CapIntersactionWithContEqn2})$ is proved similarly, needing only the
additional remark that for $\hat{x}\in\hat{X}$,
$\alpha^{\\#}\cap_{t}\hat{x}\in\mathbb{Z}_{2}\langle\hat{x}\rangle_{\hat{x}\in\hat{X}}\oplus
Pos^{*}(\hat{X})$, which follows by the same reasoning as above, using that
$b(\hat{x})\geq 0$ when $\mu(\hat{x})$ lies in the Morse range. ∎

The following proposition is essentially tautological.

######  Proposition 6.25.

Let $(H,J)$ be Floer regular, $t\in S^{1}$, and suppose that $p\in\Sigma$ is
$(H,J;t)$-generic (for the point class in homology). Then $p\in
W(\hat{x},\hat{y})$ implies that $\mu(\hat{x})-\mu(\hat{y})\geq 2$, and
$\hat{y}\in\text{supp }(p\cap_{t}\hat{x})$ if and only if $(t,p)\in
W(\hat{x},\hat{y})$.

Combining the above with Corollary 6.7, allows us to conclude

######  Corollary 6.26.

Suppose that $\mu(\hat{x})=2k+1$ for $k\in\mathbb{Z}$ and $p$ is
$(H,J;t)$-generic, then $\hat{y}\in\text{supp }(p\cap_{t}\hat{x})$ if and only
if there exists an open neighbourhood of $(t,p)\in S^{1}\times\Sigma$ which is
foliated by leaves of $\mathcal{F}^{\hat{x},\hat{y}}$.

Recall from Section 6.1 that $W(\hat{x},\hat{y})=\\{(t,u(s,t))\in
S^{1}\times\Sigma:u\in\widetilde{\mathcal{M}}(\hat{x},\hat{y};H,J)\\}$, and
write $\mathcal{W}(\hat{X})$ for the union of all $W(\hat{x},\hat{y})$ where
$\hat{x}\in\hat{X}_{(1)}$, $\hat{y}\in\hat{X}_{(-1)}$.

######  Lemma 6.27.

Let $(H,J)$ be Floer regular and $\hat{X}\in murm(H)$, then
$\mathcal{W}(\hat{X})$ is open and dense in $S^{1}\times\Sigma$.

###### Proof.

Let
$(\mathcal{H},\mathbb{J})\in\mathscr{H\mspace{-8.0mu}J}^{\hat{X}}(f,J^{-};H,J)$.
We fix $t\in S^{1}$ arbitrarily and let $p\in\Sigma$ be
$(\mathcal{H},\mathbb{J};t)$-generic. We let $\sigma\in CF_{1}(f,J^{-})$
represent the fundamental class $[\Sigma]\in QH_{2}(\Sigma)\simeq HF_{1}(f)$,
and we note that we must have $[p\cap_{t}\sigma]=[pt]\in QH_{0}(\Sigma)$,
which is in particular not $0$. But Lemma 6.22 implies that
$(\pi^{\hat{X}}\circ h_{\mathcal{H}})_{*}$ is injective on homology, and so

$\displaystyle 0\neq(\pi^{\hat{X}}\circ h_{\mathcal{H}})(p\cap_{t}\sigma)$
$\displaystyle=\pi^{\hat{X}}(p\cap_{t}(\pi^{\hat{X}}\circ
h_{\mathcal{H}})(\sigma)),$

and this implies that for every such generic $p$, there must exist some
$\hat{x}\in\text{supp }(\pi^{\hat{X}}\circ
h_{\mathcal{H}})(\sigma)\subseteq\hat{X}_{(1)}$ and some
$\hat{y}\in\hat{X}_{(-1)}$ such that $\hat{y}\in\text{supp }p\cap_{t}\hat{x}$.
Hence for every $t\in S^{1}$, there is a generic set of $p\in\Sigma$ such that
$p$ is $(\mathcal{H},\mathbb{J};t)$-generic and so $(t,p)$ lies inside the
open $3$-dimensional connecting submanifold $W(\hat{x},\hat{y})$ for some such
$\hat{x},\hat{y}\in\hat{X}$, which proves the lemma. ∎

######  Lemma 6.28.

Let $(H,J)$ be Floer regular, and $\hat{X}\subseteq\widetilde{Per}_{0}({H})$
any unlinked capped braid. Then for any
$\hat{\gamma}\in\widetilde{Per}_{0}({H})$ such that
$\mathcal{M}(\hat{x}^{+},\hat{\gamma};H,J)\times\mathcal{M}(\hat{\gamma},\hat{x}^{-};H,J)\neq\emptyset$
for some $\hat{x}^{+}\in\hat{X}_{(1)}$, $\hat{x}^{-}\in\hat{X}_{(-1)}$,
$\hat{X}$ and $\hat{\gamma}$ are unlinked.

###### Proof.

Suppose that $\hat{X}\cup\\{\hat{\gamma}\\}$ is linked, then Lemma 6.12
implies that $\hat{\gamma}\in Pos^{*}(\hat{X})$, and so
$\ell(\hat{\gamma},\hat{x})>0$ for some $\hat{x}\in\hat{X}$, but then
Proposition 6.3 implies that $\ell(\hat{x}^{-},\hat{x})>0$, which contradicts
the assumption that $\hat{X}$ is unlinked. ∎

Inductively applying Lemma 6.28 yields

######  Corollary 6.29.

Suppose that $(H,J)$ is Floer regular, and let
$\hat{X}\subseteq\widetilde{Per}_{0}({H})$ be such that $\hat{X}$ is unlinked
and $\hat{X}=\hat{X}_{(1)}\cup\hat{X}_{(-1)}$, then $\hat{X}$ and
$\hat{\Upsilon}$ are unlinked, where $\hat{\Upsilon}$ is the capped braid
consisting of all $\hat{\gamma}\in\widetilde{Per}_{0}({H})$ with
$\mu(\hat{\gamma})=0$ such that
$\mathcal{M}(\hat{x}^{+},\hat{\gamma};H,J)\times\mathcal{M}(\hat{\gamma},\hat{x}^{-};H,J)\neq\emptyset$,
$\hat{x}^{\pm}\in\hat{X}$.

Finally, we are ready to prove the existence of the advertised foliation. We
write

$\displaystyle\mathcal{M}^{H,J}(\hat{X})$
$\displaystyle:=\bigcup_{\hat{x},\hat{y}\in\hat{X}}\mathcal{M}(\hat{x},\hat{y};H,J).$

Note that we include the case where $\hat{x}=\hat{y}$ in the above union. In
such a case $[u]\in\mathcal{M}(\hat{x},\hat{x};H,J)$ may simply be identified
with the loop $x$.

######  Theorem 6.30 (Existence part of Theorem 6.23).

Let $(H,J)$ be a non-degenerate Floer pair, and $\hat{X}\in murm(H)$, then the
collection of submanifolds
$\mathcal{F}^{\hat{X}}:=\cup_{[u]\in\mathcal{M}^{H,J}(\hat{X})}\\{\text{im}\;\check{u}\\}$
forms a Stefan-Sussmann foliation of $S^{1}\times\Sigma$.

###### Proof.

We adapt a strategy used in [11] that shows that the foliation
$\widetilde{\mathcal{F}}^{\hat{X}}$ with leaves given by the graphs
$\tilde{u}$ of all the $u\in\widetilde{\mathcal{M}}(\hat{x},\hat{y};H,J)$,
$\hat{x},\hat{y}\in\hat{X}$, is a smooth $2$-dimensional foliation if
$\mathbb{R}\times S^{1}\times\Sigma$, from which it follows immediately that
$\mathcal{F}^{\hat{X}}$ is a Steffan-Sussmann foliation. Indeed, in this
event, $\mathcal{F}^{\hat{X}}$ integrates the distribution
$\mathcal{D}^{\hat{X}}=\check{\pi}_{*}\widetilde{\mathcal{D}}^{\hat{X}}$,
where $\widetilde{\mathcal{D}}^{\hat{X}}$ is the distribution integrated by
$\widetilde{\mathcal{F}}^{\hat{X}}$, and this realizes $\mathcal{D}^{\hat{X}}$
in a way that is manifestly smooth in the sense of generalized distributions
(see Definition 7.2).

To see that $\widetilde{\mathcal{F}}^{\hat{X}}$ is a smooth foliation, note
that by Lemma 6.27, the set $\mathcal{W}(\hat{X})$ is open and dense in
$S^{1}\times\Sigma$. This implies, by the $\mathbb{R}$-invariance of solutions
to Equation 1, that the set of points $\widetilde{W}(\hat{X})$ lying on the
graph $\tilde{u}$ of some $u\in\widetilde{\mathcal{M}}(\hat{x},\hat{y};H,J)$,
$\hat{x}\in\hat{X}_{(1)},\hat{y}\in\hat{X}_{(-1)}$ is open and dense in
$\mathbb{R}\times S^{1}\times\Sigma$. Consequently, the partition
$\bigcup\widetilde{\mathcal{F}}^{\hat{x},\hat{y}}$, where the union runs over
all $\hat{x}\in\hat{X}_{(1)}$, $\hat{y}\in\hat{X}_{(-1)}$, gives a smooth
foliation of an open, dense set of $\mathbb{R}\times S^{1}\times\Sigma$.
Consequently we may argue just as in [11] in the paragraphs following the
proof of lemma $6.10$ (p. 231-232); all of the remaining leaves in
$\widetilde{\mathcal{F}}^{\hat{X}}$ are graphs of constant orbits or of
cylinders $u$ which connect orbits of index difference equal to $1$. In either
case, by standard compactness theorems of Floer theory those graphs which form
the leaves of the foliation of $\widetilde{\mathcal{W}}(\hat{X})$ converge
modulo reparametrization in the $C^{\infty}_{loc}$-topology either to the
graphs $(s,t)\mapsto(s,t,x(t))$ of the orbits $x$ for $\hat{x}\in\hat{X}$, or
to graphs of cylinders connecting orbits of index difference $1$, which come
in pairs
$(u,v)\in\widetilde{\mathcal{M}}(\hat{x},\hat{\gamma};H,J)\times\widetilde{\mathcal{M}}(\hat{\gamma},\hat{y};H,J)$,
$\hat{x}\in\hat{X}_{(1)},\hat{\gamma}\in\hat{X}_{(0)},\hat{y}\in\hat{X}_{(-1)}$.
By Corollary 6.29 and Lemma 6.28, the capped braid formed by the collection of
all the $\hat{\gamma}\in\widetilde{Per}_{0}({H})$ on which such pairs break
are unlinked with $\hat{X}$, and so lie in $\hat{X}$ by maximality.
Consequently, the graphs of such broken trajectories cannot intersect, nor can
they intersect any leaf of $\widetilde{\mathcal{F}}^{\hat{X}}$ in the dense
set $\widetilde{\mathcal{W}}(\hat{X})$. Since every point in $\mathbb{R}\times
S^{1}\times\Sigma$ lies in the closure of $\widetilde{\mathcal{W}}(\hat{X})$,
every such point much lie on the graph of an orbit in $X$ or the graph of such
a broken cylinder. It follows that the union of all the leaves in
$\widetilde{\mathcal{F}}^{\hat{X}}$ thus fits together into a smooth foliation
on all of $\mathbb{R}\times S^{1}\times\Sigma$, and so the theorem follows. ∎

####  6.3.1 $\mathcal{F}^{\hat{X}}$ as negative gradient flow-lines of the
restricted action functional

For $\hat{X}\in murm(H)$, denote

$\displaystyle\mathfrak{M}_{\hat{X}}=\mathfrak{M}_{\hat{X};H,J}$
$\displaystyle:=\\{\hat{\alpha}\in\widetilde{\mathcal{L}_{0}}(\Sigma):\exists\hat{x},\hat{y}\in\hat{X},\exists
u\in\widetilde{\mathcal{M}}(\hat{x},\hat{y};H,J),\;\text{such that
}\hat{u}_{s}=\hat{\alpha},\text{for some }s\in\mathbb{R}\\}$

######  Proposition 6.31.

The map $Ev:S^{1}\times\mathfrak{M}_{\hat{X}}\rightarrow S^{1}\times\Sigma$,
given by $Ev(t,\hat{\alpha})=(t,\alpha(t))$ is a diffeomorphism.

###### Proof.

The generalized distribution
$\mathcal{D}^{\hat{X}}_{(t,u(s,t))}=\langle\partial_{s}u,\partial_{t}\oplus\partial_{u}\rangle$
which is integrated by $\mathcal{F}^{\hat{X}}$ contains the one-dimensional
distribution
$\mathcal{D}^{\mathfrak{M}}_{(t,u(s,t))}=\langle\partial_{t}\oplus\partial_{t}u\rangle$,
which is easily seen to be smooth near the singular fibers by employing the
local model for leaves of a Stefan-Sussmann foliation of Section 7.
Consequently, $\mathcal{D}^{\mathfrak{M}}$ is a smooth foliation which
integrates precisely to the graphs of the maps $\alpha:S^{1}\rightarrow\Sigma$
for $\hat{\alpha}\in\mathfrak{M}_{\hat{X}}$. This is obviously equivalent to
the proposition. ∎

######  Definition 6.32.

For $\hat{X}\in murm(H)$, define the ($\hat{X}$-)restricted action functional
$A^{\hat{X}}\in C^{\infty}(S^{1}\times\Sigma)$ by
$A^{\hat{X}}:=\mathcal{A}_{H}\circ Ev^{-1}$. Additionally, for each $t\in
S^{1}$, we define $A^{\hat{X}}_{t}:=\iota_{t}^{*}A^{\hat{X}}$, where
$\iota_{t}:\Sigma\hookrightarrow S^{1}\times\Sigma$ is the inclusion of the
fiber over $t\in S^{1}$.

Note that each $A^{\hat{X}}_{t}$ is automatically Morse, since the Hessian of
$A^{\hat{X}}_{t}$ at $x(t)$ for $\hat{x}\in\hat{X}$ obviously inherits the
non-degeneracy of the Hessian of $\mathcal{A}_{H}$ at $\hat{x}$. In fact, our
construction clearly identifies Floer trajectories connecting orbits in
$\hat{X}$ with negative gradient flow lines of the $A^{\hat{X}}_{t}$, giving
us Morse models for the foliation $\mathcal{F}^{\hat{X}}$.

######  Proposition 6.33.

If $(H,J)$ is Floer regular, $\hat{X}\in murm(H)$ and $\epsilon>0$ is
sufficiently small, then for every $t\in S^{1}$, and every
$\hat{x},\hat{y}\in\hat{X}$, there is a natural identification
$\widetilde{\mathcal{M}}(\hat{x},\hat{y};H,J)\cong\widetilde{\mathcal{M}}(\hat{x},\hat{y};\epsilon
A^{\hat{X}}_{t},J_{t})$ given by $u(s,t)\mapsto u(\epsilon s,t)$.

######  Corollary 6.34.

Let $(H,J)$ be Floer regular and $\hat{X}\in murm(H)$. Then for every $t\in
S^{1}$, and any $\epsilon>0$ sufficiently small,
$CF_{*}(\hat{X};H,J)\cong(C^{Morse}(A^{\hat{X}}_{t},g_{J_{t}})\otimes\Lambda_{\omega})_{*+1}\cong
CF_{*}(\epsilon A^{\hat{X}},J_{t})$.

####  6.3.2 $\mathcal{F}^{\hat{X}}$ as a positively transverse foliation

Each regular leaf of the foliation $\mathcal{F}^{\hat{X}}$ arises naturally as
the image of an embedding $\hat{u}:\mathbb{R}\times S^{1}\hookrightarrow
S^{1}\times\Sigma$ for $u$, the standard orientation on the cylinder induces
the orientation $\partial_{s}u\wedge\partial_{t}u$ on each regular leaf, so we
may view $\mathcal{F}^{\hat{X}}$ in a natural way as an oriented singular
foliation.

######  Definition 6.35.

Let $\mathcal{F}$ be an oriented codimension $1$ Steffan-Sussmann foliation of
an oriented $d$-dimensional manifold $(M^{d},o_{M})$. We will say that a
smooth path $\alpha:[0,1]\rightarrow M$ is positively transverse to
$\mathcal{F}$ if the following dichotomy holds, either

  1. 1.

$\alpha$ is contained in a singular leaf of $\mathcal{F}$, or

  2. 2.

for every $t\in[0,1]$, $\\{(\partial_{t}\alpha)_{t},v_{1},\ldots,v_{d-1}\\}$
is an oriented basis for $(T_{\alpha(t)}M,o_{M})$, where
$\\{v_{1},\ldots,v_{d-1}\\}$ is an oriented basis for the tangent space of the
regular leaf of $\mathcal{F}$ passing through $\alpha(t)$.

######  Definition 6.36.

Let $\mathcal{F}$ be an oriented codimension $1$ Steffan-Sussmann foliation on
an oriented $d$-dimensional manifold $(M^{d},o_{M})$ and let
$X\in\mathcal{X}(M)$ be a vector field generating an isotopy
$(\phi_{t}^{X})_{t\in\mathbb{R}}$. We say that $X$ (or
$(\phi_{t}^{X})_{t\in\mathbb{R}}$) is positively transverse to $\mathcal{F}$
if every integral curve of $X$ is positively transverse to $\mathcal{F}$.

######  Proposition 6.37.

Let $(H,J)$ be Floer regular, $\hat{X}\in murm(H)$, and
$\check{X}_{H}:=\partial_{t}\oplus X_{H}\in\mathcal{X}(S^{1}\times\Sigma)$,
then $\mathcal{F}^{\hat{X}}$ is positively transverse to $\check{X}_{H}$.

###### Proof.

As the singular leaves of $\mathcal{F}^{\hat{X}}$ are orbits of
$\check{X}_{H}$, it suffices to consider points $(t,p)\in S^{1}\times\Sigma$
lying on regular leaves. In such a case, since $u$ solves Equation 1, the
basis formed by
$\\{\check{X}_{H},\partial_{s}\check{u},\partial_{t}\check{u}\\}$ is easily
seen to be orientation-equivalent to the basis
$\\{\partial_{t},\partial_{s}u,J_{t}\partial_{s}u\\}$, which is a positively
oriented basis, as $J_{t}\in\mathcal{J}(\Sigma,\omega)$ for all $t\in S^{1}$.
∎

The previous proposition tells us that to any non-degenerate Hamiltonian $H$
and each $\hat{X}\in murm(H)$, we may associate a foliation on
$S^{1}\times\Sigma$ with respect to which the graph of the isotopy is well-
behaved in a certain sense. However, if we’re willing to modify the isotopy by
a contractible loop, then we can in fact do better and obtain a positively
transverse singular foliation on $\Sigma$ itself.

To see this, consider the distribution
$\mathcal{D}^{\mathfrak{M}}_{(t,u(s,t))}=\langle\partial_{t}\oplus\partial_{t}u\rangle$
introduced in the proof of Proposition 6.31. As noted therein,
$\mathcal{D}^{\mathfrak{M}}$ integrates to a smooth $1$-dimensional foliation
by the graphs of the loops $t\mapsto u_{s}(t)$ for
$\hat{u}_{s}\in\mathfrak{M}_{\hat{X}}$. This induces a natural loop of
diffeomorphisms $(\psi^{\hat{X}}_{t})_{t\in S^{1}}$ given by sliding the fiber
$\\{0\\}\times\Sigma$ along the foliation which integrates
$\mathcal{D}^{\mathfrak{M}}$. In other words, we have the isotopy
$\psi^{\hat{X}}_{t}(p)=u_{p}(s,t)$, $t\in S^{1}$, where
$u_{p}\in\widetilde{\mathcal{M}}(\hat{x},\hat{y};H,J)$,
$\hat{x},\hat{y}\in\hat{X}$, is any Floer cylinder such that $u_{p}(s,0)=p$.
It follows from Corollary 6.7 and the fact that if $\hat{x}=\hat{y}$ then
$u(s,t)=x(t)$ that $\psi^{\hat{X}}$ is well-defined.

######  Proposition 6.38.

$\psi:=(\psi^{\hat{X}}_{t})_{t\in S^{1}}$ is a contractible loop.

###### Proof.

$\psi$ defines a loop of diffeomorphisms based at the identity by
construction. If the genus of $\Sigma$ is strictly greater than $1$, then the
proposition follows from the fact that $Diff_{0}(\Sigma)$ is contractible.

If $\Sigma=\mathbb{T}^{2}$, $Diff_{0}(\Sigma)$ has as a strong deformation
retract the group of diffeomorphisms given by the action of the torus on
itself. In particular, contractible loops in $Diff_{0}(\mathbb{T}^{2})$ are
precisely the loops having contractible orbits. Note that for any
$\hat{x}=[x,w]\in\hat{X}$, $\psi^{\hat{X}}_{t}(x(0))=x(t)$, and so
$\psi^{\hat{X}}$ has contractible periodic orbits, and hence is contractible
if $\Sigma=\mathbb{T}^{2}$.

In the case that $\Sigma=S^{2}$, it is an easy exercise to see that if
$(f_{t})_{t\in S^{1}}$ is a loop based at the identity in $Diff(S^{2})$, then
the homotopy class of $(f_{t})_{t\in S^{1}}$ in $\pi_{1}(Diff(S^{2}))$ is
classified by the parity of the winding number of the loop

$\displaystyle v(t)$ $\displaystyle:=((\Phi_{\hat{z}}^{t})^{-1}\circ
Df_{t}\circ\Phi_{\hat{z}}^{t})(v_{0}),\;t\in S^{1}$

for any $v_{0}\in\mathbb{R}^{2}\setminus\\{0\\}$, where
$\hat{z}\in\widetilde{\mathcal{L}_{0}}(S^{2})$ is any capped $1$-periodic
orbit of $(f_{t})_{t\in S^{1}}$ and

$\displaystyle\Phi_{\hat{z}}:S^{1}\times\mathbb{R}^{2}$
$\displaystyle\rightarrow z^{*}TS^{2}$

is any trivialization of the tangent bundle along $z$ which extends over the
chosen capping disk. Since the capped loops $\hat{x}\in\hat{X}$ are all capped
$1$-periodic orbits of $\psi^{\hat{X}}$ by construction, it suffices to fix
some $\hat{x}=[x,w]\in\hat{X}$ and to compute the winding number of some
vector in $T_{x(0)}S^{2}$ under the linearized flow of $\psi^{\hat{X}}$ along
$x$. Suppose without loss of generality that $\mu(\hat{x};H)=1$ and let
$\xi_{1},\xi_{2}\in x_{0}^{*}T\Sigma$ be a basis of eigenvectors of $A_{x,J}$
with winding number $0$ relative the capping $w$, then by the construction of
$\psi^{\hat{X}}$, $(D\psi_{t})(\xi_{i}(0))=\xi_{i}(t)$ for $t\in S^{1}$ by the
asymptotic estimates of Theorem 3.8. Consequently the linearized winding of
$\psi^{\hat{X}}$ along $x$ relative the capping $w$ is even, and so
$\psi^{\hat{X}}$ is a contractible loop. ∎

Note that $\mathcal{F}^{\hat{X}}$ is everywhere transverse to the fibers
$\\{t\\}\times\Sigma$ of $S^{1}\times\Sigma$, and so may be viewed as an
$S^{1}$-family of (singular) foliations on $\Sigma$. Let us write
$\mathcal{F}^{\hat{X}}_{t}$ for the foliation obtained on $\Sigma$ by
intersecting $\mathcal{F}^{\hat{X}}$ with $\\{t\\}\times\Sigma$.

######  Theorem 6.39.

Let $(H,J)$ be a Floer regular pair, $\hat{X}\in murm(H)$, then the orbits of
the isotopy $(\psi^{\hat{X}})^{-1}\circ\phi^{H}$ are positively transverse to
the foliation $\mathcal{F}^{\hat{X}}_{0}$

###### Proof.

Writing $\psi=\psi^{\hat{X}}$, observe that the vector field
$(Z_{t})_{t\in[0,1]}$ which generates the isotopy $\psi^{-1}\circ\phi^{H}$ is
easily computed via the chain rule as
$(Z_{t})_{u(s,0)}=(\psi_{t}^{-1})_{*}(X^{H}_{t}-\partial_{t}u)_{u(s,t)}$. Note
moreover that the definition of $\psi_{t}$ implies that

$\displaystyle(\psi_{t})_{*}(\partial_{s}u)_{u(s,0)})$
$\displaystyle=(\partial_{s}u)_{u(s,t)}$

for all $u\in\mathfrak{M}^{\hat{X}}$ and all $(s,t)\in\mathbb{R}\times S^{1}$.
Consequently, because for all $t\in S^{1}$ $\psi_{t}$ is an orientation-
preserving diffeomorphism, we see that

$\displaystyle
sgn\;\omega_{u(s,0)}(Z_{t},\partial_{s}u)=sgn\;(\psi_{t}^{*}\omega)(Z_{t},\partial_{s}u)=sgn\;\omega_{u(s,t)}(X^{H}_{t}-\partial_{t}u,\partial_{s}u)$
$\displaystyle=sgn\;\omega_{u(s,t)}(-J_{t}\partial_{s}u,\partial_{s}u)$

from which the claim follows. ∎

Theorem B is an immediate consequence of Proposition 6.33 and the preceding
Theorem.

###  6.4 Consequences for the structure of Hamiltonian isotopies

######  Definition 6.40.

For $(H,J)$ non-degenerate and $\hat{X}\in murm(H)$, we define the Piexoto
graph of $\mathcal{F}^{\hat{X}}$ to be the directed graph
$\Gamma(\mathcal{F}^{\hat{X}})$ whose vertex set is $\hat{X}$ and such that
there is a directed edge from $\hat{x}$ to $\hat{y}$ only if
$\mu(\hat{x})-\mu(\hat{y})=1$, and in this case there is an edge from
$\hat{x}$ to $\hat{y}$ for each element in $\mathcal{M}(\hat{x},\hat{y};H,J)$.

######  Remark.

Note that since $\mathcal{F}^{\hat{X}}_{t}$ may be realized as the singular
foliation obtained by the negative gradient flow of $(A^{\hat{X}},g_{J_{t}})$,
$\Gamma(\mathcal{F}^{\hat{X}})$ may be naturally identified with the Piexoto
graph (see [23]) of $(A^{\hat{X}}_{t},g_{J_{t}})$

######  Definition 6.41.

Let $(H,J)$ be non-degenerate and $\hat{X}\in murm(H)$. To any capped loop
$\hat{\gamma}\in\widetilde{\mathcal{L}_{0}}(\Sigma)$ such that $(\gamma,x)$ is
a braid for all $\hat{x}\in\hat{X}$, we may define the linking cochain
$\ell_{\hat{\gamma}}(\hat{x}):=\ell(\hat{\gamma},\hat{x})\in\mathbb{Z}$ for
any $\hat{x}\in V(\Gamma(\mathcal{F}^{\hat{X}}))=\hat{X}$, as well as the
intersection cochain
$I_{\gamma}:E(\Gamma(\mathcal{F}^{\hat{X}}))\rightarrow\mathbb{Z}$, where
$I_{\gamma}(u)$ counts the signed intersection number of (some transverse
perturbation of) the maps $\check{u}(s,t)=(t,u(s,t))$ and
$\check{\gamma}(t)=(t,\gamma(t))$.

The following relation between these two quantities is immediate from the
definition of the homological linking number.

######  Proposition 6.42.

Let $(H,J)$ be non-degenerate and $\hat{X}\in murm(H)$. For any capped loop
$\hat{\gamma}\in\widetilde{\mathcal{L}_{0}}(\Sigma)$ such that $(\gamma,x)$ is
a braid for all $\hat{x}\in\hat{X}$, we have $I_{\gamma}=\delta\ell_{\gamma}$
(ie. $I_{\gamma}(u)=\ell_{\gamma}(\hat{y})-\ell_{\gamma}(\hat{x})$, where
$u\in\widetilde{\mathcal{M}}(\hat{x},\hat{y};H,J)$).

For any Hamiltonian $H$, we write $H^{\sharp m}:=H\\#\ldots\\#H$ for the
$m$-fold concatenated Hamiltonian which generates $(\phi^{H}_{t})^{\sharp m}$.
For $\hat{x}=[x(t),w(se^{2\pi it})]\in\widetilde{\mathcal{L}_{0}}(\Sigma)$, we
write $\hat{x}^{\sharp m}:=[x(mt),w(se^{2\pi imt})]$, and for
$\hat{X}=\\{\hat{x}_{i}\\}_{i=1}^{k}$ we write $\hat{X}^{\sharp
m}:=\\{\hat{x}_{i}^{\sharp m}\\}_{i=1}^{k}$.

Note that the fact that the isotopy
$(\varphi_{t})_{t\in[0,1]}:=((\psi^{\hat{X}})_{t}^{-1}\circ\phi^{H}_{t})_{t\in[0,1]}$
is positively transverse to the singular foliation $\mathcal{F}^{\hat{X}}_{0}$
on $\Sigma$ implies that for all $m\in\mathbb{Z}_{>0}$, its $m$-fold
concatenation $(\varphi_{t}^{\sharp m})_{t\in[0,1]}$ is positively transverse
to $\mathcal{F}^{\hat{X}}_{0}$. As a consequence, if
$\hat{\gamma}\in\widetilde{\mathcal{L}_{0}}(\Sigma)$ is such that
$(\gamma,x^{\sharp m})$ is a braid for all $\hat{x}^{\sharp
m}\in\hat{X}^{\sharp m}$, then we have that

$\displaystyle I_{\gamma}(u)=\ell_{\hat{\gamma}}(\hat{y}^{\sharp
m})-\ell_{\hat{\gamma}}(\hat{x}^{\sharp m}),$

where $u\in\widetilde{\mathcal{M}}(\hat{x},\hat{y};H,J)$. We obtain as an
immediate corollary

######  Corollary 6.43.

Let $(H,J)$ be non-degenerate, $\hat{X}\in murm(H)$ and let
$\hat{\gamma}\in\widetilde{Per}_{0}({H^{\sharp k}})$ for
$k\in\mathbb{Z}_{>0}$, then $I_{\hat{\gamma}}$ is non-negative on every edge
of $\Gamma(\mathcal{F}^{\hat{X}})$. Moreover, for every
$\hat{x},\hat{z}\in\hat{X}$,
$\ell(\hat{x},\hat{\gamma})<\ell(\hat{z},\hat{\gamma})$ if and only if
$\check{\gamma}(t)\in W(\hat{x},\hat{z})$ for some $t\in S^{1}$.

######  Definition 6.44.

For a Hamiltonian $H$, we will say that a capped braid
$\hat{X}\subseteq\widetilde{Per}_{0}({H})$ is strongly linking if for any
$\hat{\gamma}\in\widetilde{Per}_{0}({H})$, $\ell(\hat{\gamma},\hat{x})=0$ for
all $\hat{x}\in\hat{X}$ implies that $\hat{\gamma}\in\hat{X}$. Denote by
$usl(H)$ the collection of all $\hat{X}\subseteq\widetilde{Per}_{0}({H})$ such
that $\hat{X}$ is both unlinked and strongly linking.

Clearly, $usl(H)\subseteq mu(H)$.

######  Theorem 6.45 (Theorem C).

Let $H$ be non-degenerate. If $\hat{X}\in murm(H)$, then $\hat{X}^{\sharp
m}\in usl(H^{\sharp m})$ for all $m\in\mathbb{Z}_{m>0}$. In particular, every
$\hat{X}\in murm(H)$ is maximally unlinked as a subset of
$\widetilde{Per}_{0}({H})$.

###### Proof.

Let $\hat{X}\in murm(H)$, fix some $J$ such that $(H,J)$ is Floer regular, and
suppose for a contradiction that
$\hat{\gamma}\in\widetilde{Per}_{0}({H^{\sharp k}})\setminus\hat{X}^{\sharp
m}$ but $\ell(\hat{x}^{\sharp m},\hat{\gamma})=0$ for all $\hat{x}^{\sharp
m}\in\hat{X}^{\sharp m}$. We may in fact suppose that
$\hat{\gamma}\not\in\pi_{2}(\Sigma)\cdot\hat{X}^{\sharp m}$, since if
$\hat{\gamma}=A\cdot\hat{x}$ for some $A\in\pi_{2}(\Sigma)$, then for any
$\hat{y}^{\sharp m}\in\hat{X}^{\sharp m}$, $\hat{x}^{\sharp
m}\neq\hat{y}^{\sharp m}$, Proposition 2.18 implies
$\ell(\hat{\gamma},\hat{y}^{\sharp m})=\ell(\hat{\gamma},\hat{y}^{\sharp
m})+\frac{c_{1}(A)}{2}=\frac{c_{1}(A)}{2}$, since $\hat{X}$ is unlinked, which
easily implies that $\hat{X}^{\sharp m}$ is unlinked as well.

So we may as well assume that $\gamma\neq x$ for any $\hat{x}^{\sharp
m}\in\hat{X}^{\sharp m}$. Since $\mathcal{F}^{\hat{X}}$ foliates
$S^{1}\times\Sigma$, it is necessary that $\check{\gamma}(0)\in
W(\hat{x},\hat{y})$ for some $\hat{x},\hat{y}\in\hat{X}$ and so by Corollary
6.43 implies that $\ell(\hat{x}^{\sharp m},\hat{\gamma})<\ell(\hat{y}^{\sharp
m},\hat{\gamma})$, a contradiction. ∎

####  6.4.1 Comparison to Le Calvez’s theory of transverse foliations

######  Definition 6.46.

Let $I=(\phi_{t})_{t\in[0,1]}$ be an isotopy of homeomorphisms based at the
identity. We define

$\displaystyle Fix(I)$ $\displaystyle:=\bigcap_{t\in[0,1]}Fix(\phi_{t}),$

and we say that $I$ is a maximal isotopy if for every $x\in Fix(\phi_{1})$,
the loop $t\mapsto\phi_{t}(x)$ is not contractible in $\Sigma\setminus
Fix(I)$.

######  Definition 6.47 (cf. [17]).

Given a continuous oriented singular $1$-dimensional foliation $\mathcal{F}$
of $\Sigma$, a continuous path $\gamma:[0,1]\rightarrow\Sigma$ is said to be
positively transverse to $\mathcal{F}$ if its image is disjoint from
$Sing(\mathcal{F})$ and for each $t_{0}\in[0,1]$, there exists an orientation
preserving homeomorphism of a neighbourhood of $\gamma(t_{0})$ to a
neighbourhood of $0\in\mathbb{R}^{2}$ which sends $\mathcal{F}$ to the
standard vertical foliation of $\mathbb{R}^{2}$, oriented downward, and sends
$\gamma$ to a map whose $x$-coordinate is increasing in a neighbourhood of
$t_{0}$.

In [15], Le Calvez developed a theory which associates to any maximal isotopy
$I=(\phi_{t})_{t\in[0,1]}$, an oriented singular continuous $1$-dimensional
foliation $\mathcal{G}^{I}$ on $\Sigma$ having singular points on precisely
the points $x\in Fix(I)$, and moreover having the property that the dynamics
of $I$ are homotopically positively transverse to the leaves of
$\mathcal{G}^{I}$, in the following sense

######  Definition 6.48.

We say that an isotopy $I$ homotopically positively transverse to an oriented
singular continuous foliation $\mathcal{G}$ if, for all $x\in\Sigma\setminus
Fix(I)$, the path $x\mapsto\phi_{t}(x)$ is homotopic relative endpoints inside
of $\Sigma\setminus Fix(I)$ to a path which is positively transverse to
$\mathcal{G}$.

######  Remark.

Le Calvez and those working with his theory use the term ‘positively
transverse’ rather than ‘homotopically positively transverse’ as we have used
here. We introduce this term here merely to disambiguate between the other
notion of positive transversality (for smooth isotopies) which we have already
employed in this paper.

The reader will surely notice the similarity of Le Calvez’s result to the
results expounded in the previous section. Let us compare them more closely on
the domain of overlap of these two theories, namely, the case where $\phi_{1}$
is a smooth, non-degenerate Hamiltonian diffeomorphism.

As we have shown, to any non-degenerate pair $(H,J)$ and any $\hat{X}\in
murm(H)$, we may associate a new isotopy
$I^{\hat{X}}:=((\psi^{\hat{X}}_{t})^{-1}\circ\phi^{H}_{t})_{t\in[0,1]}$ having

$\displaystyle Fix(I)$ $\displaystyle=\bigcup_{\hat{x}\in\hat{X}}\\{x(0)\\}.$

The fact that $murm(H)\subset mu(H)$ (by Theorem 6.45) implies that this
isotopy is maximal, and we know that it is positively transverse (not simply
homotopically positively transverse) to the foliation
$\mathcal{F}^{\hat{X}}_{0}$ of $\Sigma$. This implies positive transversality
in the sense of Le Calvez, but is — at least prima facie — a rather stronger
condition, in that positive transversality in Le Calvez’s sense allows us to
homotope the orbits of the maximal isotopy, individually, in order to achieve
positive transversality in the usual sense, whereas here, every non-trivial
orbit of $((\psi^{\hat{X}}_{t})^{-1}\circ\phi^{H}_{t})$ is already positively
transverse in the usual sense. Said another way, if $\hat{X}$ is a trivial
capped braid, so that the isotopy $(\phi^{H}_{t})_{t\in[0,1]}$ is already a
maximal isotopy, then it is already homotopically positively transverse to
$\mathcal{F}^{\hat{X}}_{0}$ because $(\psi^{\hat{X}})^{-1}$ provides us with a
choice of homotopy for each orbit $t\mapsto\phi^{H}_{t}(x)$ in such a way that
these homotopies fit together in a coherent way so as to be induced by
composition with a contractible loop. Thus, the transverse singular foliations
constructed by Floer theory provide instances of Le Calvez-type foliations
which enjoy a certain sort of ‘uniform’ homotopic transversality which makes
them particularly nice. This raises the following potentially interesting
question for the theory of Le Calvez-type foliations

######  Question 1.

If $\phi\in Homeo_{0}(\Sigma)$ and $I=(\phi_{t})_{t\in[0,1]}$ is a maximal
isotopy ending at $\phi$, does there always exist a singular foliation
$\mathcal{G}$ of Le Calvez-type together with a contractible loop
$(\psi_{t})_{t\in S^{1}}\subset Homeo(\Sigma)$ (based at the identity), such
that for all $x\in\Sigma\setminus Sing(\mathcal{G})$, the paths
$t\mapsto(\psi_{t}^{-1}\circ\phi_{t})(x)$ are positively transverse to
$\mathcal{G}$?

In [15], Le Calvez also observed that in the case of Hamiltonian
homeomorphisms satisfying some minor niceness requirements of their fixed
point sets which always applies in our setting, the foliations
$\mathcal{F}^{I}$ constructed by his methods are always gradient-like, in the
sense that they admit discrete Lyapunov functions which are decreasing along
the leaves of the foliation. These Lyapunov functions are, essentially,
winding numbers with the maximally unlinked set of orbits used to construct
the foliation. Thus, we can see the results of our previous section as
recovering Le Calvez’s theory in the smooth case, on the proviso that
$\hat{X}\in murm(H)$, and offering certain refinements on the structure of
these fixed points along with a certain strengthening of the transversality
condition.

We should note, however, that in general $murm(H)\neq mu(H)$, and so even in
the smooth case, Le Calvez’s method provides foliations which do not arise via
our methods. In fact, it is not difficult to show that for every $\hat{X}\in
murm(H)$, the isotopy $t\mapsto
I=((\psi^{\hat{X}}_{t})^{-1}\circ\phi^{H}_{t})_{t\in[0,1]},t\in[0,1]$ is
torsion-low in the sense of Yan in [36]. Conversely, it follows essentially
directly from Proposition $1.5$ in [36] (together with remarking on the
relationship between the winding number of the linearization at a fixed point
and its Conley-Zehnder index — see Remark $17$ in [12], for instance) that if
$I^{\prime}=(\phi^{H}_{t})_{t\in[0,1]}$ is a smooth torsion-low isotopy with
$\phi^{H}_{1}$ non-degenerate, then

$\displaystyle\hat{X}$ $\displaystyle:=\bigcup_{x\in
Fix(I^{\prime})}\\{[x,x]\\}\subset\widetilde{Per}_{0}({H})$

is maximally unlinked relative the Morse range. Consequently, the theory
developed in this section recovers precisely the ‘torsion-low’ portion of Le
Calvez’s theory from a Floer theoretic perspective. Note that this discussion
also answers Question 1 in the affirmative when $\phi\in Ham(\Sigma,\omega)$
is non-degenerate and $I$ is low-torsion. If the techniques developed in this
paper could be extended to general maximally unlinked collections of capped
orbits and not just those which are maximally unlinked relative the Morse
range (note that this will require dealing with higher-dimensional Floer
moduli spaces, since in general there is no reason that a pair of unlinked
capped orbits should have Conley-Zehnder indices differing by at most $2$),
then we would be able to answer Question 1 in the affirmative for every $I$
when $\phi\in Ham(\Sigma,\omega)$ is non-degenerate. It therefore seems
reasonable to suggest the following

######  Conjecture.

If $\phi\in Ham(\Sigma,\omega)$, then the answer to Question 1 is affirmative.

##  7 Appendix: Stefan-Sussmann Foliations

It will be convenient in what follows to have some explicit language with
which to speak about singular foliations. To that end, we will make use of
some elementary notions from the theory of Stefan-Sussmann foliations (cf. [3]
and the references therein) which will be suitable to our purposes.

As in the non-singular theory, one may think of (singular) foliations as being
partitions of the ambient space into integral submanifolds of some
(generalized) distributions. To that end, let

$\displaystyle\mathcal{G}^{k}(M)$ $\displaystyle\rightarrow M$

denote the $k$-Grassmannian of $M$, having fiber $Gr(k,T_{x}M)$ over $x\in M$,
and let

$\displaystyle\mathcal{G}^{*}(M)$
$\displaystyle:=\sqcup_{k=0}^{n}\mathcal{G}^{k}(M),$

where $n=\dim M$, denote the total Grassmannian of $M$.

######  Definition 7.1.

A (generalized) distribution on a manifold $M$ is a section

$\displaystyle D:M$ $\displaystyle\rightarrow\mathcal{G}^{*}(M).$

A local section $X:M\rightarrow TM$ is said to belong to $D$ if $X(x)\in D(x)$
for all $x\in dom(X)$. The set of all smooth local sections
$X\in\mathcal{X}_{loc}(M)$ belonging to $D$ is denoted by $\Delta_{D}$.

######  Definition 7.2.

A generalized distribution $D$ is said to be smooth if for every $x\in M$,

$\displaystyle D(x)$
$\displaystyle=\text{span}\;\langle\;X(x)\;\rangle_{X\in\Delta_{D}}.$

Naturally, we will want to define objects which integrate these generalized
smooth distributions. To that end, following [33], we shall define

######  Definition 7.3.

A (smooth) $k$-leaf of $M$ is a subset $L\subset M$ equipped with a
differentiable structure $\sigma$ such that

  1. 1.

$(L,\sigma)$ is a connected $k$-dimensional immersed submanifold of $M$ and

  2. 2.

for any continuous map $f:N\rightarrow M$ such that $f(N)\subset L$ and $N$ a
locally connected topological space, we have that

$\displaystyle f:N$ $\displaystyle\rightarrow(L,\sigma)$

is continuous.

######  Definition 7.4.

A $(C^{\infty})$-singular (Stefan-Sussmann) foliation of $M$ is a partition
$\mathcal{F}$ of $M$ into smooth leaves of $M$ such that for every $x\in M$,
there exists a local smooth chart

$\displaystyle\varphi:U$
$\displaystyle\xrightarrow{\cong}\mathcal{O}(x)\subset M$

from $U\subset\mathbb{R}^{n}$ an open neighbourhood of $0\in\mathbb{R}^{n}$,
such that

  1. 1.

$U=V\times W$ for $V$ an open neighbourhood of $0$ in $\mathbb{R}^{k}$ and $W$
an open neighbourhood of $0$ in $\mathbb{R}^{n-k}$, where $k$ is the dimension
of the smooth leaf $L_{x}\in\mathcal{F}$ containing $x$.

  2. 2.

$\varphi(0,0)=x$.

  3. 3.

For any leaf $L\in\mathcal{F}$,

$\displaystyle L\cap\varphi(U\times W)$ $\displaystyle=\varphi(U\times l),$

where $l:=\\{w\in W:\;\varphi(0,w)\in L\\}$.

######  Definition 7.5.

A smooth generalized distribution $D$ is said to be integrable if for every
$x\in M$ there exists an immersed submanifold $L\subset M$, such that

  1. 1.

$x\in L$, and

  2. 2.

$T_{y}L\subset D(y)$ for all $y\in L$.

Such an immersed submanifold is called an integral submanifold of $D$.

The main point is the following (due to [33])

######  Theorem 7.6.

If $D$ is a smooth integrable generalized distribution and $\mathcal{F}_{D}$
is the partition of $M$ formed by taking the collection maximal connected
integral submanifolds of $D$, then $\mathcal{F}_{D}$ is a smooth singular
Stefan-Sussmann foliation.

For a singular foliation $\mathcal{F}$, we let

$\displaystyle d(-,\mathcal{F}):M$ $\displaystyle\rightarrow\mathbb{Z}_{\geq
0}$ $\displaystyle x$ $\displaystyle\mapsto\dim L_{x}$

denote the function which keeps track of the dimension of the leaf of
$\mathcal{F}$ passing through $x\in M$. It’s not hard to see that
$d(-,\mathcal{F})$ is lower semi-continuous.

######  Definition 7.7.

A smooth singular foliation is said to have codimension $k$ if

$\displaystyle n-k$ $\displaystyle=\max_{x\in M}d(x,\mathcal{F}).$

For a codimension $k$ smooth singular foliation, we define the domain of
$\mathcal{F}$ to be

$\displaystyle dom(\mathcal{F})$ $\displaystyle:=\\{x\in
M:d(x,\mathcal{F})=n-k\\},$

while we define the singular set of $\mathcal{F}$ to be

$\displaystyle sing(\mathcal{F})$ $\displaystyle:=M\setminus
dom(\mathcal{F}).$

A leaf of $\mathcal{F}$ is said to be regular if it is of maximal dimension,
otherwise it is said to be singular. $\mathcal{F}$ is said to be oriented if
every regular leaf of $\mathcal{F}$ is in addition equipped with an
orientation, and the local charts about points on the regular leaves may be
taken to be orientation-preserving.

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