assured that these orbits are disjoint.

###### Proof.

See [Abo15, Lemma 1.2.13]. ∎

Combining the two claims above, we conclude that for a generic tuple $\bm{H}$
properties 2 and 3 are satisfied.

For a collection of linear Hamiltonians one may also define a chain complex

$(SC_{\kappa,\text{psym}}^{*}(M;\bm{H},J_{t}),d)$

by choosing a consistent Floer data on
$\widehat{\mathscr{H}}^{\bm{\mu},\bm{\mu}^{\prime}}_{\kappa,\chi}$ adapted to
$\bm{H}$. The proofs of the regularity and compactness of the associated
moduli spaces are similar to those of Theorems 2.33 and 2.38 and are less
technical.

We define a preorder on collections of linear Hamiltonians as follows:

######  Definition 2.45.

Given tuples $\bm{H}^{+}$ and $\bm{H}^{-}$ of linear Hamiltonians of slopes
$b^{+}$ and $b^{-}$, respectively, we say that

(2.97) $\bm{H}^{+}\preceq\bm{H}^{-}$

if $b^{+}\leq b^{-}$.

Given two collections $\bm{H}^{+}\preceq\bm{H}^{-}$ we construct the
continuation map

(2.98) $\mathfrak{c}\colon
SC_{\kappa,\text{psym}}^{*}(M;\bm{H}^{+},K^{+},J^{+})\to
SC_{\kappa,\text{psym}}^{*}(M;\bm{H}^{-},K^{-},J^{-})$

where $(K^{+},J^{+})$ is some consistent Floer data compatible with
$\bm{H}^{+}$ and $(K^{-},J^{-})$ is compatible with $\bm{H}^{-}$.

First of all, we set $H_{j}^{m}(s,t)$ to satisfy

(2.99) $\displaystyle H_{j}^{m}(s,t)=b_{s}r\text{, on the symplectization end
for }r\geq b^{-},$ (2.100) $\displaystyle H_{j}^{m}(s,t)=(H_{j}^{m})^{+}\text{
for }s\gg 0\text{ and }H_{j}^{m}(s,t)=(H_{j}^{m})^{-}\text{ for }s\ll 0,$

where $b_{s}$ is a value interpolating between $b^{+}$ and $b^{-}$ and can be
regarded as a smooth function $\mathbb{R}\to[b^{+},b^{-}]$. Similarly, we set
$J_{t}^{s}$ to be $s$-dependent almost complex structure on $\hat{M}$
coinciding with $J_{t}^{\pm}$ near $\pm\infty$.

These choices essentially fix Floer data on
$\widetilde{\mathcal{H}}^{\bm{\mu},\bm{\mu}^{\prime}}_{\kappa,\chi}$ for
$\chi=0$. Notice that this data is no longer $\mathbb{R}$-invariant, hence
such data does not induce a choice of data on
$\mathcal{H}^{\bm{\mu},\bm{\mu}^{\prime}}_{\kappa,\chi}$.

######  Definition 2.46.

We say that a choice of Floer data $(\tilde{K},\tilde{J})$ for all
$\widetilde{\mathcal{H}}^{\bm{\mu},\bm{\mu}^{\prime}}_{\kappa,\chi}$ is a
_continuation Floer data_ for the pair $(\bm{H}^{+},J^{+})$ and
$(\bm{H}^{-},J^{-})$ if it satisfies:

  1. (1)

for $\chi=0$ it coincides with Floer data induced by
$H^{\text{tot}}_{m,j}(s,\cdot)$, $J_{t}^{s}$ as above;

  2. (2)

$H_{(S,\pi)}(z)=b_{s}r$ on the symplectization end, where $z\in S$ and
$\pi_{\mathbb{R}}(z)=s$ for any
$(S,\pi)\in\widetilde{\mathcal{H}}^{\bm{\mu},\bm{\mu}^{\prime}}_{\kappa,\chi}$;

  3. (3)

$\tilde{K}_{(S,\pi)}(z)=K_{(S,\pi)}^{+}(z)$ for $z\in S$ such that
$\pi_{\mathbb{R}}(z)\geq 1$, similarly
$\tilde{K}_{(S,\pi)}(z)=K_{(S,\pi}^{-}(z)$ for $z\in S$ such that
$\pi_{\mathbb{R}}(z)\leq-1$, for any
$(S,\pi)\in\widetilde{\mathcal{H}}^{\bm{\mu},\bm{\mu}^{\prime}}_{\kappa,\chi}$;

  4. (4)

$\tilde{J}_{(S,\pi)}(z)=J^{+}(z)$ for $z\in S$ such that
$\pi_{\mathbb{R}}(z)\geq 1$; similarly
$\tilde{J}_{(S,\pi)}(z)=J_{(S,\pi)}^{-}(z)$ for $z\in S$ such that
$\pi_{\mathbb{R}}(z)\leq-1$, for any
$(S,\pi)\in\widetilde{\mathcal{H}}^{\bm{\mu},\bm{\mu}^{\prime}}_{\kappa,\chi}$.

We notice that since $\mathcal{H}^{\bm{\mu},\bm{\mu}^{\prime}}_{\kappa,\chi}$
is a quotient of
$\widetilde{\mathcal{H}}^{\bm{\mu},\bm{\mu}^{\prime}}_{\kappa,\chi}$ the wnb
groupoid $\mathscr{H}^{\bm{\mu},\bm{\mu}^{\prime}}_{\kappa,\chi}$ may be
adapted to introduce a branched structure
$\widetilde{\mathscr{H}}^{\bm{\mu},\bm{\mu}^{\prime}}_{\kappa,\chi}$
associated with
$\widetilde{\mathcal{H}}^{\bm{\mu},\bm{\mu}^{\prime}}_{\kappa,\chi}$.

Given tuples of orbits
$\bm{x}^{\pm}\in\mathcal{P}_{\text{psym}}(\bm{H}^{\pm})$, and a choice of
continuation Floer data $(\tilde{K},\tilde{J})$ we define a _continuation
moduli space_ $\mathscr{K}^{\chi}(\tilde{K};\tilde{J};\bm{x}^{+};\bm{x}^{-})$
to be the groupoid with elements of
$\Big{(}\mathscr{K}^{\chi}(\tilde{K};\tilde{J};\bm{x}^{+};\bm{x}^{-})\Big{)}_{0}$
consisting of pairs:

(2.101)
$\Big{(}(S,\pi,\bm{p}^{\pm},\bm{a}^{\pm},B)\in\left(\widetilde{\mathscr{H}}^{\bm{\mu},\bm{\mu}^{\prime}}_{\kappa,\chi}\right)_{0},\,u=(\pi,v)\colon
S\to\mathbb{R}\times S^{1}\times\hat{M}\Big{)}$

such that

(2.102) $\begin{dcases}(dv-\tilde{Y})^{0,1}=0\text{ with respect to
}\tilde{J}_{(S,\pi)};\\\
\lim_{s\to+\infty}v\circ\epsilon_{i}^{+}(s,\cdot)=x_{i}^{+}(\cdot);\\\
\lim_{s\to-\infty}v\circ\epsilon_{i}^{-}(s,\cdot)=x_{i}^{-}(\cdot).\end{dcases}$

The morphisms
$\left(\mathscr{K}^{\chi}(\bm{x}^{+};\bm{x}^{-};\tilde{K};\tilde{J})\right)_{1}$
and the weight function are induced from
$\widetilde{\mathcal{H}}^{\bm{\mu},\bm{\mu}^{\prime}}_{\kappa,\chi}$; see the
discussion in Section 2.9.

######  Proposition 2.47.

For a generic choice of continuation Floer data $(\tilde{K},\tilde{J})$ the
moduli spaces $\mathscr{K}^{\chi}(\tilde{K};\tilde{J};\bm{x}^{+};\bm{x}^{-})$
are wnb groupoids of dimension

(2.103) $(n-2)\chi+|\bm{x}^{-}|-|\bm{x}^{+}|.$

###### Proof.

We refer the reader to the proof of Theorem 2.33 for details. ∎

We now define the continuation map $\mathfrak{c}$ by counting curves in the
moduli space above, i.e., given
$\bm{x}^{+}\in\mathcal{P}_{\text{psym}}(\bm{H}^{+})$ we set

(2.104)
$\mathfrak{c}([\bm{x}^{+}])=\sum_{\begin{subarray}{c}\bm{x}^{-}\in\mathcal{P}_{\text{psym}}(\bm{H}^{-}),\,u\in\mathscr{K}^{\chi}(\tilde{K};\tilde{J};\bm{x}^{-};\bm{x}^{+})_{H}\\\
(n-2)\chi+|\bm{x}^{-}|-|\bm{x}^{+}|=0,\,\chi<0\end{subarray}}\frac{\mathfrak{c}_{u}([\bm{x}^{+}])}{N_{\bm{\mu}(\bm{x}^{-})}},$

where $[\bm{x}^{+}]\in o_{\bm{x}^{+}}^{\mathbb{Q}}$ is some generator and
$\mathfrak{c}_{u}$ is the morphism on orientation lines associated with $u$,
as in (2.85):

(2.105) $\mathfrak{c}_{u}\colon o_{\bm{x}^{+}}\rightarrow
o_{\bm{x}^{-}}[(2-n)\chi].$

We point out that one again should pass to Kuranishi replacements for $n\geq
3$ whenever necessary as in Section 2.9.

######  Lemma 2.48.

The map $\mathfrak{c}$ is a chain map.

###### Proof.

We claim that $\mathscr{K}^{\chi}(\tilde{K};\tilde{J};\bm{x}^{-};\bm{x}^{+})$
with $(n-2)\chi+|\bm{x}^{-}|-|\bm{x}^{+}|=1$ admits a compactification (in the
sense of (2.89)) with boundary
$\partial|\mathscr{K}^{\chi}(\bm{x}^{+};\bm{x}^{-};\tilde{K};\tilde{J})|_{H}$
covered by images of

(2.106) $\bigsqcup_{\chi_{1},\chi_{2};\,\bm{x}^{-}_{0}\in
SC^{*}_{\kappa}(M,\bm{H}^{-})}\mathscr{K}^{\chi_{1}}(\tilde{K};\tilde{J};\bm{x}^{+};\,\bm{x}^{-}_{0})\times\mathscr{M}^{\chi_{2}}(K^{-};J^{-};\bm{x}^{-}_{0};\bm{x}^{-}),$

where

$\chi_{1}+\chi_{2}=\chi,$
$(n-2)\chi_{1}+|\bm{x}^{-}_{0}|-|\bm{x}^{+}|=(n-2)\chi_{2}+|\bm{x}^{-}|-|\bm{x}^{-}_{0}|-1=0;$

and

(2.107) $\bigsqcup_{\chi_{1},\chi_{2};\,\bm{x}^{+}_{0}\in
SC^{*}_{\kappa}(M,\bm{H}^{+})}\mathscr{M}^{\chi_{1}}(K^{+};J^{+};\bm{x}^{+};\bm{x}^{+}_{0})\times\mathscr{K}^{\chi_{2}}(\tilde{K};\tilde{J};\bm{x}^{+}_{0};\bm{x}^{-}),$

where

$\chi_{1}+\chi_{2}=\chi,$
$(n-2)\chi_{1}+|\bm{x}^{+}_{0}|-|\bm{x}^{+}|-1=(n-2)\chi_{2}+|\bm{x}^{-}|-|\bm{x}^{+}_{0}|=0.$

The proof is similar to that of Theorem 2.38 but easier since for linear
Hamiltonians all orbits are contained in the compact region. Each such
building has the associated weight $N_{\bm{\mu}(\bm{x}^{0})}$. We leave it to
the reader to verify that compositions of morphisms on orientation lines
associated with (2.106) and (2.107) for given $o_{\bm{x}^{+}}^{\mathbb{Q}}$
and $o_{\bm{x}^{-}}^{\mathbb{Q}}[(2-n)\chi]$ differ by $(-1)$ (also compare
with the proof of Theorem 2.41).

Taking weights into account we conclude that

(2.108) $\mathfrak{c}\circ d-d\circ\mathfrak{c}=0.$

∎

######  Lemma 2.49.

The continuation map $\mathfrak{c}$ is independent of the choice of
continuation Floer data $(\tilde{K},\tilde{J})$.

###### Proof.

We omit the proof here and refer the reader to [Abo15, Lemma 1.6.13] for the
heuristic of the argument that can be adapted in the context of Hurwitz
spaces. ∎

######  Lemma 2.50.

Given three tuples of Hamiltonians satisfying
$\bm{H}^{+}\preceq\bm{H}^{\text{\char 73}}\preceq\bm{H}^{-}$, there is a
commutative diagram

$SC_{\kappa}^{*}(\hat{M};\bm{H}^{+})$$SC_{\kappa}^{*}(\hat{M};\bm{H}^{\text{\char
73}})$$SC_{\kappa}^{*}(\hat{M};\bm{H}^{-})$$\mathfrak{c}_{+\to\text{\char
73}}$$\mathfrak{c}_{+\to-}$$\mathfrak{c}_{\text{\char 73}\to-}$

###### Proof.

Let $(\tilde{K}^{+},\tilde{J}^{+})$ and $(\tilde{K}^{-},\tilde{J}^{-})$ be
Floer data defining continuation maps $\mathfrak{c}_{+\to\text{\char 73}}$ and
$\mathfrak{c}_{{}^{\prime}\to-}$ respectively. We pick the following
collection of Floer data $(\tilde{K}^{R},\tilde{J}^{R})$ for $R>0$ on all
$\widetilde{\mathcal{H}}^{\bm{\mu},\bm{\mu}^{\prime}}_{\kappa,\chi}$,
satisfying that for any
$(S,\pi)\in\widetilde{\mathcal{H}}^{\bm{\mu},\bm{\mu}^{\prime}}_{\kappa,\chi}$:

  1. (1)

$\tilde{K}^{R}_{(S,\pi)}(z)=\tilde{K}^{+}_{(S,\pi^{-R})}(z)$ for $z\in S$ such
that $\pi_{\mathbb{R}}(z)\geq 0$ where $\pi^{-R}=\pi+(-R,0)$; similarly
$\tilde{K}^{R}_{(S,\pi)}(z)=\tilde{K}^{-}_{(S,\pi^{R})}(z)$ for $z\in S$ such
that $\pi_{\mathbb{R}}(z)\leq 0$, where $\pi^{R}=\pi+(R,0)$;

  2. (2)

$\tilde{J}^{R}_{(S,\pi)}(z)=\tilde{J}^{+}_{(S,\pi^{-R})}(z)$ for $z\in S$ such
that $\pi_{\mathbb{R}}(z)\geq 0$; similarly
$\tilde{J}^{R}_{(S,\pi)}(z)=\tilde{J}^{-}_{(S,\pi^{R})}(z)$ for $z\in S$ such
that $\pi_{\mathbb{R}}(z)\leq 0$.

We notice that for $z$ satisfying $-R\leq\pi_{\mathbb{R}}(z)\leq R$ the Floer
data as above coincides with some $(K^{\text{\char 73}},J^{\text{\char 73}})$
defining differential on $SC_{\kappa}^{*}(\hat{M},\bm{H}^{\text{\char 73}})$.

The data $(\tilde{K}^{R},\tilde{J}^{R})$ also defines the continuation map
$\mathfrak{c}_{+\to-}$ on cohomology. Now we fix $\bm{x}^{\pm}\in
SC_{\kappa}^{*}(\hat{M},\bm{H}^{\pm})$ and consider a pair $(u^{+},u^{-})$
with

$u^{+}\in\mathscr{K}^{\chi_{1}}(\tilde{K}^{+};\tilde{J}^{+};\bm{x}^{+};\,\bm{x}^{\text{\char
73}}_{0})\text{ and
}u^{-}\in\mathscr{K}^{\chi_{2}}(\tilde{K}^{-};\tilde{J}^{-};\bm{x}^{\text{\char
73}}_{0};\,\bm{x}^{-}),\text{ where}$
$(n-2)\chi_{1}+|\bm{x}^{\prime}_{0}|-|\bm{x}^{+}|=(n-2)\chi_{2}+|\bm{x}^{-}|-|\bm{x}^{\text{\char
73}}_{0}|=0.$

We claim then that for $R\gg 0$ there is a glued curve

$u^{+}\\#_{R}u^{-}\in\mathscr{K}^{\chi}(\tilde{K}^{R};\tilde{J}^{R};\bm{x}^{+};\,\bm{x}^{-}),$

where $\chi=\chi_{1}+\chi_{2}$. Moreover, we claim that for $R\gg 0$ all
elements of
$|\mathscr{K}^{\chi}(\tilde{K}^{R};\tilde{J}^{R};\bm{x}^{+};\,\bm{x}^{-})|$
are obtained in this fashion. The only nuance is that this map is not
injective and the curve $u^{+}\\#_{R}u^{-}$ as above is obtained in
$N_{\bm{\mu}(\bm{x}^{\prime}_{0})}$ many ways in the same way as in the
description of boundary degenerations in Theorem 2.38. Therefore, there is a
bijection between elements of the finite set
$|(\tilde{K}^{R};\tilde{J}^{R};\bm{x}^{+};\,\bm{x}^{-})|$ taken with weights
as above and

$\bigsqcup_{\chi_{1},\chi_{2};\,\bm{x}^{\prime}_{0}\in
SC^{*}_{\kappa}(M,\bm{H}^{\prime})}|\mathscr{K}^{\chi_{1}}(\tilde{K}^{+};\tilde{J}^{+};\bm{x}^{+};\,\bm{x}^{\text{\char
73}}_{0})|\times|\mathscr{K}^{\chi_{2}}(\tilde{K}^{-};\tilde{J}^{-};\bm{x}^{\text{\char
73}}_{0};\bm{x}^{-})|.$

This implies

(2.109) $\mathfrak{c}_{+\to-}=\mathfrak{c}_{\text{\char
73}\to-}\circ\mathfrak{c}_{+\to\text{\char 73}}$

∎

######  Corollary 2.51.

For $\bm{H}$ and $\bm{H}^{\text{\char 73}}$ of the same slope, their
symplectic cohomologies are isomorphic via the continuation map:

$\mathfrak{c}\colon
SH_{\kappa}^{*}(\hat{M},\bm{H})\xrightarrow{\cong}SH_{\kappa}^{*}(\hat{M},\bm{H}^{\text{\char
73}}).$

Lemmas 2.49 and 2.50 allow one to define Heegaard Floer symplectic homology as
a direct limit:

(2.110)
$SH_{\kappa,\text{psym}}^{*}(\hat{M})=\lim_{\mathfrak{c}}SH_{\kappa}^{*}(\hat{M},\bm{H})$

At last, one may restrict to a sequence $\\{\bm{H}^{m}\\}_{m\in\mathbb{N}}$
with unbounded slopes, i.e.,

######  Lemma 2.52.

For any sequence of tuples of linear Hamiltonians $\bm{H}^{m}$ with slopes
$b_{k}$ satisfying $\lim_{k\to\infty}b_{k}=+\infty$, there is an isomorphism

(2.111)
$\lim_{i\to\infty}SH_{\kappa}^{*}(\hat{M},\bm{H}^{i})=SH_{\kappa}^{*}(\hat{M})$

###### Proof.

We omit the proof which follows the lines of [Abo15, Lemma 1.6.17]. ∎

######  Theorem 2.53.

Given a tuple
$\bm{H}^{\text{tot}}\in{\mathcal{\bm{H}}}_{S^{1}}^{\kappa}(\hat{M})$ and
$J_{t}\in\mathcal{J}(\hat{M})$ , there is an isomorphism

(2.112) $SH_{\kappa,\text{psym}}^{*}(\hat{M};H^{\text{tot}},J)\cong
SH_{\kappa,\text{psym}}^{*}(\hat{M})$

###### Sketch of the proof..

Here, we only sketch the proof and refer the reader for a more detailed
treatment in case of $\kappa=1$ to [Rit13, Appendix 3]. Set $\bm{H}^{k}$ equal
to the tuple of linear Hamiltonians of slope $k$ and equal to
$\bm{H}^{\text{tot}}$ on the complement of $(k,+\infty)\times\partial
M\subset\hat{M}$. Let us denote by
$SC_{\kappa}^{*}(\bm{H}^{\text{tot}};\mathcal{A}>c)$ the subcomplex of
$SC_{\kappa}^{*}(\hat{M};\bm{H}^{\text{tot}})$ generated by tuples of orbits
$\bm{x}$ with action $\mathcal{A}(\bm{x})>c$. Note that for $c\ll-k$ we may
naturally define a continuation map by interpolating between
$\bm{H}^{\text{tot}}$ and $\bm{H}^{k}$ which on the level of cohomology takes
the form

$SH_{\kappa}^{*}(\bm{H}^{k})\to
SH_{\kappa}^{*}(\bm{H}^{\text{tot}};\mathcal{A}>c).$

Additionally, for $k^{\prime}\gg-c$ there is natural cochain map induced by
inclusion at the level of tuples of orbits

$SC_{\kappa}^{*}(\bm{H}^{\text{tot}};\mathcal{A}>c)\to
SC_{\kappa}^{*}(\bm{H}^{k^{\prime}}).$

Composing the above for $k^{\prime}\gg-c\gg k$ we get

(2.113) $SH_{\kappa}^{*}(\bm{H}^{k})\rightarrow
SH_{\kappa}^{*}(\bm{H}^{\text{tot}};\mathcal{A}>c)\rightarrow
SH_{\kappa}^{*}(\bm{H}^{k^{\prime}})$

and clearly, this composition is the continuation map.

On the other hand, for $-c^{\prime}\gg k^{\prime}\gg-c$ the composition of
maps

(2.114) $SC_{\kappa}^{*}(\bm{H}^{\text{tot}};\mathcal{A}>c)\rightarrow
SC_{\kappa}^{*}(H^{k^{\prime}})\rightarrow
SC_{\kappa}^{*}(\bm{H}^{\text{tot}};\mathcal{A}>c^{\prime})$

is the natural inclusion. The claim then follows by Lemma 2.52 and since

$SH_{\kappa}^{*}(\bm{H}^{\text{tot}})=\lim_{c\to-\infty}SH_{\kappa}^{*}(\bm{H}^{\text{tot}};\mathcal{A}>c).$

∎

As a corollary, we get that
$SH^{*}_{\kappa,\text{psym}}(\hat{M};\bm{H},J_{t};(H,F,J)$ is independent of
the choice of
$\bm{H}^{\text{tot}}\in{\mathcal{\bm{H}}}_{S^{1}}^{\kappa}(\hat{M})$ and
$J_{t}\in\mathcal{J}(\hat{M})$ and the choice of Floer data $(H,F,J)$ that we
made in our construction. It is also well known that the standard symplectic
cohomology $SH^{*}_{\kappa=1}(\hat{M})$ defined as a direct limit is an
invariant under _Liouville isomorphisms_ ; see [Sei08, Section 3], [Rit10,
Theorem 8]. The same argument works for any $\kappa$ and is similar to the one
we provide in the proof of Theorem 2.53. Hence we conclude that Theorem 1.1
holds true for $SH^{*}_{\kappa,\text{psym}}(M)$.

###  2.13. Heegaard Floer symplectic cohomology with unsymmetrized orbit
tuples

In this section, we give another version of Heegaard Floer symplectic
cohomology $SC_{\kappa,\text{unsym}}(\hat{M})$ which we expect to produce
closely related chain complex to $SC_{\kappa,\text{psym}}(\hat{M})$ even at
the level of cohomology. On the chain level
$SC_{\kappa,\text{unsym}}(\hat{M})$ will have more generators in every degree
than $SC_{\kappa,\text{psym}}(\hat{M})$.

First, we fix some notation. For a given permutation
$\sigma\in\mathfrak{S}_{\kappa}$ we assign its cycle decomposition
$\sigma=c_{1}(\sigma)\circ\ldots\circ c_{\ell(\sigma)}(\sigma)$, with
$\bm{\mu}(\sigma)=(\mu_{1}(\sigma),\ldots,\mu_{\ell(\sigma)}(\sigma)$ being a
cycle type (i.e., $\mu_{i}(\sigma)$ is the length of cycle $c_{i}(\sigma)$) of
$\sigma$ and $\ell(\sigma)$ denotes the length of the partition
$\bm{\mu}(\sigma)$. We apply here the following order on cycles: we say that
the cycle $c>c^{\prime}$ for two cycles $c,c^{\prime}\in\mathfrak{S}_{\kappa}$
if $c$ is longer than $c^{\prime}$ or if the least element permuted by $c$ is
smaller than the least element permuted by $c^{\prime}$. Note that for a given
permutation $\sigma\in\mathfrak{S}_{\kappa}$ the above relation provides a
total order on the elements of the cyclic decomposition of $\sigma$ and
$c_{i}(\sigma)$ in the above is assumed to be the $i$-the largest cycle in the
decomposition of $\sigma$ under this relation.

Let $H_{0}\in\mathcal{H}(\hat{M})$ and $F_{0}\in
C^{\infty}(S^{1}\times\hat{M})$ satisfy 1 and 2.

We then choose $\kappa$ different functions $F_{i}\colon
S^{1}\times\hat{M}\to\mathbb{R}$ for $i=1,\ldots,\kappa$ satisfying

  1. (H1’)

each $F_{i}\colon\mathbb{R}/\mathbb{Z}\times\hat{M}\to\mathbb{R}$ is a non-
negative function such that $|F_{i}-F_{0}|$ is absolutely bounded by a small
constant and vanishes away from the locus of Hamiltonian orbits of
$H_{0}^{\text{tot}}=H_{0}+F_{0}$ of all integer periods, and for
$t\in[0,\delta]\cup[1-\delta,1]\subset S^{1}$ it holds that
$F_{0}(t)=F_{i}(t)$;

  2. (H2’)

for any cyclic permutation $c=(j_{1},\ldots,j_{k})\in\mathfrak{S}_{\kappa}$ we
define $F_{c}$ to be the function
$F_{c}\colon\mathbb{R}/k\mathbb{Z}\times\hat{M}\to\mathbb{R}$ such that
$F_{c}(i-1+t)=F_{j_{i}}(t)$ for $t\in[0,1)$ and $i\in Z$ such that $1\leq
i\leq k$ (note that $F_{c}$ is smooth by 1); moreover, it is required that any
time-$k$ orbit of $H_{c}^{\text{tot}}=H_{0}+F_{c}$ is non-degenerate and lies
in a small neighborhood of the corresponding time-$k$ orbit of
$H_{0}^{\text{tot}}$;

  3. (H3’)

Hamiltonian time-$k$ orbits for different $H^{\text{tot}}_{c}$’s are disjoint
and embedded.

We leave it to the reader to verify that such collections
$\\{F_{i}\\}_{i=1}^{\kappa}$ exist and are sufficiently generic (see also
Section 2.2).

Consider the set $\mathcal{P}_{\text{unsym}}$ of ordered $\kappa$-tuples of
Hamiltonian chords defined as

(2.117)
$\displaystyle\mathcal{P}_{\text{unsym}}=\left\\{\bm{x}=(x_{1},\ldots,x_{\kappa})\left|\begin{array}[]{ll}\text{$x_{i}\colon[0,1]\to\hat{M}$
is the integral flow of $\phi^{t}_{H^{\text{tot}}_{i}}$,}\\\
\text{$x_{i}(1)=x_{\sigma(i)}(0)$ for some permutation
$\sigma\in\mathfrak{S}_{\kappa}$.}\end{array}\right.\right\\}$

For each $\bm{x}\in\mathcal{P}_{\text{unsym}}$ we would denote the associated
permutation by $\sigma(\bm{x})\in\mathfrak{S}_{\kappa}$. For a given $\bm{x}$
with $\sigma=\sigma(\bm{x})$ to each cycle $c_{i}(\sigma)$ with
$i=1,\ldots,\ell(\sigma)$ we associate a loop $x^{c_{i}(\sigma)}$ in $\hat{M}$
formed of chords $x_{j}$ with $j$ such that $c_{i}(\sigma)(j)\neq j$. Clearly,
$x^{c_{i}(\sigma)}$ is a Hamiltonian orbit with respect to
$H^{\text{tot}}_{c_{i}(\sigma)}$. See Figure 2.13 for an example. We would
also use the notation $\ell(\bm{x})$ for $\ell(\sigma)$.

Figure 6. A $\kappa$-tuple ($\kappa=5$) of Hamiltonian chords
$\bm{$}asanelementof$P_unsym$,where$H_i${}^{\prime}sstandforperturbedHamiltonianfunctions.Thepermutationassociatedwiththisgeneratorisgivenby$(1)(2)(345)$inthecyclicformand$x$iscomprisedofloops$x_(1),
x_(2), x_(345).

######  Remark 2.54.

Notice that the presentation above allows one to view $\bm{x}$ as a loop in
the configuration space $\mathrm{Conf}_{\kappa}(\hat{M})$.

As in the partially symmetrized case, to an orbit $x^{c_{i}(\sigma)}$ we
associate a differential operator

(2.118) $D_{x^{c_{i}(\sigma)}}\colon W^{1,2}(\mathbb{C},\mathbb{C}^{n})\to
L^{2}(\mathbb{C},\mathbb{C}^{n})$

following [Abo10, Section C.6], the orientation line of $x^{c_{i}(\sigma)}$

(2.119) $o_{x^{c_{i}(\sigma)}}=\text{det}(D_{x^{c_{i}(\sigma)}})$

and the _orientation line_ $o_{\bm{x}}$ of
$\bm{x}\in\mathcal{P}_{\text{unsym}}$ is given as follows

(2.120) $o_{\bm{x}}\coloneqq o_{x^{c_{1}(\sigma)}}\otimes\ldots\otimes
o_{x^{c_{\ell(\sigma)}(\sigma)}}.$

Regarding $o_{\bm{x}}$ as a $\mathbb{Z}$-module we set

(2.121) $o_{\bm{x}}^{\mathbb{Q}}=\mathbb{Q}\otimes_{\mathbb{Z}}o_{\bm{x}}$

Note that each orbit $x^{c_{i}(\sigma)}$ has a well-defined Conley-Zehnder
index $CZ(x^{c_{i}(\sigma)})$. We then define the grading of $\bm{x}$ by

(2.122) $|\bm{x}|=\kappa n-CZ(x^{c_{1}(\sigma)})-\ldots-
CZ(x^{c_{\ell(\sigma)}(\sigma)}).$

We set

(2.123)
$V_{\kappa,\text{unsym}}=\bigoplus_{\bm{x}\in\mathcal{P}_{\text{psym}}}o_{\bm{x}}^{\mathbb{Q}}.$

It is a graded vector space with

$V^{i}_{\kappa,\text{unsym}}=\bigoplus_{|\bm{x}|=i}o_{\bm{x}}^{\mathbb{Q}}.$

Consider a formal variable $\hbar$ with grading $|\hbar|=2-n$.

The Heegaard Floer symplectic cochain complex in the unsymmetrized case then
has as its underlying graded abelian group

(2.124)
$SC_{\kappa,\text{unsym}}^{*}(\hat{M})=\bigoplus_{m\in\mathbb{Z}}SC_{\kappa,\text{unsym}}^{m}(\hat{M})$

with

(2.125)
$SC_{\kappa,\text{unsym}}^{m}(\hat{M})=\sum_{i+(2-n)j=m}V^{i}_{\kappa,\text{psym}}\hbar^{j}=\sum_{i+(2-n)j=m}V^{i}_{\kappa,\text{psym}}[(n-2)j].$

The group $SC_{\kappa,\text{unsym}}^{m}(\hat{M})$ may be regarded as a graded
vector subspace of $V_{\kappa,\text{unsym}}[[\hbar]]$. We notice that
$SC^{*}_{\kappa,\text{unsym}}$ is a graded $\mathbb{Q}[h]$-module (but it is
not a $\mathbb{Q}[[\hbar]]$-module). In particular, multiplication by $\hbar$
is a $(2-n)$-degree map on $SC^{*}_{\kappa,\text{psym}}(\hat{M})$.

For each $\kappa$-tuple of chords
$\bm{x}=(x_{1},\ldots,x_{\kappa})\in\mathcal{P}_{\text{unsym}}$, we define the
action of $\bm{x}$ via

(2.126) $\mathcal{A}(\bm{x})=\sum^{\kappa}_{i=1}\mathcal{A}(x_{i}).$

where the action of chord $x_{i}$ is given by the expression

(2.127)
$\mathcal{A}(x_{i})=\int^{1}_{0}x_{i}^{*}\lambda-\int^{1}_{0}H^{\text{tot}}_{i}(t,x_{i}(t))dt,$

Now we outline how to define a differential on
$SC^{*}_{\kappa,\text{unsym}}(\hat{M})$ following the strategy we applied in
previous sections to $SC^{*}_{\kappa,\text{psym}}(\hat{M})$.

Pick two permutations $\sigma,\sigma^{\prime}\in\mathfrak{S}_{\kappa}$.

######  Definition 2.55.

The _Hurwitz space_
$\widetilde{\mathcal{H}}^{\sigma,\sigma^{\prime}}_{\kappa,\chi}$ is the space
of ismorphism classes _branched covers_ , where a branched cover $(S,\pi\colon
S\leavevmode\nobreak\ \longrightarrow\leavevmode\nobreak\
\mathbb{R}\leavevmode\nobreak\ \times\leavevmode\nobreak\
S^{1},\bm{p}^{+},\bm{a}^{+},\bm{p}^{-},\bm{a}^{-},B)$ is the following
collection of data: (1) A Riemann surface $S$ without boundary of Euler
characteristic $\chi$ punctured at positive punctures
$\bm{p}^{+}=\\{p^{+}_{1},\ldots,p^{+}_{l(\bm{\mu})}\\}$ and negative punctures
$\bm{p}^{-}=\\{p^{-}_{1},\ldots,p^{-}_{l(\bm{\mu}^{\prime})}\\}$. (2) For each
positive puncture $p_{i}^{+}$ we pick $\mu_{i}(\sigma)$ asymptotic markers
$a_{i1}^{+},\ldots a_{i\mu_{i}(\sigma)}\in\mathcal{S}^{1}_{p_{i}^{+}}S$. We
assume the total order on the positive asymptotic markers
$a_{1}^{+},\ldots,a_{\kappa}^{+}$ such that the asymptotic markers belonging
to $\mathcal{S}^{1}_{p_{i}^{+}}S$ appear on this circle in the order
prescribed by the cycle $c_{i}(\sigma)$ (note that
$\mathcal{S}^{1}_{p_{i}^{+}}S$ posses the induced orientation from $S)$. There
is an analogous choice of negative asymptotic markers
$a_{1}^{-},\ldots,a_{\kappa}^{-}$ satisfying similar conditions with respect
to $\sigma^{\prime}$. (3) A holomorphic branched cover $\pi\colon
S\longrightarrow\mathbb{R}\times S^{1}$ with simple branched points away from
$\pm\infty$. At $+\infty$ (resp. at $-\infty$) the branching partial is given
by $\bm{\mu}(\sigma)$ (resp. by $\bm{\mu}(\sigma^{\prime})$), i.e., a point
$p_{i}^{+}$ (resp. a point $p_{i}^{-}$) is a ramification point of
multiplicity $\mu_{i}(\sigma)$ (resp. multiplicity $\mu_{i}(\sigma^{\prime})$)
of an extended map $\overline{\pi}\colon\overline{S}\longrightarrow S^{2}$. We
additionally require that the map induced by $\pi$ from
$\mathcal{S}^{1}_{p_{i}^{+}}\overline{S}$ to $\mathcal{S}^{1}_{+\infty}S^{2}$
sends all associated with $p_{i}^{+}$ asymptotic markers to the marker
$1\in\mathcal{S}^{1}_{+\infty}S^{2}$, together with similar requirements for
negative punctures of $S$. (4) An ordered tuple $B$ of points $q_{1},\ldots
q_{b}$ in $\mathbb{R}\times S^{1}$ of simple branching, where $b=-\chi$.

Two branched covers $(S,\pi,\bm{p}^{\pm},\bm{a}^{\pm},B)$ and
$(S^{\prime},\pi^{\prime},\bm{p}^{\prime\pm},\bm{a}^{\prime\pm},B^{\prime})$
are said to be _isomorphic_ if there exists a biholomorphism
$\alpha\colon\overline{S}\rightarrow\overline{S^{\prime}}$ such that
$\alpha(p^{\pm}_{i})=p_{i}^{\prime\pm}$,
$\alpha(a^{\pm}_{i})=a_{i}^{\prime\pm}$ and $\pi^{\prime}\circ\alpha=\pi$.

Without going into much detail, we claim that there is a compactification
$\overline{\mathcal{H}}^{\sigma,\sigma^{\prime}}_{\kappa,\chi}$ containing a
subset
$\widehat{\mathcal{H}}^{\sigma,\sigma^{\prime}}_{\kappa,\chi}\subset\overline{\mathcal{H}}^{\sigma,\sigma^{\prime}}_{\kappa,\chi}$
of branched buildings with smooth levels, see Section 2.5. Furthermore, one
may introduce a branched structure
$(\widehat{\mathscr{H}}^{\sigma,\sigma^{\prime}}_{\kappa,\chi},\Lambda)$ on
$\widehat{\mathcal{H}}^{\sigma,\sigma^{\prime}}_{\kappa,\chi}$ following the
strategy of Section 2.6. We highlight here one of the main differences.

Recall that a partition $\bm{\mu}\vdash\kappa$, which we regard as a cycle
type, corresponds to the unique conjugacy class in $\mathfrak{S}_{\kappa}$.
Then the analogue of codimension $1$ boundary degeneration described in
Definition (2.20) is denoted by

(2.128)
${\mathcal{H}^{\sigma,\bm{\mu}}_{\kappa,\chi_{1}}\times_{\bm{m}}\mathcal{H}^{\bm{\mu},\sigma^{\prime}}_{\kappa,\chi_{2}}}.$

Essentially, it is a space of height $2$ buildings with levels glued along
$\ell(\bm{\mu})$ nodes with matching markers $\bm{m}$ and branching profile
over the corresponding node given by $\bm{\mu}$. Then the analogue of the
forgetful covering (2.51), which is essential for the construction of
$\widehat{\mathscr{H}}^{\bm{\mu},\bm{\mu}^{\prime}}_{\kappa,\chi}$, is given
by

(2.129)
$bf_{\text{unsym}}^{1}\colon\bigsqcup_{\sigma^{\prime\prime}\colon\bm{\mu}(\sigma^{\prime\prime})=\bm{\mu}}{\mathcal{H}^{\sigma,\sigma^{\prime\prime}}_{\kappa,\chi_{1}}\times\mathcal{H}^{\sigma^{\prime\prime},\sigma^{\prime}}_{\kappa,\chi_{2}}}\to{\mathcal{H}^{\sigma,\bm{\mu}}_{\kappa,\chi_{1}}\times_{\bm{m}}\mathcal{H}^{\bm{\mu},\sigma^{\prime}}_{\kappa,\chi_{2}}}$

We note that $bf_{\text{unsym}}^{1}$ is a $\kappa!$-sheeted cover as for given
matching $\bm{m}$ there are exactly $\kappa!$ ways to assign asymptotic
markers with such matching (in the sense of (2.53)). Then the weight function
$\Lambda$ takes value $\frac{1}{\kappa!}$ on each element of
${\mathcal{H}^{\sigma,\sigma^{\prime\prime}}_{\kappa,\chi_{1}}\times\mathcal{H}^{\sigma^{\prime\prime},\sigma^{\prime}}_{\kappa,\chi_{2}}}\times(R,+\infty]\subset(\widehat{\mathscr{H}}^{\sigma,\sigma^{\prime}}_{\kappa,\chi})^{\vee}_{H}.$

Following the lines of Section 2.8 one may introduce consistent cylindrical
ends and finite cylinders data on
$\mathscr{H}^{\sigma,\sigma^{\prime}}_{\kappa,\chi}$. This may be further
utilized to introduce consistent Floer data $(H,F,J)$ on
$\mathscr{H}^{\sigma,\sigma^{\prime}}_{\kappa,\chi}$. We note that for any
$[S,\pi]\in\mathscr{H}^{\sigma,\sigma^{\prime}}_{\kappa,\chi}$ the function
$H^{\text{tot}}_{(S,\pi)}=H_{(S,\pi)}+F_{(S,\pi)}$ restricts to
$H^{\text{tot}}_{c}$ on a given cylinder (away from the collar) with assigned
cycle $c$.

Given such consistent Floer data and two tuples
$\bm{x},\bm{x}^{\prime}\in\mathcal{P}_{\text{unsym}}$ with
$\sigma=\sigma(\bm{x})$ and $\sigma^{\prime}=\sigma(\bm{x}^{\prime})$, we
define
$\widetilde{\mathscr{M}}_{\text{unsym}}^{\chi}(H^{\text{tot}};J;\bm{x};\bm{x}^{\prime})$
to be the set of pairs

(2.130)
$\Big{(}[(S,\pi,\bm{p}^{\pm},\bm{a}^{\pm},B)]\in\left(\mathscr{H}^{\sigma,\sigma^{\prime}}_{\kappa,\chi}\right)_{0},\,u=(\pi^{\prime},v):S\to\mathbb{R}\times
S^{1}\times\hat{M}\Big{)}$

such that

(2.131) $\begin{dcases}[S,\pi^{\prime}]=[S,\pi];\\\ (dv-
Y_{(S,\pi)})^{0,1}=0,\text{ with respect to }J_{(S,\pi)};\\\
\lim_{s\to+\infty}v\circ\delta_{i}^{+}(s,\cdot)=x^{c_{i}(\sigma)}(\cdot)\text{
for }i=1,\ldots,\ell(\sigma);\\\
\lim_{s\to-\infty}v\circ\delta_{i}^{-}(s,\cdot)=x^{c_{i}(\sigma^{\prime})}(\cdot)\text{
for }i=1,\ldots,\ell(\sigma^{\prime}).\end{dcases}$

We set
$\mathscr{M}_{\text{unsym}}^{\chi}(H^{\text{tot}};J;\bm{x};\bm{x}^{\prime})$
to be the quotient of
$\widetilde{\mathscr{M}}_{\text{unsym}}^{\chi}(H^{\text{tot}};J;\bm{x};\bm{x}^{\prime})$
under the following equivalence relation. We say that two pairs

$\Big{(}[(S_{1},\pi_{1},\bm{p}^{\pm},\bm{a}^{\pm},B)],\,u=(\pi^{\prime}_{1},v_{1})\Big{)}\text{
and
}\Big{(}[(S_{2},\pi_{2},\bm{p}^{\pm},\bm{a}^{\pm},B)],\,u=(\pi^{\prime}_{2},v_{2})\Big{)}$

are equivalent if $[S_{1},\pi_{1}]=[S_{2},\pi_{2}]$ and $v_{1}=v_{2}$.

######  Proposition 2.56.

For a generic Floer data on
$\mathscr{H}^{\sigma,\sigma^{\prime}}_{\kappa,\chi}$ the moduli space
$\widetilde{\mathscr{M}}_{\text{unsym}}^{\chi}(H^{\text{tot}};J;\bm{x};\bm{x}^{\prime})$
and
$\mathscr{M}_{\text{unsym}}^{\chi}(H^{\text{tot}};J;\bm{x};\bm{x}^{\prime})$
are wnb groupoids. Moreover, the dimension of
$\mathscr{M}_{\text{unsym}}^{\chi}(H^{\text{tot}};J;\bm{x};\bm{x}^{\prime})$
is given by

$\text{dim}_{\mathbb{R}}\mathscr{M}_{\text{unsym}}^{\chi}(H^{\text{tot}};J;\bm{x};\bm{x}^{\prime})=(n-2)\chi+|\bm{x}^{\prime}|-|\bm{x}|-1.$

The proof of this proposition is similar to that of Theorem 2.33.

In case when
$\text{dim}_{\mathbb{R}}\mathscr{M}_{\text{unsym}}^{\chi}(H^{\text{tot}};J;\bm{x};\bm{x}^{\prime})=1$
there is an induced morphism

(2.132) $\partial_{u}\colon o_{\bm{x}}\to
o_{\bm{x}^{\prime}}\bigl{[}(2-n)\chi\bigr{]},$

similar to (2.86).

The differential on $SC_{\kappa,\text{unsym}}(\hat{M})$ is then defined by the
formula

(2.133)
$d[\bm{x}]=\sum_{\begin{subarray}{c}\bm{x}^{\prime},\chi\colon|\bm{x}^{\prime}|-|\bm{x}|=1,\\\
u\in\mathscr{M}_{\text{unsym}}^{\chi=0}(\bm{x};\bm{x}^{\prime})\end{subarray}}\partial_{u}([\bm{x}])+\sum_{\begin{subarray}{c}\bm{x}^{\prime},\chi\colon(n-2)\chi+|\bm{x}^{\prime}|-|\bm{x}|=1,\\\
u\in\mathscr{M}_{\text{unsym}}^{\chi}(H^{\text{tot}};J;\bm{x};\bm{x}^{\prime})\end{subarray}}\frac{\partial_{u}([\bm{x}])}{\kappa!},$

where $[x]$ is a generator in $o_{\bm{x}}^{\mathbb{Q}}$ for
$\bm{x}\in\mathcal{P}_{\text{unsym}}$.

The proof of the fact that the above forms a chain complex is based on the
following

######  Proposition 2.57.

For a generic consistent choice of $(H,F,J)$, given
$(n-2)\chi+|\bm{x}^{\prime}|-|\bm{x}|\leq 2$, the wnb groupoid
$\mathscr{M}_{\text{unsym}}^{\chi}(H^{\text{tot}};J;\bm{x};\bm{x}^{\prime})$
admits a compactification
$\overline{\mathscr{M}}_{\text{unsym}}^{\chi}(H^{\text{tot}};J;\bm{x};\bm{x}^{\prime})$.
In case $(n-2)\chi+|\bm{x}^{\prime}|-|\bm{x}|=2$ the boundary takes the form
$\bigsqcup_{\begin{subarray}{c}\chi_{1}+\chi_{2}=\chi,\\\
(n-2)\chi_{1}+|\bm{x}^{\prime\prime}|-|\bm{x}|=(n-2)\chi_{2}+|\bm{x}^{\prime}|-|\bm{x}^{\prime\prime}|=1\end{subarray}}\mathscr{M}_{\text{unsym}}^{\chi_{1}}(H^{\text{tot}};J;\bm{x};\bm{x}^{\prime\prime})\times\mathscr{M}_{\text{unsym}}^{\chi_{2}}(H^{\text{tot}};J;\bm{x}^{\prime\prime};\bm{x}^{\prime}).$

The proof is similar to that of Theorem 2.38 and left to the reader. We then
state the following

######  Theorem 2.58.

The differential $d$ makes $SC^{*}_{\kappa,\text{unsym}}(\hat{M})$ into a
chain complex. We conclude this section by pointing out that the unsymmetrized
version can also be defined via tuples of linear Hamiltonians as in Section
2.12 and, therefore, $SH_{\kappa,\text{unsym}}(\hat{M})$ is also an invariant
of the Liouville domain $M$.

###  2.14. Orientation lines conventions

Here we introduce some basics on orientation lines that we use to have all
chain complexes and chain maps appearing in this text well-defined over
$\mathbb{Q}$. We use the conventions from [Abo15] with slight adjustements.

Given a finite dimensional real vector space $V$, its determinant line is the
$1$-dimensional $\mathbb{Z}$-graded real vector space
$\operatorname{det}V=\Lambda^{\operatorname{dim}(V)}V$ supported in degree
$\operatorname{dim}(V)$.

To any $1$-dimensional graded real vector space $\delta$ we associate an
orientation line $|\delta|$, which is the rank $1$ graded free abelian group
generated by the two possible orientations of $\delta$, modulo the relation
that their sum vanishes. The orientation line $|\delta|$ is supported in the
same degree as $\delta$. When $\delta=\operatorname{det}V$ we denote its
orientation line by $|V|$. Given an orientable manifold $M$ we will denote by
$|M|$ the orientation line generated by the orientations of $M$ and supported
in degree $\operatorname{dim}_{\mathbb{R}}M$.

Given two vector spaces $U$ and $V$, let $W=U\oplus V$. There is then an
isomorphism

$|U|\otimes|V|\cong|W|$

given by first choosing a basis corresponding to a given orientation of $U$
and then extending it to the basis of $W$ by adding (on the right) a basis of
$V$ corresponding to a given orientation of $V$. Notice that this procedure
gives rise to the _Koszul sign_ :

(2.134)
$|W|\cong|U|\otimes|V|\cong(-1)^{\operatorname{dim}(U)\cdot\operatorname{dim}(V)}|V|\otimes|U|\cong(-1)^{\operatorname{dim}(U)\cdot\operatorname{dim}(V)}|W|.$

More generally, for an exact sequence of vector spaces

$0\to V\to W\to U\to 0$

which splits, we assign an isomorphism

(2.135) $|U|\otimes|V|\cong|W|.$

We note that this convention is different from the one used in [Abo15]. We are
motivated by the following example. Let $N$ be an orientable submanifold of an
orientable manifold $M$. Let us denote by $\eta$ the normal bundle $TM/TN$
over $N$. There is then a short exact sequence

$0\to TN\to TM|_{N}\to\eta\to 0.$

The standard convention for inducing an orientation on $N$ from an orientation
on $M$ is to first set a positive orientation on $\eta$ (which in some cases
is straightforward as $\eta$ could be a trivial bundle) and then declare an
orientation on $N$ to be positive if a basis with such orientation extends a
basis with positive orientation on $\eta$ to a basis with positive orientation
on $M$. In other words,

$|\eta|\otimes|N|\cong|M|.$

Given a $\mathbb{Z}$-graded line $\ell$ (rank $1$ free abelian group), its
dual line $\ell^{-1}=$ $\operatorname{Hom}_{\mathbb{Z}}(\ell,\mathbb{Z})$ is,
by definition, supported in the opposite degree as $\ell$. Here, we use the
convention that $\mathbb{Z}$ is supported in degree $0$. There is a canonical
isomorphism $\ell^{-1}\otimes\ell\cong\mathbb{Z}$ induced by evaluation.
According to (2.134) there is an isomorphism
$(-1)^{\operatorname{deg}(\ell)}\ell\otimes\ell^{-1}\cong\mathbb{Z}$.

We will denote by $\zeta$ the trivial rank $1$ free abelian group (i.e., with
prescribed isomorphism with $\mathbb{Z}$) supported in degree $1$.

Given a $\mathbb{Z}$-graded rank $1$ free abelian group $\ell$, we denote by
$\ell[k]$ the rank $1$ group obtained by shifting the degree down by
$k\in\mathbb{Z}$, i.e., $\ell[k]$ is supported in degree
$\operatorname{deg}(\ell)-k$. We will use the agreement

(2.136) $\displaystyle\ell[-k]\cong\ell\otimes\zeta^{k},\text{ for }k>0,$
$\displaystyle\ell[-k]\cong\zeta^{-k}\otimes\ell,\text{ for }k<0.$

For any such rank $1$ free abelian group $\ell$ we associate

(2.137) $\ell^{\mathbb{Q}}=\ell\otimes_{\mathbb{Z}}\mathbb{Q},$

which is a $\mathbb{Z}$-graded $\mathbb{Q}$-vector space supported in
$\operatorname{deg}(\ell)$.

##  3\. The free multiloop complex

In this section, we introduce a certain Morse chain complex which is intended
to be a counterpart of the unsymmetrized version of HFSH. We note that this
complex is the closed string analogue of the Morse complex introduced in
[HKTY]. Throughout this section, $Q$ is a closed orientable $n$-manifold.

###  3.1. Free multiloops

The analogue of $SH_{\kappa}(T^{*}Q)$ on the Morse side should on the chain
level be based on time-$1$ free loops in the configuration space
$\operatorname{UConf}_{\kappa}(Q)$. To model this, we consider $\kappa$-strand
multiloops on $Q$. We start by recalling the setup of [AS10] for the Morse
complex of paths alone.

Consider the space of continuous time-1 paths on $Q$ with free endpoints:

$\Omega(Q)=\\{\gamma\in C^{0}([0,1],Q)\\}.$

We denote the subset of $\Omega(Q)$ in the class $W^{1,2}$ by $\Omega^{1}(Q)$.

Fix a generic Riemannian metric $g$ on $Q$. Let

$L=L(t,q,v)\colon[0,1]\times TQ\to\mathbb{R}$

be a _Lagrangian function_ which is the _Fenchel dual_ of a Hamiltonian $H$
that we introduced in Section 2.13.

We then define the Lagrangian action functional
$\mathcal{A}_{L_{i}}\colon\Omega^{1}(Q)\to\mathbb{R}$ by

(3.1)
$\mathcal{A}_{L}(\gamma)=\int_{0}^{1}L(t,\gamma(t),\dot{\gamma}(t))\,dt.$

Let $\varphi^{t}_{L}$ be the integral curve on $Q$ satisfying the Euler-
Lagrangian equation:

(3.2)
$\frac{d}{dt}\partial_{v}L(t,\gamma(t),\dot{\gamma}(t))=\partial_{q}L(t,\gamma(t),\dot{\gamma}(t)).$

The critical points of $\mathcal{A}_{L}$ are exactly the solutions of (3.2).
For $L$ as one of the functions above, the Lagrangian action functional
$\mathcal{A}_{L}$ is not of class $C^{2}$, unless $L$ is a polynomial of
degree at most two on each fiber of $TQ$, hence in general one may not
construct a Morse complex on $\Omega(Q)$ by means of the gradient vector field
of $\mathcal{A}_{L}$. To overcome this obstacle in [AS09] the authors
introduce the notion of pseudo-gradient vector field $X$ for such functions on
Hilbert manifolds and prove the existence of pseudo-gradients for Lagrangian
action functional. The following summarizes the key properties of
$\mathcal{A}_{L}$ and its pseudo-gradient vector fields (see [AS09, Theorem
4.1], [AS10, Appendix A.1]):

######  Theorem 3.1.

  1. (1)

For a generic choice of $L$ all critical points of $\mathcal{A}_{L}$ are non-
degenerate.

  2. (2)

The action functional $\mathcal{A}_{L}$ is bounded below, and there exists a
smooth pseudo-gradient vector field $X$ on $\Omega^{1}(Q)$, i.e.,

    1. (PG1)

$X$ is forward-complete, i.e., the flow $\Phi(t,x)$ is defined for all $t>0$
and $x\in\Omega^{1}(Q)$;

    2. (PG2)

$\mathcal{A}_{L}$ is a Lyapunov function for $X$;

    3. (PG3)

$X$ is a Morse vector field whose singular points (= critical points of
$\mathcal{A}_{L}$) have finite Morse index;

    4. (PG4)

the pair $(X,\mathcal{A}_{L})$ satisfies the Palais-Smale condition;

    5. (PG5)

X satisfies the Morse-Smale condition, i.e., for each pair of critical points
$x,y\in\Omega^{1}(Q)$, the unstable manifold of $x$ is transverse to the
stable manifold of $y$.

  3. (3)

Any sufficiently small perturbation of $X$ which coincides with $X$ on a
neighborhood of the critical points of $\mathcal{A}_{L}$, is still a pseudo-
gradient (i.e., satisfies (PG1)-(PG5)).

Let $\mathcal{K}_{L}$ be a set of small perturbations of $X$ which are pseudo-
gradients (one may restrict their attention to those vector fields which have
_Whitney distance_ from $X$ less than $1$; see [AS10, AM06]).

For the generalization of the above to the set of what we call multiloops,
choose Lagrangians $L_{i}$ to be _Fenchel duals_ (see [AS10] for the precise
definition) of $H^{\text{tot}}_{i}$’s that we picked in Section 2.13 for
$H_{0}=\frac{1}{2}\lvert p\rvert^{2}$.

By the assumption 1 on the $H_{i}^{\text{tot}}$ it follows that for $\delta$
such that$0<\delta<1/2$ as in 1

(3.3) $L_{i}(t,\cdot,\cdot)=L_{0}(t,\cdot,\cdot)\,\text{ for
}\,t\in[0,\delta]\cup[1-\delta,1].$

To each permutation $\sigma\in\mathfrak{S}_{\kappa}$ we associate the set
$\Omega^{1}_{\sigma}(Q)$ of $\kappa$-tuples of multipaths on $Q$ which satisfy

(3.4)
$\displaystyle\Omega^{1}_{\sigma}(Q)=\left\\{\bm{\gamma}=(\gamma_{1},\ldots,\gamma_{\kappa})\in\Omega^{1}(Q)^{\kappa}\,|\gamma_{i}(1)=\gamma_{\sigma(i)}(0)\right\\}.$

We call elements of such a set _free multiloops_ since each such multipath
consists of $\ell(\sigma)$ loops.

We denote by $\bm{L}$ the collection $(L_{1},\ldots,L_{\kappa})$ and define
the Lagrangian action functional on $\Omega^{1}_{\sigma}(Q)$ by

(3.5)
$\mathcal{A}_{\bm{L}}(\bm{\gamma})=\sum_{i=1}^{\kappa}\mathcal{A}_{L_{i}}(\gamma_{i}).$

One can also view $\Omega^{1}_{\sigma}(Q)$ as set of paths
$\bm{\gamma}\in\Omega^{1}(Q^{\kappa})$ with boundary conditions

(3.6) $(\bm{\gamma}(0),\bm{\gamma}(1))\in R_{\sigma}\subset Q^{\kappa}\times
Q^{\kappa},$

where $R_{\sigma}=\left\\{(\bm{x},\bm{y})\in Q^{\kappa}\times
Q^{\kappa}\,|\,x_{\sigma(i)}=y_{i}\,\right\\}$.

The critical points of $\mathcal{A}_{\bm{L}}$ form a subset
$\mathcal{P}_{\sigma}^{L}\subset\Omega^{1}_{\sigma}(Q)$ defined as

(3.7)
$\displaystyle\mathcal{P}_{\sigma}^{\bm{L}}=\left\\{\bm{\gamma}\in\Omega^{1}_{\sigma}(Q)\,|\,\text{$\gamma_{i}\colon[0,1]\to
Q$ is the integral flow of $\varphi^{t}_{L_{i}}$}\right\\}.$

When there is no ambiguity, we also denote $(\bm{\gamma},\sigma)$ simply by
$\bm{\gamma}$. We denote the permutation element $\sigma$ associated to
$\bm{\gamma}$ by $\sigma({\bm{\gamma}})$. If we write $\sigma({\bm{\gamma}})$
in its cyclic form, then each cycle corresponds to a loop in $Q$. We denote
the number of cycles associated with $\bm{\gamma}$ by $m({\bm{\gamma}})$. We
further require that for all $\bm{\gamma}\in\mathcal{R}_{\kappa}$

  1. (1)

$\gamma_{i}(t)\neq\gamma_{j}(t)$ for all $t$ and $i\neq j$;

  2. (2)

each $\gamma_{i}$ is a non-degenerate critical point of $\mathcal{A}_{L_{i}}$.

The following lemma follows immediately from Theorem 3.1.

######  Lemma 3.2.

For a generic choice of metric and
$\\{H_{1}^{\text{tot}},\ldots,H_{\kappa}^{\text{tot}}\\}$, the above
conditions can be achieved. Moreover, $\mathcal{A}_{\bm{L}}$ is a Morse
function on $\Omega^{1}_{\sigma}(Q)$ such that the set of critical points of
$\mathcal{A}_{\bm{L}}$ is $\mathcal{P}_{\sigma}^{\bm{L}}$, all of which are
non-degenerate and have finite Morse indices.

Clearly for
$\bm{\gamma}=(\gamma_{1},\ldots,\gamma_{\kappa})\in\mathcal{P}^{\bm{L}}_{\sigma}$
the Morse index $|\bm{\gamma}|$ can be computed as a sum of Morse indices of
$\gamma_{i}$:

(3.8)
$|\bm{\gamma}|=I_{L_{1}}(\gamma_{1})+\ldots+I_{L_{\kappa}}(\gamma_{\kappa})$

We denote the set of all multiloops by

(3.9)
$\Lambda^{1}_{\kappa}(Q)=\bigsqcup_{\sigma\in\mathfrak{S}_{\kappa}}\Omega^{1}_{\sigma}(Q)$

and the set of all critical points of $\bm{L}$ by

(3.10)
$\mathcal{P}^{\bm{L}}_{\kappa}=\bigsqcup_{\sigma\in\mathfrak{S}_{\kappa}}\mathcal{P}_{\sigma}^{\bm{L}}.$

###  3.2. The free multiloop complex

We first recall the definition of the Morse chain complex
$CM_{*}\left(\Lambda^{1}(Q)\right)$ in the standard $\kappa=1$ setting
following [AS06, AS10]. The chains are given by the graded vector space over
$\mathbb{Q}$ generated by the critical points of $\mathcal{A}_{L}$. By Theorem
3.1 there exists a pseudo-gradient vector field $X$. The Morse differential is
then given by counting index $1$ _pseudo-gradient trajectories_ of
$\mathcal{A}_{L}$ associated with the flow of $X$

(3.11) $\frac{\partial\gamma}{\partial s}=X(\gamma),$

where we are parametrizing paths in $\Lambda^{1}(Q)$ by the variable $s$.

Let $\widetilde{\mathcal{M}}\left(\gamma,\gamma^{\prime};X\right)$ be the
moduli space of pseudo-gradient trajectories from $\gamma$ to
$\gamma^{\prime}$ with respect to $X$ and
$\mathcal{M}\left(\gamma,\gamma^{\prime};X\right)=\widetilde{\mathcal{M}}\left(\gamma,\gamma^{\prime};X\right)/\mathbb{R}$
be the quotient by the $\mathbb{R}$-translation. Then we define

$\begin{gathered}\partial\colon CM_{*}\left(\Lambda^{1}(Q)\right)\rightarrow
CM_{*}\left(\Lambda^{1}(Q)\right)\\\
\partial\gamma=\sum_{|\gamma^{\prime}|=|\gamma|-1}\\#\mathcal{M}\left(\gamma,\gamma^{\prime};X\right)\cdot\gamma^{\prime}.\end{gathered}$

The homology of this complex is independent of the choice of
$X\in\mathcal{K}_{L}$ according to [AS10]. More generally, given a critical
point $\gamma\in\mathcal{P}^{L}$ one may define the _stable_ and _unstable_
manifolds of $\gamma$ with respect to $X$:

(3.12) $\displaystyle W^{s}(\gamma;X)$
$\displaystyle\coloneqq\\{\tilde{\gamma}\in\Lambda^{1}(Q)\mid\varphi_{X}^{t}(\tilde{\gamma})\rightarrow\gamma\text{
for }t\rightarrow+\infty\\},$ $\displaystyle W^{u}(\gamma;X)$
$\displaystyle\coloneqq\\{\tilde{\gamma}\in\Lambda^{1}(Q)\mid\varphi_{X}^{t}(\tilde{\gamma})\rightarrow\gamma\text{
for }t\rightarrow-\infty\\}.$

Note that by condition 23 the unstable manifold is always finite-dimensional.
For the above manifolds, the vector field $X$ will be suppressed from the
notation below when it is understood or unimportant.

For $\kappa\geq 1$ and $\bm{\gamma}\in\Omega_{\sigma}^{1}(Q)$, there is a
straightforward definition of a pseudo-gradient trajectory on
$\Omega_{\sigma}^{1}(Q)$ given by

$\frac{\partial\bm{\gamma}}{\partial
s}=\bm{X}(\gamma)\coloneqq\prod_{i=1}^{\kappa}X_{i}\left(\gamma_{i}\right).$

The Morse cohomology whose differential is defined by counting pseudo-gradient
multi-trajectories between critical points as above gives nothing but
$\ell(\sigma)$ copies of the Morse homology $HM_{*}\left(\Omega^{1}(Q)\right)$
of the loop space of $Q$. In order to mimic higher genus curves appearing in
the Floer differential (2.91) we need to modify the definition of the Morse
differential so that, given a gradient trajectory $\bm{\gamma}(s)$ starting at
$\bm{\gamma}$, whenever there exists $s$ such that $\bm{\gamma}(s)$ crosses
the big diagonal of $\text{Sym}^{\kappa}(Q)$, additional trajectories that
bifurcate from the original one are created.

Specifically, given two multiloops $\bm{\gamma}\in\Omega^{1}_{\sigma}(Q)$ and
$\bm{\gamma}^{\prime}\in\Omega^{1}_{\sigma^{\prime}}(Q)$ belonging to
$\mathcal{P}_{\kappa}^{\bm{L}}$ and a choice of $\ell+1$ tuples
$\bm{X}^{0},\ldots\bm{X}^{\ell}$ each consisting of pseudo-gradient vector
fields $X^{j}_{i}\in\mathcal{K}_{L_{i}}$ we define
$\widetilde{\mathcal{P}}(\bm{\gamma},\bm{\gamma}^{\prime};\bm{X}^{0},\ldots,\bm{X}^{\ell})$
as follows:

######  Definition 3.3.

A _piecewise pseudo-gradient_ trajectory
$\bm{\Gamma}\in\widetilde{\mathcal{P}}(\bm{\gamma},\bm{\gamma}^{\prime};l)$ is
a tuple

$\bm{\Gamma}=\left((\Gamma_{0},\ldots,\Gamma_{\ell}),\bm{s},\bm{\sigma},\bm{\rho}\right),$

where $\bm{s}=(s_{0},\ldots,s_{\ell+1})$ with $s_{0}=+\infty$,
$s_{\ell+1}=-\infty$ and $+\infty>s_{1}>\ldots>s_{l}>-\infty$,
$\bm{\sigma}=(\sigma_{0},\ldots,\sigma_{\ell})$ and
$\bm{\rho}=(\rho_{1},\ldots,\rho_{\ell})$ are two collections of permutations
in $\mathfrak{S}_{\kappa}$. We require that:

  1. (1)

$\Gamma_{i}=(\Gamma_{i}(\cdot)_{1},\ldots,\Gamma_{i}(\cdot)_{\kappa})\colon[s_{i+1},s_{i}]\to\Omega^{1}_{\sigma_{i}}(Q)$
is a continuous map.

  2. (2)

$\Gamma_{i}^{\prime}(s)=\bm{X}^{i}(\Gamma_{i}(s))$.

  3. (3)

$\Gamma_{0}(+\infty)=\bm{\gamma}$,
$\Gamma_{\ell}(-\infty)=\bm{\gamma}^{\prime}$.

  4. (4)

For each $i\colon 0\leq i\leq\ell-1$ there is a transposition
$(a_{i}b_{i})\in\mathfrak{S}_{\kappa}$ such that

(3.13)
$(\rho^{-1}_{i+1}\circ\sigma_{i+1}\circ\rho_{i+1})\circ\sigma_{i}^{-1}=(\sigma_{i}(a_{i})\sigma_{i}(b_{i})).$

  5. (5)

For $a_{i},b_{i}\in\\{1,\ldots,\kappa\\}$ satisfying (3.13) for
$i=0,\ldots,\ell-1$ there exists $t_{i}\in(0,1)$, so that

$\displaystyle\Gamma_{i+1}(s_{i+1})_{\rho_{i+1}(a_{i})}(t)=\left\\{\begin{array}[]{ll}\text{$\Gamma_{i}(s_{i+1})_{a_{i}}(t)$
for $t\in[0,t_{i+1}]$}\\\ \text{$\Gamma_{i}(s_{i+1})_{b_{i}}(t)$ for
$t\in[t_{i+1},1]$}\end{array},\right.$
$\displaystyle\Gamma_{i+1}(s_{i+1})_{\rho_{i+1}(b_{i})}(t)=\left\\{\begin{array}[]{ll}\text{$\Gamma_{i}(s_{i+1})_{b_{i}}(t)$
for $t\in[0,t_{i+1}]$}\\\ \text{$\Gamma_{i}(s_{i+1})_{a_{i}}(t)$ for
$t\in[t_{i+1},1]$}\end{array}.\right.$

  6. (6)

$\Gamma_{i+1}(s_{i+1})_{\rho_{i+1}(j)}=\Gamma_{i+1}(s_{i+1})_{j}$ for $j\neq
a_{i},b_{i}$.

Figure 7. The image of $\bm{\Gamma}$ viewed as the graph in
$Q\times\mathbb{R}_{s}$, where $\kappa=3$ and
$\bm{s}_{\bm{t},\bm{\sigma},\bm{\rho}}$ with
$\sigma_{i}=\begin{pmatrix}1&2&3\\\ 1&3&2\end{pmatrix}$,
$\sigma_{i+1}=\begin{pmatrix}1&2&3\\\ 2&3&1\end{pmatrix}$,
$\rho_{i+1}=\begin{pmatrix}1&2&3\\\ 3&1&2\end{pmatrix}$ and
$(a_{i}b_{i})=(12)$. The blue arcs labelled by $j=1,\ldots,\kappa$ denote the
image of $\Gamma(s)_{j}(0)$. Each such $\Gamma(\cdot)_{j}(0)$ is allowed to
have singularities at switching points.

Note that with $\bm{\Gamma}$ there is an associated sequence $\bm{t}$ of
$t_{i}$’s corresponding to “moments of switching” as in (5).

######  Proposition 3.4.

For a generic choice of collections of pseudo-gradients
$\bm{X}^{0},\ldots,\bm{X}^{\ell}$ and any $\bm{\gamma},\bm{\gamma}^{\prime}\in
CM_{*}(\Lambda^{1}_{\kappa}(Q))$, the space
$\widetilde{\mathcal{P}}(\bm{\gamma},\bm{\gamma}^{\prime};\bm{X}^{0},\ldots,\bm{X}^{\ell})$
is a smooth manifold of dimension
$|\bm{\gamma}|-|\bm{\gamma}^{\prime}|-(n-2)\ell$. Furthermore, given some
orientations of unstable manifolds of $\bm{\gamma}$ and $\bm{\gamma}^{\prime}$
there is a canonical orientation on
$\widetilde{\mathcal{P}}(\bm{\gamma},\bm{\gamma}^{\prime};\bm{X}^{0},\ldots,\bm{X}^{\ell})$.

This proposition is a consequence of the discussion that follows. For
$\\{a,b\\}\subset\\{1,\ldots\kappa\\}$ consider the subset
$\Delta_{ab}\Omega^{1}_{\sigma}(Q)\subset\Omega^{1}_{\sigma}(Q)$ given by

(3.14)
$\Delta_{ab}\Omega^{1}_{\sigma}(Q)=\\{\bm{\gamma}\in\Omega^{1}_{\sigma}(Q)\,|\,\text{
there exists }t\in(0,1)\text{ s.t. }\gamma_{a}(t)=\gamma_{b}(t)\\}.$

Let us consider the submersion

$\displaystyle ev^{\kappa}_{[0,1]}\colon[0,1]\times\Omega^{1}_{\sigma}\to
Q^{\kappa},$
$\displaystyle(t,\gamma_{1},\ldots,\gamma_{\kappa})\mapsto(\gamma_{1}(t),\ldots,\gamma_{\kappa}(t)).$

Then $\Delta_{ab}\Omega^{1}_{\sigma}(Q)$ is obtained from the inverse image of

$\Delta_{ab}Q^{\kappa}=\\{(q_{1},\ldots,q_{\kappa})\in
Q^{\kappa}\,|\,q_{a}=q_{b}\\}\subset Q^{\kappa}$

under $ev^{\kappa}_{[0,1]}$ by projecting to $\Omega^{1}_{\sigma}(Q)$.
Therefore, $\Delta_{ab}\Omega^{1}_{\sigma}(Q)$ is a codimension $(n-1)$
submanifold of the Hilbert manifold $\Omega^{1}_{\sigma}(Q)$. Moreover, an
orientation on $Q$ then induces an orientation on
$\Delta_{ab}\Omega^{1}_{\sigma}(Q)$ (as it induces a coorientation on
$\Delta_{ab}Q^{\kappa}$).

The diagonal $\Delta_{ab}\Omega^{1}_{\sigma}(Q)$ has an open subset
$\Delta^{\text{sm}}_{ab}\Omega^{1}_{\sigma}(Q)\subset\Delta_{ab}\Omega^{1}_{\sigma}(Q)$
given by

(3.15)
$\Delta^{\text{sm}}_{ab}\Omega^{1}_{\sigma}(Q)=\\{\bm{\gamma}\in\Omega^{1}_{\sigma}(Q)\,|\,\text{
there exists a unique }t\in(0,1)\text{ s.t. }\gamma_{a}(t)=\gamma_{b}(t)\\}.$

For $\bm{\gamma}\in\Delta^{\text{sm}}_{ab}\Omega^{1}_{\sigma}(Q)$ we denote by
$t_{\Delta}(\bm{\gamma})$ the associated $t\in(0,1)$ for which
$\gamma_{a}(t)=\gamma_{b}(t)$. The _switching_ described in (5) of Definition
3.3 comes from the following _smooth_ map

(3.16) $\displaystyle
sw_{ab}\colon\Delta^{\text{sm}}_{ab}\Omega^{1}_{\sigma}(Q)\to\Delta^{\text{sm}}_{ab}\Omega^{1}_{(\sigma(a)\sigma(b))\circ\sigma}(Q),$
$\displaystyle\bm{\gamma}\mapsto\bm{\gamma}^{\text{sw}}=(\gamma^{\text{sw}}_{1},\ldots,\gamma^{\text{sw}}_{\kappa}),\text{
where }$
$\displaystyle\gamma_{a}^{\text{sw}}(t)=\left\\{\begin{array}[]{ll}\text{$\gamma_{a}(t)$
for $t\in[0,t_{\Delta}(\bm{\gamma})]$}\\\ \text{$\gamma_{b}(t)$ for
$t\in[t_{\Delta}(\bm{\gamma}),1]$}\end{array},\right.\quad\gamma_{b}^{\text{sw}}(t)=\left\\{\begin{array}[]{ll}\text{$\gamma_{b}(t)$
for $t\in[0,t_{\Delta}(\bm{\gamma})]$}\\\ \text{$\gamma_{a}(t)$ for
$t\in[t_{\Delta}(\bm{\gamma}),1]$}\end{array},\right.$
$\displaystyle\gamma_{i}^{\text{sw}}=\gamma_{i},\text{ for }i\neq a,b.$

We claim that $\text{sw}_{ab}$ is, moreover, a diffeomorphism. There is a
conjugation action on $\Lambda^{1}_{\kappa}(Q)$ by any
$\rho\in\mathfrak{S}_{\kappa}$ given by the following map

(3.17)
$\displaystyle\text{c}_{\rho}\colon\Lambda^{1}_{\kappa}(Q)\to\Lambda^{1}_{\kappa}(Q),$
$\displaystyle\bm{\gamma}\mapsto(\gamma_{\rho(1)},\ldots,\gamma_{\rho(\kappa)}).$

Notice that

$c_{\rho}\left(\Omega^{1}_{\sigma}(Q)\right)=\left(\Omega^{1}_{\rho\circ\sigma\circ\rho^{-1}}(Q)\right).$

We now state, without proof, two general lemmas from the Morse theory on
Hilbert manifolds. For more details we refer the reader to [AS10, Appendix
A.2] and [AM06].

######  Lemma 3.5.

  1. (1)

Let $\mathcal{M}$ be a Hilbert manifold with a pair $(X,f)$ that satisfies the
conditions of Theorem 3.1. Let $x\in\text{crit}(f)$, $N$ be an $m$-dimensional
submanifold of $\mathcal{M}$, and $\mathcal{M}_{0}$ be a codimension $m_{0}$
submanifold, such that both of them are disjoint from $\text{crit}(f)$. Then
for a generic perturbation of $X$ supported away from the set of critical
points the following sets

(3.18) $\displaystyle\mathcal{M}_{0}^{x}=W^{u}(x;X)\cap\mathcal{M}_{0},$
(3.19) $\displaystyle\mathcal{M}_{0}^{N}=W^{u}(N;X)\cap\mathcal{M}_{0}$

are submanifolds of respective dimensions $\operatorname{ind}(x)-m_{0}$ and
$(m+1)-m_{0}$.

  2. (2)

Denote by $\widetilde{\mathcal{M}}_{0}^{x}$ and
$\widetilde{\mathcal{M}}_{0}^{N}$ the stable manifolds of
$\mathcal{M}_{0}^{x}$ and $\mathcal{M}_{0}^{N}$ under $X$, respectively. Then,
if the submanifolds $W^{u}(x;X)$ and $N$ are oriented, and the submanifold
$\mathcal{M}_{0}$ is cooriented, then the manifolds
$\widetilde{\mathcal{M}}_{0}^{x}$ and $\widetilde{\mathcal{M}}_{0}^{N}$ carry
canonical orientations. This can be expressed in terms of orientation lines
(see Section 2.14) as

(3.20)
$\zeta^{-1}\otimes|\eta_{\mathcal{M}_{0}}|\otimes|\widetilde{\mathcal{M}}_{0}^{x}|\cong|W^{u}(x)|,$
(3.21)
$\zeta^{-1}\otimes|\eta_{\mathcal{M}_{0}}|\otimes|\widetilde{\mathcal{M}}_{0}^{N}|\cong|W^{u}(N)|.$

######  Lemma 3.6.

  1. (1)

Let $\varphi\colon\mathcal{M}_{1}\to\mathcal{M}_{2}$ be a smooth map between
Hilbert manifolds with $\mathcal{M}_{2}$ admitting a pair $(f_{2},X_{2})$ as
in Theorem 3.1. Then for a given smooth $m$-dimensional submanifold $N$ of
$\mathcal{M}_{1}$ and a critical point $y\in\mathcal{M}_{2}$ the set

(3.22) $N^{y}=N\cap\varphi^{-1}(W^{s}(y;X_{2}))$

is a submanifold of dimension $m-\operatorname{ind}(y)$ for generic
perturbation of $X_{2}$ supported away from the critical points.

  2. (2)

Denote by $\widetilde{N}$ and $\widetilde{N}^{y}$ the stable manifolds of $N$
and $N^{y}$, respectively. Then, if the manifold $W^{s}(y;X_{2})$ is
cooriented and the manifold $N$ is oriented, the manifold $N^{y}$ carries
canonical orientation. This can be expressed in terms of orientation lines as

(3.23) $|\eta_{W^{s}(y)}|\otimes|\widetilde{N}^{y}|\cong|\widetilde{N}|.$

######  Remark 3.7.

There is a $(-1)^{\operatorname{ind}(y)}$ sign difference between the
isomorphism in (3.23) and the one dictated by our convention for orienting a
submanifold.

###### Proof of Proposition 3.4.

Let
$\bm{\Gamma}=\left((\Gamma_{0},\ldots,\Gamma_{\ell}),\bm{s},\bm{\sigma},\bm{\rho}\right)\in\mathcal{P}(\bm{\gamma},\bm{\gamma}^{\prime};l)$
be a piecewise pseudo-gradient trajectory. Let us consider the following
sequence of manifolds (see Figure 8 for an example with $\ell=2$):

$W^{0}_{\bm{\sigma},\bm{\rho}}=W^{u}(\bm{\gamma};\bm{X}^{0})\cap\Delta^{\text{sm}}_{a_{0}b_{0}}\Omega^{1}_{\sigma_{0}}(Q),$
$W^{i+1}_{\bm{\sigma},\bm{\rho}}=W^{u}(c_{\rho_{i+1}}\circ
sw_{a_{i}b_{i}}(W^{i}_{\bm{\sigma},\bm{\rho}});\bm{X}^{i+1})\cap\Delta^{\text{sm}}_{a_{i+1}b_{i+1}}\Omega^{1}_{\sigma_{i+1}}(Q)\text{
for }i=0,\ldots,\ell-2,$
$W^{\ell}_{\bm{\sigma},\bm{\rho}}=W^{\ell-1}_{\bm{\sigma},\bm{\rho}}\cap(c_{\rho_{\ell}}\circ
sw_{a_{\ell-1}b_{\ell-1}})^{-1}\left(W^{s}(\bm{\gamma}^{\prime};\bm{X}^{\ell})\right).$

Let us denote by $\widetilde{W}^{i}_{\bm{\sigma},\bm{\rho}}$ the manifold
consisting of all points on piecewise pseudo-gradient trajectories connecting
$\bm{\gamma}$ and $W^{i}_{\bm{\sigma},\bm{\rho}}$.

Figure 8. A cartoon depiction of
$\mathcal{P}(\bm{\gamma},\bm{\gamma}^{\prime};2)$ with $n=3,|\bm{\gamma}|=3$
and $|\bm{\gamma}^{\prime}|=0$. A sample piecewise pseudo-gradient trajectory
$\bm{\Gamma}=\leavevmode\nobreak\
(\Gamma_{0},\Gamma_{1},\Gamma_{2})\in\mathcal{P}(\bm{\gamma},\bm{\gamma}^{\prime};2)$
is depicted in turquoise. Portions of manifolds
$W^{u}(\bm{\gamma};\bm{X}^{0}),W^{u}(c_{\rho_{1}}\circ
sw_{a_{0}b_{0}}(W^{0}_{\bm{\sigma},\bm{\rho}});\bm{X}^{1})$ and
$W^{s}(\bm{\gamma}^{\prime};\bm{X}^{2})$ are given in red. We also depict
$W^{0}_{\bm{\sigma},\bm{\rho}}$ and $W^{1}_{\bm{\sigma},\bm{\rho}}$ in
turquoise. The various diagonals in $\Lambda^{1}_{\kappa}(Q)$ are given in
green.

Then from Lemma 3.5 it follows that
$\widetilde{W}^{0},\ldots,\widetilde{W}^{\ell-1}$ are smooth manifolds and
each $W^{i}$ has dimension

$|\bm{\gamma}|-(n-2)(i+1).$

Since each diagonal
$\Delta^{\text{sm}}_{a_{i}b_{i}}\Omega^{1}_{\sigma_{0}}(Q)$ has a natural
coorientation induced by the orientation of $Q$, by Lemma 3.5 there is a
canonical orientation on each of
$\widetilde{W}^{0},\ldots,\widetilde{W}^{\ell-1}$ given an orientation of
$W^{u}(\bm{\gamma};\bm{X}^{0})$. More explicitly there are isomorphisms

(3.24) $|\widetilde{W}^{0}|\cong\zeta^{2-n}\otimes o_{\bm{\gamma}},$ (3.25)
$|\widetilde{W}^{i+1}|\cong\zeta^{2-n}\otimes|W^{i}|.$

In the above, we use the isomorphism

(3.26)
$|\eta_{\Delta_{a_{i}b_{i}}^{\text{sm}}\Omega^{1}_{\sigma_{i+1}}(Q)}|\cong\zeta^{n-1},$

given by the choice of orientation on $Q$.

By Lemma 3.6 the set $\widetilde{W}^{\ell}$ is then a smooth manifold of
dimension

$|\bm{\gamma}|-|\bm{\gamma}^{\prime}|-(n-2)\ell$

with a canonical orientation induced by coorientation of
$W^{s}(\bm{\gamma}^{\prime};\bm{X}^{\ell})$ and the orientation of
$W^{\ell-1}$ as above, namely,

(3.27)
$o_{\bm{\gamma}^{\prime}}\otimes|\widetilde{W}^{\ell}|\cong|\widetilde{W}^{\ell-1}|.$

We note that $\bm{\Gamma}\in\widetilde{W}^{\ell}$. Finally, we notice that
$\widetilde{\mathcal{P}}(\bm{\gamma},\bm{\gamma}^{\prime};\ell)$ is a disjoint
union of these $\widetilde{W}^{\ell}$ for all possible pairs
$(\bm{\sigma},\bm{\rho})$. ∎

Note that there is an $\mathbb{R}$-action on
$\widetilde{\mathcal{P}}(\bm{\gamma},\bm{\gamma}^{\prime};\ell)$ by the
simultaneous shift of $(s_{1},\ldots,s_{l})$ and a simultaneous shift on all
of the pseudo-gradient trajectories of a given piecewise pseudo-gradient
trajectory. We denote the quotient by
$\mathcal{P}(\bm{\gamma},\bm{\gamma}^{\prime};\ell)$. Given
$\Gamma\in\widetilde{\mathcal{P}}(\bm{\gamma},\bm{\gamma}^{\prime};\ell)$, we
denote its class in $\mathcal{P}(\bm{\gamma},\bm{\gamma}^{\prime};\ell)$ by
$[\Gamma]$.

We are now ready to define the Morse free multiloop complex. For a critical
multiloop $\bm{\gamma}\in\mathcal{P}^{\bm{L}}_{\kappa}$ we associate rank-$1$
abelian group $o_{\bm{\gamma}}$ generated by orientations of
$W^{u}(\bm{\gamma};\bm{X})$ and we set as before

(3.28)
$o_{\bm{\gamma}}^{\mathbb{Q}}=\mathbb{Q}\otimes_{\mathbb{Z}}o_{\bm{\gamma}}.$

Then we set

(3.29)
$V^{\kappa}=\bigoplus_{\bm{\gamma}\in\mathcal{P}_{\kappa}^{\bm{L}}}o_{\bm{\gamma}}^{\mathbb{Q}},$

and

(3.30)
$V^{\kappa}_{i}=\bigoplus_{|\bm{\gamma}|=i}o_{\bm{\gamma}}^{\mathbb{Q}}.$

######  Definition 3.8.

The Morse chain free multiloop complex
$CM_{*}\left(\Lambda^{1}_{\kappa}(Q)\right)$ is a graded vector subspace of
$W_{\kappa}\llbracket\hbar\rrbracket$ given by

(3.31)
$CM_{m}\left(\Lambda^{1}_{\kappa}(Q)\right)=\sum_{i+(n-2)j=m}V_{i}^{\kappa}\hbar^{j}=\sum_{i+(n-2)j=m}V_{i}^{\kappa}[(2-n)j].$

By composing isomorphisms in equations (3.24), (3.25), (3.27) and permuting
the terms (respecting the Koszul sign rule; see Section 2.14) we get a
canonical isomorphism

(3.32)
$\big{|}\widetilde{\mathcal{P}}(\bm{\gamma},\bm{\gamma}^{\prime};\ell)\big{|}^{-1}\otimes
o_{\bm{\gamma}}\cong o_{\bm{\gamma}^{\prime}}\otimes\zeta^{(n-2)\ell}\cong
o_{\bm{\gamma}^{\prime}}\bigl{[}(2-n)\ell\bigr{]}.$

When $|\bm{\gamma}|-|\bm{\gamma}^{\prime}|-(n-2)\ell=1$, given a piecewise
pseudo-gradient trajectory $[\bm{\Gamma}]\leavevmode\nobreak\
\in\leavevmode\nobreak\ \mathcal{P}(\bm{\gamma},\bm{\gamma}^{\prime};\ell)$,
the above isomorphism can be rewritten as

(3.33) $|\mathbb{R}\partial_{s}|^{-1}\otimes o_{\bm{\gamma}}\cong
o_{\bm{\gamma}^{\prime}}\bigl{[}(2-n)\ell\bigr{]}.$

Fixing the upward orientation (arising from the positive gradient direction),
which corresponds to the opposite orientation on $\mathbb{R}$, induces a
morphism

(3.34) $\partial_{[\bm{\Gamma}]}\colon o_{\bm{\gamma}}^{\mathbb{Q}}\to
o_{\bm{\gamma}^{\prime}}^{\mathbb{Q}}\bigl{[}(2-n)\ell\bigr{]}.$

We extend this map to any generator $[\bm{\gamma}]\hbar^{j}$ of
$CM_{*}\bigl{(}\Lambda^{1}_{\kappa}(Q)\bigr{)}$ by tensoring the above
isomorphism with $\zeta^{(n-2)j}$ on the right:

(3.35) $\partial_{[\bm{\Gamma}]}\colon
o_{\bm{x}}\otimes\zeta^{(n-2)j}\to\otimes
o_{\bm{x}^{\prime}}\bigl{[}(2-n)\ell\bigr{]}\otimes\zeta^{(n-2)j}\cong
o_{\bm{x}^{\prime}}\bigl{[}(2-n)(\ell+j)\bigr{]}.$

We may now define the differential on Morse free multiloop complex
$CM_{*}\bigl{(}\Lambda^{1}_{\kappa}(Q)\bigr{)}$ by

(3.36) $\displaystyle\partial\colon CM_{*}(\Lambda^{1}_{\kappa}(Q))\to
CM_{*-1}(\Lambda^{1}_{\kappa}(Q)),$
$\displaystyle\partial([\bm{\gamma}]\hbar^{j})=\sum_{\begin{subarray}{c}\ell\geq
0,|\bm{\gamma}^{\prime}|=|\bm{\gamma}|-(n-2)\ell-1,\\\
[\bm{\Gamma}]\in\mathcal{P}(\bm{\gamma},\bm{\gamma}^{\prime};\ell)\end{subarray}}\partial_{[\bm{\Gamma}]}([\bm{\gamma}]\hbar^{j}).$

where $[\bm{\gamma}]\hbar^{j}$ is some generator of
$o_{\bm{\gamma}}^{\mathbb{Q}}[(2-n)j]$ for $j\geq 0$. We devote the following
section to showing that this is indeed a chain complex.

For the later use we define the _reflected_ cochain complex
$CM^{*}_{-}(\Lambda^{1}_{\kappa}(Q))$ as follows. First, set

$V_{\kappa,-}^{i}=\bigoplus_{|\bm{\gamma}|=-i}(o_{\bm{\gamma}}^{-1})^{\mathbb{Q}}.$

Then

$CM^{m}_{-}(\Lambda^{1}_{\kappa}(Q))=\sum_{-(i+(n-2)j)=m}V^{i}_{\kappa,-}[(n-2)j].$

Applying our conventions (2.136) we notice that

$o_{\bm{\gamma}}^{-1}[(n-2)j]\cong\zeta^{(2-n)j}\otimes
o_{\bm{\gamma}}^{-1}\cong(o_{\bm{\gamma}}[(2-n)j])^{-1}.$

Then clearly the isomorphism (3.33) also induces a morphism

(3.37) $d_{[\bm{\Gamma}]}\colon(o_{\bm{\gamma}}^{-1})^{\mathbb{Q}}\to
o_{\bm{\gamma}}^{-1}[(n-2)j]^{\mathbb{Q}}.$

The differential on the reflected Morse miltiloop complex is then given by

(3.38) $\displaystyle\partial\colon CM^{*}_{-}(\Lambda^{1}_{\kappa}(Q))\to
CM^{*+1}_{-}(\Lambda^{1}_{\kappa}(Q)),$ $\displaystyle
d([\bm{\gamma}_{-}]\hbar^{-j})=\sum_{\begin{subarray}{c}\ell\geq
0,|\bm{\gamma}^{\prime}|=|\bm{\gamma}|-(n-2)\ell-1,\\\
[\bm{\Gamma}]\in\mathcal{P}(\bm{\gamma},\bm{\gamma}^{\prime};\ell)\end{subarray}}d_{[\bm{\Gamma}]}([\bm{\gamma}]\hbar^{-j}).$

At last, we claim that there is a natural reflection chain map between the two
complexes, inducing a homology isomorphism

(3.39) $\mathcal{I}\colon CM_{-}^{*}\bigl{(}\Lambda^{1}_{\kappa}(Q)\bigr{)}\to
CM_{-*}\bigl{(}\Lambda^{1}_{\kappa}(Q)\bigr{)},$

which one obtains by applying the isomorphism $\ell\cong\ell^{-1}$ for
orientation lines and inspecting Koszul sign rule, see Section 2.14.

###  3.3. Compactness

Since a piecewise gradient trajectory is not action-decreasing, the
nonemptiness of $\mathcal{P}(\bm{\gamma},\bm{\gamma}^{\prime};\ell)$ does not
imply
$\mathcal{A}_{\bm{L}}(\bm{\gamma})\geq\mathcal{A}_{\bm{L}}(\bm{\gamma}^{\prime})$.
However, we can still show that
$\mathcal{A}_{\bm{L}}(\bm{\gamma}^{\prime})-\mathcal{A}_{\bm{L}}(\bm{\gamma})$
is $C^{0}$-bounded, so for each $\ell\geq 0$,
$\mathcal{P}(\bm{\gamma},\bm{\gamma}^{\prime};\ell)$ is empty for all but
finitely many $\bm{\gamma}^{\prime}$. Therefore, the chain complex
$CM^{*}_{\kappa}(\Lambda Q)$ is locally finite.

######  Lemma 3.9.

Fix a generic choice of a collection of pseudo-gradient vector field tuples
$\bm{X}^{0},\ldots,\bm{X}^{\ell}$, given
$|\bm{\gamma}|-|\bm{\gamma}^{\prime}|-(n-2)\ell\leq 2$, the space
$\mathcal{P}(\bm{\gamma},\bm{\gamma}^{\prime};\ell)$ admits a
compactification.

###### Sketch of the proof.

Given $(2-n)l+|\bm{\gamma}^{\prime}|-|\bm{\gamma}|\leq 2$,
$\mathcal{P}(\bm{\gamma},\bm{\gamma}^{\prime};l)$ is a smooth manifold of
dimension $\leq 1$ by Lemma 3.4. We need to consider all possible limiting
behaviors of a sequence of elements in
$\mathcal{P}_{*}(\bm{\gamma},\bm{\gamma}^{\prime};l)$. The proof is similar to
[HKTY, Lemma A.9]. We sketch the cases similar to [HKTY, Lemma A.9] and
emphasize the difference.

Let

$[\Gamma^{r}]=[((\Gamma^{r}_{0},\ldots,\Gamma^{r}_{l}),(\bm{s}^{r},\bm{\sigma}^{r},\bm{\rho}^{r})]$

be a sequence of piecewise gradient trajectories in
$\mathcal{P}(\bm{\gamma},\bm{\gamma}^{\prime};l)$ parametrized by
$r\in\mathbb{N}$. We can assume that the premutation data
$((\sigma_{1}^{r},\ldots,\sigma_{\ell}^{r}),(\rho^{r}_{1},\ldots,\rho^{r}_{\ell}))=(\bm{\sigma},\bm{\rho})$
is constant for this sequence. As $r\to\infty$, if $[\Gamma^{r}]$ does not
limit to an interior element of
$\mathcal{P}_{*}(\bm{\gamma},\bm{\gamma}^{\prime};l)$, then there are four
possibilities:

(1) _$s^{r}_{i}-s^{r}_{i+1}\to+\infty$ for some $1\leq i\leq l-1$_. This
corresponds to breaking along $\Gamma^{r}_{i}$. Breaking along
$\Gamma^{r}_{0}$ or $\Gamma^{r}_{l}$ is similar. In the limit $r\to\infty$,
$\Gamma_{r}$ degenerates to a broken piecewise pseudo-gradient trajectory
which lies in
$\mathcal{P}(\bm{\gamma},\bm{\gamma}^{\prime\prime};\ell_{1})\times\mathcal{P}(\bm{\gamma}^{\prime\prime},\bm{\gamma}^{\prime};\ell_{2})$,
where $\ell_{1}+\ell_{2}=\ell$. The sum of the expected dimensions of
$\mathcal{P}_{*}(\bm{\gamma},\bm{\gamma}^{\prime\prime};\ell_{1})$ and
$\mathcal{P}(\bm{\gamma}^{\prime\prime},\bm{\gamma}^{\prime};\ell_{2})$ is
equal to $|\bm{\gamma}|-|\bm{\gamma}^{\prime}|-(n-2)\ell-2$. Therefore, if
$|\bm{\gamma}|-|\bm{\gamma}^{\prime}|-(n-2)\ell\leq 1$,
$s^{r}_{i}-s^{r}_{i+1}\to+\infty$ does not occur for a generic choice of
pseudo-gradient vector fields. If
$|\bm{\gamma}|-|\bm{\gamma}^{\prime}|-(n-2)\ell=2$, this is the required
degeneration for the differential on $CM^{*}_{\kappa}(\Lambda_{\kappa}^{1}Q)$
to be well-defined.

(2) _$s^{r}_{i}-s^{r}_{i+1}\to 0$ for some $1\leq i\leq\ell-1$_. In the limit
$r\to\infty$, two switchings occur at the same $s$. This case is discussed in
[HKTY, Lemma A.9]. We refer the reader to Figure 9 for an illustration.

Figure 9. The image of $\sqcup_{i,s}\Gamma_{i}(s)$ viewed as the graph in
$Q\times\mathbb{R}_{s}$. The blue arc labelled by $j$ denotes the image of
$\Gamma_{i}(s)_{j}(0)$. As $r\to\infty$, $s^{r}_{i}-s^{r}_{i+1}\to 0$ as shown
by (A); the limit is shown by (B), where two switchings occur simultaneously;
the family (A) is paired up with another family (C). Note that the allowance
of all permutation sequences in Definition 3.3 is necessary for the existence
of (C).

(3) Generically, a 1-parameter family of piecewise pseudo-gradient
trajectories may undergo a birth-death bifurcation. This is also discussed in
[HKTY, Lemma A.9] and we omit the proof here.

(4) _$t_{i+1}^{r}\to 0$ for some $i=0,\ldots,\ell-1$_. Let us assume that
$t_{i}\to 0^{+}$, meaning that all these times are in $[0,\epsilon)$ for $i\gg
0$ and a small $\epsilon>0$. For the limit of the paths
$\Gamma^{r}_{i}(s_{i+1})_{a_{i}}$ and $\Gamma^{r}_{i}(s_{i+1})_{b_{i}}$ we
have that $\Gamma^{\infty}_{i}(s_{i+1})_{a_{i}}$ and
$\Gamma^{\infty}_{i}(s_{i+1})_{b_{i}}$ intersect at $t_{i+1}^{\infty}=0$.

We consider a subset of the closure
$\overline{\Delta}_{ab}\Omega^{1}_{\sigma}(Q)\subset\Omega^{1}_{\sigma}(Q)$
given by

(3.40)
$\Delta^{\text{sm},0,1}_{ab}\Omega^{1}_{\sigma}(Q)=\\{\bm{\gamma}\in\Omega^{1}_{\sigma}(Q)\,|\,\text{
there exists a unique }t\in[0,1]\text{ s.t. }\gamma_{a}(t)=\gamma_{b}(t)\\}$

which can be also given as a disjoint union

$\Delta^{\text{sm},0,1}_{ab}\Omega^{1}_{\sigma}(Q)=\Delta^{\text{sm}}_{ab}\Omega^{1}_{\sigma}(Q)\sqcup\Delta^{\text{sm},0}_{ab}\Omega^{1}_{\sigma}(Q)\sqcup\Delta^{\text{sm},1}_{ab}\Omega^{1}_{\sigma}(Q)\text{,
where }$
$\Delta^{\text{sm},0}_{ab}\Omega^{1}_{\sigma}(Q)=\\{\bm{\gamma}\in\Delta^{\text{sm},0,1}_{ab}\Omega^{1}_{\sigma}(Q)\,|\,\gamma_{a}(0)=\gamma_{b}(0)\\},$
$\Delta^{\text{sm},1}_{ab}\Omega^{1}_{\sigma}(Q)=\\{\bm{\gamma}\in\Delta^{\text{sm},0,1}_{ab}\Omega^{1}_{\sigma}(Q)\,|\,\gamma_{a}(1)=\gamma_{b}(1)\\}.$

Notice that both $\Delta^{0}_{ab}\Omega^{1}_{\sigma}(Q)$ and
$\Delta^{1}_{ab}\Omega^{1}_{\sigma}(Q)$ are codimension $n$ submanifolds.
Additionally, we point out that

(3.41)
$\Delta^{\text{sm},0}_{ab}\Omega^{1}_{\sigma}(Q)=\Delta^{\text{sm},1}_{\sigma^{-1}(a)\sigma^{-1}(b)}\Omega^{1}_{\sigma}(Q).$

With these notations we then have
$\Gamma_{i}^{\infty}(s_{i+1})\in\Delta^{0}_{a_{i}b_{i}}\Omega^{1}_{\sigma_{i}}(Q)$.
Using notations from the proof of Proposition 3.4 we consider the following
set

$W^{i}_{\bm{\sigma},\bm{\rho},i_{0,1}}=W^{u}(c_{\rho_{i}}\circ
sw_{a_{i-1}b_{i-1}}(W^{i-1}_{\bm{\sigma},\bm{\rho}});\bm{X}^{i})\cap\Delta^{\text{sm},0,1}_{a_{i}b_{i}}\Omega^{1}_{\sigma_{i}}(Q).$

Replacing $W^{i}_{\bm{\sigma},\bm{\rho}}$ with this set we then obtain
$W^{\ell}_{\bm{\sigma},\bm{\rho},i_{0,1}}$ following the procedure in the
proof of Proposition 3.4. Note that $\bm{\Gamma}^{\infty}$ corresponds to a
point of $W^{\ell}_{\bm{\sigma},\bm{\rho},i_{0}}\subset
W^{\ell}_{\bm{\sigma},\bm{\rho},i_{0,1}}$. Consider a new collection
$\bm{\rho}^{\prime}$ with $\rho^{\prime}_{j}=\rho_{j}$ for $j\neq i+1$ and
$\rho^{\prime}_{i+1}$ satisfying:

(3.42)
$\rho^{\prime}_{i+1}(a_{i})=\rho_{i+1}(b_{i}),\,\rho^{\prime}_{i+1}(b_{i})=\rho_{i+1}(a_{i}),\,\rho^{\prime}_{i+1}(c)=\rho_{i+1}(c)\text{
for }c\neq a_{i},b_{i}.$

Expediting (3.13) and (3.42) one can see that

$(\rho^{\prime-1}_{i+1}\circ\sigma_{i+1}\circ\rho^{\prime}_{i+1})\circ\sigma_{i}^{-1}=(a_{i}b_{i}).$

Hence, $W^{\ell}_{\bm{\sigma},\bm{\rho}^{\prime}}$ and
$W^{\ell}_{\bm{\sigma},\bm{\rho}^{\prime},i_{0,1}}$ are well-defined. Now
applying (3.41) we have

$\bm{\Gamma}^{\infty}\in
W^{\ell}_{\bm{\sigma},\bm{\rho}^{\prime},i_{1}}\subset
W^{\ell}_{\bm{\sigma},\bm{\rho}^{\prime},i_{0,1}}.$

Finally, since $W^{\ell}_{\bm{\sigma},\bm{\rho}^{\prime},i_{0,1}}$ is of
dimension $1$ as $W^{\ell}_{\bm{\sigma},\bm{\rho},i_{0,1}}$ for generic choice
of pseudo-gradient vector fields $\bm{X}^{i}$’s, there must be a
$1$-dimensional family of pseudo-gradient trajectories in
$W^{\ell}_{\bm{\sigma},\bm{\rho}^{\prime}}$ limiting to $\bm{\Gamma}^{\infty}$
(see Figure 10 for an example of such a family near $s_{i+1}$). Therefore, we
may represent $\bm{\Gamma}^{\infty}$ as a limit of a sequence
$\bm{\Gamma}^{-r}\in W^{\ell}_{\bm{\sigma},\bm{\rho}^{\prime}}.$ The
constructed above sequence corresponds to $t_{i+1}^{-r}\to 1$ which implies
that such singular behaviors are paired up and do not constitute part of the
boundary of $\mathcal{P}(\bm{\gamma},\bm{\gamma}^{\prime};\ell)$.

We notice that the choice of $\bm{\rho}^{\prime}$ in the argument above is the
main reason why in Definition 3.3 we have introduced permutation data
$\bm{\rho}$.

Figure 10. On the left side trajectory $[\Gamma^{r}]$ is depicted near
$s_{i+1}$. On the right side, we depict trajectory $[\Gamma^{-r}]$ that is
constructed in the proof.

Finally, we note that the compactification of
$\mathcal{P}(\bm{\gamma},\bm{\gamma}^{\prime};\ell)$ exists since
$\bm{\gamma}$ is of finite Morse index. ∎

As an immediate corollary of Lemma 3.9 we obtain

######  Proposition 3.10.

$\partial^{2}=0$.

###### Proof.

By Lemma 3.9, $\\#\mathcal{P}(\bm{\gamma},\bm{\gamma}^{\prime};\ell)$ is
finite whenever $|\bm{\gamma}^{\prime}|=|\bm{\gamma}|-(n-2)\ell-1$. We also
point out that for a given $\gamma$ the space
$\\#\mathcal{P}(\bm{\gamma},\bm{\gamma}^{\prime};\ell)$ is empty for all but
finitely many $\bm{\gamma}^{\prime}$ by the Palais-Smale condition 24. Given
$|\bm{\gamma}|-|\bm{\gamma}^{\prime}|-(n-2)\ell=2$, applying the continuation
procedure described in Lemma 3.9 the boundary of the compactification of
$\mathcal{P}(\bm{\gamma},\bm{\gamma}^{\prime};l)$ may be regarded as a finite
number of broken pseudo-gradient trajectories, i.e.,

$\partial\overline{\mathcal{P}}(\bm{\gamma},\bm{\gamma}^{\prime};\ell)=\bigsqcup_{\begin{subarray}{c}\ell_{1}+\ell_{2}=\ell,\\\
|\bm{\gamma}|-|\bm{\gamma}^{\prime\prime}|=|\bm{\gamma}^{\prime\prime}|-|\bm{\gamma}^{\prime}|=(n-2)\ell+1\end{subarray}}\mathcal{P}(\bm{\gamma},\bm{\gamma}^{\prime\prime};\ell_{1})\times\mathcal{P}(\bm{\gamma}^{\prime\prime},\bm{\gamma}^{\prime};\ell_{2}).$

To finish the proof it is enough to compare the composition map
$\partial_{[\bm{\Gamma}_{2}]}\circ\partial_{[\bm{\Gamma}_{1}]}$ and the map
induced by boundary orientation. That is,

(3.43)
$|\mathbb{R}\partial_{s}|^{-1}\otimes\big{|}\mathcal{P}(\bm{\gamma},\bm{\gamma}^{\prime};\ell)\big{|}^{-1}\otimes
o_{\bm{\gamma}}\cong o_{\bm{\gamma}^{\prime}}\bigl{[}(2-n)\ell\bigr{]},$
(3.44)
$|\mathbb{R}\partial_{s}\bm{\Gamma}_{2}|^{-1}\otimes|\mathbb{R}\partial_{s}\bm{\Gamma}_{1}|^{-1}\otimes
o_{\bm{\gamma}}\cong|\mathbb{R}\partial_{s}\bm{\Gamma}_{2}|^{-1}\otimes
o_{\bm{\gamma}^{\prime\prime}}\bigl{[}(2-n)\ell_{1}\bigr{]}\cong
o_{\bm{\gamma}^{\prime}}\bigl{[}(2-n)\ell\bigr{]}.$

Now it remains to notice that $\partial_{s}\bm{\Gamma}_{1}$ with the preferred
orientation as in (3.33) is an outward pointing vector. Therefore,
$\partial|_{o_{\bm{\gamma}}}^{2}=0$. We leave it to the reader to verify that
$\partial|_{o_{\bm{\gamma}}[(2-n)j]}^{2}=0$ for any $j>0$. ∎

We write $CM_{*}(\Lambda^{1}_{\kappa}(Q);\bm{L},\bm{X})$ and
$HM_{*}(\Lambda^{1}_{\kappa}(Q);\bm{L},\bm{X})$ to indicate the specific
choice of $\bm{L}=\\{L_{1},\ldots,L_{\kappa}\\}$ and pseudo-gradient vector
fields $\bm{X}$.

######  Proposition 3.11.

The homology groups $HM_{*}(\Lambda^{1}_{\kappa}(Q);\bm{L},\bm{X})$ do not
depend on the choice of $\bm{L}=\\{L_{1},\ldots,L_{\kappa}\\}$ and pseudo-
gradient vector fields $\bm{X}$.

###### Proof.

According to [AS10] changing $\bm{L}$ or pseudo-gradient fields $\bm{X}$
induces chain homotopy at the level of standard Morse chain complexes. Hence,
if we were considering only pseudo-gradient trajectories with $\ell=0$ the
statement would immediately follow. We then claim that showing that changing
one of $L_{i}$’s or $\bm{X}_{j}$’s alone induces a chain homotopy in the
presence of switchings is similar to the above-mentioned line of the argument.
∎

##  4\. The isomorphism between the Heegaard Floer symplectic cohomology and
the homology of the free multiloop complex

In this section, we define a chain map from the Floer complex to the Morse
complex

$\mathcal{F}\colon SC^{*}_{\kappa,\text{unsym}}(T^{*}Q)\to CM_{\kappa
n-*}^{\kappa}(\Lambda Q),$

by counting rigid elements in a mixed moduli space which combines holomorphic
curves and piecewise gradient trajectories. We establish an analogue of the
Viterbo isomorphism theorem in the context of Heegaard Floer symplectic
cohomology by showing that $\mathcal{F}$ is a quasi-isomorphism.

In this section, we assume that $Q$ is an orientable closed manifold with
vanishing second _Stiefel-Whitney class_

$w_{2}(TQ)\leavevmode\nobreak\ \in\leavevmode\nobreak\
H^{2}(T^{*}Q;\mathbb{Z}/2).$

This allows one to fix a spin structure on $Q$ regarded as a Lagrangian
submanifold in $T^{*}Q$. We also note that the approach in this section is
parallel to [HKTY, Section 3].

We point out that we work here with the unsymmetrized version of HFSH
introduced in Section 2.13 and sometimes we abuse notations and do not specify
that we work in this context as it is understood throughout this section.

###  4.1. The half-cylinder Hurwitz space

We start by introducing the space of branched covers of a half-cylinder in the
context of Section 2.13.

######  Definition 4.1.

Let $\sigma,\sigma_{0}\in\mathfrak{S}_{\kappa}$. For given $\chi>0$ define
$\mathcal{H}^{\sigma,\sigma_{0},\geq 0}_{\kappa,\chi}$ to be the set of tuples

$\left(\pi\colon S\to\mathbb{R}_{\geq 0}\times
S^{1},\bm{p}^{+},\bm{a}^{+},\bm{P}^{0},\bm{a}^{0},B\right)\text{, where}$

  1. (1)

$S$ is a Riemann surface of Euler characteristic $\chi$ with positive
punctures $\bm{p}^{+}=\\{p^{+}_{1},\ldots,p^{+}_{l(\bm{\mu})}\\}$ and boundary
components $\bm{P}^{0}=\\{P^{0}_{1},\ldots,P^{0}_{\ell(\sigma_{0})}\\}$.
Denote by $\overline{S}$ the compactification of $S$ such that
$S=\overline{S}\setminus\bm{p}^{+}$.

  2. (2)

$\bm{a}^{+}=(a_{1}^{+},\ldots,a_{\kappa}^{+})$ is a collection of positive
asymptotic markers at positive punctures $p_{i}^{+}$ such that for each
$p_{i}^{+}$ there are exactly $\mu_{i}(\sigma)$ asymptotic markers
$a_{i_{1}}^{+},\ldots,a_{i_{\mu_{i}}}^{+}\in\mathcal{S}^{1}_{p_{i}^{+}}S=(T_{p_{i}^{+}}S\setminus\\{0\\})/\mathbb{R}_{>0}$,
and these are exactly the indices permuted by $c_{i}(\sigma)$ abd they appear
on $\mathcal{S}^{1}_{p_{i}^{+}}$ in the order prescribed by the cycle
$c_{i}(\sigma)$.

Analogously we choose marked points, which we simply call _markers_ , on
$\bm{P}^{0}$ such that for each boundary component $P^{0}_{i}$ there are
exactly $\mu_{i}(\sigma_{0})$ markers.The permutation on the set of markers
induced by walking along $P_{i}^{0}$ in the boundary orientation direction is
required to be equal to $c_{i}(\sigma_{0})$.

  3. (3)

$\pi\colon S\longrightarrow\mathbb{R}_{\geq 0}\times S^{1}$ is a holomorphic
branched cover with simple branched points away from $+\infty$ and boundary
$\partial S=\bm{P}^{0}$. At $+\infty$ the branching profile is given by
$\bm{\mu}$ and for each boundary component $P^{0}_{i}$ the restriction
$\pi|_{P^{0}_{i}}$ is the $\mu_{i}^{0}$-cover of $\\{0\\}\times S^{1}$.

  4. (4)

The map induced by $\pi$ from $\mathcal{S}^{1}_{p_{i}^{+}}S$ to
$\mathcal{S}^{1}_{+\infty}(\mathbb{R}_{\geq 0}\times S^{1})$ sends all
asymptotic markers assigned to $p_{i}^{+}$ to the asymptotic marker
$1\in\mathcal{S}^{1}_{+\infty}(\mathbb{R}_{\geq 0}\times S^{1})$. The markers
$a_{i}^{0}$’s are all sent to $(0,0)\in R_{\geq 0}\times S^{1}$ under $\pi$.

  5. (5)

$B=(q_{1},\ldots q_{b})$, is an ordered collection of points in
$\mathbb{R}_{>0}\times S^{1}$ of simple branching, where $b=-\chi$.

Figure 11. The tuple $((s,\pi),\bm{s}_{\bm{t},\bm{\sigma},\bm{\rho}})$ above
is an element of $\mathcal{T}^{\sigma,\sigma_{0},\sigma^{\prime}}_{-3,\ell}$
for $\kappa=4$ with $\sigma=(1342)$, $\sigma_{0}=(14)(23)$, and we do not
specify $\sigma^{\prime}$.

Following the approach of Section 2.5 we define a portion
$\widehat{\mathcal{H}}^{\sigma,\sigma_{0},\geq 0}_{\kappa,\chi}$ of the
compactification $\overline{\mathcal{H}}^{\sigma,\sigma_{0},\geq
0}_{\kappa,\chi}$ (which we will not specify here). Namely, an element of
$\widehat{\mathcal{H}}^{\sigma,\sigma_{0},\geq 0}_{\kappa,\chi}$ may have
several levels of covers of the cylinder $\mathbb{R}\times S^{1}$ above the
cover of half cylinder. That is

######  Definition 4.2.

The space of _smooth half-cylindrical buildings_
$\widehat{\mathcal{H}}^{\sigma,\sigma_{0},\geq 0}_{\kappa,\chi}$ consists of

$[(\overrightarrow{C}=(C_{0},\ldots,C_{n}),S=S_{0}\cup\ldots\cup
S_{n},(\bm{v}^{n-1},\ldots,\bm{v}^{0}),(\bm{m}^{n-1},\ldots,\bm{m}^{0}),\pi]$

where

  1. (1)

$\pi|_{S_{0}}\colon S_{0}\to\mathbb{R}_{\geq 0}\times S^{1}$ is a branched
cover of a half cylinder with data of markers on the boundary, but without
asymptotic markers. In other words, it comes from an element of
${\mathcal{H}^{\sigma^{\prime},\sigma_{0},\geq 0}_{\kappa,\chi_{2}}}$ for some
$\sigma^{\prime}\in\mathfrak{S}_{\kappa}$ by forgetting asymptotic markers;

  2. (2)

$\bm{m}^{0}$ is a tuple of matching markers corresponding to the nodes
$\bm{v}^{0}$ connecting $S_{0}$ and $S_{1}$ (see (4) of Definition 2.5 for
more details);

  3. (3)

$[(\overrightarrow{C}_{>0}=(C_{n},\ldots,C_{1}),S_{>0}=S_{n}\cup\ldots\cup
S_{1},(\bm{v}^{n-1},\ldots,\bm{v}^{1}),(\bm{m}^{n-1},\ldots,\bm{m}^{1}).\pi|_{S_{>0}}]$
comes from an element of
$\widehat{\mathcal{H}}^{\sigma,\sigma^{\prime}}_{\kappa,\chi_{1}}$ with
forgotten asymptotic markers at negative punctures of $C_{1}$ (see Definition
2.5).

  4. (4)

It holds that $\chi_{1}+\chi_{2}=\chi$.

Similarly to
$\widehat{\mathcal{H}}^{\bm{\mu},\bm{\mu}^{\prime}}_{\kappa,\chi}$ (see
Definition 2.20) a codimension $1$ boundary of
$\widehat{\mathcal{H}}^{\sigma,\sigma_{0},\geq 0}_{\kappa,\chi}$ can be given
as the images under natural embeddings of spaces

(4.1) $\bigsqcup_{\begin{subarray}{c}\chi_{1}+\chi_{2}=\chi,\\\
\sigma^{\prime}\in\mathfrak{S}_{\kappa}\end{subarray}}\mathcal{H}^{\sigma,\sigma^{\prime}}_{\kappa,\chi_{1}}\times_{\bm{m}}\mathcal{H}^{\sigma^{\prime},\sigma_{0},\geq
0}_{\kappa,\chi_{2}}.$

###  4.2. Floer data for the half-cylinder Hurwitz space

We set up the perturbation data on the half-cylinder Hurwitz space following
Section 2.8. First, we note that there is a wnbc groupoid
$\widehat{\mathscr{H}}^{\sigma,\sigma_{0},\geq 0}_{\kappa,\chi}$ built via
construction similar to the one given in Section 2.6.

As in the Section 2.8 there is a smooth choice of positive cylindrical ends
$\\{\delta^{i}_{+}\\}_{i=1}^{i=\ell(\bm{\sigma})}$ and finite cylinders
$\delta^{j}_{i,r}$ over the whole
$\widehat{\mathscr{H}}^{\sigma,\sigma_{0},\geq 0}_{\kappa,\chi}$.

A Floer datum on $(S,\pi)\in(\widehat{\mathscr{H}}^{\sigma,\sigma_{0},\geq
0}_{\kappa,\chi})^{\vee}_{H}$ (here $(S,\pi)$ is assumed to be away from the
boundary) is a collection $(H_{(S,\pi)},F_{(S,\pi)},J_{(S,\pi)})$, where

  1. (1)

$H_{(S,\pi)}\colon S\to\mathcal{H}(\hat{M})$ is a domain-dependent Hamiltonian
function, required to coincide with $H_{0}$ on cylindrical ends and finite
cylinders and is equal to $0$ near the boundary $\partial S$;

  2. (2)

a domain-dependent function $F_{(S,\pi)}\in C^{\infty}(\hat{M})$ on $S$ such
that conditions given in 2 are satisfied (we note that on a cylinder with
assigned cycle $c\in\mathfrak{S}_{\kappa}$ the function $F$ has to coincide
with $F_{c}$ away from the collars) and additionally $F$ vanishes near
$\partial S$;

  3. (3)

a domain-dependent almost complex structure $J_{(S,\pi)}\colon
S\to\mathcal{J}_{\hat{M}}$ coinciding with $J_{t}$ as in (2.21) at each
cylindrical end and at each finite cylinder

As in the case of Hurwitz spaces, there exists a choice of _consistent Floer
data_ $(H,F,J)$ on $\widehat{\mathscr{H}}^{\sigma,\sigma_{0},\geq
0}_{\kappa,\chi}$ in the sense of Section 2.8. As before we will use the
notation $H^{\text{tot}}=H+F$.

###  4.3. The mixed moduli space

Consider $\bm{x}\in\mathcal{P}_{\text{unsym}}(T^{*}Q)$,
$\bm{\gamma}\in\mathcal{P}_{\kappa}^{\bm{L}}$ and a permutation
$\sigma_{0}\in\mathfrak{S}_{\kappa}$. We set $\sigma=\sigma(\bm{x})$ and
$\sigma^{\prime}=\sigma(\bm{\gamma})$.

We define the mixed moduli space
$\mathcal{T}^{\sigma_{0}}_{\chi,\ell}(H^{\text{tot}},J,\bm{X};\bm{x},\bm{\gamma})$
to be the space of tuples

(4.2) $\left((S,\pi\colon S\to\mathbb{R}_{\geq 0}\times
S^{1}),u=(\pi,v),\bm{\Psi}\right),$

where $(S,\pi)\in\mathcal{H}^{\sigma,\sigma_{0},\geq 0}_{\kappa,\chi}$, and
$u$ is a map

$u\colon S\to\mathbb{R}_{\geq 0}\times S^{1}\times T^{*}Q,$

and $\bm{\Psi}=((\Psi_{0},\ldots,\Psi_{\ell}),\bm{s},\bm{\sigma},\bm{\rho})$
is a collection of maps

$\Psi_{i}\colon[s_{i+1},s_{i}]\to\Omega^{1}_{\sigma_{i}}(Q),$

where $\bm{s}=(s_{0},\ldots,s_{\ell+1})$ with $s_{0}=0$, $s_{\ell+1}=-\infty$
and $0>s_{1}>\ldots>s_{\ell}>-\infty$, $\bm{\sigma}$ and $\bm{\rho}$ are two
collections of permutations in $\mathfrak{S}_{\kappa}$ with a leading term of
$\bm{\sigma}$ being exactly the $\sigma_{0}$ as above (here we allow ourselves
slight abuses of notation).

The map $u=(\pi,v)$ satisfies

(4.3) $\begin{dcases}(dv-Y_{(S,\pi)})^{0,1}=0,\text{ with respect to
}J_{(S,\pi)};\\\
\lim_{s\to+\infty}v\circ\delta_{i}^{+}(s,\cdot)=x^{c_{i}(\sigma)}(\cdot)\text{
for }i=1,\ldots,\ell(\sigma);\\\ \operatorname{Im}(v|_{\partial S})\subset
Q.\end{dcases}$

Here $Y_{(S,\pi)}$ is $1$-form associated $H^{\text{tot}}_{(S,\pi)}$ as in
(2.73). We point out that as in Lemma 2.31 the map $u$ can be regarded as a
pseudo-holomorphic map with respect to the almost complex structure
$\underline{J}^{H^{\text{tot}}}_{(S,\pi)}$ defined as in Section 2.8.

Let us denote the space of pairs $\left((S,\pi),u\right)$ satisfying (4.3) by
$\mathcal{H}^{\sigma,\sigma_{0},\geq
0}_{\kappa,\chi}(H^{\text{tot}},J;\bm{x})$. Then there is a natural
_evaluation map_

(4.4) $\mathcal{EV}\colon\mathcal{H}^{\sigma,\sigma_{0},\geq
0}_{\kappa,\chi}(H^{\text{tot}},J;\bm{x})\to\Omega^{1}_{\sigma_{0}}(Q)$

given by sending a pair $\left((S,\pi),u\right)$ to the restriction of $u$ to
$\partial S$.

We then require for collection $\Psi$ to satisfy the following:

  1. (1)

$\Psi_{0}(0)=\mathcal{EV}(u)$, $\Psi_{l}(-\infty)=\bm{\gamma}$.

  2. (2)

$\Psi_{i}^{\prime}(s)=\bm{X}^{i}(\Psi_{i}(s))$.

  3. (3)

For each $i$ such that $0\leq i\leq\ell-1$ there is a transposition
$(a_{i}b_{i})\in\mathfrak{S}_{\kappa}$ such that

$(\rho^{-1}_{i+1}\circ\sigma_{i+1}\circ\rho_{i+1})\circ\sigma_{i}^{-1}=(\sigma_{i}(a_{i})\sigma_{i}(b_{i})).$

  4. (4)

For $a_{i},b_{i}\in\\{1,\ldots,\kappa\\}$ satisfying the above for
$i=0,\ldots,\ell-1$ there exists $t_{i}\in(0,1)$, so that

$\displaystyle\Psi_{i+1}(s_{i+1})_{\rho_{i+1}(a_{i})}(t)=\left\\{\begin{array}[]{ll}\text{$\Psi_{i}(s_{i+1})_{a_{i}}(t)$
for $t\in[0,t_{i+1}]$}\\\ \text{$\Psi_{i}(s_{i+1})_{b_{i}}(t)$ for
$t\in[t_{i+1},1]$}\end{array},\right.$
$\displaystyle\Psi_{i+1}(s_{i+1})_{\rho_{i+1}(b_{i})}(t)=\left\\{\begin{array}[]{ll}\text{$\Psi_{i}(s_{i+1})_{b_{i}}(t)$
for $t\in[0,t_{i+1}]$}\\\ \text{$\Psi_{i}(s_{i+1})_{a_{i}}(t)$ for
$t\in[t_{i+1},1]$}\end{array}.\right.$

  5. (5)

$\Psi_{i+1}(s_{i+1})_{\rho_{i+1}(j)}=\Psi_{i+1}(s_{i+1})_{j}$ for $j\neq
a_{i},b_{i}$.

######  Lemma 4.3.

For a tuple of Hamiltonian orbits $\bm{x}\in\mathcal{P}_{unsym}(T^{*}Q)$, a
multiloop $\bm{\gamma}\in\mathcal{P}_{\kappa}^{\bm{L}}$, and a generic choice
of Floer data on $\mathcal{H}^{\sigma,\sigma_{0},\geq 0}_{\kappa,\chi}$, and a
generic collection $\bm{X}$ of pseudo-gradient vector fields $\bm{X}$, the
space
$\mathcal{T}^{\sigma_{0}}_{\chi,\ell}(H^{\text{tot}},J,\bm{X};\bm{x},\bm{\gamma})$
is a smooth manifold of dimension

$\kappa n-|\bm{x}|-|\bm{\gamma}|-(n-2)(\ell-\chi).$

Moreover, given an orientation of $Q$ and a spin structure on the pull-back of
tangent bundle $TQ$ to $T^{*}Q$ there is an induced orientation
$\mathcal{T}^{\sigma_{0}}_{\chi,\ell}(H^{\text{tot}},J,\bm{X};\bm{x},\bm{\gamma})$.

###### Proof.

First, we show that $\mathcal{H}^{\sigma,\sigma_{0},\geq
0}_{\kappa,\chi}(H^{\text{tot}},J;\bm{x})$ is a smooth manifold along the
lines of the argument in Theorem 2.33 (compare with [Abo11, Section 3] for the
$\kappa=1$ case). For index calculation, we glue the operator $D_{u}$
associated with a given element of $\mathcal{H}^{\sigma,\sigma_{0},\geq
0}_{\kappa,\chi}(H^{\text{tot}},J;\bm{x})$ to the operator $D_{\bm{x}}$
associated with a tuple of Hamiltonian orbits $\bm{x}$, obtained by the direct
product of operators $D_{x^{c_{i}(\sigma)}}$ (see (2.118)) to obtain an
operator $D_{u}\\#D_{\bm{x}}$ from a surface with no punctures. Then

$\operatorname{ind}(D_{u}\\#D_{\bm{x}})=n(\chi+\ell(x))$

by [Sei08a, Proposition 11.13] and

$\operatorname{ind}(D_{\bm{x}})=n(\ell(\bm{x})-\kappa)-|\bm{x}|.$

Hence

$\operatorname{ind}(D_{u})=\kappa n-|\bm{x}|+n\chi$

and then one concludes as in Lemma 2.35 that the dimension of
$\mathcal{H}^{\sigma,\sigma_{0},\geq
0}_{\kappa,\chi}(H^{\text{tot}},J;\bm{x})$ is equal to

(4.5) $\kappa n-|\bm{x}|+(n-2)\chi.$

Given a spin structure on $TQ$ regarded as a subbundle of $TT^{*}Q$, following
[Sei08a, Proposition 11.13] and [Abo11, Appendix A], there are isomorphisms

(4.6) $\big{|}\mathcal{H}^{\sigma,\sigma_{0},\geq
0}_{\kappa,\chi}(H^{\text{tot}},J;\bm{x})\big{|}\cong
o_{\bm{x}}^{-1}\otimes\big{|}\mathcal{H}^{\sigma,\sigma_{0},\geq
0}_{\kappa,\chi}\big{|}\otimes|Q|^{\ell(\sigma_{0})}\otimes\zeta^{\kappa
n+n(\chi-\ell(\sigma_{0})}\cong\zeta^{(n-2)\chi}\otimes
o_{\bm{x}}^{-1}\otimes\zeta^{\kappa n}.$

For the later isomorphism, we use the fixed orientation on $Q$ to get
trivialization $|Q|\cong\zeta^{n}$ and the standard orientation on
$\mathcal{H}^{\sigma,\sigma_{0},\geq 0}_{\kappa,\chi}$ to get
$|\mathcal{H}^{\sigma,\sigma_{0},\geq 0}_{\kappa,\chi}|\cong\zeta^{-2\chi}$;
we note that in the second isomorphism, there is a Koszul sign involved that
we do not mention explicitly.

Next, we claim that for generic Floer data, the image of the space
$\mathcal{H}^{\sigma,\sigma_{0},\geq
0}_{\kappa,\chi}(H^{\text{tot}},J;\bm{x})$ under the evaluation map
$\mathcal{EV}$ is a smooth submanifold of $\Omega^{1}_{\sigma_{0}}(Q)$. It is
implied by [AS06, Theorem 3.8] and more generally by the discussion in [AS10,
Section 5]. Finally, one may conclude the proof by applying the argument of
Proposition 3.4 to show that for a generic choice of pseudo-gradient vector
fields $\bm{X}^{i}$ the piecewise pseudo-gradient trajectories connecting the
image of $\mathcal{H}^{\sigma,\sigma_{0},\geq
0}_{\kappa,\chi}(H^{\text{tot}},J;\bm{x})$ under $\mathcal{EV}$ with
$\bm{\gamma}$ constitute a smooth manifold.

As for the orientation on
$\mathcal{T}^{\sigma_{0}}_{\chi,\ell}(H^{\text{tot}},J,\bm{X};\bm{x},\bm{\gamma})$,
one may argue as in the proof of Proposition 3.4 that

(4.7)
$o_{\bm{\gamma}}\otimes\big{|}\mathcal{T}^{\sigma_{0}}_{\chi,\ell}(H^{\text{tot}},J,\bm{X};\bm{x},\bm{\gamma})\big{|}\cong\zeta^{(2-n)\ell}\otimes\big{|}\mathcal{H}^{\sigma,\sigma_{0},\geq
0}_{\kappa,\chi}(H^{\text{tot}},J;\bm{x})\big{|}.$

By combining (4.6) and (4.7) one gets an orientation on
$\mathcal{T}^{\sigma_{0}}_{\chi,\ell}(H^{\text{tot}},J,\bm{X};\bm{x},\bm{\gamma})$
induced by a choice of a generator of $o_{\bm{x}}$. ∎

######  Lemma 4.4.

For a generic choice of Floer data on $\mathcal{H}^{\sigma,\sigma_{0},\geq
0}_{\kappa,\chi}$, and a collection $\bm{X}$ of pseudo-gradient vector fields,
and an arbitrary tuple of Hamiltonian orbits
$\bm{x}\in\mathcal{P}_{\text{unsym}}(T^{*}Q)$ and $m\geq 0$, the space

$\underset{\ell-\chi=m,\,\sigma_{0}\in\mathfrak{S}_{\kappa}}{\bigsqcup}\mathcal{T}^{\sigma_{0}}_{\chi,\ell}(H^{\text{tot}},J,\bm{X};\bm{x},\bm{\gamma})$

is empty for all but finitely many
$\bm{\gamma}\in\mathcal{P}_{\kappa}^{\bm{L}}$.

###### Proof.

We omit the proof as it follows from the Palais-Smale condition 24, see [AS06]
for details. ∎

From now on we will omit the auxiliary data in the notation for the mixed
moduli spaces.

_Asside on Kuranishi replacements._ As in Section 2.9, for $n\geq 3$ a
sequence of elements in the mixed moduli space may converge to an pair with
Floer side admitting ghost bubbles. More precisely, a sequence of curves
$(u_{i},\Psi_{(i)})$, $i=1,2,\dots$, in
$\mathcal{T}^{\sigma_{0}}_{\chi,\ell}(H^{\text{tot}},J,\bm{X};\bm{x},\bm{\gamma})$,
after passing to a subsequence, can limit to

$(u_{\infty}=u_{\infty,1}\cup u_{\infty,2},\Psi_{(\infty)}),$

where:

  * (i)

the main part $u_{\infty,1}$ is the union of components that are not ghost
bubbles; each component of the $u_{\infty,1}$ is somewhere injective; and

  * (ii)

$u_{\infty,2}$ is a union of ghost bubbles, i.e., locally constant maps where
the domain is a possibly disconnected compact Riemann surface (possibly with
boundary) and $u_{\infty,2}$ maps to points on
$\operatorname{Im}(u_{\infty,1})$.

To remedy this we introduce we introduce Kuranishi replacements
$\mathcal{T}^{\sigma_{0},\sharp}_{\chi,\ell}(H^{\text{tot}},J,\bm{X};\bm{x},\bm{\gamma})$
satisfying similar requirements to 1-3 (see [HKTY, Section 3.2.2] for more
specifics). As before, we will pass to Kuranishi replacements whenever
neccessary and drop $\sharp$ from the notation.

Before diving into the discussion of the compactification of mixed moduli
spaces of dimension $1$ we introduce two new types of boundary behavior. Let
us fix tuples $\bm{x}\in\mathcal{P}_{\text{unsym}}(T^{*}Q)$ and
$\bm{\gamma}\in\mathcal{P}^{\bm{L}}_{\kappa}$ satisfying for some $\chi\leq
0,\ell\geq 0$:

$\kappa n-|\bm{x}|-1=|\bm{\gamma}|+(n-2)(\ell-\chi).$

######  Definition 4.5.

_The Type I singular mixed moduli space_
$\mathcal{T}^{\sigma_{0},\mathrm{I}}_{\chi,\ell}(\bm{x},\bm{\gamma})$ is a
space of tuples

$\left((S,\pi),u,\bm{\Psi}\right)$

as in (4.2) with the following differences:

  1. (1)

$(S,\pi)$ represents a point in
$\overline{\mathcal{H}}^{\sigma,\sigma_{0},\geq 0}_{\kappa,\chi}$, such that
(only) one of the branch points in $B$ has $s$-coordinate equal to $0$;

  2. (2)

$\bm{\Psi}=\left((\Psi_{1},\ldots,\Psi_{\ell}),\bm{s}=(s_{0},s_{1},\ldots,s_{\ell+2}),\bm{\sigma}=(\sigma_{1},\ldots,\sigma_{\ell}),\bm{\rho}=(\rho_{2},\ldots,\rho_{\ell+1})\right)$
is a tuple (note the different number of elements in each tuple) with
conditions that $s_{0}=s_{1}$ and $\sigma_{1}=\sigma_{0}$.

(a) Figure 12. Approaching a Type I singularity.

######  Remark 4.6.

Heuristically, a Type I boundary singularity is a result of a ramification
point approaching the boundary and adding a new switching point on the Morse
side at $s=0$ (see Figure 12). The number of switching points of the curves in
$\mathcal{T}^{\sigma_{0},\mathrm{I}}_{\chi,\ell}(\bm{x},\bm{\gamma})$ is equal
to $\ell+1$. We point out here that $s_{1}$ is not a genuine switching point
since there is no naturally assigned permutation $\rho_{1}$ with it and the
pseudo-gradient flows from $s_{1}=s_{0}$ to $s_{2}$ are prescribed by the old
$\sigma_{0}$.

######  Definition 4.7.

_The Type II singular mixed moduli space_
$\mathcal{T}^{\sigma_{0},\mathrm{II}}_{\chi,\ell}(\bm{x},\bm{\gamma})$ is the
space of tuples

$\left((S,\pi),u,\bm{\Psi}\right)$

as in (4.2) with the following differences:

  1. (1)

$(S,\pi\colon S\to\mathbb{R}_{\geq 0}\times S^{1})$ is a branched cover with a
set of branched points consisting of $|B|=-\chi+1$ points with (only) one of
these points having $s$-coordinate equal to $0$;

  2. (2)

$\bm{\Psi}=\left((\Psi_{1},\ldots,\Psi_{\ell}),\bm{s}=(s_{0},s_{1},\ldots,s_{\ell+1}),\bm{\sigma}=(\sigma_{0},\sigma_{1},\ldots,\sigma_{\ell}),\bm{\rho}=(\rho_{1},\ldots,\rho_{\ell})\right)$
is a tuple with conditions that $s_{0}=s_{1}$, and for markers
$a^{0}_{a_{0}}$, $a^{0}_{b_{0}}$ such that positive arcs connecting these
markers to the next (in counterclockwise order) markers on $\partial S$
contain the ramification point it holds that

$(\rho^{-1}_{1}\circ\sigma_{1}\circ\rho_{1})\circ\sigma_{0}^{-1}=(\sigma_{0}(a_{0})\sigma_{0}(b_{0})).$

(a) Figure 13. Approaching a Type II singularity.

######  Remark 4.8.

A Type II boundary singularity is a result of $s_{1}$ approaching $s_{0}$
accompanied by the creation of a new ramification point on $S$ (as well as a
branch point on the target; see Figure 13 for a schematic picture). The Euler
characteristic of curves in
$\mathcal{T}^{\sigma_{0},\mathrm{II}}_{\chi,\ell}(\bm{x},\bm{\gamma})$ is
equal to $\chi-1$.

######  Remark 4.9.

Following discussion of Remarks 4.6 and 4.8, we note that given a curve in
$\mathcal{T}^{\sigma_{0},\mathrm{I}}_{\chi,\ell}(\bm{x},\bm{\gamma})$ we may
obtain a curve in
$\mathcal{T}^{\sigma_{0}^{\prime},\mathrm{II}}_{\chi+1,\ell+1}(\bm{x},\bm{\gamma})$
by arbitrary adding $\sigma_{0}^{\prime}$ and $\rho_{1}$ to the data in
Definition 4.5 satisfying condition (2) in Definition 4.7. Hence one
geometrical curve of
$\mathcal{T}^{\sigma_{0},\mathrm{I}}_{\chi,\ell}(\bm{x},\bm{\gamma})$
corresponds to $\kappa!$ curves of

$\bigsqcup_{\sigma_{0}^{\prime}\in\mathfrak{S_{\kappa}}}\mathcal{T}^{\sigma_{0}^{\prime},\mathrm{II}}_{\chi+1,\ell+1}(\bm{x},\bm{\gamma}).$

This phenomenon is crucial for the map defined via counting mixed curves to be
a chain map.

######  Lemma 4.10.

For a generic choice of consistent Floer data on
$\widehat{\mathscr{H}}^{\sigma,\sigma_{0},\geq 0}_{\kappa,\chi}$ and a
collection of pseudo-gradient vector fields $\bm{X}$, a tuple of Hamiltonian
orbits $\bm{x}\in\mathcal{P}_{\text{unsym}}(T^{*}Q)$ and a multiloop
$\bm{\gamma}\in\mathcal{P}_{\kappa}^{\bm{L}}$ satisfying

(4.8) $\kappa n-|\bm{x}|-|\bm{\gamma}|-(n-2)(\ell-\chi)=1,$

the wnb groupoid
$\mathscr{T}^{\sigma_{0}}_{\chi,\ell}(H^{\text{tot}},J,\bm{X};\bm{x},\bm{\gamma})(U)$
admits a compactification
$\overline{\mathscr{T}}^{\sigma_{0}}_{\chi,\ell}(H^{\text{tot}},J,\bm{X};\bm{x},\bm{\gamma})$.

###### Sketch of the proof.

As in the proof of 2.38 one may show that there is a portion of the boundary
corresponding to the vertical stretching, and we leave it to the reader to
fill in the details. Points of this type on the boundary of
$\overline{\mathscr{T}}^{\sigma_{0}}_{\chi,\ell}(H^{\text{tot}},J,\bm{X};\bm{x},\bm{\gamma})$
correspond to elements of

(4.9) $\bigsqcup_{\begin{subarray}{c}\bm{x}^{\prime}\in
SC^{*}_{\kappa}(T^{*}Q),\\\
\chi_{1}+\chi_{2}=\chi\end{subarray}}\mathscr{M}^{\chi_{1},\text{ind}=1}(\bm{x},\bm{x}^{\prime})\times\mathcal{T}^{\sigma_{0},\text{ind}=0}_{\chi_{2},\ell}(\bm{x}^{\prime},\bm{\gamma})$

counted with weights $\frac{1}{\kappa!}$.

The second type of boundary degeneration corresponds to stretching on the
Morse side and is treated in Lemma 3.9. These points of the boundary are
elements of

(4.10) $\bigsqcup_{\begin{subarray}{c}\bm{\gamma}^{\prime}\in CM_{\kappa
n-*}(\Lambda^{1}_{\kappa}(Q)),\\\
\ell_{1}+\ell_{2}=\ell\end{subarray}}\mathcal{T}^{\sigma_{0},\text{ind}=0}_{\chi,\ell_{1}}(\bm{x},\bm{\gamma}^{\prime})\times\mathcal{P}^{\text{ind}=1}(\bm{\gamma}^{\prime},\bm{\gamma};\ell_{2}).$

One last case that we have to analyze is the boundary nodal degeneration with
a possible occurrence of almost ghost bubbles attached along the boundary. Let
us first treat this scenario. Namely, assume that $(u_{i},\Psi_{i})$ limits to
$(u_{\infty}=u_{\infty,1}\cup u_{\infty,2},\Psi_{\infty})$, where
$u_{\infty,2}$ is null-homologous (see 2) has domain attached to the domain of
$u_{\infty,1}$ along $k$ boundary nodes. We then have

$\operatorname{ind}u_{\infty}=\operatorname{ind}u_{i}+k(n-2)$

by (4.5). On the other hand, the appearance of a boundary node along the
Lagrangian $\\{0\\}\times S^{1}\times Q$ is a codimension $(n+1)-2\cdot 1=n-1$
condition since the Lagrangian is $(n+1)$-dimensional and the boundary of the
domain is mapped to a real $1$-dimensional curve. In total, we get a dimension
drop of

$k(n-1)-k(n-2)=k.$

Hence, $u_{\infty,2}$ can be attached only along the boundary node $1$ and it
must have only $1$ boundary component. This component can not be a disk, since
there have to be at least two ramification points on it. Therefore,
$\operatorname{ind}(u_{\infty,2})<0$ and it violates 3 and 2.

Finally, from the above analysis, it follows that there could be no more than
$1$ boundary node. If no ghost bubbles appear, this leaves us with the two
singular types of boundary points
$\mathcal{T}^{\sigma_{0},\mathrm{I}}_{\chi,\ell}(\bm{x},\bm{\gamma})$ and
$\mathcal{T}^{\sigma_{0},\mathrm{II}}_{\chi,\ell}(\bm{x},\bm{\gamma})$. ∎

###  4.4. Chain map

To define a morphism $\mathcal{F}$

(4.11) $\mathcal{F}\colon SC^{*}_{\kappa,\text{unsym}}(T^{*}Q)\to CM_{\kappa
n-*}\bigl{(}\Lambda^{1}_{\kappa}(Q)\bigr{)},\\\ $

we first consider the canonical isomorphism

(4.12) $o_{\bm{\gamma}}[(2-n)(\ell-\chi)]\otimes
o_{\bm{x}}\cong\big{|}\mathcal{T}^{\sigma_{0}}_{\chi,\ell}(\bm{x},\bm{\gamma})\big{|}^{-1}\otimes\zeta^{\kappa
n}.$

When the dimension of the mixed moduli space is $0$ the above provides the
isomorphism

(4.13)
$o_{\bm{x}}\cong\bigl{(}o_{\bm{\gamma}}[(2-n)(\ell-\chi)]\bigr{)}^{-1}\otimes\zeta^{\kappa
n},$

which induces a morphism

(4.14) $f_{[u,\bm{\Psi}]}\colon
o_{\bm{x}}\rightarrow\bigl{(}o_{\bm{\gamma}}[(2-n)(\ell-\chi)]\bigr{)}^{-1}[-\kappa
n].$

We extend it to $o_{\bm{x}}[(n-2)j]$ for $j>0$ by simply multiplying the above
with $\zeta^{(2-n)j}$ on the left. We first define $\mathcal{F}^{-}\colon
SC^{*}_{\kappa}(T^{*}Q)\to
CM_{-}^{*}\bigl{(}\Lambda^{1}_{\kappa}(Q)\bigr{)}[-\kappa n]$ as above to be

(4.15)
$\mathcal{F}^{-}([\bm{x}]\hbar^{j})=\sum_{\begin{subarray}{c}|\bm{\gamma}|=\kappa
n-|\bm{x}|-(n-2)(\ell-\chi),\\\ \ell\geq 0,\chi\leq
0,\,\sigma_{0}\in\mathfrak{S}_{\kappa},\\\
(u,\bm{\Psi})\in\mathcal{T}^{\sigma_{0}}_{\chi,\ell}(\bm{x},\bm{\gamma})\end{subarray}}(-1)^{|\bm{x}|+(n-2)j}\cdot\frac{f_{(u,\bm{\Psi})}([x]\hbar^{j})}{(\kappa!)^{\ell}}.$

The morphism $\mathcal{F}$ is then obtained by composing $\mathcal{F}^{-}$
with a $\kappa n$-shift of $CM_{-}^{*-\kappa
n}\bigl{(}\Lambda^{1}_{\kappa}(Q)\bigr{)}[-\kappa n]$ and the reflection chain
map $\mathcal{I}\colon CM_{-}^{*}\bigl{(}\Lambda^{1}_{\kappa}(Q)\bigr{)}\to
CM_{-*}\bigl{(}\Lambda^{1}_{\kappa}(Q)\bigr{)}$; see (3.39).

By Lemma 4.10, $\mathcal{T}^{\sigma_{0}}_{\chi,\ell}(\bm{x},\bm{\gamma})$ is
finite, hence the maps $\mathcal{F}^{-}$ and $\mathcal{F}$ are well-defined.

######  Theorem 4.11.

The map $\mathcal{F}^{-}$ is a chain map.

###### Proof.

Figure 14. The neighborhood of a nodal degeneration. Each point on the thick
lines $A$ and $B$ denotes the 1-dimensional family of image of the evaluation
map $\mathcal{EV}(u)$ for some $u$ defined above. The intersection of $A$ and
$B$ represents $\mathcal{EV}(u_{*})$ where $u_{*}$ is a nodal curve. The blue
arcs denote pseudo-gradient trajectories. The green arc denotes
$\Delta^{\text{sm}}_{a_{0}b_{0}}\Omega^{1}_{\sigma_{0}}(Q)$. The entire
quadrilateral denotes the image of the gradient trajectories starting from
$A$.

We analyze the codimension 1 boundary of
$\mathcal{T}^{\sigma_{0}}_{\chi,\ell}(\bm{x},\bm{\gamma})$. A special
codimension 1 phenomenon is the nodal degeneration along the zero section $Q$
which corresponds to a point $\left((S,\pi_{*}),u_{*},\bm{\Psi}_{*}\right)$ of
the Type II singular space
$\mathcal{T}^{\sigma_{0},\mathrm{II}}_{\chi+1,\ell+1}(\bm{x},\bm{\gamma})$.
This point can also be regarded (by dropping $\sigma_{0}$ and $\rho_{1}$ from
the $\bm{\Psi}$ data) as a point of the Type I singular mixed moduli space
$\mathcal{T}^{\sigma_{1},\mathrm{I}}_{\chi,\ell}(\bm{x},\bm{\gamma})$. Here
$\sigma_{1}$ is a permutation from $\bm{\sigma}$-data of $\bm{\Psi}_{*}$. We
give a sketch of the proof that there is a $1$-dimensional family of points in
$\mathcal{T}^{\sigma_{1}}_{\chi,\ell}(\bm{x},\bm{\gamma})$ converging to such
a point.

For $\left((S,\pi_{*}),u_{*},\bm{\Psi}\right)$ as above, we assume that the
multiloop $\mathcal{EV}\left(u_{*}\right)\in\Omega^{1}_{\sigma_{0}}(Q)$ is
sufficiently generic. Let
$R\left(\mathcal{EV}\left(u_{*}\right)\right)\in\Omega^{1}_{\sigma_{1}}(Q)$ be
the switching of the multiloop $\mathcal{EV}\left(u_{*}\right)$. Refer to
Figure 3.2, which gives a schematic description of a neighborhood of
$\mathcal{EV}\left(u_{*}\right)\in\Lambda^{1}_{\kappa}(Q)$; in the figure
$\mathcal{EV}\left(u_{*}\right)$ is identified with
$R\left(\mathcal{EV}\left(u_{*}\right)\right)$.

Let

  1. (i)

$B$ be the set of
$\mathcal{EV}\left(u_{\mathrm{I}}\right),u_{\mathrm{I}}\in\mathcal{H}^{\sigma,\sigma_{1},\geq
0}_{\kappa,\chi}(\bm{x})$, such that there is a piecewise smooth pseudo-
gradient trajectory $\bm{\Psi}$ from $\mathcal{EV}\left(u_{\mathrm{I}}\right)$
to $\bm{\gamma}$;

  2. (ii)

$A$ be the set of
$\mathcal{EV}\left(u_{\mathrm{II}}\right),u_{\mathrm{II}}\in\mathcal{H}^{\sigma,\sigma_{0},\geq
0}_{\kappa,\chi+1}(\bm{x})$; and

  3. (iii)

$A^{\prime}\subset A$ be the subset consisting of
$\mathcal{EV}\left(u_{\mathrm{II}}\right),u_{\mathrm{II}}\in\mathcal{H}^{\sigma,\sigma_{0},\geq
0}_{\kappa,\chi+1}(\bm{x})$, for which there is a piecewise smooth pseudo-
gradient trajectory $\bm{\Psi}$ from
$\mathcal{EV}\left(u_{\mathrm{II}}\right)$ to $\bm{\gamma}$.

We have $u_{*}\in\mathcal{H}^{\sigma,\sigma_{0},\geq
0}_{\kappa,\chi+1}(\bm{x}),\mathcal{EV}\left(u_{*}\right)\in A$, and
$R\left(\mathcal{EV}\left(u_{*}\right)\right)\in\partial B$.

_Step 1 (Description of $\mathcal{H}^{\sigma,\sigma_{0},\geq
0}_{\kappa,\chi+1}$ and $\mathcal{H}^{\sigma,\sigma_{1},\geq
0}_{\kappa,\chi}(\bm{x})$ near $u_{*}$)_ If
$\operatorname{dim}\mathcal{H}^{\sigma,\sigma_{1},\geq
0}_{\kappa,\chi}(\bm{x})=b+1$, then

$\operatorname{dim}\mathcal{H}^{\sigma,\sigma_{0},\geq
0}_{\kappa,\chi+1}(\bm{x})=b+1+(n-2),$

by (4.5).

The following theorem generalizes the gluing result of Ekholm and Shende
[ES19, Section 4.4.4]. Let $U\left(u_{*}\right)$ be a neighborhood of $u_{*}$
in $\mathcal{H}^{\sigma,\sigma_{0},\geq 0}_{\kappa,\chi+1}(\bm{x})$ and let

$Z=\\{u_{\mathrm{II}}\in\mathcal{H}^{\sigma,\sigma_{0},\geq
0}_{\kappa,\chi+1}(\bm{x})\mid u_{\mathrm{II}}\text{ has a node along
}s=0\\}.$

######  Theorem 4.12.

There exists a neighborhood of $Z\cap U(u_{*})$ in
$\mathcal{H}^{\sigma,\sigma_{0},\geq 0}_{\kappa,\chi+1}(\bm{x})$ of the form
$D^{b}\times D^{n-1}$, such that $Z\cap U(u_{*})=D^{b}\times\\{0\\}$.
Moreover, $Z\cap U(u_{*})$ can be viewed as an open subset of
$\partial\overline{\mathcal{H}}^{\sigma,\sigma_{1},\geq
0}_{\kappa,\chi}(\bm{x})$.

###### Sketch of proof of Theorem 4.12.

We convert the desired perturbations of a nodal curve into a smooth one into a
commonly used SFT-type gluing result (see, for example, [FOOO] which treats
holomorphic curves before/after Lagrangian surgery). “Zooming in” near the
intersection point, we consider the following local model:

  1. (1)

the ambient symplectic/almost complex manifold is $\mathbb{C}^{n}$ with the
standard symplectic form,

  2. (2)

the Lagrangian boundary condition is the standard
$\mathbb{R}^{n}\subset\mathbb{C}^{n}$, and

  3. (3)

the punctures of the finite energy holomorphic curves under consideration
limit to chords (in a Morse-Bott family) from $S^{n-1}$ to itself, where
$S^{n-1}\subset S^{2n-1}$ is the Legendrian boundary at infinity of
$\mathbb{R}^{n}$.

We consider pairs of half-planes of the form

$\begin{gathered}P_{1}\left(z_{3},\ldots,z_{n-1}\right)\coloneqq\left\\{\left(z_{1},\ldots,z_{n-1}\right)\in\mathbb{C}^{n-1}\mid
z_{2}=z_{1},z_{1}\in\mathbb{H},z_{3},\ldots,z_{n-1}\in\mathbb{R}\right\\},\\\
P_{2}\coloneqq\left\\{\left(z_{1},\ldots,z_{n-1}\right)\in\mathbb{C}^{n-1}\mid
z_{2}=-z_{1},z_{1}\in\mathbb{H},z_{3}=\cdots=z_{n-1}=0\right\\}.\end{gathered}$

For the pair $P_{1}(0,\ldots,0)$ and $P_{2}$ we consider gluing along the node
using model

$C_{c}\coloneqq\left\\{\left(z_{1},z_{2},z_{3},\ldots,z_{n-1}\right)\in\mathbb{C}^{n-1}\mid
z_{1}\in\mathbb{H},z_{2}^{2}-z_{1}^{2}=c,\,z_{3}=\ldots=z_{n-1}=0\right\\}_{c\in\mathbb{R}_{\geq
0}}.$

Writing

$i_{\dot{F}_{z_{3},\ldots,z_{n-1};c}}\colon\dot{F}_{z_{3},\ldots,z_{n-1};c}\to\mathbb{C}^{n-1}$

for the holomorphic embedding with image $(P_{1}(z_{3},\ldots,z_{n-1}),P_{2})$
whenever one of $z_{i}$’s is not equal to $0$ and with the image $C_{c}$
otherwise. We also take a double cover

$\pi_{z_{3},\ldots,z_{n-1};c}\colon\dot{F}_{z_{3},\ldots,z_{n-1};c}\rightarrow\mathbb{C}$

with image $\mathbb{H}$. The map

$\left(i_{\dot{F}_{z_{3},\ldots,z_{n-1};c}},\pi_{z_{3},\ldots,z_{n-1};c}\right)\colon\dot{F}_{z_{3},\ldots,z_{n-1};c}\rightarrow\mathbb{C}^{n-1}\times\mathbb{C}$

is the holomorphic map to $\mathbb{C}^{n}$ with boundary on $\mathbb{R}^{n}$.

Half-planes are Morse-Bott regular, since they can be doubled into standard
complex lines in $\mathbb{C}^{n-1}$ and a curve is regular if and only if its
double is; the same also holds for $C_{c}$. Finally, we leave it to the reader
to verify the necessary SFT-type gluing result for finite energy holomorphic
curves with Lagrangian boundary in this Morse-Bott case. ∎

_Step 2 (Descriptions of $A$ and $B$)_ Now suppose that having a gradient
trajectory that goes to $\bm{\gamma}$ is a codimension $b$ condition for some
$b\geq 0$. Recalling the definitions of $A$ and $B$ from earlier in the proof,
$\operatorname{dim}B=1$. Theorem 4.12 and the genericity of the evaluation map
$\mathcal{EV}$ give:

######  Claim 4.13.

There exists an identification
$A\cap\mathcal{EV}\left(U\left(u_{*}\right)\right)\simeq D^{b}\times D^{n-1}$
with respect to which
$\mathcal{EV}\left(u_{*}\right)=(0,0),\,\mathcal{EV}\left(Z\cap
U\left(u_{*}\right)\right)=D^{b}\times\\{0\\}$, and
$R\left(\mathcal{EV}\left(u_{*}\right)\right)$ is the unique point on
$\partial B$ on a neighborhood of
$R\left(\mathcal{EV}\left(u_{*}\right)\right)$.

_Step 3 (Description of $A^{\prime}$)_

######  Claim 4.14.

Under the identification from Claim 3.5,
$A^{\prime}\cap\mathcal{EV}\left(U\left(u_{*}\right)\right)=\\{0\\}\times r$,
where $r$ is a radial ray in $D^{n-1}$ emanating from $0$.

###### Proof of Claim 4.14.

By transversality for mixed moduli, we know that $A^{\prime}$ is a
$1$-dimensional manifold (see Lemma 4.3). By definition, for any point of
$A^{\prime}$ finite time flow of $\bm{X}^{0}$ hits a point of
$\Delta^{\text{sm}}_{a_{0}b_{0}}\Omega^{1}_{\sigma_{0}}(Q)$ and
$\mathcal{EV}(u_{*})\in\Delta^{\text{sm}}_{a_{0}b_{0}}\Omega^{1}_{\sigma_{0}}(Q)$.
Since $\bm{X}^{0}$ is transversal to
$\Delta^{\text{sm}}_{a_{0}b_{0}}\Omega^{1}_{\sigma_{0}}(Q)$ along
$W^{u}(A^{\prime};\bm{X}^{0})\cap\Delta^{\text{sm}}_{a_{0}b_{0}}\Omega^{1}_{\sigma_{0}}(Q)$
the point $u_{*}$ has to be a boundary point of $A^{\prime}$. This implies the
claim. ∎

This finishes the proof of the fact that any point of
$\mathcal{T}^{\sigma_{0},\mathrm{II}}_{\chi+1,\ell+1}(\bm{x},\bm{\gamma})$ is
paired by continuation with a specific point of
$\mathcal{T}^{\sigma_{1},\mathrm{I}}_{\chi,\ell}(\bm{x},\bm{\gamma})$.

Conversely, one may show in a similar way that for a $1$-dimensional family
converging to a point of
$\mathcal{T}^{\sigma_{0},\mathrm{I}}_{\chi,\ell}(\bm{x},\bm{\gamma})$ and any
$\rho,\sigma_{0}^{\prime}\in\mathfrak{S}_{\kappa}$ as in Remark 4.9 there is a
continuation via a family of curves in
$\mathcal{T}^{\sigma_{0}^{\prime}}_{\chi+1,\ell+1}(\bm{x},\bm{\gamma}).$

Together, the above two statements imply that counting points of singular type
I moduli spaces
$\mathcal{T}^{\sigma_{0},\mathrm{I}}_{\chi,\ell}(\bm{x},\bm{\gamma})$ with
weights $\kappa!$ and the signs induced by boundary orientation of
$\mathcal{T}^{\sigma_{0}}_{\chi,\ell}(\bm{x},\bm{\gamma})$ gives the opposite
of the sum of points of
$\mathcal{T}^{\sigma_{0},\mathrm{II}}_{\chi+1,\ell+1}(\bm{x},\bm{\gamma})$
with signs. That is

(4.16)
$\kappa!\cdot\sum_{\begin{subarray}{c}\sigma_{0}\in\mathfrak{S}_{\kappa},\\\
(u_{\mathrm{I}},\bm{\Psi}_{\mathrm{I}})\in\mathcal{T}^{\sigma_{0},\mathrm{I}}_{\chi,\ell}(\bm{x},\bm{\gamma})\end{subarray}}\varepsilon^{\mathrm{I}}_{(u_{\mathrm{I}},\bm{\Psi}_{\mathrm{I}})}+\sum_{\begin{subarray}{c}\sigma_{0}\in\mathfrak{S}_{\kappa},\\\
(u_{\mathrm{II}},\bm{\Psi}_{\mathrm{II}})\in\mathcal{T}^{\sigma_{0},\mathrm{II}}_{\chi+1,\ell+1}(\bm{x},\bm{\gamma})\end{subarray}}\varepsilon^{\mathrm{II}}_{(u_{\mathrm{II}},\bm{\Psi}_{\mathrm{II}})}=0.$

According to Lemma 4.10, the remaining codimension-$1$ degenerations
correspond to stretching along the $s$-direction, and correspond to composing
$\mathcal{F}^{-}$ with differentials of the Heegaard Floer symplectic cochain
complex and of the Morse multiloop complex, see (4.9) and (4.10).

Now, by counting elements of the boundaries of the spaces
$\mathcal{T}^{\sigma_{0}}_{\chi,\ell}(\bm{x},\bm{\gamma})$ for all
$\ell,-\chi\geq 0$ and $\sigma_{0}\in\mathfrak{S}_{\kappa}$ with signs induced
by boundary orientations and respective weights $\frac{1}{(\kappa!)^{\ell}}$
we obtain

(4.17) $(-1)^{\kappa n}\partial\circ\mathcal{F}^{-}-(-1)^{\kappa
n}\mathcal{F}^{-}\circ d=0,$

where vanishing of terms corresponding to points of singular moduli spaces
follows from (4.16). It only remains to show that boundary orientation of
$\overline{\mathcal{T}}^{\sigma_{0}}_{\chi,\ell}(\bm{x},\bm{\gamma})$ indeed
provides the sign as stated above.

Let us consider a pair

$(u,(v,\bm{\Psi}))\in\mathscr{M}^{\chi_{1}}(\bm{x},\bm{x}^{\prime})\times\mathcal{T}^{\sigma_{0}}_{\chi_{2},\ell}(\bm{x}^{\prime},\bm{\gamma})\subset\partial\overline{\mathcal{T}}^{\sigma_{0}}_{\chi,\ell}(\bm{x},\bm{\gamma})\text{
with }$ (4.18) $|\bm{x}^{\prime}|-|\bm{x}|+(n-2)\chi_{1}=1,\,\kappa
n-|\bm{x}^{\prime}|-|\bm{\gamma}|-(n-2)(\ell-\chi_{2})=0\text{ and
}\chi_{1}+\chi_{2}=\chi,$

and compare the map $f_{(v,\bm{\Psi})}\circ d_{u}$ with the map
$o_{\bm{x}}\to(o_{\bm{\gamma}}[(2-n)(\ell-\chi)])^{-1}[-\kappa n]$ induced by
boundary orientation at this pair. We notice that under the condition $\kappa
n-|\bm{x}|-|\bm{\gamma}|-(n-2)(\ell-\chi)=1$ the isomorphism (4.12) is
equivalent to

(4.19)
$(-1)^{|\bm{\gamma}|+(n-2)(\ell-\chi)+1}\big{|}\mathcal{T}^{\sigma_{0}}_{\chi,\ell}(\bm{x},\bm{\gamma})\big{|}\otimes
o_{\bm{x}}\cong\bigl{(}o_{\bm{\gamma}}[(2-n)(\ell-\chi)]\bigr{)}^{-1}\otimes\zeta^{\kappa
n}.$

Recalling (2.85) we note that composition $f_{(v,\bm{\Psi})}\circ d_{u}$ is
given by isomorphisms

(4.20) $|\mathbb{R}\partial_{s}u|\otimes o_{\bm{x}}\cong
o_{\bm{x}^{\prime}}[(2-n)\chi_{1}]\cong\bigl{(}o_{\bm{\gamma}}[(2-n)(\ell-\chi)]\bigr{)}\otimes\zeta^{\kappa
n}.$

Clearly, $\partial_{s}u$ is an outward pointing vector on
$\partial\overline{\mathcal{T}}^{\sigma_{0}}_{\chi,\ell}(\bm{x},\bm{\gamma})$,
which implies that
$|\mathbb{R}\partial_{s}u|\cong\big{|}\mathcal{T}^{\sigma_{0}}_{\chi,\ell}(\bm{x},\bm{\gamma})\big{|}$.
We also point out that

(4.21) $|\bm{\gamma}|+(n-2)(\ell-\chi)+1=\kappa
n+1-|\bm{x}^{\prime}|-(n-2)\chi_{1},$

which follows from (4.18). Now recalling the sign in front of
$f_{(v,\bm{\Psi})}$ in (4.15) we conclude that the map induced by (4.19) is
equal to the restriction of $-(-1)^{\kappa n}\mathcal{F}^{-}\circ d$.

As for the other type of boundary contributions we consider

$((u,\bm{\Psi}),[\bm{\Gamma}])\in\mathcal{T}^{\sigma_{0}}_{\chi,\ell_{1}}(\bm{x},\bm{\gamma}^{\prime})\times\mathcal{P}(\bm{\gamma}^{\prime},\bm{\gamma};\ell_{2})\text{
with}$ (4.22) $\kappa
n-|\bm{x}|-|\bm{\gamma}^{\prime}|-(n-2)(\ell_{1}-\chi_{2})=0,\,|\bm{\gamma}^{\prime}|-|\bm{\gamma}|-(n-2)\ell_{2}=1,\text{
and }\ell_{1}+\ell_{2}=\ell.$

The composition $\partial_{[\bm{\Gamma}]}\circ f_{(u,\Psi)}$ is then given by
the composition

(4.23) $(-|\mathbb{R}\partial_{s}\bm{\Gamma}|)\otimes
o_{\bm{x}}\cong(-|\partial_{s}\bm{\Gamma}|)\otimes\big{(}o_{\bm{\gamma}^{\prime}}[(2-n)(\ell_{1}-\chi)]\big{)}^{-1}[-\kappa
n]\cong\big{(}o_{\bm{\gamma}^{\prime}}[(2-n)(\ell-\chi)]\big{)}^{-1}\otimes[-\kappa
n]$

The vector $-\partial_{s}\bm{\Gamma}$ is outward pointing, hence
$-|\mathbb{R}\partial_{s}\bm{\Gamma}|\cong\big{|}\mathcal{T}^{\sigma_{0}}_{\chi,\ell}(\bm{x},\bm{\gamma})\big{|}$,
which means that $\partial_{[\bm{\Gamma}]}\circ f_{(u,\Psi)}$ differs from the
map induced by boundary orientation only by a term
$(-1)^{|\bm{\gamma}|+(n-2)(\ell-\chi)+1}$. Now, noting that
$|\bm{\gamma}|+(n-2)(\ell-\chi)+1=\kappa n-|\bm{x}|$, we conclude that
boundary induced map coincides with $(-1)^{\kappa
n}\partial\circ\mathcal{F}^{-}$. ∎

######  Remark 4.15.

We point out that our proof of Theorem 4.11 is based on the proof of the
analogous statement for HDHF that is worked out in [HKTY].

###  4.5. The isomorphism

For $\kappa=1$, the complex $SC_{\kappa,\text{unsym}}^{*}(T^{*}Q)$ coincides
with the symplectic chain complex $SC^{*}(T^{*}Q)$ computing the symplectic
cohomology of $T^{*}Q$, whereas the complex $CM^{*}_{\kappa}(\Lambda Q)$
coincides with the Morse chain complex $CM^{*}(\Lambda)$. Originally due to
Viterbo [Vit95, Vit99], the homology groups of these two chain complexes are
known to be isomorphic. In [Abo11] Abouzaid gives a refined proof of this fact
accounting for the twisting of the symplectic homology depending on the second
Stiefel-Whitney class of $Q$. We claim that the map constructed there (see
[Abo11, Section 3.2]) coincides with $\mathcal{F}$ on the level of homology.
First, we notice that the morphism $\mathcal{V}$ in [Abo11] on the chain level
has as its target _generic_ (in Morse-theoretic sense) singular chains
$C_{*}(\Lambda Q)$ of the loop space. We claim that $\mathcal{F}$ is chain
homotopic to a composition of $\mathcal{V}$ with a map $\mathcal{A}_{L}$ from
generic singular chains to Morse complex given by counting rigid flows from
singular chains to critical points of the appropriate Morse function (see,
e.g. [HL99, Section 2]; we also refer the reader to a similar discussion in
the case of based loop space in [Abo12, Section 5]). Therefore, we conclude:

######  Theorem 4.16.

For $\kappa=1$, $\mathcal{F}$ induces a homology isomorphism.

######  Remark 4.17.

In the context of linear Hamiltonians the map constructed in [Abo15, Section
4] is heuristically the same map that we construct for quadratic at infinity
Hamiltonians. One may argue that the limit of the maps given there is equal to
$\mathcal{F}$.

We also point out that the morphism $\mathcal{EV}$ is proved to be an
isomorphism in [Abo11] by showing that it is an inverse of the map constructed
in [AS06] and is known to be an isomorphism. It would be of its own interest
to construct an analogue of this morphism for HFSH.

######  Lemma 4.18.

The restriction of $\mathcal{F}$ to $\hbar=0$ is an isomorphism.

###### Proof.

The proof is similar to that of [HTY ̵a, Lemma 6.5]. We only sketch the
difference here. Recall that with each tuple
$\bm{x}\in\mathcal{P}_{\text{unsym}}(T^{*}Q)$ (resp. $\bm{\gamma}\in
CM^{*}_{\kappa}(\Lambda Q)$) there is an associated permutation
$\sigma(\bm{x})\in\mathfrak{S}_{\kappa}$ (resp.
$\sigma(\bm{\gamma})\in\mathfrak{S}_{\kappa}$). By definition the restriction
of $SC^{*}_{\kappa,\text{unsym}}(T^{*}Q)$ to $\hbar=0$ is split according to
permutation type:

$(SC^{*}_{\kappa}(T^{*}Q)|_{\hbar=0},d^{0})=\bigoplus_{\sigma\in
S_{\kappa}}(SC^{*}_{\kappa}(T^{*}Q)|_{\hbar=0,\sigma},d^{0}),$

where $SC^{*}_{\kappa}(T^{*}Q)|_{\hbar=0,\sigma}$ is generated by tuples of
Hamiltonian orbits with associated permutation equal to $\sigma$. Similarly,

$(CM_{*}^{\kappa}(\Lambda Q)|_{\hbar=0},\partial_{0})=\bigoplus_{\sigma\in
S_{\kappa}}(CM_{*}^{\kappa}(\Lambda Q)|_{\hbar=0,\sigma},\partial_{0}).$

Note that $\mathcal{F}|_{\hbar=0}$ also respects the splitting according to
permutation type. It suffices to show

(4.24) $\mathcal{F}|_{\hbar=0}\colon
SC^{*}_{\kappa}(T^{*}Q)|_{\hbar=0,\sigma}\to CM^{\kappa}_{\kappa n-*}(\Lambda
Q)|_{\hbar=0,\sigma}$

is an isomorphism. Note that for any generator $[\bm{x}]\in
SC^{*}_{\kappa}(T^{*}Q)|_{\hbar=0,\sigma}$ with $[\bm{x}]\in
o_{\bm{x}}^{\mathbb{Q}}$ for some
$\bm{x}\in\mathcal{P}_{\text{unsym}}(T^{*}Q)$ the count
$\mathcal{F}|_{\hbar=0}(\bm{x})$ has non-zero entries only for elements in
$o_{\bm{\gamma}}^{\mathbb{Q}}$ with $\bm{\gamma}$ such that
$\sigma(\bm{\gamma})=\sigma(\bm{x})$. We denote by
$\gamma^{c_{1}(\sigma)},\ldots,\gamma^{c_{\ell(\sigma)}(\sigma)}$ the loops
constituting $\bm{\gamma}$. Then we have the following.

$\mathcal{T}^{\text{Id}}_{0,0}(\bm{x},\bm{\gamma})\simeq\prod_{i=1}^{i=\ell(\sigma)}\mathcal{T}_{0,0}(x^{c_{i}(\sigma)},\gamma^{c_{i}(\sigma)}).$

It then suffices to show (4.24) holds when $\sigma$ is a cycle, which is given
by Theorem 4.16. ∎

######  Lemma 4.19.

Given cochain complex $(C^{*},d^{0})$ and a chain complex
$(D_{*},\partial_{0})$. Let $F$ be a degree $m$ chain map

$F\colon(\widetilde{C}^{*},d)\to(\widetilde{D}_{*},\partial)$

between associated complexes as in Construction 2.39 given by

$F=F_{0}+F_{1}\hbar+F_{2}\hbar^{2}+\ldots.$

If $F_{0}\colon(C^{*},d^{0})\to(D_{*},\partial_{0})$ is a quasi-isomorphism,
then $F$ is a quasi-isomorphism as well.

###### Proof.

For simplicity, we replace $(D_{*},\partial_{0})$ with the associated cochain
complex $(D_{s}^{*},\partial^{0})$ obtained by first “reversing arrows” and
replacing the degree with its negative and then shifting the degree by $m$.
Similarly, we obtain $(\widetilde{D}_{s}^{*},\partial)$. Then
$F_{s}\colon(\widetilde{C}^{*},d)\to(\widetilde{D}_{s}^{*},\partial)$ is a
chain map of degree $0$.

Now, we notice that

$Cone(F_{s})|_{\hbar=0}=Cone(F_{0}).$

We recall that a chain map is a quasi-isomorphism if and only if its cone is
acyclic. According to the conditions of the lemma, $Cone(F_{0})$ is acyclic,
and hence $Cone(F_{s})|_{\hbar=0}$ is acyclic. It remains to show that it
implies that $Cone(F_{s})$ is acyclic. From now on, denote this cochain
complex via $(E^{*},d_{E})$. Let

$v=v_{0}+v_{1}\hbar+v_{2}\hbar^{2}+\ldots$

be a cocycle of degree $k$ in $E^{k}$. Here $|v_{i}|=k+i(n-2)$. Let us show
that there is $w\in E^{k-1}$ such that $d_{E}w=v$. First, since
$(E^{*}|_{\hbar=0},d_{E}^{0})$ is a quasi-isomorphism, there is $w_{0}\in
E^{k-1}|_{\hbar=0}$ such that

(4.25) $d_{E}^{0}w_{0}=v_{0}.$

Then we may write

$v-d_{E}w_{0}=\tilde{v}_{1}\hbar+\tilde{v}_{2}\hbar+\ldots.$

Clearly, $v-d_{E}w_{0}$ is a cocycle, and since multiplication by $\hbar$ is a
chain injective map, hence $\tilde{v}=\tilde{v}_{1}+\tilde{v}_{2}\hbar+\ldots$
is a cocycle. Arguing as in (4.25) there is a $w_{1}\in E^{k+(n-2)-1}$
satisfying

$d_{E}^{0}w_{1}=\tilde{v}_{1}.$

Clearly, one has

$v-d_{E}(w_{0}+w_{1}\hbar)=\tilde{\tilde{v}}_{2}\hbar^{2}+\ldots.$

Repeating the procedure above one obtains a sequence $w_{0},w_{1},\ldots,$
such that

$d_{E}(w_{0}+w_{1}\hbar+w_{2}\hbar^{2}+\ldots)=v.$

∎

######  Theorem 4.20.

$\mathcal{F}$ induces an isomorphism on homology.

###### Proof.

This statement follows from Lemmas 4.18 and 4.19. ∎

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