulas we present below, however, do not
hold for small integers $p$. which yields that $\iota_{*}([\Lambda])$ is of
infinite order in $\pi_{1}\big{(}\overline{\Ham}(\bb F_{2k+1}^{\mu})\big{)}$
for any positive, rational number $\mu$. Notice that Theorem 1.1 and the facts
(F2) to (F6) from Proposition 2 follow from this proposition.

The graph below illustrates Proposition 3.

$\frac{1}{2}$
---
$1$
---
$\mu$
---
$1$
---
$p=7,8,9$
---
$p=10,11,12$
---
$p=3(n-1)+k$,
---
$k=1,2,3$
$p\equiv 2\\!\mod 4$
---
$p\equiv 1\\!\mod 2$
---

$\mu+1$
---

$-2\mu+3$
---

$2$
---
$3$
---
$4$
---
$n$
---
The values of $\wt\gamma([\Lambda^{p}])$ as functions
---
of the symplectic form $\omega^{\prime}_{\mu}$.

######  Remark 3.2.

It is interesting to compare the first item in Proposition 3 with the results
of Ostrover [Ost06]. Indeed, he establishes that for $\mu>\frac{1}{2}$, the
quantum homology $QH_{4}(\F^{\mu}_{2k+1})$ is a field, while it _splits_ into
field summands for $\mu<\frac{1}{2}$. Moreover, he proves that the restriction
of $\Gamma$ to any field summand is bounded. Our result shows that $\Gamma$ is
however not bounded on the whole quantum homology in the case
$\mu<\frac{1}{2}$.

It remains to prove Proposition 3 which follows from intermediate computations
whose results are collected below, depending on the value of $\mu$.

$\displaystyle\begin{array}[]{c|c|c}&\nu(u^{-p})&\nu(u^{p})\\\
\hline\cr\phantom{\Big{(}}\\!\\!\\!0<\mu\leq\tfrac{1}{2}&\big{(}p-2\big{\lfloor}\frac{p-1}{3}\big{\rfloor}\big{)}\mu+\big{(}\big{\lfloor}\frac{p-1}{3}\big{\rfloor}+1\big{)}&-(p-2)\mu\\\
\hline\cr\phantom{\Big{(}}\\!\\!\\!\tfrac{1}{2}<\mu\leq
1&\big{(}p-2\big{\lfloor}\frac{p+2}{4}\big{\rfloor}\big{)}\mu+\big{(}\big{\lfloor}\frac{p+2}{4}\big{\rfloor}+1\big{)}&-\big{(}p-2\big{\lfloor}\frac{p+1}{4}\big{\rfloor}\big{)}\mu-\big{(}\big{\lfloor}\frac{p+1}{4}\big{\rfloor}-1\big{)}\\\
\hline\cr\phantom{\Big{(}}\\!\\!\\!1<\mu\phantom{\leq
3}&\big{\lfloor}\frac{p+1}{2}\big{\rfloor}\mu+\big{(}\big{\lfloor}\frac{p}{4}\big{\rfloor}+1\big{)}&-\big{\lfloor}\frac{p}{2}\big{\rfloor}\mu-\big{\lfloor}\frac{p-1}{4}\big{\rfloor}\end{array}$

These formulas are proved by induction. Let us first focus on the latter two
cases.

We assume $\mu>\tfrac{1}{2}$.

We wish to show that for $p$ big enough, $\nu(u^{-p})$ and $\nu(u^{p})$ are of
the form specified in the table above, depending on the sign of $\mu-1$. The
values of $\nu(u^{-p})$ will be extracted from the proof of the following
lemma.

######  Lemma 3.3.

For all integers $q\geq 3$, there are polynomials $P_{i}^{q}$ in $t$, so that

$u^{-4q}=u_{0}\,P_{0}^{q}(t)+u_{1}\,P_{1}^{q}(t)+u\,P_{2}^{q}(t)+\mathbb{1}P_{3}^{q}(t).$

These polynomials are of the form

$\displaystyle P_{0}^{q}(t)$ $\displaystyle=\alpha_{0}^{q}\,t^{2q\mu+(q+1)}$
$\displaystyle+\;\mbox{lower order terms}$ $\displaystyle P_{1}^{q}(t)$
$\displaystyle=\alpha_{1}^{q}\,t^{(2q-1)\mu+(q+1)}$
$\displaystyle+\;\mbox{lower order terms}$ $\displaystyle P_{2}^{q}(t)$
$\displaystyle=\alpha_{2}^{q}\,t^{(2q-1)\mu+(q+1)}$
$\displaystyle+\;\mbox{lower order terms}$ $\displaystyle P_{3}^{q}(t)$
$\displaystyle=\alpha_{3}^{q}\,t^{2q\mu+q}$ $\displaystyle+\;\mbox{lower order
terms}$

where all coefficients $\alpha_{i}^{q}$ are positive.

Its proof consists of tedious but straigthforward computations. First, notice
that the relations (9), give $u^{-1}=u_{0}\,t^{\mu+1}$ and the following
multiplication table

$\begin{array}[]{c||c|c|c|c}a&u_{0}&u_{1}&u&\mathbb{1}\\\
\hline\cr\phantom{\Big{(}}\\!\\!a\ast
u^{-1}&u_{1}&u\,t^{\mu}+\mathbb{1}&\mathbb{1}&u_{0}\,t^{\mu+1}\end{array}$
(17)

Next, compute explicitly $u^{-4q}$ for $q=3$ as initialization:

$\displaystyle u^{-12}$
$\displaystyle=3u_{0}\,t^{6\mu+4}+3u_{1}\,t^{5\mu+4}+u\,t^{5\mu+4}+\mathbb{1}(t^{6\mu+3}+t^{4\mu+4})\,$

which is of the form specified in Lemma 3.3 since $\mu\geq\frac{1}{2}$
(otherwise $P_{3}^{3}$ would have valuation $4\mu+4$).

Finally, assume $u^{-4q}$ has the expression given in the lemma and compute
the 4 next powers of $u$. Up to lower order terms, we get (to ease the reading
we denote $P^{q}_{i}(t)$ simply by $P_{i}$):

$\displaystyle u^{-4q-1}$
$\displaystyle=u_{0}P_{3}t^{\mu+1}+u_{1}P_{0}+uP_{1}t^{\mu}+\mathbb{1}(P_{1}+P_{2})$
(18) $\displaystyle u^{-4q-2}$
$\displaystyle=u_{0}(P_{1}+P_{2})t^{\mu+1}+u_{1}P_{3}t^{\mu+1}+uP_{0}t^{\mu}+\mathbb{1}(P_{0}+P_{1}t^{\mu})$
$\displaystyle u^{-4q-3}$
$\displaystyle=u_{0}(P_{0}+P_{1}t^{\mu})t^{\mu+1}+u_{1}(P_{1}+P_{2})t^{\mu+1}+uP_{3}t^{2\mu+1}+\mathbb{1}(P_{0}+P_{3}t)t^{\mu}$

This yields for $u^{-4(q+1)}$:

$\displaystyle P_{0}^{q+1}(t)$ $\displaystyle=P_{3}^{q}(t)\cdot
t^{2\mu+2}+P_{0}^{q}(t)\cdot t^{2\mu+1}$ $\displaystyle+\mbox{ l.o.t.}$
$\displaystyle P_{1}^{q+1}(t)$ $\displaystyle=P_{0}^{q}(t)\cdot
t^{\mu+1}+P_{1}^{q}(t)\cdot t^{2\mu+1}$ $\displaystyle+\mbox{ l.o.t.}$
$\displaystyle P_{2}^{q+1}(t)$ $\displaystyle=(P_{1}^{q}(t)+P_{2}^{q}(t))\cdot
t^{2\mu+1}$ $\displaystyle+\mbox{ l.o.t.}$ $\displaystyle P_{3}^{q+1}(t)$
$\displaystyle=P_{3}^{q}(t)\cdot t^{2\mu+1}+(P_{1}^{q}(t)+P_{2}^{q}(t))\cdot
t^{\mu+1}$ $\displaystyle+\mbox{ l.o.t.}$

whose respective leading order terms are

$\displaystyle P_{0}^{q+1}:\quad$
$\displaystyle(\alpha_{0}^{q}+\alpha_{3}^{q})\,t^{2(q+1)\mu+(q+1)+1}$
$\displaystyle P_{1}^{q+1}:\quad$
$\displaystyle(\alpha_{0}^{q}+\alpha_{1}^{q})\,t^{(2(q+1)-1)\mu+((q+1)+1)}$
$\displaystyle P_{2}^{q+1}:\quad$
$\displaystyle(\alpha_{1}^{q}+\alpha_{2}^{q})\,t^{(2(q+1)-1)\mu+((q+1)+1)}$
$\displaystyle P_{3}^{q+1}:\quad$
$\displaystyle\alpha_{3}^{q}\,t^{2(q+1)\mu+(q+1)}$

since all the coefficients $\alpha_{i}^{q}$ are positive and since
$\mu>\tfrac{1}{2}$ so that $(P_{1}^{q}(t)+P_{2}^{q}(t))\cdot t^{\mu+1}$ only
consists of lower order terms of $P_{3}^{q+1}$.

This proves the lemma. The formula giving $\nu(u^{-p})$ now follows from the
lemma together with the valuation of intermediate powers of $u^{-1}$ which
easily follow from the computations (18). Indeed, depending on the sign of
$\mu-1$, we get:

$\displaystyle\begin{array}[]{c|c|c}&\tfrac{1}{2}<\mu\leq
1&1<\mu\phantom{\Big{(}}\\\
\hline\cr\phantom{\Big{(}}p=4q\phantom{+1}&\lx@intercol\hfil
2q\mu+(q+1)\hfil\lx@intercol\\\
\hline\cr\phantom{\Big{(}}p=4q+1&\lx@intercol\hfil(2q+1)\mu+(q+1)\hfil\lx@intercol\\\
\hline\cr\phantom{\Big{(}}p=4q+2&2q\mu+(q+2)&(2q+1)\mu+(q+1)\\\
\hline\cr\phantom{\Big{(}}p=4q+3&(2q+1)\mu+(q+2)&2(q+1)\mu+(q+1)\end{array}$
(24)

We now turn to the valuation of positive powers of $u$. Symmetrically to the
negative powers, $\nu(u^{p})$ will be extracted from the proof of the
following lemma.

######  Lemma 3.4.

For all integers $q\geq 2$, there are polynomials $Q_{i}^{q}$ in $t$, so that

$u^{4q}=u_{0}\,Q_{0}^{q}(t)+u\,Q_{2}^{q}(t)+u_{3}\,Q_{3}^{q}(t)+\mathbb{1}Q_{4}^{q}(t).$

These polynomials are of the form

$\displaystyle Q_{0}^{q}(t)$ $\displaystyle=-\beta_{0}^{q}\,t^{-2q\mu-(q-1)}$
$\displaystyle+\;\mbox{lower order terms}$ $\displaystyle Q_{2}^{q}(t)$
$\displaystyle=-\beta_{2}^{q}\,t^{-(2q+1)\mu-(q-1)}$
$\displaystyle+\;\mbox{lower order terms}$ $\displaystyle Q_{3}^{q}(t)$
$\displaystyle=\beta_{3}^{q}\,t^{-(2q+1)\mu-(q-1)}$
$\displaystyle+\;\mbox{lower order terms}$ $\displaystyle Q_{4}^{q}(t)$
$\displaystyle=\beta_{4}^{q}\,t^{-2q\mu-q}$ $\displaystyle+\;\mbox{lower order
terms}$

where all coefficients $\beta_{i}^{q}$ are positive.

Notice that we changed the basis of the cohomology, this yields easier
computations and a slight notational discrepancy. The relevant multiplication
table now is

$\begin{array}[]{c||c|c|c|c}a&u_{0}&u&u_{3}&\mathbb{1}\\\
\hline\cr\phantom{\Big{(}}\\!\\!a\ast u&\bb
1\,t^{-\mu-1}&u_{3}\,t^{-\mu}&u_{0}-u_{3}\,t^{-\mu}&u\end{array}$ (25)

We initialize the induction with $u^{4q}$ for $q=2$:

$\displaystyle
u^{8}=-u_{0}\,(2t^{-4\mu-1}+t^{-6\mu})-u\,t^{-5\mu-1}+u_{3}\,(3t^{-5\mu-1}+t^{-7\mu})+\mathbb{1}(t^{-4\mu-2}+t^{-6\mu-1})$

which is seen to be of the form specified by Lemma 3.4, again, because
$\mu\geq\tfrac{1}{2}$.

We now assume $u^{4q}$ has the expression given above and compute the 4 next
powers of $u$. To ease the reading we denote $Q^{q}_{i}(t)$ simply by $Q_{i}$
and we introduce the notation $Q_{2,3}$ for $Q_{2}-Q_{3}$. We get, up to lower
order terms:

$\displaystyle u^{4q+1}$
$\displaystyle=u_{0}Q_{3}+uQ_{4}+u_{3}Q_{2,3}t^{-\mu}+\mathbb{1}Q_{0}t^{-\mu-1}$
(26) $\displaystyle u^{4q+2}$
$\displaystyle=u_{0}Q_{2,3}t^{-\mu}+uQ_{0}t^{-\mu-1}+u_{3}(Q_{4}t^{-\mu}-Q_{2,3}t^{-2\mu})+\mathbb{1}Q_{3}t^{-\mu-1}$
$\displaystyle u^{4q+3}$
$\displaystyle=u_{0}(Q_{4}t^{-\mu}-Q_{2,3}t^{-2\mu})+uQ_{3}t^{-\mu-1}$
$\displaystyle\hskip
56.9055pt+u_{3}(Q_{0}t^{-2\mu-1}-Q_{4}t^{-2\mu}+Q_{2,3}t^{-3\mu})+\mathbb{1}Q_{2,3}t^{-2\mu-1}$

This yields for $u^{4(q+1)}$ (again, up to lower order terms):

$\displaystyle Q_{0}^{q+1}(t)$ $\displaystyle=Q_{0}^{q}(t)\cdot
t^{-2\mu-1}-Q_{4}^{q}(t)\cdot t^{-2\mu}+(Q_{2}^{q}(t)-Q_{3}^{q}(t))\cdot
t^{-3\mu}$ $\displaystyle Q_{2}^{q+1}(t)$
$\displaystyle=(Q_{2}^{q}(t)-Q_{3}^{q}(t))\cdot t^{-2\mu-1}$ $\displaystyle
Q_{3}^{q+1}(t)$ $\displaystyle=-Q_{0}^{q}(t)\cdot
t^{-3\mu-1}+Q_{3}^{q}(t)\cdot t^{-2\mu-1}+Q_{4}^{q}(t)\cdot
t^{-3\mu}-(Q_{2}^{q}(t)-Q_{3}^{q}(t))\cdot t^{-4\mu}$ $\displaystyle
Q_{4}^{q+1}(t)$ $\displaystyle=Q_{4}^{q}(t)\cdot
t^{-2\mu-1}-(Q_{2}^{q}(t)-Q_{3}^{q}(t))\cdot t^{-3\mu-1}$

whose respective leading order terms are

$\displaystyle Q_{0}^{q+1}:\quad$
$\displaystyle-(\beta_{0}^{q}+\beta_{4}^{q})\,t^{-2(q+1)\mu-((q+1)-1)}$
$\displaystyle Q_{2}^{q+1}:\quad$
$\displaystyle-(\beta_{2}^{q}+\beta_{3}^{q})\,t^{-(2(q+1)+1)\mu-((q+1)-1)}$
$\displaystyle Q_{3}^{q+1}:\quad$
$\displaystyle(\beta_{0}^{q}+\beta_{3}^{q}+\beta_{4}^{q})\,t^{-(2(q+1)+1)\mu-((q+1)-1)}$
$\displaystyle Q_{4}^{q+1}:\quad$
$\displaystyle\beta_{4}^{q}\,t^{-2(q+1)\mu-(q+1)}$

since all the coefficients $\beta_{i}^{q}$ are positive and since
$\mu>\tfrac{1}{2}$.

This proves Lemma 3.4. The expression of $\nu(u^{p})$ then follows from it,
together with the valuation of intermediate powers of $u$ which easily follow
from the computations (26). Namely, we get

$\displaystyle\begin{array}[]{c|c|c}&\tfrac{1}{2}<\mu\leq
1&1<\mu\phantom{\Big{(}}\\\
\hline\cr\phantom{\Big{(}}p=4q\phantom{+1}&\lx@intercol\hfil-2q\mu-(q-1)\hfil\lx@intercol\\\
\hline\cr\phantom{\Big{(}}p=4q+1&-(2q+1)\mu-(q-1)&-2q\mu-q\\\
\hline\cr\phantom{\Big{(}}p=4q+2&-2(q+1)\mu-(q-1)&-(2q+1)\mu-q\\\
\hline\cr\phantom{\Big{(}}p=4q+3&\lx@intercol\hfil-(2q+1)\mu-q\hfil\lx@intercol\end{array}$
(32)

Together with the expression of $\nu(u^{-p})$ given by Table (24), this
concludes the proof of Proposition 3 in the case $\mu>\tfrac{1}{2}$.

We assume $0<\mu\leq\tfrac{1}{2}$.

The sketch of the proof in this case is similar, the main lemmas only need a
few adjustments.

For negative powers of $u$, the situation is perfectly similar to the previous
cases, except that the results depend on the rest of the Euclidean division of
$p$ by $3$ rather than $4$.

######  Lemma 3.5.

For all integers $q\geq 3$, there are polynomials $P_{i}^{q}$ in $t$, so that

$u^{-3q-1}=u_{0}\,P_{0}^{q}(t)+u_{1}\,P_{1}^{q}(t)+u\,P_{2}^{q}(t)+\mathbb{1}P_{3}^{q}(t).$

These polynomials are of the form

$\displaystyle P_{0}^{q}(t)$ $\displaystyle=t^{(q+1)\mu+(q+1)}$
$\displaystyle+\;\mbox{lower order terms}\phantom{\,.}$ $\displaystyle
P_{1}^{q}(t)$ $\displaystyle=\tfrac{(q-1)(q-2)}{2}\,t^{(q+2)\mu+q}$
$\displaystyle+\;\mbox{lower order terms}\phantom{\,.}$ $\displaystyle
P_{2}^{q}(t)$ $\displaystyle=(q-1)\,t^{(q+2)\mu+q}$
$\displaystyle+\;\mbox{lower order terms}\phantom{\,.}$ $\displaystyle
P_{3}^{q}(t)$ $\displaystyle=q\,t^{(q+1)\mu+q}$ $\displaystyle+\;\mbox{lower
order terms}\,.$

We use the multiplication table (17) to compute

$u^{-10}=u_{0}\,t^{4\mu+4}+u_{1}\,t^{5\mu+3}+2u\,t^{5\mu+3}+3\bb
1\,t^{4\mu+3}\,$

which is indeed of the expected form.

The expressions of the following 3 powers $u^{-3q-2}$, $u^{-3q-3}$, and
$u^{-3(q+1)-1}$ are given from $u^{-3q-1}$ by equation (18). From this we
conclude that

$\displaystyle P_{0}^{q+1}:\quad$ $\displaystyle t^{(q+2)\mu+(q+2)}$
$\displaystyle+\mbox{ l.o.t.}$ $\displaystyle P_{1}^{q+1}:\quad$
$\displaystyle\big{(}\tfrac{(q-1)(q-2)}{2}+(q-1)\big{)}\,t^{(q+3)\mu+(q+1)}$
$\displaystyle+\mbox{ l.o.t.}$ $\displaystyle P_{2}^{q+1}:\quad$
$\displaystyle q\,t^{(q+3)\mu+(q+1)}$ $\displaystyle+\mbox{ l.o.t.}$
$\displaystyle P_{3}^{q+1}:\quad$ $\displaystyle(q+1)\,t^{(q+1)\mu+(q+1)}$
$\displaystyle+\mbox{ l.o.t.}$

since $\mu\leq\frac{1}{2}$. Notice that this yields the expected expression of
$P_{1}^{q+1}$ since $\tfrac{(q-1)(q-2)}{2}$ is the sum of all integers between
$1$ and $q-2$.

This concludes the proof of the lemma, from which we immediately get that
$\nu(u^{-3q-1})=(q+1)\mu+(q+1)$. The other necessary valuations can be
extracted from (18):

$\displaystyle\nu(u^{-3q-2})=(q+2)\mu+(q+1)\;\mbox{ and
}\;\nu(u^{-3q-3})=(q+3)\mu+(q+1)\,.$ (33)

For positive powers of $u$, the situation is somehow easier since we can
directly compute the valuation of each power of $u$. The relevant lemma is the
following.

######  Lemma 3.6.

For all integers $p\geq 5$, there are polynomials $Q_{i}^{p}$ in $t$, so that

$u^{p}=u_{0}\,Q_{0}^{p}(t)+u\,Q_{2}^{p}(t)+u_{3}\,Q_{3}^{p}(t)+\mathbb{1}Q_{4}^{p}(t).$

These polynomials are of the form

$\displaystyle Q_{0}^{p}(t)$ $\displaystyle=(-1)^{p+1}\,t^{-(p-2)\mu}$
$\displaystyle+\;\mbox{lower order terms}\phantom{\,.}$ $\displaystyle
Q_{2}^{p}(t)$ $\displaystyle=(-1)^{p+1}\,t^{-(p-3)\mu-1}$
$\displaystyle+\;\mbox{lower order terms}\phantom{\,.}$ $\displaystyle
Q_{3}^{p}(t)$ $\displaystyle=(-1)^{p}\,t^{-(p-1)\mu}$
$\displaystyle+\;\mbox{lower order terms}\phantom{\,.}$ $\displaystyle
Q_{4}^{p}(t)$ $\displaystyle=(-1)^{p}\,t^{-(p-2)\mu-1}$
$\displaystyle+\;\mbox{lower order terms}\,.$

Using the multiplication table (25), we compute

$u^{5}=u_{0}t^{-3\mu}+ut^{-2\mu-1}-u_{3}t^{-4\mu}-\bb 1t^{-3\mu-1}$

and we compute $u^{p+1}$ from $u^{p}$

$u^{p+1}=u_{0}\,Q_{3}^{p}(t)+u\,Q_{4}^{p}(t)+u_{3\,}(Q_{2}^{p}(t)-Q_{3}^{p}(t))\,t^{-\mu}+\bb
1\,Q_{0}^{p}(t)\,t^{-\mu-1}+\mbox{ l.o.t.}$

Notice that $Q_{3}^{p+1}$ is of the expected form because
$\mu\leq\frac{1}{2}$.

This proves the lemma and, together with (33), end the proof of Proposition 3
in the last remaining case, for $0<\mu\leq\frac{1}{2}$. $\square$

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VH: Sorbonne Université and Université de Paris, CNRS, IMJ-PRG, F-75005 Paris,
France & Institut Universitaire de France.

E-mail address<EMAIL_ADDRESS>
AJ: Heidelberg University, Institut für Mathematik, Excellence Cluster
STRUCTURES, 69120 Heidelberg, Germany

E-mail address<EMAIL_ADDRESS>
RL: Université Paris-Saclay, CNRS, Laboratoire de mathématiques d’Orsay,
91405, Orsay, France.

E-mail address<EMAIL_ADDRESS>