claim is not explicitly stated in [Las79] but can be observed from Lashof’s
construction.

Moreover, the correspondence $\varphi\mapsto\alpha$ induces a bijection
between stable isotopy classes of stable $G$-smoothings of $M$ and stable
isotopy classes of stable $G$-vector bundle reductions of $T_{\mu}M$.

Our construction relies on the following relative version of Lashof’s theorem.

######  Theorem 6.10.

Let $G$, $M$, $E$, and $\varphi$ be as in Theorem 6.9. Let $C\subset M$ be a
$G$-invariant closed set and $U\subset M$ be a $G$-invariant open neighborhood
of $C$. Suppose

  1. (1)

$U$ is equipped with a $G$-smoothing $\alpha_{0}$ and $E|_{U}$ is equipped
with a smooth $G$-equivariant vector bundle structure.

  2. (2)

$\varphi|_{U}:(E|_{U})_{\mu}\to T_{\mu}U$ is a smooth microbundle reduction
and is isotopic to the microbundle reduction $TU_{\alpha_{0}}\to T_{\mu}U$.

Then there exists an orthogonal $G$-representation ${\bm{R}}$ and a
$G$-smoothing $\alpha$ on $M\times{\bm{R}}$ such that over a $G$-invariant
open neighborhood $U^{\prime}\subset U$ of $C$,
$(U^{\prime}\times{\bm{R}})_{\alpha}$ is diffeomorphic to the product
$(U^{\prime})_{\alpha_{0}}\times{\bm{R}}$.

###### Proof.

See Appendix B. ∎

####  6.1.3. Topological submersions

Now we start to prepare for smoothing the K-chart lift of the Hamiltonian
Floer flow category following a generalization of the strategy used in
[AMS21]. First, we discuss topological submersions which are relevant for
constructing lifts of tangent microbundles.

######  Definition 6.11.

Let $M$ be an ${\mathbb{A}}$-space and $B$ be a smooth
${\mathbb{A}}^{\prime}$-manifold. Let $\pi:M\to B$ be a continuous map
covering a map between the posets ${\mathbb{A}}\to{\mathbb{A}}^{\prime}$.

  1. (1)

Let $p\in M$ and $b=\pi(p)\in B$. A product neighborhood of $p$ is a pair
$(W,\iota)$ where $W$ is an open neighborhood of $p$ (denoting
$W|_{b}=\pi^{-1}(b)\cap W$) and $\iota:W\to W|_{b}\times\pi(W)$ is an
homeomorphism satisfying

     * •

$\pi(W)$ is an open neighborhood of $b$;

     * •

$\pi\circ\iota^{-1}:W|_{b}\times\pi(W)\to\pi(W)$ is the projection to
$\pi(W)$;

     * •

$\iota|_{W|_{b}}:W|_{b}\to W|_{b}\times\\{b\\}$ is the identity map.

  2. (2)

The map $\pi$ is called a topological submersion if every point $p\in M$
admits a product neighborhood and each fiber is a topological manifold (of a
fixed dimension) without boundary. In this case, the vertical tangent
microbundle $T_{\mu}^{\rm vt}M$ is the microbundle

$\textstyle{M\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Delta}$$\textstyle{M\times_{B}M\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p_{1}}$$\textstyle{M}$

where $p_{1}:M\times_{B}M\to M$ is the projection to the first factor, viewing
the fiber product $M\times_{B}M$ as a subset of $M\times M$.

######  Remark 6.12.

Our primary examples of topological submersions come from the forgetful map
$\pi_{pq}:V_{pq}\to B_{pq}$ in (5.39). It is important to notice that the
fibers are indeed topological manifolds without boundary, because the complex
structure on the domain of the element $(\Sigma,u,F,\eta_{1},\dots,\eta_{d})$
has been fixed.

Now we include group actions. Let $G$ be a compact Lie group. Recall that a
$G$-action on an ${\mathbb{A}}$-space is a continuous $G$-action which
preserve each stratum. We assume that $G$ acts continuously on $M$ and
smoothly on $B$, and that $\pi:M\to B$ is $G$-equivariant. Then for each $g\in
G$, $p\in M$, and a product neighborhood $(W,\iota)$ of $p$,
$(g(W),g_{*}\iota)$ is a product neighborhood of $g(p)$, where

$g_{*}\iota(w)=g(\iota(g^{-1}(w))).$

Here $g:W|_{b}\times\pi(W)\to W|_{g(b)}\times\pi(g(W))$ is the diagonal
action.

######  Definition 6.13.

[AMS21, Definition 4.19] For each $p\in M$, denote by $G_{p}\subset G$ the
stabilizer of $p$. A $G_{p}$-invariant product neighborhood of $p$ is a
product neighborhood $(W,\iota)$ of $p$ which is $G_{p}$-invariant, i.e.,
$(g(W),g_{*}\iota)=(W,\iota)$ for any $g\in G_{p}$.

We say that the $G$-equivariant map $\pi:M\to B$ is a $G$-equivariant
topological submersion if every point $p\in M$ admits a $G_{p}$-invariant
product neighborhood. We say that the $G$-action is fiberwise locally linear
if for each $b\in B$, the action of $G_{b}\subset G$ on $M_{b}$ is locally
linear.

######  Remark 6.14.

Any $G$-equivariant vector bundle $p:E\to B$ is a fiberwise locally linear
equivariant topological submersion.

Now we recall how to construct smoothings of the total space of a topological
submersion with a fiberwise smooth structure.

######  Definition 6.15.

[AMS21, Definition 4.27] Let $G$ be a compact Lie group. Let $M$ be a
topological manifold with a $G$-action. Let $B$ be a smooth $G$-manifold and
$\pi:M\to B$ be a $G$-equivariant topological submersion.

  1. (1)

A fiberwise smooth structure on $M$ (with respect to the map $\pi$) is a
choice of smooth structures (by definition a maximal $C^{\infty}$-compatible
atlas) on all fibers of $\pi$.

  2. (2)

Given a fiberwise smooth structure on $M$, two product neighborhoods
$\iota_{i}:W_{i}\to W_{i}|_{b_{i}}\times\pi(W_{i})$, $i=1,2$ are $C_{\rm
loc}^{1}$-compatible if for each $p\in W_{1}\cap W_{2}$ there exists a product
neighborhood $\iota:W\to W|_{b}\times\pi(W)$, $b=\pi(p)$ satisfying the
following conditions. For each $v\in\pi(W)$, the definition of product
neighborhoods (see Definition 6.11) implies that the map

$\iota_{v}:W|_{v}\to\\{v\\}\times W|_{b}$

is a homeomorphism. Consider the family of maps

$\eta_{v}:W|_{b}\to W_{i}|_{b_{i}},\
w\mapsto\Pi_{i}\big{(}\iota_{i}(\iota_{v}^{-1}(w))\big{)}$

where $\Pi_{i}:W_{i}|_{b_{i}}\times\pi(W_{i})\to W_{i}|_{b_{i}}$ is the
projection. Then for each $v$ and $i=1,2$, $\eta_{v}$ is a smooth map from
$W|_{b}$ to $W_{i}|_{b_{i}}$ and varies with $v$ continuously with respect to
the $C_{\rm loc}^{1}$-topology.

  3. (3)

A fiberwise smooth $C_{\rm loc}^{1}$ $G$-bundle is a $G$-equivariant
topological submersion $\pi:M\to B$ together with a collection
$(\iota_{i})_{i\in I}$ of $C_{\rm loc}^{1}$-compatible $G_{p}$-invariant
product neighborhoods around a collection of points $(p_{i})_{i\in I}$ whose
domains cover $M$.

######  Remark 6.16.

Despite our previous discussions work under the setting that $M$ is an
${\mathbb{A}}$-manifold, the above notions is defined for $M$ being a manifold
without boundary. In practice, when we perform smoothing inductively, we apply
Theorem 6.10 to the top stratum of a topological ${\mathbb{A}}$-manifold $M$.
Therefore, it suffices to set up certain parts of the theory for manifolds
without boundary or corners.

####  6.1.4. Fiberwise submersions

A fiberwise smooth $C_{\rm loc}^{1}$-bundle $\pi:M\to B$ has a well-defined
vertical tangent bundle, denoted by $T^{\rm vt}M$, by patching up the tangent
bundles of the fibers, and an isotopy class of $G$-vector bundle lift

$T^{\rm vt}M\to T_{\mu}^{\rm vt}M$

by using a $G$-invariant fiberwise Riemannian metric (see [AMS21, Lemma
4.29]).

Now suppose $\pi:M\to B$ is a fiberwise smooth $C_{\rm loc}^{1}$ $G$-bundle.
There is a well-defined $G$-microbundle

$\pi^{*}T_{\mu}B\oplus T^{\rm vt}_{\mu}M,$

a well-defined $G$-vector bundle

$\pi^{*}TB\oplus T^{\rm vt}M,$

and a well-defined $G$-isotopy class of vector bundle lifts

$(\pi^{*}TB\oplus T^{\rm vt}M)_{\mu}\to\pi^{*}T_{\mu}B\oplus T^{\rm
vt}_{\mu}M$ (6.4)

constructed using the exponential map after further choosing a Riemannian
metric on $B$. Note that both the space of fiberwise Riemannian metrics on $M$
varying continuously over $B$, and the space of Riemannian metrics are
contractible. Therefore, all these choices induce the same isotopy class of
the vector bundle reduction for the direct sum $\pi^{*}T_{\mu}B\oplus T^{\rm
vt}_{\mu}M$. To qualify the condition for stable $G$-smoothing (see Theorem
6.9), one only needs to identify the right hand side with the tangent
microbundle of the total space $M$. However this identification is not
canonical even up to isotopy of microbundle isomorphisms. This is what we need
to construct during the smoothing process.

######  Remark 6.17.

Different data for constructing the splitting (6.4) from [AMS21, Lemma 4.24,
Proposition 4.26] can in fact be interpolated with each other, so the
splittings (6.4) arising from this way should be isotopic to each other.
Keeping track of these choices are important for showing that the “invariants”
constructed in this paper are indeed independent of all the auxiliary choices,
but we do not need such invariance results for our applications.

######  Definition 6.18.

[AMS21, Definition 4.22] Suppose $\pi:M\to B$ is a $G$-equivariant topological
submersion. Let $W\subset M$ be a $G$-invariant open subset. A $G$-equivariant
fiberwise submersion along $W$ is a continuous map $\phi:\widetilde{W}\to M$
where $\widetilde{W}\subset W\times W$ is an open neighborhood of the diagonal
satisfying the following properties.

  1. (1)

$\phi_{q,b}:=\phi|_{\widetilde{W}\cap(\\{q\\}\times M_{b})}$ is a
homeomorphism onto an open subset of $M_{\pi(q)}$ for all $b\in\pi(W)$ and
$q\in W$.

  2. (2)

$\phi_{q,\pi(q)}$ sends each point
$(q,q^{\prime})\in\widetilde{W}\cap(\\{q\\}\times M_{\pi(q)})$ to
$q^{\prime}$.

  3. (3)

$\widetilde{W}$ is a $G$-invariant set with respect to the diagonal $G$-action
on $W\times W$ and $\phi$ is a $G$-equivariant map.

######  Remark 6.19.

Suppose $M$ is smooth and $\pi:M\to B$ is a smooth submersion. Then one can
construct a fiberwise submersion using a $G$-invariant Riemannian metric and
the exponential map in the horizontal direction.

######  Remark 6.20.

A $G$-equivariant fiberwise submersion along $W=M$ provides a $G$-microbundle
morphism

$\tau:T_{\mu}M\to T_{\mu}^{\rm vt}M$

which is defined as

$M\times
M\supseteq\widetilde{M}\ni(p,p^{\prime})\mapsto(p,\phi(p,p^{\prime}))$

which is the identity morphism on the sub-microbundle $T_{\mu}^{\rm
vt}M\subset T_{\mu}M$. On the other hand, there is a canonical $G$-microbundle
morphism

$P:T_{\mu}M\to\pi^{*}T_{\mu}B,\
(p,p^{\prime})\mapsto(p,(\pi(p),\pi(p^{\prime}))).$

Hence the $G$-equivariant fiberwise submersion induces an isomorphism of
microbundles

$P\oplus\tau:T_{\mu}M\to\pi^{*}T_{\mu}B\oplus T^{\rm vt}_{\mu}M.$

If $\pi$ is a fiberwise smooth $C_{\rm loc}^{1}$ $G$-bundle, then a
$G$-equivariant fiberwise submersion along $M$ induces a well-defined
$G$-isotopy class of vector bundle lift

$(\pi^{*}TB\oplus T^{\rm vt}M)_{\mu}\cong T_{\mu}M.$

######  Lemma 6.21.

Let $C\subset M$ be a $G$-invariant closed set and $U\subset M$ be a
$G$-invariant open neighborhood of $C$. Let $D\subset M$ be another
$G$-invariant closed set. Suppose $\phi:\widetilde{U}\to M$ is a
$G$-equivariant fiberwise submersion along $U$. Then there exists a
$G$-invariant open neighborhood $W$ of $C\cup D$ and a $G$-equivariant
fiberwise submersion $\psi:\widetilde{W}\to M$ along $W$ which coincides with
$\phi$ in a small neighborhood of the diagonal of $C\times C$ in $U\times U$.

###### Proof.

This is a restatement of the extension lemma [AMS21, Lemma 4.24] and a
relative version of [AMS21, Proposition 4.25]. ∎

###  6.2. Pre-smoothing operations and stable complex structures

This subsection can be viewed as a continuation of Section 5, in which we
explore the more refined information of the K-charts
$K_{pq}=(V_{pq},E_{pq},G_{pq},S_{pq})$. These properties will be used in the
construction of smoothing. Furthermore, we also present materials relevant for
building the normal complex structure.

####  6.2.1. Topological submersions

We have the following analogue of [AMS21, Corollary 6.28, 6.29].

######  Proposition 6.22.

The natural forgetful map

$\pi_{pq}:V_{pq}\to B_{d_{pq}}$

is a $G_{pq}$-equivariant topological submersion and has the structure of a
$C_{\rm loc}^{1}$ fiberwise smooth $G_{pq}$-bundle.

###### Proof.

This statement essentially a reformulation of the standard gluing construction
in Floer theory which is presented for instance, in [Par16, Appendix C]. The
slight difference is, here we are in a Morse–Bott situation because the
asymptotic operator of elements in the moduli space
${\mathcal{M}}_{J_{\tilde{\Psi}},H}(k_{1},\ldots,k_{d})$ has nontrivial kernel
along the ${\mathcal{C}}$ and ${\mathcal{E}}_{i}$ direction. However, this
does not introduce any problem to the argument.

As a first step, let us show that $\pi_{pq}$ is a $G_{pq}$-equivariant
topological submersion. Consider a point $x\in V_{pq}$ represented by
$(u,\Sigma,F,0)$. Using the sheared almost complex structure and Gromov’s
graph trick, we can view $(u,\Sigma,F,\eta_{1},\dots,\eta_{d})$ as a pseudo-
holomorphic stable cylinder contained in the $0$-section of
${\mathcal{E}}_{d}\to M\times{\mathcal{C}}$. Recall that the framing $F$
defines a holomorphic map $\iota_{F}:\Sigma\to\mathbb{CP}^{d}$ of degree $d$.
Let us choose $d^{\prime}=(d+2)d$ generic hyperplanes
$H_{1},\ldots,H_{d^{\prime}}\subset\mathbb{CP}^{d}$ which intersect the image
of $\underline{\smash{\mu}}$ transversely at points different from nodes and
markings, and choose

$z_{i}\in v(\Sigma)\cap H_{i},\ i=1,\ldots,d^{\prime}.$

Then the map $[v]\mapsto(z_{1},\ldots,z_{d^{\prime}})$ is a local
diffeomorphism from ${\mathcal{F}}{}_{0,2}^{\mathbb{R}}(d)$ to an open subset
$j:U\subset\overline{\mathcal{M}}{}_{0,2+d^{\prime}}^{\mathbb{R}}$. These
intersection points play the role as local stabilizing divisors. Namely, we
choose the preimage of the divisors $H_{1},\dots,H_{d^{\prime}}$ under the
projection map ${\mathcal{E}}_{d}\to
M\times{\mathcal{C}}\to{\mathcal{C}}\to{\mathbb{C}P}^{d}$ to be the local
stabilizing divisors. By keeping track of the positions of the marked points
obtained from intersecting the framing map with these divisors, we obtain map

$\mu_{H}:V_{pq}\to\overline{\mathcal{M}}{}_{0,2+d^{\prime}}^{\mathbb{R}}$

defined over a neighborhood of $x\in V_{pq}$.

Let $K_{\rm map}$ be the kernel of the linearized Cauchy–Riemann operator at
$(u,\iota_{F},0)$ (without deforming the underlying map to $\mathbb{CP}^{d}$).
The gluing map

$g:U\times K\to V_{pq}$

defined by [Par16, Equation (C.10.3)] fit into the following diagram:

$\vbox{ \lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
17.67424pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{U\times
K\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
22.61862pt\raise 5.1875pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-0.8264pt\hbox{$\scriptstyle{g}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 45.04022pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx<EMAIL_ADDRESS>0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-0.82361pt\hbox{$\scriptstyle{{\rm pr}_{U}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 0.0pt\raise-30.9261pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
45.04022pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{V_{pq}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
54.82716pt\raise-20.3797pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-0.82361pt\hbox{$\scriptstyle{\mu_{H}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern
54.82716pt\raise-28.84608pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx<EMAIL_ADDRESS>0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{U\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
22.77191pt\raise-46.74829pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-1.62779pt\hbox{$\scriptstyle{j}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern
41.67424pt\raise-40.75941pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
41.67424pt\raise-40.75941pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\overline{\mathcal{M}}{}_{0,2+d^{\prime}}^{\mathbb{R}}}$}}}}}}}\ignorespaces}}}}\ignorespaces}.$

According to [Par16, Section C.12], $g$ is a homeomorphism onto its image. The
commutativity of the above diagram follows from the definition [Par16,
Equation (C.10.4)]. Furthermore, the restriction of the gluing map $g$ to a
fiber $\\{a\\}\times K$ is smooth, see [Par16, Section C.9] (this has already
been explored in Section 5). Note that the topological submersion property
holds over a $G_{pq}$-invariant open neighborhood of $S_{pq}^{-1}(0)$ in
$V_{pq}$ by the transversality assumption. Up to restricting to such an open
subset, we see that $\pi_{pq}:V_{pq}\to B_{pq}$ is indeed a
$G_{pq}$-equivariant topological submersion.

It remains to see that $\pi_{pq}$ actually has a $C^{1}_{\rm loc}$
$G_{pq}$-bundle structure. Note that the gluing construction of
$J$-holomorphic curves is based on applying a Newton–Picard iteration scheme
to a pre-glued curve. By [Par16, Proposition B.11.1, C.11.1], the $L^{2}$-norm
of the honest solution near the ends of the neck region controls all
$C^{k}$-norms through the region. With this in mind, the rest of the argument
follows from [AMS21, Corollary 6.29]. ∎

As a consequence, we see that the tangent microbundle of the interior of
$V_{pq}$ admits a vector bundle lift by (6.4) and Lemma 6.21. This suffices
for us to construct a $G_{pq}$-equivariant smoothing on the interior of
$V_{pq}$, but it falls short of providing a smoothing of $V_{pq}$ as a
manifold with corners. This problem will be solved on the outer-collaring
$V_{pq}^{+}$ by performing an inductive argument.

####  6.2.2. Vertical stable complex structures

Following [AB21, Section 11], we describe how to construct a stabilization of
the vertical tangent bundle $T^{\rm vt}V_{pq}$ of the $C^{1}_{\rm loc}$
$G_{pq}$-bundle $\pi_{pq}:V_{pq}\to B_{pq}$ such that the resulting vector
bundle admits a complex structure. We also explain how to use these stable
complex structures to construct normal complex structures on derived orbifold
charts.

For any $1$-periodic orbit $\underline{\smash{p}}$ of $H$, fix a unitary
trivialization of the complex vector bundle $\underline{\smash{p}}^{*}TM\to
S^{1}$. Moreover, we choose a complex linear connection $\nabla$ defined on
the pullback of $TM$ under the projection map $S^{1}\times M\to M$, such that
its restriction to the graph of any $1$-periodic orbit is induced by the
chosen trivializations. For a $1$-periodic orbit $\underline{\smash{p}}$,
using the projection ${\rm pr}_{S^{1}}:{\mathbb{R}}\times S^{1}\to S^{1}$, we
consider the complex vector bundle

$(\underline{\smash{p}}\circ{\rm pr}_{S^{1}})^{*}TM\to{\mathbb{R}}\times
S^{1}.$

Denote by
$B_{H,\underline{\smash{p}}}(t):S^{1}\to\text{End}(\underline{\smash{p}}^{*}TM)$
the section obtained by differentiating the flow of $X_{H_{t}}$, i.e.,

$B_{H,\underline{\smash{p}}}(t)(W)=\nabla_{W}X_{H_{t}},\ \forall W\in
T_{\underline{\smash{p}}(t)}M.$

Using a smooth cut-off function, we can consider the Cauchy–Riemann operator

$\nabla^{0,1}_{\underline{\smash{p}}}:\Omega^{0}((\underline{\smash{p}}\circ{\rm
pr}_{S^{1}})^{*}TM)\to\Omega^{0,1}((\underline{\smash{p}}\circ{\rm
pr}_{S^{1}})^{*}TM)$

such that

$\nabla^{0,1}_{\underline{\smash{p}}}=\left\\{\begin{array}[]{rr}(\nabla-
B_{H,\underline{\smash{p}}}\otimes dt)_{J}^{0,1}\text{ near }-\infty\\\
\nabla^{0,1}\text{ near }+\infty.\end{array}\right.$

It is a standard fact that $\nabla^{0,1}_{\underline{\smash{p}}}$ defines an
elliptic operator. Therefore, we can find a finite dimensional complex vector
space $V_{\underline{\smash{p}}}^{-}$ and a linear map
$\lambda_{\underline{\smash{p}}}:V_{\underline{\smash{p}}}^{-}\to\Omega^{0,1}((\underline{\smash{p}}\circ{\rm
pr}_{S^{1}})^{*}TM)$ such that the map

$\nabla^{0,1}_{\underline{\smash{p}}}\oplus\lambda_{\underline{\smash{p}}}:\Omega^{0}((\underline{\smash{p}}\circ{\rm
pr}_{S^{1}})^{*}TM)\oplus
V_{\underline{\smash{p}}}^{-}\to\Omega^{0,1}((\underline{\smash{p}}\circ{\rm
pr}_{S^{1}})^{*}TM)$

is surjective. This map is called the asymptotic operator of
$\underline{\smash{p}}$. We fix a choice of $V_{\underline{\smash{p}}}^{-}$
and $\lambda_{\underline{\smash{p}}}$, and denote the kernel of the above map
by $V_{\underline{\smash{p}}}^{+}$. The virtual vector space
$(V_{\underline{\smash{p}}}^{+},V_{\underline{\smash{p}}}^{-})$ will be used
to construct a stable complex structure on $T^{\rm vt}V_{pq}$.

######  Lemma 6.23.

For any $p,q\in T^{\rm Floer}$ such that $\overline{\mathcal{M}}{}^{\rm
Floer}_{pq}\neq\emptyset$, there exists a complex vector bundle $I^{\rm
vt}_{pq}\to V_{pq}$ such that there exists an homotopy between the vector
bundles

$V_{\underline{\smash{q}}}^{+}\oplus T^{\rm vt}V_{pq}\oplus
V_{\underline{\smash{p}}}^{-}\cong V_{\underline{\smash{q}}}^{-}\oplus I^{\rm
vt}_{pq}\oplus V_{\underline{\smash{p}}}^{+}.$ (6.5)

###### Proof.

We explain the construction in the case of a single-layered thickening. The
vertical tangent bundle of $V_{pq}$ can be identified with the kernel of the
linearized operator associated with (5.25) with the framing $F$ being kept
fixed. To be more precise, using the model as in the proof of Theorem 5.41,
given an element in $V_{pq}$ with representative $(\Sigma,u,\mu,\eta)$, the
deformation operator over each component $\Sigma_{\alpha}$ is of the form

$\displaystyle W^{1,p}(\Sigma_{\alpha},u^{*}TM)\oplus
W^{1,p}(\Sigma_{\alpha},(u,\mu)^{*}E(k))\to
L^{p}(\Lambda^{0,1}(\Sigma_{\alpha},u^{*}TM))\oplus
L^{p}(\Lambda^{0,1}(\Sigma_{\alpha},u^{*}TM))$
$\displaystyle(\xi,\eta^{\prime})\mapsto(D_{u}\xi+P(\eta^{\prime}),D_{E}\eta^{\prime}).$

By taking the direct sum of these linear operators ranging over all the
components of $\Sigma$, and impose the usual matching condition in the case of
nodal curves, we indeed identify $T^{\rm vt}V_{pq}$ over $(\Sigma,u,\mu,\eta)$
with the kernel of a Fredholm operator which is schematically written as

$D^{\rm
vt}_{u,\mu,\eta}:{\mathcal{E}}_{u,\mu,\eta}\to{\mathcal{F}}_{u,\mu,\eta}.$

Now we consider the concatenation of the three operators $D^{\rm
vt}_{u,\mu,\eta}\oplus\nabla^{0,1}_{\underline{\smash{q}}}\oplus\lambda_{\underline{\smash{q}}}\oplus
0$

${\mathcal{E}}_{u,\mu,\eta}\oplus(\Omega^{0}((\underline{\smash{q}}\circ{\rm
pr}_{S^{1}})^{*}TM)\oplus V_{\underline{\smash{q}}}^{-})\oplus
V_{\underline{\smash{p}}}^{-}\to{\mathcal{F}}_{u,\mu,\eta}\oplus\Omega^{0,1}((\underline{\smash{q}}\circ{\rm
pr}_{S^{1}})^{*}TM).$

This operator is Fredholm and surjective, with kernel identified with

$T^{\rm vt}V_{pq}\oplus V^{+}_{\underline{\smash{q}}}\oplus
V_{\underline{\smash{p}}}^{-}.$

We will construct a homotopy of this operator following the proof of [AB21,
Proposition 11.30]. To this end, introduce an auxialiary moduli space
$\tilde{V}_{pq}\to V_{pq}$. It is described by adding an additional marked
point on the compactified lateral line
${\mathbb{R}}\cup\\{\pm\infty\\}\cong[0,1]$ of the cylindrical components and
the map $\tilde{V}_{pq}\to V_{pq}$ is the associated forgetful map. The fiber
of the universal family over $\tilde{V}_{pq}$ is the same as the fiber of
$V_{pq}$ when the newly-added marked point $\neq\pm\infty$; when it is given
by $\pm\infty$, we insert a cylinder between the consecutive two cylindrical
components which is mapped to the common asymptotic orbit. In particular,
$\tilde{V}_{pq}\to V_{pq}$ is a fiber bundle with fiber $\cong[0,1]$. For an
element $(\Sigma,u,\mu,\eta)\in V_{pq}$, the domain of the universal family
over $(\Sigma,u,\mu,\eta)\times\\{1\\}$ coincides with the domain of $D^{\rm
vt}_{u,\mu,\eta}\oplus\nabla^{0,1}_{\underline{\smash{q}}}\oplus\lambda_{\underline{\smash{q}}}\oplus
0$.

Write

$D^{\rm vt}_{u,\mu,\eta}=D^{{\rm vt},{\mathbb{C}}}_{u,\mu,\eta}+Y^{\rm
vt}_{u,\mu,\eta}$

where $D^{{\rm vt},{\mathbb{C}}}_{u,\mu,\eta}$ is a complex linear
Cauchy–Riemann operator and $Y^{\rm vt}_{u,\mu,\eta}$ is a $0$-th order
differential operator which is complex anti-linear. Then we can consider
another operator $D^{{\rm
vt},{\mathbb{C}}}_{u,\mu,\eta}\oplus\nabla^{0,1}_{\underline{\smash{p}}}\oplus\lambda_{\underline{\smash{p}}}\oplus
0$

${\mathcal{E}}_{u,\mu,\eta}\oplus(\Omega^{0}((\underline{\smash{p}}\circ{\rm
pr}_{S^{1}})^{*}TM)\oplus V_{\underline{\smash{p}}}^{-})\oplus
V_{\underline{\smash{q}}}^{-}\to{\mathcal{F}}_{u,\mu,\eta}\oplus\Omega^{0,1}((\underline{\smash{p}}\circ{\rm
pr}_{S^{1}})^{*}TM).$

If this Fredholm operator is surjective, then there exists a finite
dimensional complex vector space $(I^{\rm vt}_{pq})_{u,\mu,\eta}$ such that
the kernel of this operator is

$(I^{\rm vt}_{pq})_{u,\mu,\eta}\oplus V_{\underline{\smash{p}}}^{+}\oplus
V_{\underline{\smash{q}}}^{-}.$

Note that using the position of the additional marked point on the lateral
line $\tilde{V}_{pq}\to V_{pq}$ and a cut-off function defined on the domains
of the universal family over $\tilde{V}_{pq}$ whose differential has support
on the horizontal component where the marked point lives on, we can construct
a homotopy between two Fredholm operators

$D^{\rm
vt}_{u,\mu,\eta}\oplus\nabla^{0,1}_{\underline{\smash{q}}}\oplus\lambda_{\underline{\smash{q}}}\oplus
0\text{ and }D^{{\rm
vt},{\mathbb{C}}}_{u,\mu,\eta}\oplus\nabla^{0,1}_{\underline{\smash{p}}}\oplus\lambda_{\underline{\smash{p}}}\oplus
0$

which varies smoothly as we vary $(\Sigma,u,\mu,\eta)$ along the fiber of
$V_{pq}\to B_{pq}$.

Moreover, $D^{{\rm vt},{\mathbb{C}}}_{u,\mu,\eta}$ can be made surjective if
the $k$ as from $E(k)$ is chosen to be large enough. This fact can be derived
from exactly the same proof of [AMS21, Proposition 6.26]. Therefore the
statement is proved by defining $I^{\rm vt}_{pq}$ to be the vector bundle over
$V_{pq}$ by assembling $(I^{\rm vt}_{pq})_{u,\mu,\eta}$. ∎

######  Corollary 6.24.

If the almost free action of $G_{pq}$ on $V_{pq}$ is smooth, then the induced
orbifold vector bundle $T^{\rm vt}V_{pq}$ on the quotient orbifold
$V_{pq}/G_{pq}$ has a normal complex structure.

###### Proof.

This follows from Lemma 6.23 and the follwing fact. For $p\in{\mathcal{P}}$
(also for $q\in{\mathcal{P}}$), the trivial vector bundles
$V_{\underline{\smash{p}}}^{+}\times V_{pq}\to V_{pq}$ and
$V_{\underline{\smash{p}}}^{-}\times V_{pq}\to V_{pq}$ can be viewed as
$G_{pq}$-equivariant vectors for which $G_{pq}$ acts trivially on the factor
$V_{\underline{\smash{p}}}^{\pm}$. In particular, we can use
$V_{\underline{\smash{p}}}^{\pm}$ to stabilize the space $V_{pq}$. Because the
induced $G_{pq}$-action on $V_{\underline{\smash{p}}}^{\pm}\times V_{pq}\to
V_{pq}$ is trivial on the first factor, they do not affect the _normal_
directions to the union of $G_{pq}$-orbits with the same isotropy group of
both the tangent bundle of $V_{pq}$. Therefore the statement follows from
taking the nontrivial part of the representations of (6.5) after taking the
$G_{pq}$-quotient. ∎

####  6.2.3. Stabilizing the K-chart lift

In the equivariant smoothing theorem of Lashof [Las79] a smooth structure can
only be obtained after stabilizing the manifold. Given a $G$-manifold $M$, a
stabilization of $M$ is $M\times{\bm{R}}$ where ${\bm{R}}$ is an orthogonal
$G$-space.181818We could try to extend the theory to the case of nontrivial
bundles but that is not completely necessary. This operation fits into the
stabilization of Kuranishi charts. As we are dealing with infinitely many
Kuranishi charts associated to a K-chart lift of the Floer category $T^{\rm
Floer}$ (or its outer-collaring), once we stabilize certain charts (and their
products) and equipped them with smooth structures, a “larger” chart needs to
be stabilized by a larger representation in order to include the stabilization
of products corresponding to its strata. Thus we need to introduce the
following concept which describes how a system of stabilizations should fit
together.

######  Definition 6.25.

Let $T^{\mathcal{P}}$ be a flow category over ${\mathcal{P}}$ and suppose the
collection $\\{K_{pq}=(G_{pq},V_{pq},E_{pq},S_{pq})\\}_{p<q}$ of Kuranishi
charts with the collection $\\{{\bm{\iota}}_{prq}\\}_{p<r<q}$ of K-chart
embeddings define a K-chart lift of $T^{\mathcal{P}}$ (see Definition 5.9). A
stabilization of consists of the following objects.

  1. (1)

For each $T_{pq}$ a finite-dimensional orthogonal $G_{pq}$-representation
${\bm{R}}_{pq}$. Denote by $R_{pq}$ the trivial $G_{pq}$-equivariant vector
bundle $V_{pq}\times{\bm{R}}_{pq}$. Then for each stratum $\alpha=pr_{1}\cdots
r_{l}q$, define the $G_{\alpha}$-equivariant vector bundle $R_{\alpha}$ over
$V_{\alpha}=V_{pr_{1}}\times\cdots V_{r_{l}q}$ to be

$R_{pr_{1}}\boxplus\cdots\boxplus R_{r_{l}q}.$ (6.6)

  2. (2)

For any pair of strata $\alpha\leq\beta$, an equivariant bundle embedding

$\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
8.81894pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{R_{\alpha}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
16.65257pt\raise 6.51389pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-1.34723pt\hbox{$\scriptstyle{\phi_{\beta\alpha}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 32.81894pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
0.0pt\raise-31.51945pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
32.81894pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{R_{\beta}|_{\partial^{\alpha}V_{\beta}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
48.29727pt\raise-30.20277pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx<EMAIL_ADDRESS>0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{V_{\alpha}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
35.24007pt\raise-41.35277pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
35.24007pt\raise-41.35277pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\partial^{\alpha}V_{\beta}}$}}}}}}}\ignorespaces}}}}\ignorespaces.$

Assume that these data satisfy the following compatibility condition.

  1. (A)

For each triple of strata $\alpha\leq\beta\leq\gamma$ one has
$\phi_{\gamma\beta}\circ\phi_{\beta\alpha}=\phi_{\gamma\alpha}$.

  2. (B)

The bundle embeddings are induced from products. Namely, given a stratum
$\alpha=pr_{1}\cdots r_{l}q\in{\mathbb{A}}^{\mathcal{P}}_{pq}$, there exist
equivariant isometric linear embeddings

$\phi_{r_{i}r_{i+1}}:R_{r_{i}r_{i+1}}\hookrightarrow R_{pq}\text{ for
}i=0,\dots,l$

with respect to the group embeddings $G_{r_{i}r_{i+1}}\hookrightarrow G_{pq}$
such that

$\phi_{pq,\alpha}=\phi_{pq,pr_{1}}\times\cdots\times\phi_{pq,r_{l}q},$

and $\phi_{\beta\alpha}$ for a general pair $\beta\leq\alpha$ is constructed
similarly using the corresponding factorization.

Given a stabilization containing $\\{{\bm{R}}_{pq}\\}$ and
$\\{\phi_{\beta\alpha}\\}$ we can define another K-chart lift for
$T^{\mathcal{P}}$. Its collection of Kuranishi charts are

$\widehat{K}_{pq}:=(G_{pq},\widehat{V}_{pq},\widehat{E}_{pq},\widehat{S}_{pq}):={\rm
Stab}_{R_{pq}}(K_{pq});$

its collection of chart embeddings

$\widehat{\bm{\iota}}_{prq}:\widehat{K}_{pr}\times\widehat{K}_{rq}\to\partial^{prq}\widehat{K}_{pq}$

are defined as follows. Given $(x_{pr},e_{pr})\in\widehat{V}_{pr}$,
$(x_{rq},e_{rq})\in\widehat{V}_{rq}$, define

$\widehat{\iota}_{prq}((x_{pr},e_{pr}),(x_{rq},e_{rq}))=(\iota_{prq}(x_{pr},x_{rq}),\phi_{pq,prq}(e_{pr},e_{rq}))$

which is clearly equivariant with respect to the group map $G_{pr}\times
G_{rq}\to G_{pq}$.

######  Lemma 6.26.

The data $\\{\widehat{K}_{pq}\\}_{p<q}$ with
$\\{\widehat{\bm{\iota}}_{prq}\\}_{p<r<q}$ is a K-chart lift of
$T^{\mathcal{P}}$.

###### Proof.

This follows directly from the definition. ∎

We need to equip a stabilization certain extra structures.

######  Definition 6.27.

Suppose the K-chart lift ${\mathfrak{K}}^{\mathcal{P}}$ is equipped with a
collar structure. Then a stabilization is called collared if there are
equivariant bundle isomorphisms

$\textstyle{R_{pq}|_{\partial^{\alpha}V_{pq}}\times[0,\epsilon)^{{\mathbb{F}}_{\alpha}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{R_{pq}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\partial^{\alpha}V_{pq}\times[0,\epsilon)^{{\mathbb{F}}_{\alpha}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{V_{pq}}$

such that the bundle embeddings are collared, i.e.,

$\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
38.32526pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{R_{\beta}|_{\partial^{\alpha}V_{\beta}}\times[0,\epsilon)^{{\mathbb{F}}_{\alpha}\setminus{\mathbb{F}}_{\beta}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
62.32526pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
0.0pt\raise-32.54001pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
62.32526pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{R_{\gamma}|_{\partial^{\alpha}V_{\gamma}}\times[0,\epsilon)^{{\mathbb{F}}_{\alpha}\setminus{\mathbb{F}}_{\beta}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
100.43977pt\raise-31.87332pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx<EMAIL_ADDRESS>0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{R_{\beta}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
85.29071pt\raise-42.37332pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
85.29071pt\raise-42.37332pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{R_{\gamma}|_{\partial^{\beta}V_{\gamma}}}$}}}}}}}\ignorespaces}}}}\ignorespaces.$

Note that although our stabilization is constructed using product bundles, the
stabilization maps are not necessarily induced from the linear maps between
the representations. Therefore, the above definition is not entirely
superfluous.

We will also use not-necessarily-trivial inner products on those trivial
bundles $R_{pq}$ in the construction.

######  Definition 6.28.

Let ${\mathfrak{K}}^{\mathcal{P}}$ be a K-chart lift of $T^{\mathcal{P}}$
equipped with a collar structure and a collared stabilization
${\mathfrak{R}}$. A collared system of inner products on ${\mathfrak{R}}$ is a
collection of $G_{pq}$-equivariant inner products on the bundle $R_{pq}$ such
that

  1. (1)

All the embeddings $\phi_{\beta\alpha}:R_{\alpha}\to
R_{\beta}|_{\partial^{\alpha}V_{\beta}}$ are isometric.

  2. (2)

The collar isomorphisms are isometric.

Suppose ${\mathfrak{R}}$ is equipped with a collared system of inner products.
Then define

$R_{\beta\alpha}\subset R_{\beta}|_{V_{\alpha}}$

to be the orthogonal complement of $R_{\alpha}$ in $R_{\beta}|_{V_{\alpha}}$.
Then by basic linear algebra one can see that whenever
$\alpha\leq\beta\leq\gamma$, as subbundles of $R_{\gamma}|_{V_{\alpha}}$, one
has

$R_{\gamma\alpha}=R_{\gamma\beta}|_{V_{\alpha}}\oplus R_{\beta\alpha}.$

Then the original scaffolding can be extended to include the extra pieces
$R_{\beta\alpha}$. Indeed, define

$\widehat{F}_{\beta\alpha}:=\pi^{*}F_{\beta\alpha}\oplus\pi^{*}R_{\beta\alpha}$

where $\pi$ denotes (temporarily) the projection $V_{pq}\times{\bm{R}}_{pq}\to
V_{pq}$ and the induced projection $V_{\alpha}\times{\bm{R}}_{\alpha}\to
V_{\alpha}$. The stabilization described in Proposition 5.64 is also extended
to a map

${\rm
Stab}_{\widehat{F}_{pq,\alpha}}(\widehat{V}_{\alpha})\to\partial^{\alpha}\widehat{V}_{pq}.$

After taking the equivariantization, we obtain a map

$G_{pq}\times_{G_{\alpha}}({\rm
Stab}_{\widehat{F}_{pq,\alpha}}(\widehat{V}_{\alpha}))\to\widehat{V}_{pq}$

which defines an equivariant open embedding. We would like to remark that the
orthogonal complement of the $G_{pq}$-equivariantization of $R_{pq,\alpha}\to
V_{\alpha}$ in $R_{pq}|_{G_{pq}(V_{\alpha})}$ is not necessarily a product
bundle. Therefore, some extra care needs to be taken in the smoothing process.

####  6.2.4. An induction scheme of stabilization

In the smoothing process, the construction of successive stabilizations is
elementary but complicated. In general, the stabilizations are constructed
inductively, similar to many other inductive constructions. We would like to
describe how the induction can be carried out. We first look at the simple
case when a moduli space has strata of codimension at most 2. Assume $p<q$ in
${\mathcal{P}}^{\rm Floer}$ and let $V_{pq}$ be the domain of a global
Kuranishi chart of $\overline{\mathcal{M}}{}_{pq}^{\rm Floer}$. In the
following discussion, we assume that the tangent microbundle of the underlying
thickened moduli space of the relevant K-charts admits a vector bundle lift,
which is the necessary condition for us to apply the smoothing theory.
Moreover, in the discussion of extending the smoothing relatively, we assume
that the lifts are compatible in a suitable sense. The construction of vector
bundle lifts and the precise meaning of compatibility will be detailed in
Section 6.4.

Note that we also need to endow the vector bundle $E_{pq}\to V_{pq}$ with a
smooth structure after suitable stabilization. However, this is rather
straightforward after smoothing $V_{pq}$: we can simply approximate the
classifying map of $E_{pq}$ by a smooth map which stays in the same homotopy
class.

The case with no boundary or corner. When $V_{pq}$ has no boundary or corner,
then it is the initial step of the induction. One can find a trivial
$G_{pq}$-bundle $V_{pq}\times{\bm{R}}_{pq}$ induced by a suitable
representation $R_{pq}$ needed for stable smoothing.

The case with only codimension one boundary strata. When $V_{pq}$ has only
codimension one boundary strata, without loss of generality, suppose there is
only one boundary stratum $V_{prq}$. Using the induction hypothesis, suppose
${\bm{R}}_{pr}$ and ${\bm{R}}_{rq}$ have been given. Consider the
representation ${\bm{R}}_{pr}$ resp. ${\bm{R}}_{rq}$ of $G_{pr}$ resp.
$G_{rq}$. By Frobenius reciprocity (see [Sep07, Theorem 7.47]), there exist
orthogonal representations ${\bm{R}}_{pr}^{pq}$ resp. ${\bm{R}}_{rq}^{pq}$ of
$G_{pq}$ which contains $G_{pr}$ resp. $G_{rq}$ as subrepresentations of
$G_{pr}$ resp. $G_{rq}$. Then define

${\bm{R}}_{pq}={\bm{R}}_{pr}^{pq}\oplus{\bm{R}}_{rq}^{pq}$

which is an orthogonal $G_{pq}$-representation.

Now we define the bundle embeddings required for stabilization. We introduce
more notations. Over the product $V_{prq}=V_{pr}\times V_{rq}$, define

$\displaystyle\ R_{pr\to pq}:=R_{pr}\boxplus\\{0\\},\ $ $\displaystyle\
R_{rq\to pq}:=\\{0\\}\boxplus R_{rq}.$

Then

$R_{prq}=R_{pr\to pq}\oplus R_{rq\to pq}.$

We would like the bundle embedding $\phi_{pq,prq}$ to be the sum of

$\displaystyle\ \phi_{pr\to pq}:R_{pr\to pq}\to
R_{pq}|_{\partial^{prq}V_{pq}},\ $ $\displaystyle\ \phi_{rq\to pq}:R_{rq\to
pq}\to R_{pq}|_{\partial^{prq}V_{pq}}.$

Indeed, $\phi_{pr\to pq}$ and $\phi_{rq\to pq}$ are the ones canonically
induced from the linear inclusions

$\displaystyle\
{\bm{R}}_{pr}\hookrightarrow{\bm{R}}_{pr}^{pq}\hookrightarrow{\bm{R}}_{pq},\ $
$\displaystyle\
{\bm{R}}_{rq}\hookrightarrow{\bm{R}}_{rq}^{pq}\hookrightarrow{\bm{R}}_{pq}.$

Then all the requirements of Definition 6.25 are automatically satisfied. In
addition, for the purpose of stable smoothing, one can also take an additional
direct sum to ${\bm{R}}_{pq}$ by another orthogonal $G_{pq}$-space
${\bm{R}}_{pq,0}$.

Note that in this final step, we need to apply the relative smoothing result
Theorem 6.10. Indeed, the space $R_{prq}$ admits a smooth structure by taking
the product of smooth structures on $R_{pr}$ and $R_{rq}$, whose existence is
based on the induction hypothesis. Then the equivariantization
$G_{pq}(R_{prq})$ has a smooth structure. Denote by $G_{pq}(R_{prq}^{\perp})$
the equivariantization of the orthogonal complement of $R_{prq}$ in
$R_{pq}|_{V_{prq}}$. Using the projection map $R_{prq}\to V_{prq}$, the vector
bundle $G_{pq}(R_{prq}^{\perp})$ can be pulled back to $G_{pq}(R_{prq})$ and
the total space of this vector bundle can be endowed with a smooth structure,
by approximating the classifying map. This in turn equips
$R_{pq}|_{\partial^{prq}V_{pq}}$ with a smooth structure. Using the collar
structure

$\partial^{prq}V_{pq}\times[0,\epsilon)\to V_{pq}$

which extends to a collar structure of the stabilized charts, we see that an
open neighborhood of $R_{pq}|_{\partial^{prq}V_{pq}}$ has a smooth structure
using the product decomposition

$R_{pq}|_{\partial^{prq}V_{pq}}\times[0,\epsilon).$

By replacing the open interval $[0,\epsilon)$ by a closed interval of the form
$[0,\frac{\epsilon}{2}]$, a suitable compatibility between the vector bundle
lifts guarantees that we can apply Theorem 6.10 to obtain the $G_{pq}$
orthogonal representation ${\bm{R}}_{pq,0}$ which induces a stabilized
smoothing for $V_{pq}$ extending the previously constructed stabilized
smoothings on $V_{pr}$ and $V_{rq}$.

######  Remark 6.29.

We hope that the above arguments showcase the importance of applying Frobenius
reciprocity, approximations of classifying maps of equivariant vector bundles,
and the relative smoothing result in our inductive construction. Moreover, as
we can see from above, outer-collaring conveniently provide us with an
automatic smoothing near the boundary stratum. Additionally, it is important
to keep in mind that the compatibility between the vector bundle lifts of
tangent microbundles of the total space and boundary stratum is crucial for
the application of Theorem 6.10.

The case with only codimension one or two strata. Now we consider the case
when ${\mathbb{A}}_{pq}^{\mathcal{P}}=\\{pq,prq,psq,prsq\\}$. Suppose the
representations

${\bm{R}}_{pr},\ {\bm{R}}_{rs},\ {\bm{R}}_{sq},{\bm{R}}_{ps},{\bm{R}}_{rq}$

are chosen so that they induce compatible stabilized smoothings except the top
stratum $V_{pq}$. Define the bundles $R_{pr\to pq}$ etc. in a way similar to
the previous case. Suppose we have also defined bundle embeddings

$\phi_{pr\to ps},\ \phi_{rs\to ps},\ \phi_{rs\to rq},\ \phi_{sq\to rq}.$

In addition, we make the following assumptions, which basically says that for
$ps$ and $rq$, the stabilization data are constructed as in the previous case.

  1. (1)

${\bm{R}}_{ps}$ is the direct sum

${\bm{R}}_{ps}={\bm{R}}_{pr}^{ps}\oplus{\bm{R}}_{rs}^{ps}\oplus{\bm{R}}_{ps,0}$

of orthogonal $G_{ps}$-spaces such that ${\bm{R}}_{pr}^{ps}$ resp.
${\bm{R}}_{rs}^{ps}$ contains ${\bm{R}}_{pr}$ resp. ${\bm{R}}_{rs}$ as
subrepresentations of $G_{pr}$ resp. $G_{rs}$. Similarly, there is a direct
sum

${\bm{R}}_{rq}={\bm{R}}_{rs}^{rq}\oplus{\bm{R}}_{sq}^{rq}\oplus{\bm{R}}_{rq,0}$

of orthogonal $G_{rq}$-spaces such that ${\bm{R}}_{rs}^{rq}$ resp.
${\bm{R}}_{sq}^{rq}$ contains ${\bm{R}}_{rs}$ resp. ${\bm{R}}_{sq}$ as
subrepresentations of $G_{rs}$ resp. $G_{sq}$.

  2. (2)

The bundle embedding

$\phi_{pr\to ps}:R_{pr\to ps}\to R_{ps}|_{\partial^{prs}V_{ps}}$

is induced from the linear inclusion
${\bm{R}}_{pr}\hookrightarrow{\bm{R}}_{pr}^{ps}\hookrightarrow{\bm{R}}_{ps}$.
Similar requirement applies to other bundle embeddings.

Now we define an orthogonal $G_{pq}$-space ${\bm{R}}_{pq}$. There are two
codimension one strata, $prq$ and $psq$. For the $G_{pr}$-space
${\bm{R}}_{pr}$, by Frobenius reciprocity, there is a $G_{pq}$-orthogonal
space ${\bm{R}}_{pr}^{pq}$ which contains ${\bm{R}}_{pr}$ as a
subrepresentation of $G_{pr}$. We choose similarly ${\bm{R}}_{rq}^{pq}$,
${\bm{R}}_{ps}^{pq}$, ${\bm{R}}_{sq}^{pq}$. For the purpose of stable
smoothing, we choose another $G_{pq}$-orthogonal space ${\bm{R}}_{pq,0}$,
which is constructed similarly as the case of one boundary stratum by applying
relative smoothing. Then define

${\bm{R}}_{pq}:=\Big{(}{\bm{R}}_{pr}^{pq}\oplus{\bm{R}}_{rq}^{pq}\Big{)}\oplus\Big{(}{\bm{R}}_{ps}^{pq}\oplus{\bm{R}}_{sq}^{pq}\Big{)}\oplus{\bm{R}}_{pq,0}.$

We rewrite this decomposition as

${\bm{R}}_{pq}={\bm{R}}_{pq,r}\oplus{\bm{R}}_{pq,s}\oplus{\bm{R}}_{pq,0}.$
(6.7)

Then we want to define the bundle embeddings into

$R_{pq}:=R_{pq,r}\oplus R_{pq,s}\oplus R_{pq,0}.$

This time the bundle embeddings are not purely induced from linear maps
between the representations.

  1. (1)

We define $\phi_{pr\to pq}$ and $\phi_{sq\to pq}$. We only describe
$\phi_{pr\to pq}$ in detail. With respect to (6.7), we write

$\phi_{pr\to pq}=\phi_{pr,r}\oplus\phi_{pr,s}\oplus 0.$

We define $\phi_{pr,r}:R_{pr\to pq}\to R_{pq,r}$ to be the one induced from
the linear inclusion

${\bm{R}}_{pr}\hookrightarrow{\bm{R}}_{pr}^{pq}\hookrightarrow{\bm{R}}_{pq,r}.$

For $\phi_{pr,s}:R_{pr\to pq}\to R_{pq,s}$, notice that it is defined over
$V_{prq}$. We first define it over $\partial^{prsq}V_{prq}$, which is induced
from the existing bundle map

$\phi_{pr\to ps}:R_{pr\to ps}\to R_{ps}$

and its equivariantization and stabilization (by the bundle associated to the
scaffolding), composed with the linear inclusion

$R_{ps}\hookrightarrow R_{ps}^{pq}\hookrightarrow R_{pq,s}.$

To extend to the whole $V_{prq}$, we use the collar structure and a cut-off
function on $V_{prq}$ which is supported in the collar region near
$\partial^{prsq}V_{prq}$ and which only depends on the collar coordinate.

  2. (2)

We define $\phi_{rs\to pq}$. Again, we write $\phi_{rs\to
pq}=\phi_{rs,r}\oplus\phi_{rs,s}\oplus 0$. The first component $\phi_{rs,r}$
is induced from the bundle map

$\phi_{rs\to rq}:R_{rs\to rq}\to R_{rq}$

composed with the linear inclusion
${\bm{R}}_{rq}\hookrightarrow{\bm{R}}_{rq}^{pq}\hookrightarrow{\bm{R}}_{pq,r}$;
the second component $\phi_{rs,s}$ is induced from the bundle map

$\phi_{rs\to ps}:R_{rs\to ps}\to R_{ps}$

composed with the linear inclusion
${\bm{R}}_{ps}\hookrightarrow{\bm{R}}_{ps}^{pq}\hookrightarrow{\bm{R}}_{pq,s}$.

  3. (3)

The definitions of $\phi_{ps\to pq}$ and $\phi_{rq\to pq}$ is similar to the
case of $\phi_{pr\to pq}$ and $\phi_{sq\to pq}$. We only describe in detail
the bundle map $\phi_{ps\to pq}$ (which should be defined over $V_{psq}$). We
write $\phi_{ps\to pq}$ as $\phi_{ps,r}\oplus\phi_{ps,s}\oplus 0$. The second
component $\phi_{ps,s}$ is naturally induced from the linear inclusion
${\bm{R}}_{ps}\hookrightarrow{\bm{R}}_{ps}^{pq}\hookrightarrow{\bm{R}}_{pq,s}$,
For the first component $\phi_{ps,r}$, we decompose it further as

$\phi_{ps,r}=\phi_{ps,pr}\oplus\phi_{ps,rs}.$

We first define it over the boundary stratum $\partial^{prsq}V_{psq}$ and then
use a cut-off function to turn it off as we leave this stratum towards the
interior of $V_{psq}$. Consider the composition of orthogonal projections

${\bm{R}}_{ps}\to{\bm{R}}_{pr}^{ps}\oplus{\bm{R}}_{rs}^{ps}\to{\bm{R}}_{pr}\oplus{\bm{R}}_{rs}$

which induces a bundle map

$R_{ps\to pq}|_{V_{prsq}}\to R_{pr\to pq}|_{V_{prsq}}\oplus R_{rs\to pq}.$

Then using the linear inclusions ${\bm{R}}_{pr}\to{\bm{R}}_{pr}^{pq}$ and the
map $R_{rs\to rq}\to R_{rq}\to R_{rq}^{pq}$, one defines the map $\phi_{ps,r}$
over the closed set $V_{prsq}$. One can then equivariantize, stabilize to
define it over $\partial^{prsq}V_{psq}$. Lastly, use a cut-off function to
extend this component to a neighborhood.

######  Remark 6.30.

We put effort on defining the embeddings maps between charts in the above
discussions, which contains certain distinct features than the case with fewer
strata because the bundle maps are no longer linear. We would like to remark
that the compatibility between the stabilized smoothings of $V_{prq}$ and
$V_{psq}$ near their common stratum $V_{prsq}$ is already guaranteed by the
inductive nature of our construction.

The above inductive strategy, especially the case with codimension two strata,
indicates that it seems difficult to construct a stabilization of K-chart
lifts of a flow category using purely linear maps between representations.
This brings in an extra layer of complexity in the following smoothing process
because we would like the bundle embeddings also to be smooth.

###  6.3. The main theorem about smoothing

We state the main theorem on smoothing the outer-collared K-chart lift of
$(T^{\rm Floer})^{+}$. In the following, a smoothing on a
${\mathbb{A}}$-manifold $V$ equipped with a collar structure (Definition 3.26)
is called a collared smoothing if the smoothing is equal to the product of the
standard smooth structure on $[0,\epsilon)^{{\mathbb{F}}_{\alpha}}$ and a
smoothing on $\partial^{\alpha}V$ over the collar region
$\partial^{\alpha}V\times[0,\epsilon)^{{\mathbb{F}}_{\alpha}}$. The same
notion is also used for a smoothing of a vector bundle.

######  Theorem 6.31.

Given the outer-collared K-chart lift ${\mathfrak{K}}^{+}$ of the outer-
collared flow category $(T^{{\rm Floer}})^{+}$ equipped with a collared
scaffolding (see Proposition 5.64), there exist the following objects. We
revome the “$+$” to make the notations more succinct.

  1. (1)

A stabilization of ${\mathfrak{K}}$ (see Definition 6.25), given by a
collection of orthogonal $G_{pq}$-spaces ${\bm{R}}_{pq}$ and a collection of
bundle embeddings.

  2. (2)

A collared $G_{pq}$-smoothing on
$\widehat{V}_{pq}:=V_{pq}\times{\bm{R}}_{pq}$.

  3. (3)

A collared bundle $G_{pq}$-smoothing on the $O(n)$-bundle
$\widehat{O}_{pq}\to\widehat{V}_{pq}$, where $\widehat{O}_{pq}$ is the
pullback of $O_{pq}$ under the natural projection $\widehat{V}_{pq}\to
V_{pq}$.

  4. (4)

A collared $G_{pq}$-invariant inner product on the trivial bundle $R_{pq}\to
V_{pq}$ inducing an inner product on the vector bundle
$\widehat{R}_{pq}\to\widehat{V}_{pq}$, where $\widehat{R}_{pq}$ is the
pullback of $R_{pq}$ under the natural projection $\widehat{V}_{pq}\to
V_{pq}$.

  5. (5)

A collared $G_{pq}$-smoothing on the $O(n)$-bundles $\widehat{R}_{pq}$.

These objects satisfy the following conditions.

  1. (A)

The smoothing of $\widehat{V}_{pq}$ induces a smooth bundle structure
$\widehat{Q}_{pq}$ as it is induced from the $G_{pq}$-representation
${\bm{Q}}_{pq}$. Then the smooth structures on $\widehat{O}_{pq}$ and
$\widehat{R}_{pq}$ induce a smooth structure on the obstruction bundle

$\widehat{E}_{pq}=\widehat{O}_{pq}\oplus\widehat{Q}_{pq}\oplus\widehat{R}_{pq}.$

  2. (B)

Boundary smoothings are given by products and stabilizations. More precisely,
this means the following. For any stratum $\alpha=pr_{1}\cdots r_{l}q$ of
${\mathbb{A}}^{\rm Floer}_{pq}$, the smoothings on $\widehat{V}_{pr_{1}}$,
$\ldots$, $\widehat{V}_{r_{l}q}$ induce a smooth structure on the prodcut
$\widehat{V}_{\alpha}=\widehat{V}_{pr_{1}}\times\cdots\times\widehat{V}_{r_{l}q}$
and its equivariantization $\widehat{V}_{\alpha}^{\sim}$; the smooth bundle
structures on $\widehat{O}_{pr_{1}}$, $\ldots$, $\widehat{O}_{r_{l}q}$ induce
a smooth structure on the bundle
$\widehat{O}_{\alpha}^{\sim}\to\widehat{V}_{\alpha}^{\sim}$; there is also a
canonical smooth bundle structure on $R_{\alpha}^{\sim}$ as it is induced from
trivial bundles. Then

    1. (a)

The embedding
$\widehat{V}_{\alpha}\hookrightarrow\partial^{\alpha}\widehat{V}_{pq}$ is
smooth.

    2. (b)

The bundle embedding
$\widehat{O}_{\alpha}\hookrightarrow\widehat{O}_{pq}|_{\partial^{\alpha}\widehat{V}_{pq}}$
and the bundle embedding
$\widehat{R}_{\alpha}\hookrightarrow\widehat{R}_{pq}|_{\partial^{\alpha}\widehat{V}_{pq}}$
are smooth bundle embeddings. It follows that the orthogonal complement
$\widehat{O}_{pq,\alpha}$ and the orthogonal complement
$\widehat{R}_{pq,\alpha}$ are smooth bundles over $\widehat{V}_{\alpha}$. It
follows that the bundle

$\widehat{F}_{pq,\alpha}=\widehat{O}_{pq,\alpha}\oplus\widehat{Q}_{pq,\alpha}\oplus\widehat{R}_{pq,\alpha}\to\widehat{V}_{\alpha}$

is smooth, where the smooth structure on $\widehat{Q}_{pq,\alpha}$ is induced
from the orthogonal complement of the $G_{pq}$-subrepresentation
${\bm{Q}}_{\alpha}\hookrightarrow{\bm{Q}}_{pq}$.

    3. (c)

The stabilization map

$\widehat{\theta}_{pq,\alpha}:{\rm
Stab}_{\widehat{F}_{pq,\alpha}}\Big{(}\widehat{V}_{\alpha}\Big{)}\to\partial^{\alpha}V_{pq}$

(which is the $\theta_{pq,\alpha}$ in (5.51) lifted) is a smooth embedding.

    4. (d)

Let
$\widehat{\pi}_{pq,\alpha}:\partial^{\alpha}\widehat{V}_{pq}\to\widehat{V}_{\alpha}^{\sim}$
be the projection map induced from the stabilization map above, which is
smooth. Then bundle isomorphism

$\widehat{\vartheta}_{pq,\alpha}:\widehat{\pi}_{pq,\alpha}^{*}\Big{(}\widehat{O}_{pq}|_{\widehat{V}_{\alpha}}\Big{)}\to\widehat{O}_{pq}|_{\partial^{\alpha}\widehat{V}_{pq}}$

(included in the scaffolding) is smooth.

Assuming this theorem, it is straightforward to construct a derived orbifold
lift of the flow category $(T^{\rm Floer})^{+}$ by taking group quotients.
Remove the $\widehat{\cdot}$ from the notation. Define

$C_{pq}=({\mathcal{U}}_{pq},{\mathcal{E}}_{pq},{\mathcal{S}}_{pq},\psi_{pq})$

where ${\mathcal{U}}_{pq}=V_{pq}/G_{pq}$, ${\mathcal{E}}_{pq}=E_{pq}/G_{pq}$,
${\mathcal{S}}_{pq}:{\mathcal{U}}_{pq}\to{\mathcal{E}}_{pq}$ is the induced
orbibundle section, and
$\psi_{pq}:{\mathcal{S}}_{pq}^{-1}(0)\to\overline{\mathcal{M}}{}_{pq}^{+}$ is
the induced footprint map. Moreover, for $\alpha=pr_{1}\cdots r_{l}q$, one has
the natural identification

$K_{\alpha}/G_{\alpha}\cong C_{\alpha}:=C_{pr_{1}}\times\cdots\times
C_{r_{l}q}.$

The scaffolding in the K-chart sense also descends to a scaffolding in the
D-chart sense. Indeed, for $\alpha,\beta\in{\mathbb{A}}_{pq}^{\rm Floer}$ with
$\alpha\leq\beta$, the bundle

${\mathcal{F}}_{\beta\alpha}:=F_{\beta\alpha}/G_{\alpha}\to{\mathcal{U}}_{\alpha}$

is a smooth orbifold vector bundle over ${\mathcal{U}}_{\alpha}$. Then the
stabilization map ${\rm
Stab}_{F_{\beta}\alpha}(V_{\alpha})\to\partial^{\alpha}V_{\beta}$ induces

${\bm{\theta}}_{\beta\alpha}:{\rm
Stab}_{{\mathcal{F}}_{\beta\alpha}}(C_{\alpha})\to\partial^{\alpha}C_{\beta}.$

One checkes directly that the data
$({\mathcal{F}}_{\beta\alpha},{\bm{\theta}}_{\beta\alpha})_{\alpha\leq\beta}$
form a scaffolding in the D-presentation of
$\overline{\mathcal{M}}{}_{pq}^{+}$. It should be pointed out that these
derived orbifold charts are effective due to the geometric construction.

######  Theorem 6.32.

The D-chart lift of $(T^{\rm Floer})^{+}$ induced from $\\{C_{pq}\\}$ equipped
with the compatible collar structure and scaffolding can be upgraded to an
oriented and normally complex derived orbifold lift by doing a further
stabilization of the corresponding K-chart lift.

###### Proof.

In the case of a single chart
$({\mathcal{U}}_{pq},{\mathcal{E}}_{pq},{\mathcal{S}}_{pq},\psi_{pq})$, recall
that ${\mathcal{U}}_{pq}$ is obtained from a quotient
$\widehat{V}_{pq}/G_{pq}$, such that

  1. (1)

The space $\widehat{V}_{pq}$ is equal to the product
$V_{pq}\times{\bm{R}}_{pq}$.

  2. (2)

There is a $C^{1}_{\rm loc}$ $G_{pq}$-bundle structure $\pi_{pq}:V_{pq}\to
B_{pq}$ which ensures that the tangent space of the smoothing of
$\widehat{V}_{pq}$ is isotopic to the direct sum

$R_{pq}\oplus T^{\rm vt}V_{pq}\oplus\pi^{*}TB_{pq},$ (6.8)

where we abuse the notation to denote the corresponding bundles obtained by
the pullback under the natural projection map $\widehat{V}_{pq}\to V_{pq}$.

The first simple obervation is that the stabilizing factors ${\bm{R}}_{pq}$
can be taken to be unitary representations. We can simply use
${\bm{R}}_{pq}\oplus{\bm{i}}{\bm{R}}_{pq}$ instead of a possibly real
representation ${\bm{R}}_{pq}$. Moreover, the bundle inclusions
$R_{\alpha}\oplus{\bm{i}}R_{\alpha}\hookrightarrow R_{pq}\oplus{\bm{i}}R_{pq}$
are then complex linear and the inner products on the trivial bundles
$R_{pq}\oplus{\bm{i}}R_{pq}$ become Hermitian.

We would like to do a further stabilization to the (already smoothed) K-chart
lift. Indeed, we stabilize by the $G_{pq}$-representation ${\bm{Q}}_{d_{pq}}$.
Let $\widehat{Q}_{pq}^{\prime}\to\widehat{V}_{pq}$ be the trivial bundle (to
be distinguished from the original $Q$-bundle). Notice that the bundle
inclusions $Q_{\delta}\hookrightarrow Q_{d}$ over $B_{\delta}\subset B_{d}$
for all $\delta\in{\mathbb{A}}_{d}$ is defined via a linear inclusion
${\bm{Q}}_{\delta}\to{\bm{Q}}_{d}$ the corresponding bundle. Then there are
natural bundle inclusions

$\widehat{Q}_{\alpha}^{\prime}\to\widehat{Q}_{pq}^{\prime}|_{\widehat{V}_{\alpha}}.$

As this bundle map is induced from linear map between representation spaces,
it is automatically smooth. Hence we obtained another smooth K-chart lift
whose K-charts are

$\widehat{\widehat{K}}_{pq}={\rm
Stab}_{\widehat{Q}_{pq}^{\prime}}\widehat{K}_{pq}.$

It induces a new D-chart lift.

Now we consider the normal complex structures on this new D-chart lift.
Indeed, the obstruction bundles are now all complex. Indeed,

$\widehat{\widehat{E}}_{pq}=\widehat{\widehat{O}}_{pq}\oplus\widehat{\widehat{R}}_{pq}\oplus\widehat{\widehat{Q}}_{pq}\oplus\widehat{\widehat{Q}}{}_{pq}^{\prime}$

and we give the last two summands the natural complex structure.

Now the quotient orbifold

$\widehat{\widehat{V}}_{pq}/G_{pq}$

is normally complex. Indeed, (6.8) becomes

$T\widehat{\widehat{V}}_{pq}\cong\big{(}R_{pq}\otimes{\mathbb{C}}\big{)}\oplus
T^{\rm
vt}V_{pq}\oplus(\pi_{pq}^{*}TB_{d_{pq}}\oplus\widehat{\widehat{Q}}{}_{d_{pq}}^{\prime}).$

As shown in Proposition 5.27, the quotient of the last summand above is
normally complex. Moreover, by Lemma 6.23 and Corollary 6.24, the quotient of
$T^{\rm vt}V_{pq}$ is also normally complex.

To obtain a normal complex structure on the D-chart lift, we further check:

  1. (1)

A complex structure on the scaffolding. Indeed, the original difference
bundles

$\widehat{F}_{pq,\alpha}=\widehat{O}_{pq,\alpha}\oplus\widehat{Q}_{pq,\alpha}\oplus\widehat{R}_{pq,\alpha}$

is already complex as explained below. First, the bundle
$\widehat{O}_{pq,\alpha}$ is complex because we can choose the metric on the
$O$-bundles to be Hermitian. The factor $Q_{pq,\alpha}$ is complex because it
consists of “off-diagonal” Hermitian matrices. As mentioned above, the
stabilization in the smoothing can be chosen to be complex representations,
therefore $\widehat{R}_{pq,\alpha}$ also has a complex structure. Then the
further stabilization by ${\bm{Q}}_{d_{pq}}$ gives an extra copy of
$\widehat{Q}_{pq,\alpha}$ which is complex as well.

  2. (2)

Compatibility between the information on the product boundary charts and the
information on the restriction of the chart to the boundary. This follows from
the constructions [AB21, Section 11] by incorporating the geometry of the
broken trajectories from the thick-and-thin decomposition of the domains.

  3. (3)

The compatibility with collars follows from the construction.

By examing conditions of Definition 3.36 we obtained a normal complex
structure on this (further stabilized) D-chart lift of $(T^{\rm Floer})^{+}$.

For any capped orbit $p\in T^{\rm Floer}$, we define its associated virtual
vector space to be

$(V_{p}^{+},V_{p}^{-}):=(V_{\underline{\smash{p}}}^{+},V_{\underline{\smash{p}}}^{-}),$

where the right hand side is constructed before Lemma 6.23. Then the existence
of orientation (Definition 3.43) is a classical result in Floer theory [FH93].
For a modern exposition, the reader could refer to [Par16, Section C.13]. ∎

####  6.3.1. Constructing straightening

We sketch how to construct a straightening on the D-chart lift which is
compatible with the existing collar and scaffolding structures. This
construction relies on the concrete geometric feature of the obstruction
bundles. We do not know if instead the straightening can be constructed in a
more abstract level.

We start with moduli spaces without boundary or corners. Let $C_{pq}$ be the
D-chart. Then by the method of [BX22, Lemma 3.15, Lemma 3.20], one can
construct a straightening on the pair
$({\mathcal{U}}_{pq},{\mathcal{E}}_{pq})$. Indeed, notice that there are a
sequence of obstruction bundles ${\mathcal{E}}_{pq}^{(d)}$ for $d\geq d_{pq}$.
We can actually construct, not just a connection on
${\mathcal{E}}_{pq}^{(d_{0})}$ for the lowest $d_{0}=d_{pq}$ (which is
required for straightening), but actually a sequence of metrics on all
${\mathcal{E}}_{pq}^{(d)}$ and metric connections such that the natural
inclusion
${\mathcal{E}}_{pq}^{(d)}\hookrightarrow{\mathcal{E}}_{pq}^{(d^{\prime})}$
(for $d<d^{\prime}$) preserves the metric and connection.

Inductively, suppose we have constructed such compatible structures for all
$C_{rs}$ for $d_{rs}<d_{pq}$. These structures induce metrics and metric
connections on the difference bundles ${\mathcal{F}}_{\beta\alpha}$. Now
consider the chart $C_{pq}$. Over a lowest stratum $\alpha=pr_{1}\cdots
r_{l}q$, the existing structures induce metrics and metric connections on
${\mathcal{E}}_{\beta}^{(d)}|_{{\mathcal{U}}_{\alpha}}$ for all $d\geq d_{pq}$
and proper strata between $\alpha$ and $pq$. We would like to extend the
metric and connection to ${\mathcal{E}}_{pq}^{(d)}$ over
${\mathcal{U}}_{\alpha}$ for all $d\geq d_{pq}$. Starting from the lowest
$d_{0}=d_{pq}$. Properties proved in Lemma 5.54 shows that we can first extend
the metric and metric connection to the sum ${\mathcal{E}}_{pr_{l}\to
pq}^{(d_{0})}+{\mathcal{E}}_{r_{1}q\to pq}^{(d_{0})}$. This will be compatible
with all existing metrics and connections. Then we extend (arbitrarily but
satisfying the requirement for straightening) to
${\mathcal{E}}_{pq}^{(d_{0})}|_{{\mathcal{U}}_{\alpha}}$. Then it induces a
metric and connection on the difference bundle ${\mathcal{F}}_{pq,\alpha}$.
Via the stabilization map from ${\mathcal{U}}_{\alpha}$ to
$\partial^{\alpha}{\mathcal{U}}_{pq}$, one obtains a metric on
$\partial^{\alpha}{\mathcal{U}}_{pq}$ which is straightened. Next, we can
inductively construct metrics and connections on ${\mathcal{E}}_{pq}^{(d)}$
for higher $d$. Once we finished for the lowest stratum $\alpha$, we can use
the collar structure to extend to all nearby higher strata. For a next stratum
$\beta$, we carry out the induction from $d_{0}$ above again. We omit the
details.

###  6.4. Proof of Theorem 6.31

To save notations, we remove all the superscript “+” which indicates the
outer-collaring. Instead, we keep in mind that all objects and structures have
corresponding collar structures or respect the collar structure.

####  6.4.1. Smoothing charts without boundary

Consider a moduli space $\overline{\mathcal{M}}_{pq}$ of Floer trajectories
which has no codimension one strata. Let
$K_{pq}=(G_{pq},V_{pq},E_{pq},S_{pq})$ be the global Kuranishi chart
constructed. Then $V_{pq}$ has no boundary or corner and one can apply
Lashof’s smoothing theory directly. In this case there is no outer-collaring
to the chart. The projection $\pi_{pq}:V_{pq}\to B_{pq}$ is a
$G_{pq}$-equivariant $C^{1}_{\rm loc}$ fiberwise smooth topological
submersion. Then by the existence of $G$-equivariant fiberwise submersions
(Lemma 6.21), one can choose a $G$-equivariant fiberwise submersion of
$\pi_{pq}$ which induces an isotopy class of $G_{pq}$-vector bundle reductions
of $T_{\mu}V_{pq}$. By Lashof’s theorem (Theorem 6.9) on stable
$G$-smoothings, there exists a stable $G_{pq}$-smoothing on $V_{pq}$, i.e., a
finite-dimensional orthogonal representation ${\bm{R}}_{pq}$ of $G_{pq}$ and a
$G_{pq}$-invariant smooth structure on the product

$\widehat{V}_{pq}:=V_{pq}\times{\bm{R}}_{pq}.$

Let $\widehat{\pi}_{pq}:\widehat{V}_{pq}\to B_{pq}$ be composition
$\widehat{V}_{pq}\to V_{pq}\to B_{pq}$. Then there is an isomorphism of
$G_{pq}$-equivariant vector bundles

$T\widehat{V}_{pq}\cong T^{\rm
vt}\widehat{V}_{pq}\oplus\widehat{\pi}_{pq}^{*}TB_{pq}.$

(However, the projection $\widehat{\pi}_{pq}$ may not be smooth and the fibers
may not be smooth submanifolds.)

Moreover one can give smooth structures on the obstruction bundles. Indeed,
one can inductively construct a structure of smooth equivariant $O(n)$-bundles
on $\widehat{O}_{pq}^{(d)}\to\widehat{V}_{pq}$ for all $d\geq d_{pq}$ such
that $\widehat{O}_{pq}^{(d)}$ is a smooth subbundle of
$\widehat{O}_{pq}^{(d+1)}$ by smoothing the relevant equivariant classifying
map. The bundle $\widehat{R}_{pq}\to\widehat{V}_{pq}$, which is the trivial
bundle with fiber ${\bm{R}}_{pq}$ automatically has a smooth structure. We
also use the inner product on ${\bm{R}}_{pq}$ to equip the bundle
$\widehat{R}_{pq}$ with an inner product structure, which is a smooth inner
product.

####  6.4.2. Smoothing charts with boundary and corners

Now consider a chart $K_{pq}=(G_{pq},V_{pq},E_{pq},S_{pq})$. We have assumed
that $K_{pq}$ has a collar structure which was actually constructed via outer-
collaring. We need to construct a stable smoothing which is of “product type”
near the boundary and which extends the existing stable smoothings. We first
state the induction hypothesis. As an initial remark, we induct on the energy
of the moduli spaces $d_{pq}$.

Induction Hypothesis I. Stabilization. Suppose we have the following data.

  1. (1)

An orthogonal $G_{rs}$-representation ${\bm{R}}_{rs}$ for each $rs$ with
$d_{rs}<d_{pq}$. We define a bundle

$R_{uv\to rs}^{\sim}\to\partial^{ruvs}V_{rs}\ {\rm whenever}\
ruvs\in{\mathbb{A}}_{rs}^{\rm Floer}$

as follows. Consider the product $V_{ruvs}=V_{ru}\times V_{uv}\times V_{vs}$.
Then the trivial bundle $R_{uv}=V_{uv}\times{\bm{R}}_{uv}\to V_{uv}$ is pulled
back to $V_{ruvs}$. Via the stabilization map ${\rm
Stab}_{F_{rs,ruvs}}(V_{ruvs})\hookrightarrow\partial^{ruvs}V_{rs}$ this
trivial bundle is pulled back to a trivial bundle. Then using the
$G_{rs}$-equivariantization we obtain a not-necessarily-trivial vector bundle

$R_{uv\to rs}^{\sim}\to\partial^{ruvs}V_{rs}\cong G_{rs}\times_{G_{ruvs}}({\rm
Stab}_{F_{rs,ruvs}}(V_{ruvs})).$

  2. (2)

A $G_{rs}$-equivariant bundle embedding

$\phi_{uv\to rs}:R_{uv\to rs}\to R_{rs}|_{\partial^{ruvs}V_{rs}}.$

  3. (3)

For each $\alpha=ru_{1}\cdots u_{l}s\in{\mathbb{A}}_{rs}^{\rm Floer}$, define
a bundle map

$\phi_{rs,\alpha}:R_{\alpha}:=R_{ru_{1}}\boxplus\cdots\boxplus R_{u_{l}s}\to
R_{rs}$

to be the sum of the maps $\phi_{ru_{1}\to rs},\ldots,\phi_{u_{l}s\to rs}$. We
require that the bundle map is actually a bundle embedding. This induces
bundle embeddings

$\phi_{\beta\alpha}:R_{\alpha}\to R_{\beta}\ \alpha\leq\beta,\
\alpha,\beta\in{\mathbb{A}}_{rs}^{\rm Floer}.$

We require that the collection $\\{R_{rs}\\}$ and the collection
$\\{\phi_{\beta\alpha}\\}$ satisfy the requirements for stabilization of a
K-chart lift (see Definition 6.25) up to level $d_{pq}$, i.e., the K-charts
for moduli spaces $\overline{\mathcal{M}}_{rs}^{\rm Floer}$ with
$d_{rs}<d_{pq}$ are equipped with stabilizations induced from $R_{rs}$ with
compatibility conditions satisfied.

  4. (4)

We also assume we have a $G_{rs}$-invariant inner product on the trivial
bundle $R_{rs}\to V_{rs}$ for all $d_{rs}<d_{pq}$ such that the bundle maps
$\phi_{rs,\alpha}$ are all isometric.

Moreover, we assume that the representations ${\bm{R}}_{rs}$ and bundle
embeddings $\phi_{uv\to rs}$ are of the following particular forms. (One can
see the discussion in Section 6.2.4 for why we impose the following
requirement.)

  1. (1)

We have a $G_{rs}$-equivariant orthogonal decomposition

${\bm{R}}_{rs}={\bm{R}}_{rs,-}\oplus{\bm{R}}_{rs,0}$

and a $G_{rs}$-equivariant orthogonal decomposition

${\bm{R}}_{rs,-}=\bigoplus_{r<w<s}{\bm{R}}_{rs,w}=\bigoplus_{r<w<s}{\bm{R}}_{rw}^{rs}\oplus{\bm{R}}_{ws}^{rs}.$

Intuitively, ${\bm{R}}_{rs,-}$ comes from lower stratum via Frobenius
reciprocity, and ${\bm{R}}_{rs,0}$ is chosen so that a relative smoothing
exists. Then one has the (trivial) bundles $R_{rs,-}$, $R_{rs,0}$,
$R_{rw}^{rs}$, $R_{ws}^{rs}$ over $V_{rs}$.

  2. (2)

Whenever $r<w<s$, the representations ${\bm{R}}_{rw}^{rs}$ resp.
${\bm{R}}_{ws}^{rs}$ contains ${\bm{R}}_{rw}$ resp. ${\bm{R}}_{ws}$ as
orthogonal subrepresentations of $G_{rw}$ resp. $G_{ws}$.

  3. (3)

For all $uv\prec rs$ (which implies $r\leq u<v\leq s$), abbreviate
$\phi_{uv}=\phi_{uv\to rs}$. Then the range of $\phi_{uv}$ is contained in
$R_{rs,-}$ and can be written as

$\phi_{uv}=\bigoplus_{r<w<s}\phi_{uv,w},\ \phi_{uv,w}:R_{uv\to rs}\to
R_{rs,w}.$

Moreover, for each $w$, the map $\phi_{uv,w}$ satisfies the following
conditions. We write $\phi_{uv,w}:R_{uv\to rs}\to R_{rs,w}=R_{rw}^{rs}\oplus
R_{ws}^{rs}$ as the sum $\phi_{uv,rw}\oplus\phi_{uv,ws}$.

    1. (a)

If $w\leq u$, then $\phi_{uv,rw}=0$. Moreover, over the stratum
$V_{rwuvs}\subset V_{ruvs}$, $\phi_{uv,ws}$ is induced from the bundle map

$\textstyle{R_{uv\to
ws}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi_{uv\to
ws}}$$\textstyle{R_{ws}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{V_{wuvs}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{V_{ws}}$

and the linear inclusion ${\bm{R}}_{ws}\hookrightarrow{\bm{R}}_{ws}^{rs}$.

    2. (b)

If $w\geq v$, then $\phi_{uv,ws}=0$. Moreover, over the stratum
$V_{ruvws}\subset V_{ruvs}$, $\phi_{uv,rw}$ is induced from the bundle map

$\textstyle{R_{uv\to
rw}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi_{uv\to
rw}}$$\textstyle{R_{rw}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{V_{ruvw}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{V_{rw}}$

and the linear inclusion ${\bm{R}}_{rw}\hookrightarrow{\bm{R}}_{rw}^{rs}$.

    3. (c)

If $u<w<v$, then the following is true. Consider the inclusion

${\bm{R}}_{uw}\oplus{\bm{R}}_{wv}\hookrightarrow{\bm{R}}_{uw}^{uv}\oplus{\bm{R}}_{wv}^{uv}\hookrightarrow{\bm{R}}_{uv}.$

The orthogonal projection

${\bm{R}}_{uv}\to{\bm{R}}_{uw}\oplus{\bm{R}}_{wv}$

is $G_{uwv}$-equivariant, hence induces a bundle map

$R_{uv}|_{V_{uwv}}\to R_{uw}\boxplus R_{wv}.$

On the other hand, there are bundle embeddings

$\displaystyle\ R_{uw\to rw}\hookrightarrow R_{rw}\hookrightarrow
R_{rw}^{rs},\ $ $\displaystyle\ R_{wv\to ws}\hookrightarrow
R_{ws}\hookrightarrow R_{ws}^{rs}.$

Then over the stratum $V_{ruwvs}\subset V_{ruvs}$, the map $\phi_{uv,w}$ is
the composition

$R_{\underline{\smash{uv\to rs}}}|_{V_{\underline{\smash{ruwvs}}}}\to
R_{\underline{\smash{uw\to rs}}}\oplus R_{\underline{\smash{wv\to rs}}}\to
R_{rw}^{rs}\oplus R_{ws}^{rs}.$

    4. (d)

If none of the above happends, then $\phi_{uv,w}=0$.

Moreover, the bundles and bundle maps will be added with $\widehat{\cdot}$
when lifted to the stabilizations $\widehat{V}_{rs}$. Such lifts are
canonical.

Induction Hypothesis II. Fiberwise submersion. Whenever $d_{rs}<d_{pq}$, we
have chosen a $G_{rs}$-equivariant fiberwise submersion (Definition 6.18) of
$\widehat{\pi}_{rs}:\widehat{V}_{rs}\to B_{rs}$, which is a microbundle
isomorphism

$T_{\mu}\widehat{V}_{rs}\cong T^{\rm
vt}_{\mu}\widehat{V}_{rs}\oplus\widehat{\pi}_{rs}^{*}T_{\mu}B_{d_{rs}}$

which is the canonical projection map on the second component. Together with
the structure of $C_{\rm loc}^{1}$ fiberwise smooth $G_{rs}$-bundle on
$\widehat{\pi}_{rs}$, this provides an isotopy class of vector bundle lifts

$T^{\rm vt}\widehat{V}_{rs}\oplus\widehat{\pi}_{rs}^{*}TB_{d_{rs}}\to
T_{\mu}\widehat{V}_{rs},$ (6.9)

where the isotopy class is fixed by Equation (6.4).

Induction Hypothesis III. Stable smoothings and bundle smoothings. Whenever
$d_{rs}<d_{pq}$, we have chosen a $G_{rs}$-invariant smoothing on
$\widehat{V}_{rs}:=V_{rs}\times{\bm{R}}_{rs}$ in the $G_{rs}$-isotopy class
corresponding to stable isotopy class of the vector bundle lift (6.9).
Moreover, we have chosen a structure of smooth $G_{rs}$-equivariant
$O(n)$-bundle on $\widehat{O}_{rs}^{(d)}\to\widehat{V}_{rs}$ for all $d\geq
d_{rs}$ and a structure of smooth $G_{rs}$-equivariant $O(n)$-bundle on
$\widehat{R}_{rs}$.

Before we state the conditions about these structures, we summarize a few
consequences of these induction hypotheses. First, the stable smoothings
induce, for each stratum $\alpha=pr_{1}\cdots r_{l}q$, a
$G_{\alpha}$-equivariant smooth structure on the product

$\widehat{V}_{\alpha}=\widehat{V}_{pr_{1}}\times\cdots\times\widehat{V}_{r_{k}q}$

and hence a $G_{pq}$-equivariant smooth structure on
$\widehat{V}_{\alpha}^{\sim}$ by taking the equivariantization. Second, the
smooth structures on the vector bundles induce smooth structures on the
product bundles $\widehat{O}_{\alpha}$ and $\widehat{R}_{\alpha}$. Third, as
the bundle $Q_{rs}\to V_{rs}$ is trivial, the pullback
$\widehat{Q}_{rs}\to\widehat{V}_{rs}$ is automatically smooth. Similarly, the
bundle

$\widehat{Q}_{rs,\alpha}=(\widehat{Q}_{\alpha})^{\bot}\to\widehat{V}_{\alpha}$

is smooth because it comes from a product bundle.

Now we state the conditions.

  1. (1)

The bundle embedding

$\widehat{\phi}_{\beta\alpha}:\widehat{R}_{\alpha}\to\widehat{R}_{\beta}|_{\partial^{\alpha}\widehat{V}_{\beta}}$

is smooth.191919This is not automatic as this bundle embedding is not purely
induced from linear maps between representation spaces. This implies that the
orthogonal complement

$\widehat{R}_{rs,\alpha}\to\widehat{V}_{\alpha}$

is a smooth bundle.

  2. (2)

Whenever $d^{\prime}>d\geq d_{rs}$, the natural bundle embedding
$\widehat{O}_{rs}^{(d)}\to\widehat{O}_{rs}^{(d^{\prime})}$ induced from (5.35)
is smooth.

  3. (3)

For all $d\geq d_{rs}$, the bundle embedding

$\widehat{\phi}_{uv\to rs}:\widehat{O}_{uv\to
rs}^{(d)}\to\widehat{O}_{rs}^{(d)}|_{\partial^{ruvs}V_{rs}}$

is smooth. This implies that (when $d=d_{rs}$) the orthogonal complement

$\widehat{O}_{rs,\alpha}\to V_{\alpha}$

is a smooth bundle.

  4. (4)

It follows that

$\widehat{F}_{rs,\alpha}:=\widehat{O}_{rs,\alpha}\oplus\widehat{Q}_{rs,\alpha}\oplus\widehat{R}_{rs,\alpha}$

over the smooth manifold with corners $\widehat{V}_{\alpha}$ is smooth. Then
we require that the stabilization map

$\widehat{\theta}_{rs,\alpha}:{\rm
Stab}_{\widehat{F}_{rs,\alpha}}(\widehat{V}_{\alpha})\to\partial^{\alpha}\widehat{V}_{rs}$

is a germ of diffeomorphisms onto an open set. These structures ensure that
the projection map

$\widehat{\pi}_{rs,\alpha}:\partial^{\alpha}V_{rs}\to\widehat{V}_{\alpha}^{\sim}$

is smooth.

  5. (5)

For all $d\geq d_{rs}$, the bundle isomorphism

$\widehat{\psi}_{rs,\alpha}:\widehat{\pi}_{rs,\alpha}^{*}\Big{(}\widehat{O}_{rs}^{(d)}|_{\widehat{V}_{\alpha}^{\sim}}\Big{)}\to\widehat{O}_{rs}^{(d)}|_{\partial^{\alpha}V_{rs}}$

is smooth.

  6. (6)

The existing smoothings (on both domains and bundles) respect the corner
structures. Namely, for all $rs$ with $d_{rs}<d_{pq}$, item (B) of Theorem
6.31 is satisfied.

Now we start to construct the stabilized smoothings corresponding to the
Kuranishi chart $K_{pq}$.

Inductive Construction I. We construct an orthogonal $G_{pq}$-space
${\bm{R}}_{pq}$ (which defines the trivial bundle $R_{pq}\to V_{pq}$) and
bundle embeddings

$\phi_{rs\to pq}:R_{rs\to pq}\to R_{pq}|_{\partial^{prsq}V_{pq}}$

which, together with the existing stabilizations and bundle embeddings,
satisfy the conditions required for stabilizations.

  1. (1)

Firstly, we need to find an orthogonal $G_{pq}$-representation ${\bm{R}}_{pq}$
such that the stabilization $R_{pq}=V_{pq}\times{\bm{R}}_{pq}$ receives
embeddings from $R_{\alpha}$ for all $\alpha\in{\mathbb{A}}^{\rm Floer}_{pq}$.

To start, for each $p<w<q$, consider the $G_{pw}$-representation
${\bm{R}}_{pw}$ and the $G_{wq}$-representation ${\bm{R}}_{wq}$ granted by the
induction hypothesis. By Frobenius reciprocity, there exist an orthogonal
$G_{pq}$-representation ${\bm{R}}_{pw}^{pq}$ which contains ${\bm{R}}_{pw}$ as
a subrepresentation of $G_{pw}$, and an orthogonal $G_{pq}$-representation
${\bm{R}}_{wq}^{pq}$ which contains ${\bm{R}}_{wq}$ as a subrepresentation of
$G_{wq}$. Then define

${\bm{R}}_{pq}:=\bigoplus_{p<w<q}{\bm{R}}_{pw}^{pq}\oplus{\bm{R}}_{wq}^{pq}$
(6.10)

which is an orthogonal $G_{pq}$-space. Now we define the bundle embeddings

$\phi_{rs\to pq}:R_{rs\to pq}\to R_{pq}|_{\partial^{prsq}V_{pq}}.$

We abbreviate the decomposition 6.10 as

${\bm{R}}_{pq}=\bigoplus_{w}{\bm{R}}_{pq,w}=\bigoplus_{p<w<q}{\bm{R}}_{pw}^{pq}\oplus{\bm{R}}_{wq}^{pq}.$

Abbreviate the bundle embedding to be defined by $\phi_{rs}$. Then we define
it to be the direct sum

$\phi_{rs}=\bigoplus_{w}\phi_{rs,w}=\bigoplus_{w}\phi_{rs,pw}\oplus\phi_{rs,wq}\
{\rm where}\ \phi_{rs,w}:R_{rs\to pq}\to R_{pq,w}|_{\partial^{prsq}V_{pq}}.$

In fact, for each $w$, one can define the restriction of $\phi_{rs,w}$ to the
corresponding stratum involving $w$, such as $pwrsq$, $prswq$, or $prwsq$ in
the form described in Induction Hypothesis I. These are all codimension-1
boundary strata of $prsq$. Then using the collar structure of existing objects
and a cut-off function which only depends on the collar coordinate, one can
turn off the corresponding maps $\phi_{rs,w}$ once we go away from this
corner. On the other hand, if $w=r$ or $w=s$, we just have a bundle embedding
coming from the induction hypothesis. One can check that the newly constructed
objects still satisfy the requirement for a stabilization up to level $d_{pq}$
(see Definition 6.25).

  2. (2)

Secondly, we would like to construct an inner product on $R_{pq}$.

We start from a minimal $\alpha=pr_{1}\cdots r_{l}q$. Notice that inside
$R_{pq}|_{\partial^{\alpha}V_{pq}}$, there are a collection of embedded
subbundles $R_{\beta\to pq}|_{\partial^{\alpha}V_{pq}}$ for all $\beta$ with
$\alpha\leq\beta<pq$ with an inner product equipped. We first check that, for
any pair $\beta,\beta^{\prime}$ of such strata, their inner products agree on
the overlap

$R_{\beta\to pq}\cap R_{\beta^{\prime}\to pq}\subset R_{pq}.$

In fact the intersection
$\partial^{\beta}V_{pq}\cap\partial^{\beta^{\prime}}V_{pq}$ is a deeper
stratum $\partial^{\beta\\#\beta^{\prime}}V_{pq}$. Here
$\beta\\#\beta^{\prime}$ stands for the stratum described by the word which
includes all the intermediate capped orbits in $\beta$ and $\beta^{\prime}$
between $p$ and $q$. We can in fact check from the explicit construction that

$R_{\beta\to pq}\cap R_{\beta^{\prime}\to pq}=R_{\beta\\#\beta^{\prime}\to
pq}$

Hence by induction hypothesis, the inner products agrees on overlaps.

Now there are two special codimension one strata, $\beta_{1}=pr_{1}q$ and
$\beta_{l}=pr_{l}q$. We can check that

$R_{\beta\to pq}\subset R_{\beta_{1}\to pq}+R_{\beta_{l}\to pq}.$

Hence we can define an inner product on the sum $R_{\beta_{1}\to
pq}+R_{\beta_{l}\to pq}$ which extends the existing ones. Then extend
arbitrarily to $R_{pq}$ over this stratum.

Inductively, we can use the collar structure near $\alpha$ to extend the inner
product on $\partial^{\alpha}V_{pq}$ to a neighborhood. Then the same argument
above can be applied to construct an inner product inductively on $R_{pq}$.

Inductive Construction II. We construct smooth bundle structures on the
obstruction bundles $\widehat{O}_{pq}^{(d)}\to\widehat{V}_{pq}$ for all $d\geq
d_{pq}$ over $\partial\widehat{V}_{pq}$. The construction is inductive on
strata $\alpha=pr_{1}\cdots r_{l}q$ of $pq$. We start with a minimal $\alpha$.
Then there are the vector bundles

$O_{\alpha}^{(d)}\to V_{\alpha}=V_{pr_{1}}\times\cdots\times V_{r_{l}q},\
\forall d\geq d_{pq}.$

Denote by its pullback to $\widehat{V}_{\alpha}$ by

$\widehat{O}_{\alpha}^{(d)}\to\widehat{V}_{\alpha}.$

The induction hypothesis granted a smooth $O(n)$-bundle structure on it which
is the product of each individual factor. Notice that this is a subbundle of
$\widehat{O}_{pq}^{(d)}|_{\widehat{V}_{\alpha}}$. We would like to extend this
smooth structure to a smooth structure on
$\widehat{O}_{pq}|_{\widehat{V}_{\alpha}}$.

Start with $d=d_{0}=d_{pq}$ and remember
$\widehat{O}_{pq}=\widehat{O}_{pq}^{(d_{0})}$. Consider any intermediate
stratum $\beta=ps_{1}\cdots s_{m}q$. Then one has the inclusion induced by
(5.54)

$\widehat{O}_{\alpha}^{(d_{0})}\subset\widehat{O}_{\beta}^{(d_{0})}|_{\widehat{V}_{\alpha}}\subset\widehat{O}_{pq}|_{\widehat{V}_{\alpha}}.$

Notice that by the induction hypothesis, the smooth structure on
$\widehat{O}_{\beta}^{(d_{0})}|_{\widehat{V}_{\alpha}}$ extends the smooth
structure on $\widehat{O}_{\alpha}^{(d_{0})}$. This requires that the smooth
structure on $\widehat{O}_{pq}|_{\widehat{V}_{\alpha}}$ needs to be an
extension of all such smooth structures. Indeed, consider two codimension one
strata, $pr_{1}q$ and $pr_{l}q$. Consider the two bundles

$\displaystyle\ \widehat{O}_{pr_{l}\to pq}^{(d_{0})}|_{\widehat{V}_{\alpha}},\
$ $\displaystyle\ \widehat{O}_{r_{1}q\to
pq}^{(d_{0})}|_{\widehat{V}_{\alpha}}.$

Abbreviate them temporarily by $\widehat{O}_{r_{l}}^{(d_{0})}$ and
$\widehat{O}_{r_{1}}^{(d_{0})}$ respectively. Then it is easy to see that for
any intermediate stratum $\beta$, one has

$\widehat{O}_{\beta}^{(d_{0})}|_{\widehat{V}_{\alpha}}\subset\widehat{O}_{r_{l}}^{(d_{0})}+\widehat{O}_{r_{1}}^{(d_{0})}.$
(6.11)

Then Lemma 6.33 below provides a smooth structure on the sum
$\widehat{O}_{r_{l}}^{(d_{0})}+\widehat{O}_{r_{1}}^{(d_{0})}$. Then we extend
this extension to $\widehat{O}_{pq}|_{\widehat{V}_{\alpha}}$ by choosing an
arbitrary smooth structure on its orthogonal complement.

######  Lemma 6.33.

Let $M$ be a smooth manifold and let $E\to M$ be a topological vector bundle
equipped with an inner product. Let $E_{1},E_{2}\subset E$ be subbundles such
that $E=E_{1}+E_{2}$. Moreover, suppose $E_{1}$ and $E_{2}$ are equipped with
structures of smooth vector bundles with respect to which the restriction of
the inner product is smooth, such that the intersection $E_{1}\cap E_{2}$ is
both a smooth subbundle of $E_{1}$ and a smooth subbundle of $E_{2}$. Then
there exists a unique smooth bundle structure of $E$ which extends the smooth
bundle structures on $E_{1}$ and $E_{2}$ such that the inner product is
smooth.

###### Proof.

Define a smooth vector bundle $E^{\prime}\to M$ from $E_{1}$ and $E_{2}$ as

$E^{\prime}:=E_{1}^{\prime}\oplus E_{2}^{\prime}\oplus(E_{1}\cap E_{2})$

where $E_{1}^{\prime}$ resp. $E_{2}^{\prime}$ is the orthogonal complement of
$E_{1}\cap E_{2}$ in $E_{1}$ resp. $E_{2}$. Then there is an canonical
isometric bundle isomorphism $E^{\prime}\to E$. ∎

As $\widehat{O}_{pq,\alpha}$ is the orthogonal complement of
$\widehat{O}_{\alpha}^{(d_{0})}$, the above smooth structure induces a smooth
structure on
$\widehat{O}_{pq}|_{\widehat{V}_{\alpha}}\to\widehat{V}_{\alpha}$, hence one
on $\widehat{O}_{pq,\alpha}\to\widehat{V}_{\alpha}$. On the other hand, run
the same argument as above, one can cook up a smooth $O(n)$-bundle structure
on $\widehat{R}_{pq}|_{\widehat{V}_{\alpha}}$ which extends existing ones.
Hence the orthogonal complement $\widehat{R}_{pq,\alpha}$ is a smooth
$O(n)$-bundle. The bundle $\widehat{Q}_{pq,\alpha}$, which is essentially
trivial, automatically carries a smooth structure. Hence the bundle

$\widehat{F}_{pq,\alpha}=\widehat{O}_{pq,\alpha}\oplus\widehat{Q}_{pq,\alpha}\oplus\widehat{R}_{pq,\alpha}$
(6.12)

is smooth. Then the stabilization map (see (5.51))

$\widehat{\theta}_{pq,\alpha}:{\rm
Stab}_{\widehat{F}_{pq,\alpha}}(\widehat{V}_{\alpha})\to\partial^{\alpha}\widehat{V}_{pq}$

and its equivariantization induce a smooth structure on the stratum
$\partial^{\alpha}\widehat{V}_{pq}$.

Next we can inductively construct smooth structures on the bundle
$\widehat{O}_{pq}^{(d)}|_{V_{pq,\underline{\smash{\alpha}}}}$ for all $d\geq
d_{0}=d_{pq}$. This is similar to the construction of inner products. We omit
the details. Using equivariantizations and the bundle isomorphisms coming from
the stabilization map associated to $\widehat{F}_{pq,\alpha}$, one obtains
smooth structures on the bundle
$\widehat{O}_{pq}^{(d)}|_{\partial^{\alpha}\widehat{V}_{pq}}$ for all $d\geq
d_{pq}$. Using the collar structure, we extend slightly to a neighborhood of
$\partial^{\alpha}\widehat{V}_{pq}$ inside $\widehat{V}_{pq}$.

We can carry on the induction for all strata $\alpha=pr_{1}\cdots r_{l}q$.
Suppose we have constructed a smooth structure near the boundary of
$\partial^{\alpha}\widehat{V}_{pq}$ and smooth bundle structures on
$\widehat{O}_{pq}^{(d)}$ over this neighborhood. Now start with
$d=d_{0}=d_{pq}$ and we would like to construct the smooth structure of
$\partial^{\alpha}\widehat{V}_{pq}$ and the smooth bundle structure on
$\widehat{O}_{pq}^{(d_{0})}|_{\partial^{\alpha}\widehat{V}_{pq}}$. First, the
induction hypothesis granted a smooth structure on the product
$\widehat{V}_{\alpha}$. This is compatible with all the smooth structures near
the boundary of $\partial^{\alpha}\widehat{V}_{pq}$. Then similar to above,
one can have a smooth structure on the sum (the right hand side of (6.11)).
Then extend to a smooth structure on
$\widehat{O}_{pq}|_{\partial^{\alpha}\widehat{V}_{pq}}$. The stabilization map
then grants a smooth structure on $\partial^{\alpha}\widehat{V}_{pq}$ which
extends the existing one near the boundary. The bundle isomorphism provides a
smooth bundle structure of
$\widehat{O}_{pq}|_{\partial^{\alpha}\widehat{V}_{pq}}$. Inductively, one
extends the smooth bundle structure to
$\widehat{O}_{pq}^{(d)}|_{\partial^{\alpha}\widehat{V}_{pq}}$ for all $d\geq
d_{0}$. Then use the collar structure we extend the structures slightly into
the interior of $\widehat{V}_{pq}$.

Inductive Construction III. We construct a $G_{pq}$-equivariant vector bundle
reduction of $T_{\mu}\widehat{V}_{pq}$ which is smooth near the boundary and
is in the the isotopy class of the vector bundle reduction induced from the
existing smoothings. The previous construction induces a $G_{pq}$-smoothing on
a neighborhood $N_{\epsilon}(\partial\widehat{V}_{pq})$ of the boundary, using
the smooth structures on bundles of the form (6.12) and equivariantization.
Hence there is a corresponding (isotopy class) of vector bundle lift of the
tangent microbundle $T_{\mu}\widehat{V}_{pq}$ restricted to this region. From
the construction we can see that

$T\left(N_{\epsilon}(\partial\widehat{V}_{pq})\right)\cong\left(T^{\rm
vt}\widehat{V}_{pq}\oplus\widehat{\pi}_{pq}^{*}TB_{d_{pq}}\right)|_{N_{\epsilon}(\partial\widehat{V}_{pq})}.$

On the other hand, we know that via a fiberwise submersion on
$\widehat{V}_{pq}$ one can obtain another vector bundle lift of
$T_{\mu}\widehat{V}_{pq}$ by the same vector bundle.

We claim that these two vector bundle lifts, when restricted to their common
domains, are in the same stable $G_{pq}$-isotopy class of vector bundle lifts.
We show why this is the case if $\alpha$ is a codimension 1 stratum. In this
case, we know that

$\partial^{\alpha}V_{pq}={\rm Stab}_{F_{pq,\alpha}}(V_{\alpha})$

i.e., the boundary $\partial^{\alpha}V_{pq}$ is a disk bundle over
$V_{\alpha}$. We can choose an (invariant) connection on $F_{pq,\alpha}$ to
induce a smooth splitting

$TF_{pq,\alpha}\cong T^{\rm
vt}F_{pq,\alpha}\oplus\pi_{F_{pq,\alpha}}^{*}TV_{\alpha}.$

It also induces a corresponding splitting on the microbundle level. Although
the fibers of the projection onto $B_{\delta(\alpha)}$ are no longer smooth,
the microbundle version of the above splitting can still be restricted to each
fiber. Then combining with the microbundle splitting of $T_{\mu}V_{\alpha}$
one obtains

$T_{\mu}(\partial^{\alpha}V_{pq})\cong\pi_{F_{pq,\alpha}}^{*}(F_{pq,\alpha})_{\mu}\oplus\pi_{F_{pq,\alpha}}^{*}T_{\mu}V_{\alpha}\cong
T^{\rm
vt}_{\mu}(\partial^{\alpha}V_{pq})\oplus\pi^{*}T_{\mu}B_{\delta(\alpha)}.$

Notice that this is an equivariant fiberwise submersion of
$\partial^{\alpha}V_{pq}\to\partial^{\delta(\alpha)}B_{d}$ (see Definition
6.18). The collar structure can be used to extend this fiberwise submersion
trivially into the collar region. Using Lemma 6.21 one can extend it globally
to $V_{pq}$. Then using the fiberwise smooth structure we obtained a vector
bundle lift of $T_{\mu}V_{pq}$.

On the other hand, the vector bundle lift on $V_{\alpha}$ induced from
smoothing is in the same stable isotopy class of vector bundle lifts as the
one from fiberwise smooth structure and microbundle splitting. Therefore, one
can find another orthogonal $G_{pq}$-space ${\bm{R}}_{pq}^{\prime}$ such that
the ${\bm{R}}_{pq}^{\prime}$-stabilization of the interior vector bundle lift
of $T_{\mu}\widehat{V}_{pq}$ is isotopic to the one from the boundary smooth
structure. Also notice that one can choose a $G_{pq}$-invariant continuous
cut-off function on the overlap of the two domains of the vector bundle lifts
to interpolate this two vector bundle lifts using the isotopy. Note that this
is one salient feature of the existence of collar structures: we are free to
interpolate between different data over the collar regions. Therefore, we have
obtained a vector bundle lift of $T_{\mu}\widehat{V}_{pq}$ which is smooth
near the boundary.

In general, when $\alpha$ is not a codimension 1 stratum, there are many
intermediate stratum. However, one can choose connections on all the
intermediate difference bundles $F_{\beta\alpha}$ compatibly, utilizing Lemma
5.54. This still allows us to build the interior microbundle splitting which
extends the boundary ones. We omit the details.

Inductive Construction IV. Smoothing for $pq$. In the previous steps, we have
obtained a smooth structure on a neighborhood of $\partial\widehat{V}_{pq}$, a
smooth $O(n)$-bundle structure of $\widehat{O}_{pq}^{(d)}$ over this
neighborhood, and a $G_{pq}$-equivariant vector bundle reduction

$\rho_{pq}:(T^{\rm
vt}\widehat{V}_{pq}\oplus\widehat{\pi}_{pq}^{*}B_{pq})_{\mu}\to
T_{\mu}\widehat{V}_{pq}$

which is smooth near $\partial\widehat{V}_{pq}$. Then one can apply the
relative version of Lashof’s $G$-smoothing theorem (see Appendix B and Theorem
B.3) to obtain another $G_{pq}$-orthogonal space ${\bm{R}}_{pq,0}$ and a
$G_{pq}$-smoothing on ${\rm Int}\widehat{V_{pq}}\times{\bm{R}}_{pq,0}$ which
coincides with the stabilization of the existing smoothing. Redefine
${\bm{R}}_{pq}$ by taking direct sum with ${\bm{R}}_{pq,0}$ and redefine
$\widehat{V}_{pq}$, $\widehat{O}_{pq}^{(d)}$ etc. Moreover, one can extend the
smooth structures on $\widehat{O}_{pq}^{(d)}$ inductively to the interior of
$\widehat{V}_{pq}$. Furthermore, we extend the existing inner product on the
trivial bundle $R_{pq}$ to the whole $\widehat{V}_{pq}$ smoothly.

Finally, we see that we have constructed a smoothing for any K-chart $K_{pq}$
with energy $d_{pq}$. Furthermore, they satisfy the induction hypotheses
listed at the beginning. Therefore, Theorem 6.31 holds. ∎

##  7\. Constructions for PSS, SSP, and the homotopy

In this subsection, we discuss K-charts for PSS, SSP, and continuation type
moduli spaces. The constructions here are carried out almost verbatim as the
case of moduli spaces of Floer trajectories. Therefore, our main purpose here
is to fix the notations, and only the key modifications are presented in
detail.

###  7.1. PSS bimodule and SSP bimodule

####  7.1.1. Thimbles and auxiliary moduli spaces

######  Definition 7.1.

  1. (1)

A prestable PSS thimble is a triple $(\Sigma,\Sigma_{\text{PSS}},{\bf L})$
where $\Sigma$ is a genus $0$ prestable curve with two marked points
$z_{-},z_{+}$, with $\Sigma_{\text{PSS}}$ being a distinguished horizontal
irreducible component of $\Sigma$, and ${\bf L}=({\rm L}_{i})$ where ${\rm
L}_{i}$ is a lateral line on each horizontal component lying between
$\Sigma_{\text{PSS}}$ (included!) and the marked point $z_{+}$ (see Figure 2).

Figure 2. An example of prestable PSS thimble. The red curves are the lateral
lines and the gray sphere is $\Sigma_{\rm PSS}$ on which we interpolate
between the Hamiltonian $H$ and a constant function.

  2. (2)

A prestable SSP thimble is a triple $(\Sigma,\Sigma_{\text{SSP}},{\bf L})$
where $\Sigma$ is a genus $0$ prestable curve with two marked points
$z_{-},z_{+}$, with $\Sigma_{\text{SSP}}$ being a distinguished horizontal
irreducible component of $\Sigma$, and ${\bf L}=({\rm L}_{i})$ where ${\rm
L}_{i}$ is a lateral line on each horizontal component lying between the
marked point $z_{-}$ and $\Sigma_{\text{SSP}}$.

By imposing extra marked points and stability conditions, one can define the
notion of stable marked PSS/SSP thimble as in Definition 5.12. Given an
integer $d>0$, the moduli space of stable PSS resp. SSP thimbles in
$\mathbb{CP}^{d}$ is the set of equivalence classes of objects

$(\Sigma,\Sigma_{\rm PSS},{\bf L},u),\ \ \ \ \ {\rm resp.}\ \ \ \ \
(\Sigma,\Sigma_{\rm SSP},{\bf L},u)$

where $(\Sigma,\Sigma_{\text{PSS}}/\Sigma_{\text{SSP}},{\bf L})$ is a
prestable PSS/SSP thimble and $u:\Sigma\to\mathbb{CP}^{d}$ is a stable map. We
define ${\mathcal{F}}^{\text{PSS}}_{0,2}(d)$ to be the subset of moduli spaces
of stable PSS thimbles such that the underlying map
$u:\Sigma\to\mathbb{CP}^{d}$ is not contained in any hyperplane, and
${\mathcal{F}}^{\text{SSP}}_{0,2}(d)$ can be defined similarly. These two
moduli spaces are smooth manifolds with corners. Just as Lemma 5.15, the
following statements hold.

  1. (1)

The natural forgetful map
${\mathcal{F}}^{\text{PSS}}_{0,2}(d),{\mathcal{F}}^{\text{SSP}}_{0,2}(d)\to{\mathcal{F}}_{0,2}(d)$
is smooth.

  2. (2)

The evaluation maps $\text{ev}_{\pm}$ at the marked points $z_{\pm}$ are
smooth and stratified submersive onto $\mathbb{CP}^{d}$.

  3. (3)

By intersecting with $d(d+2)$ generic hyperplanes in $\mathbb{CP}^{d}$, the
moduli spaces
${\mathcal{F}}^{\text{PSS}}_{0,2}(d),{\mathcal{F}}^{\text{SSP}}_{0,2}(d)$ are
locally diffeomorphic to a real blowup of the Deligne–Mumford space
$\overline{\mathcal{M}}_{0,d(d+2)}$.

Note that we do not necessarily equip all the horizontal components of
$(\Sigma,{\bf L})$ with a lateral line, so the real blowup in (3) is different
from the one in Lemma 5.15.

The spaces ${\mathcal{F}}_{0,2}^{\rm PSS}(d)$ and ${\mathcal{F}}_{0,2}^{\rm
SSP}(d)$ are stratified by the same poset ${\mathbb{A}}_{d}$ of ordered
partitions of $d$ (with the same depth function, see Notation 5.7). However,
the geometric meaning of an element
$\delta=(d_{0},\ldots,d_{l})\in{\mathbb{A}}_{d}$ is different. For example, in
the PSS case, the integer $d_{0}$ represents the total degree of the
components between $z_{-}$ and $\Sigma_{\rm PSS}$ (included).

######  Notation 7.2.

  1. (1)

Let $p,q$ be capped 1-periodic capped Hamiltonian orbits. Introduce the
integers

$\displaystyle\ d_{\bullet q}:={\mathcal{A}}_{H}(q)-C^{\rm PSS},\ $
$\displaystyle\ d_{p\bullet}:=C^{\rm SSP}-{\mathcal{A}}_{H}(p)$

where $C^{\rm PSS}$ resp. $C^{\rm SSP}$ is the constant chosen to define the
${\rm PSS}$ resp. ${\rm SSP}$ moduli spaces (see Subsection 4.3).

  2. (2)

Extra symmetry groups

$G^{\rm PSS}_{\bullet q}:=\\{g\in PU(d_{\bullet q}+1)\ |\
g([1,0,\ldots,0])=[1,0,\ldots,0]\in\mathbb{CP}^{d_{\bullet p}}\\}\cong
U(d_{\bullet p})$ $G_{p\bullet}^{\rm SSP}:=\\{g\in PU(d_{p\bullet}+1)\ |\
g([1,0,\ldots,0])=[1,0,\ldots,0]\in\mathbb{CP}^{d_{p\bullet}}\\}\cong
U(d_{p\bullet})$

where the isomorphisms are induced from (5.3).

  3. (3)

Homogeneous posets with partial order induced from refinement of words

${\mathbb{A}}_{\bullet q}^{\rm PSS}:=\\{\bullet r_{1}\cdots
r_{l}q|r_{1}<\cdots<r_{l}<p\text{ as objects in }T^{\rm Floer}\\}$
${\mathbb{A}}_{p\bullet}^{\rm SSP}:=\\{pr_{1}\cdots
r_{l}\bullet|p<r_{1}<\cdots<r_{l}\text{ as objects in }T^{\rm Floer}\\},$

with depth functions

${\rm depth}(\bullet r_{1}\cdots r_{l}q)=l,\ {\rm depth}(pr_{1}\cdots
r_{l}\bullet)=l.$

The concatenation of words induces maps between homogeneous posets

${\mathbb{A}}_{\bullet p}^{\rm PSS}\times{\mathbb{A}}^{\rm
Floer}_{pq}\to{\mathbb{A}}_{\bullet q}^{\rm PSS}$ ${\mathbb{A}}^{\rm
Floer}_{pq}\times{\mathbb{A}}_{q\bullet}^{\rm
SSP}\times\to{\mathbb{A}}_{p\bullet}^{\rm SSP}.$

There are natural maps of posets

$\delta^{\rm PSS}:{\mathbb{A}}_{\bullet q}^{\rm
PSS}\to{\mathbb{A}}_{d_{\bullet q}},\ \delta^{\rm
SSP}:{\mathbb{A}}_{p\bullet}^{\rm SSP}\to{\mathbb{A}}_{d_{p\bullet}}$

defined by

$\delta^{\rm PSS}(\bullet r_{1}\cdots r_{l}q):=(d_{\bullet
r_{1}},\dots,d_{r_{l}q}),\ \delta^{\rm SSP}(pr_{1}\cdots
r_{l}\bullet):=(d_{pr_{1}},\dots,d_{r_{l}\bullet}),$

such that the following diagrams commute.

$\textstyle{{\mathbb{A}}_{\bullet p}^{\rm PSS}\times{\mathbb{A}}^{\rm
Floer}_{pq}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\delta^{\rm
PSS}\times\delta}$$\textstyle{{\mathbb{A}}_{\bullet q}^{\rm
PSS}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\delta^{\rm
PSS}}$$\textstyle{{\mathbb{A}}_{d_{\bullet
p}}\times{\mathbb{A}}_{d_{pq}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathbb{A}}_{d_{\bullet
q}}}$ $\textstyle{{\mathbb{A}}^{\rm
Floer}_{pq}\times{\mathbb{A}}_{q\bullet}^{\rm
SSP}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\delta\times\delta^{\rm
SSP}}$$\textstyle{{\mathbb{A}}_{p\bullet}^{\rm
SSP}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\delta^{\rm
PSS}}$$\textstyle{{\mathbb{A}}_{d_{pq}}\times{\mathbb{A}}^{\rm
SSP}_{d_{q\bullet}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathbb{A}}_{d_{p\bullet}}}$

  4. (4)

For an index $\alpha=\bullet r_{1}\cdots r_{l}p\in{\mathbb{A}}^{\rm
PSS}_{\bullet q}$, there is an embedding of Lie groups

$G^{\text{PSS}}_{\alpha}:=G^{\text{PSS}}_{\bullet r_{1}}\times\cdots\times
G_{r_{l}q}\hookrightarrow G^{\text{PSS}}_{\bullet q}=G^{\text{PSS}}_{\bullet
q}$

as from Notation 5.7, which further induces an embedding
$G^{\text{PSS}}_{\alpha}\hookrightarrow G^{\text{PSS}}_{\beta}$ for any pair
$\alpha\leq\beta$ in ${\mathbb{A}}_{\bullet p}^{\rm PSS}$. Analogously, there
are embeddings of Lie groups $G^{\text{SSP}}_{\alpha}\hookrightarrow
G^{\text{SSP}}_{\beta}$ where $\alpha\leq\beta$ are in ${\mathbb{A}}^{\rm
SSP}_{p\bullet}$.

  5. (5)

The system of auxiliary moduli spaces

$B^{\text{PSS}}_{d}:=\big{\\{}x=[\Sigma,\Sigma_{\text{PSS}},{\bf
L},u]\in{\mathcal{F}}^{\text{PSS}}_{0,2}(d)\ |\ {\rm
ev}_{-}(x)=[1,0,\ldots,0]\in\mathbb{CP}^{d}\big{\\}},$
$B^{\text{SSP}}_{d}:=\big{\\{}x=[\Sigma,\Sigma_{\text{SSP}},{\bf
L},u]\in{\mathcal{F}}^{\text{SSP}}_{0,2}(d)\ |\ {\rm
ev}_{-}(x)=[1,0,\ldots,0]\in\mathbb{CP}^{d}\big{\\}},$

where ${\rm ev}_{-}(x)$ is the image of the evaluation map of $u$ at the
marked point $z_{-}$. Each $B^{\text{PSS}}_{d}$ resp. $B^{\text{SSP}}_{d}$ is
a smooth ${\mathbb{A}}_{d}$-manifold and has a $G_{d}$-action.

  6. (6)

For $\alpha=\bullet r_{1}\cdots r_{l}q\in{\mathbb{A}}^{\rm PSS}_{\bullet q}$
(resp. $pr_{1}\cdots r_{l}\bullet\in{\mathbb{A}}^{\rm SSP}_{p\bullet}$),
define

$B^{\rm PSS}_{\alpha}:=B^{\rm PSS}_{d_{\bullet r_{1}}}\times\cdots\times
B_{d_{r_{l}q}}$ $\text{resp. }B^{\rm
SSP}_{\alpha}:=B_{d_{pr_{1}}}\times\cdots\times B^{\rm
SSP}_{d_{r_{l}\bullet}}.$

Then $B^{\rm PSS}_{\alpha}$ (resp. $B^{\rm SSP}_{\alpha}$) has a smooth
$G_{\alpha}^{\rm PSS}$ (resp. $G_{\alpha}^{\rm SSP}$) action.

  7. (7)

Just as Equation (5.12), under the group embedding $G^{\rm
PSS}_{\alpha}\hookrightarrow G_{\bullet q}^{\rm PSS}$, there is an equivariant
embedding

$B^{\rm PSS}_{\alpha}\hookrightarrow B^{\rm PSS}_{\bullet q},$

which is part of the data of the embeddings of the form

$B^{\rm PSS}_{\alpha}\hookrightarrow B^{\rm PSS}_{\beta}$

for any $\alpha\leq\beta$ under the group embedding $G_{\alpha}^{\rm
PSS}\hookrightarrow G_{\beta}^{\rm PSS}$. This fact is also true after
replacing PSS by SSP.

####  7.1.2. Description of global charts

The following statement allows to construct framings from solutions to the
PSS/SSP equations. The proof is the same as the proof of Corollary 5.30 using
the integrality assumptions.

######  Proposition 7.3.

Let $(\Sigma,\Sigma_{\rm PSS},{\bm{L}})$ be a prestable PSS thimble, and
suppose that

$u:\Sigma\to M$

is a stable PSS thimble, i.e. the restriction of $u$ to the irreducible
component $\Sigma_{\rm PSS}$ solves the PSS equation (4.6), its restriction to
the cylindrical components on the right of $\Sigma_{\rm PSS}$ solves the Floer
equation (4.2), while its restriction to other components solves the genuine
$J$-holomorphic curve equation, and $u$ has finite automorphism group.
Consider the $2$-form $\Omega_{u,{\rm PSS}}$ defined by

$\Omega_{u,{\rm PSS}}:=\left\\{\begin{array}[]{lr}\displaystyle
u^{*}\omega-d(H_{s,t}^{\rm PSS}(u)dt)&\text{ on the distinguished component
$\Sigma_{\rm PSS}$}\\\ \displaystyle u^{*}\omega-d(H_{t}(u)dt)&\text{ on
cylindrical components on the right of $\Sigma_{\rm PSS}$}\\\ \displaystyle
u^{*}\omega&\text{ on other irreducible components.}\end{array}\right.$ (7.1)

Then there exists a holomorphic Hermitian line bundle on $\Sigma$ whose
curvature form is $-2\pi{\bm{i}}\Omega_{u,{\rm PSS}}$. Moreover, this line
bundle is unique up to isomorphism.

Similarly, if $(\Sigma,\Sigma_{\rm SSP},{\bm{L}})$ is a prestable SSP thimble
and $u:\Sigma\to M$ is a stable SSP thimble which solves the SSP equation
(4.10) on $\Sigma_{{\rm SSP}}$, there exists a Hermitian line bundle over
$\Sigma$, unique up to isomorphism, such that the curvature of its Chern
connection is equal to $-2\pi{\bm{i}}\Omega_{u,{\rm PSS}}$, where

$\Omega_{u,{\rm SSP}}:=\left\\{\begin{array}[]{lr}\displaystyle
u^{*}\omega-d(H_{s,t}^{\rm SSP}(u)dt)&\text{ on the distinguished component
$\Sigma_{\rm SSP}$}\\\ \displaystyle u^{*}\omega-d(H_{t}(u)dt)&\text{ on
cylindrical components on the left of $\Sigma_{\rm SSP}$}\\\ \displaystyle
u^{*}\omega&\text{ on other irreducible components.}\qed\end{array}\right.$

Given a stable PSS (resp. SSP) thimble, let us denote the line bundle
constructed as above by $L_{u,{\rm PSS}}$ (resp. $L_{u,{\rm SSP}}$). Now we
can introduce the concept of framed curves in the context of PSS/SSP map.

######  Definition 7.4.

Given a moduli space $\overline{\mathcal{M}}{}^{\rm PSS}_{\bullet q}$ of
stable PSS thimbles and a stratum $\alpha=\bullet r_{1}\cdots r_{l}q$ which
corresponds to thimbles breaking at the orbits $r_{1},\dots,r_{l}$, denoting
$d=d_{\bullet q}$, a framed PSS thimble (of type $\alpha$) is a tuple
$(u,\Sigma,F)$ where

  1. (1)

$\Sigma$ is a prestable PSS thimble with $l$ horizontal levels on the right of
$\Sigma_{\rm PSS}$. (see Definition 7.1).

  2. (2)

$u:\Sigma\to M$ is a smooth map whose restriction to each cylindrical
component on the right of $\Sigma_{\rm PSS}$ (included) converges to periodic
orbits prescribed by the capped orbits $p,r_{1},\dots,r_{l},q$ at $\pm\infty$
in an exponential rate, and the topological energy of these levels are
prescribed accordingly. Moreover, the $2$-form $\Omega_{u,{\rm PSS}}$ defined
in Equation (7.1) is non-negative and is strictly positive on each unstable
component of $u$.

  3. (3)

$F=(f_{0},f_{1},\ldots,f_{d})$ is basis of global sections of the line bundle
$L_{u,{\rm PSS}}$ constructed using the $2$-form (7.1). Moreover, the induced
holomorphic map

$\iota_{F,{\rm PSS}}:\Sigma\to\mathbb{CP}^{d},\
w\mapsto[f_{0}(w),\ldots,f_{d}(w)]$ (7.2)

is a stable map, which represents a point $[\iota_{F,{\rm
PSS}}]\in{\mathcal{F}}^{\rm PSS}_{0,2}(d)$. Denote by

$\tilde{\iota}_{F,{\rm PSS}}:\Sigma\to{\mathcal{C}}^{\rm
PSS}={\mathcal{C}}^{\rm PSS}_{0,2}(d)$

to be the identification between $\Sigma$ and the fiber of the universal curve
${\mathcal{C}}^{\rm PSS}_{0,2}(d)\to{\mathcal{F}}^{\rm PSS}_{0,2}(d)$ over
$[\iota_{F,{\rm PSS}}]$. This will also be called the domain map.

By replacing PSS with SSP, we obtain the definition of framed PSS thimble.
Isomorphisms of framed PSS/SSP thimbles are defined in the same way as
Definition 5.32. The compact Lie group $PU(d+1)$ acts on the space of framed
PSS/SSP thimbles by changing the framing. The stabilizer of this $PU(d+1)$
action at $(u,\Sigma,F)$ agrees with the automorphism group of the map
$u:\Sigma\to M$ if $u$ is a stable PSS/SSP thimble.

Now we can repeat the global thickening construction in the setting of PSS/SSP
maps as we did in the case for Floer trajectories. Let $p$ be a capped
$1$-periodic orbit. Fix $d=d_{\bullet q}$. Let

$k_{1}<\cdots<k_{d}$

be a sequence of positive integers. Define $V_{\bullet q}^{\rm
PSS}:=V_{\bullet q}^{\rm PSS}(k_{1},\ldots,k_{d})$ be the moduli space of
tuples

$(\Sigma,u,F,\eta_{1},\ldots,\eta_{d})$

where $\Sigma$ is a prestable PSS thimble, $u:\Sigma\to M$ is a smooth map
with topological energy $d$ converging to $q$ at the marked point $z_{+}$, $F$
is a frame of the line bundle $L_{u,{\rm PSS}}\to\Sigma$ inducing a
holomorphic map $\iota_{F,{\rm PSS}}:\Sigma\to\mathbb{CP}^{d}$ such that the
framing $F=(f_{0},\dots,f_{d})$ satisfies
$[f_{0}(z_{-}):\cdots:f_{d}(z_{-})]=[1:0:\cdots:0]$, and the corresponding
domain map $\tilde{\iota}_{F,{\rm PSS}}:\Sigma\to{\mathcal{C}}^{\rm PSS}$, and
(denoting by $\tilde{u}=(u,\tilde{\iota}_{F,{\rm PSS}}):\Sigma\to
M\times{\mathcal{C}}^{\rm PSS}$)

$\eta_{i}\in\tilde{u}^{*}E_{i}=\Gamma\left(\overline{\rm
Hom}\big{(}\iota_{F,{\rm
PSS}}^{*}T\mathbb{CP}^{d},u^{*}TM\oplus\bigoplus_{j=1}^{i-1}\tilde{u}^{*}E_{j}\big{)}\otimes\iota_{F,{\rm
PSS}}^{*}{\mathcal{O}}(k_{i})\right)\otimes\overline{H^{0}(\iota_{F,{\rm
PSS}}^{*}{\mathcal{O}}(k_{i}))_{0}}$

satisfying the following equations

$\left\\{\begin{aligned} \overline{\partial}_{J,H^{\rm
PSS}}u+\pi_{u^{*}TM}\left(\sum_{i=1}^{d}\langle\eta_{i}\rangle\circ
d\iota_{F,{\rm PSS}}\right)&\ =0,\\\
\overline{\partial}\eta_{i}+\pi_{E_{i}}\left(\sum_{j=i+1}^{d}\langle\eta_{j}\rangle\circ
d\iota_{F,{\rm PSS}}\right)&\ =0,\ i=1,\ldots,d,\end{aligned}\right.$ (7.3)

where $E_{i}$ is defined in the same way as (5.32), and the
$\overline{\partial}$-operator $\overline{\partial}_{J,H^{\rm PSS}}$ is read
off from the PSS equation 4.6.

By replacing PSS with SSP, we can similarly define the thickened moduli space
$V_{p\bullet}^{\rm SSP}=V_{p\bullet}^{\rm SSP}(k_{1},\dots,k_{d})$ for a
capped orbit $p$.

The space $V_{\bullet q}^{\rm PSS}$ (resp. $V_{p\bullet}^{\rm SSP}$) admits a
$G_{\bullet q}^{\rm PSS}=G_{d_{\bullet q}}$ (resp. $G_{p\bullet}^{\rm
SSP}=G_{d_{p\bullet}}$) action extending the action on the framing. The
obstruction bundle over $E_{\bullet q}^{\rm PSS}\to V_{\bullet q}^{\bullet}$
also has a direct sum decomposition

$E_{\bullet q}^{\rm PSS}=O_{\bullet q}^{\rm PSS}\oplus Q_{\bullet q}^{\rm
PSS}$

where $O_{\bullet q}^{\rm PSS}$ is specified by (5.34), and $Q_{\bullet
q}^{\rm PSS}$ is the trivial vector bundle from ${\bm{Q}}_{d_{\bullet q}}$ as
in (5.13). This is a $G^{\rm PSS}_{\bullet q}$-equivariant vector bundle. As
for the Kuranishi section, it is defined by

$\begin{split}S_{\bullet q}^{\rm PSS}:V_{\bullet q}^{\rm PSS}&\ \to E_{\bullet
q}^{\rm PSS}\\\ [\Sigma,u,F,\eta]&\ \mapsto(\eta,H_{F})\end{split}$

where $H_{F}\in Q_{\bullet q}^{\rm PSS}$ is represented by the Hermitian
matrix whose entries are

$\int_{\Sigma}\langle f_{i},f_{j}\rangle_{L_{u,{\rm PSS}}}\Omega_{u,{\rm
PSS}},$

where $f_{0},\dots,f_{d}$ is a basis representing the frame $F$. It is obvious
that there exists a homeomorphism

$\psi_{\bullet q}^{\rm PSS}:(S_{\bullet q}^{\rm PSS})^{-1}(0)/G^{\rm
PSS}_{\bullet q}\xrightarrow{\sim}\overline{\mathcal{M}}{}_{\bullet q}^{\rm
PSS}.$

This finishes the description of the K-chart

$K_{\bullet q}^{\rm PSS}:=K_{\bullet q}^{\rm PSS}(k_{1},\ldots,k_{d}):=(G^{\rm
PSS}_{\bullet q},V_{\bullet q}^{\rm PSS},E^{\rm PSS}_{\bullet q},S_{\bullet
q}^{\rm PSS},\psi_{\bullet q}^{\rm PSS}).$

The same description applies to the SSP moduli space and we obtain the K-chart

$K_{p\bullet}^{\rm SSP}:=K_{p\bullet}^{\rm SSP}=(G^{\rm
SSP}_{p\bullet},V_{p\bullet}^{\rm SSP},E_{p\bullet}^{\rm
SSP},S_{p\bullet}^{\rm SSP},\psi_{p\bullet}^{\rm SSP})$

for the moduli space $\overline{\mathcal{M}}{}_{p\bullet}^{\rm SSP}$.

The following statement follows from the proof of Theorem 5.47.

######  Proposition 7.5.

For fixed $k_{1},\dots,k_{d-1}$, as long as $k_{d}$ is sufficiently large, the
thickened moduli spaces $V_{\bullet q}^{\rm PSS}=V_{\bullet q}^{\rm
PSS}(k_{1},\ldots,k_{d})$ and $V_{p\bullet}^{\rm SSP}=V_{p\bullet}^{\rm
SSP}(k_{1},\ldots,k_{d})$ is regular near the zero locus of the Kuranishi
section and $E_{\bullet q}^{\rm PSS}\to V_{\bullet q}^{\rm PSS}$,
$E_{p\bullet}^{\rm SSP}\to V_{p\bullet}^{\rm SSP}$ are indeed vector bundles.
∎

####  7.1.3. The K-chart lift of the flow bimodule

We would like the system of K-charts obtained above provide a K-chart lift of
the flow bimodule $M^{\rm PSS}$ resp. $M^{\rm SSP}$. Still we describe the PSS
case with more details. Recall that the moduli space (4.7) is a
${\mathbb{A}}^{\rm PSS}_{\bullet q}$-space. Given a stratum $\alpha=\bullet
r_{1}\cdots r_{l}q$, the moduli space

$\partial^{\alpha}\overline{\mathcal{M}}{}_{\bullet q}^{\rm
PSS}=\overline{\mathcal{M}}{}_{\bullet r_{1}}^{\rm
PSS}\times\overline{\mathcal{M}}{}_{r_{1}r_{2}}^{\rm
Floer}\times\cdots\times\overline{\mathcal{M}}{}^{{\rm Floer}}_{r_{l}q}$

has a product K-chart

$K_{\alpha}^{\rm PSS}:=K_{\bullet r_{1}}^{\rm PSS}\times
K_{r_{1}r_{2}}\times\cdots\times K_{r_{l}q}$

where $K_{r_{i}r_{i+1}}$ is the K-chart constructed in (5.38) for
$\overline{\mathcal{M}}{}_{r_{i}r_{i+1}}^{\rm Floer}$. Similarly, for
$\alpha=pr_{1}\cdots r_{l}\bullet$, the moduli space

$\partial^{\alpha}\overline{\mathcal{M}}{}_{p\bullet}^{\rm
SSP}=\overline{\mathcal{M}}{}^{\rm
Floer}_{pr_{1}}\times\cdots\times\overline{\mathcal{M}}{}_{r_{l-1}r_{l}}^{\rm
Floer}\times\overline{\mathcal{M}}{}^{\rm SSP}_{r_{l}\bullet}$

has a product K-chart

$K_{\alpha}^{\rm SSP}:=K_{pr_{1}}\times\cdots\times K_{r_{l-1}r_{l}}\times
K_{r_{l}\bullet}^{\rm SSP}.$

Recall that the flow category $T^{\rm Floer}$ is endowed with a weak K-chart
lift as from Theorem 5.11 of the form

$\big{\\{}K_{pq}=(G_{pq},V_{pq},E_{pq},S_{pq},\psi_{pq})\big{\\}}_{p<q},$

together with the collection of weak K-chart embeddings

$\big{\\{}{\bm{\iota}}_{prq}:K_{pr}\times
K_{rq}\rightharpoonup\partial^{prq}K_{pq}\big{\\}}_{p<r<q}.$

######  Theorem 7.6.

For a capped $1$-periodic orbits $p$ resp. $q$, there exist a collection of
weak K-chart embeddings

$\big{\\{}{\bm{\iota}}_{\bullet rq}:K_{\bullet r}^{\rm PSS}\times
K_{rq}\rightharpoonup\partial^{\bullet rq}K_{\bullet q}^{\rm
PSS}\big{\\}}_{r<q}$ $\text{resp.
}\big{\\{}{\bm{\iota}}_{pr\bullet}:K_{pr}\times K^{\rm
SSP}_{r\bullet}\rightharpoonup\partial^{pr\bullet}K_{p\bullet}^{\rm
SSP}\big{\\}}_{p<r},$

such that the following statements are true.

  1. (1)

The following diagrams are commutative under the chart embeddings:

$\textstyle{K^{\rm PSS}_{\bullet r}\times K_{rs}\times
K_{sq}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\partial^{\bullet
rs}K_{\bullet s}^{\rm PSS}\times
K_{sq}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{K_{\bullet
r}^{\rm
PSS}\times\partial^{rsq}K_{rq}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\partial^{\bullet
rsq}K^{\rm PSS}_{\bullet q},}$ $\textstyle{K_{rs}\times K_{sp}\times K^{\rm
SSP}_{p\bullet}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\partial^{rsp}K_{rp}\times
K^{\rm
SSP}_{p\bullet}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{K_{rs}\times\partial^{sp\bullet}K^{\rm
SSP}_{s\bullet}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\partial^{\bullet
rsp}K^{\rm SSP}_{\bullet p}.}$

  2. (2)

For each stratum $\alpha=\bullet r_{1}\cdots r_{l}q\in{\mathbb{A}}_{\bullet
q}^{\rm PSS}$ resp. $\alpha=pr_{1}\cdots
r_{l}\bullet\in{\mathbb{A}}_{p\bullet}^{\rm SSP}$, define the product chart

$K_{\alpha}^{\rm PSS}:=K_{\bullet r_{1}}^{\rm PSS}\times\cdots\times
K_{r_{l}q}$ ${\rm resp.}K^{\rm SSP}_{\alpha}:=K_{pr_{1}}\times\cdots\times
K^{\rm SSP}_{r_{l}\bullet}.$

Then for any pair $\alpha\leq\beta$ in ${\mathbb{A}}_{\bullet q}^{\rm PSS}$
resp. ${\mathbb{A}}_{p\bullet}^{\rm SSP}$, there are induced product weak
embeddings

${\bm{\iota}}^{\rm PSS}_{\beta\alpha}:K^{\rm
PSS}_{\alpha}\rightharpoonup\partial^{\alpha}K^{\rm PSS}_{\beta}\text{\ resp.
}{\bm{\iota}}^{\rm SSP}_{\beta\alpha}:K^{\rm
SSP}_{\alpha}\rightharpoonup\partial^{\alpha}K^{\rm SSP}_{\beta}.$

Then the system

$\\{K^{\rm PSS}_{\alpha}\\}_{\alpha\in{\mathbb{A}}_{\bullet q}^{\rm
PSS}}\text{ resp. }\\{K^{\rm
SSP}_{\alpha}\\}_{\alpha\in{\mathbb{A}}_{p\bullet}^{\rm SSP}}$

defines a weak K-chart presentation of the ${\mathbb{A}}_{\bullet q}^{\rm
PSS}$-space $\overline{\mathcal{M}}{}_{\bullet q}^{\rm PSS}$ resp. the
${\mathbb{A}}_{p\bullet}^{\rm SSP}$-space
$\overline{\mathcal{M}}{}_{p\bullet}^{\rm SSP}$.

  3. (3)

After outer-collaring, one can match up the Kuranishi sections so that the
weak K-chart presentation is a genuine K-chart presentation.

  4. (4)

The natural projection $\pi_{\bullet q}^{\rm PSS}:V_{\bullet q}^{\rm PSS}\to
B_{d_{\bullet q}}^{\rm PSS}$ resp. $\pi_{p\bullet}^{\rm SSP}:V_{p\bullet}^{\rm
SSP}\to B_{d_{p\bullet}}^{\rm SSP}$ has a $C_{\rm loc}^{1}$ equivariant
fiberwise smooth bundle structure.

###### Sketch of proof.

The embeddings ${\bm{\iota}}_{\bullet rq}^{\rm PSS}$ and
${\bm{\iota}}_{pr\bullet}^{\rm SSP}$ are defined in a similar way as in
Section 5.4 when we embed a product of K-charts of a list of Floer moduli
spaces. It is routine to check that these embeddings satisfy the listed
commutative diagrams and give weak K-chart presentations of the corresponding
PSS and SSP moduli spaces. Notice that for the same reason, the product
Kuranishi section only matches with the boundary restriction of the larger
Kuranishi chart in the $O$-summand. We can use outer-collaring and an
interpolation to match their $Q$-summand. Lastly, the projection $\pi_{\bullet
q}^{\rm PSS}$ and $\pi_{p\bullet}^{\rm SSP}$ have the structure of $C_{\rm
loc}^{1}$ equivariant fiberwise smooth bundles for the same reason as in the
case of Floer trajectories. ∎

####  7.1.4. Scaffolding

We could state and prove a proposition similar to Proposition 5.64. However,
instead of doing the tedious thing, we only describe how the existing
scaffoldings for the K-chart lift of $(T^{\rm Floer})^{+}$ can be extended.
Notice that we will not use any object or structure we chose in Section 6.
Essentially (for the PSS case), we would like to choose subbundles

$F_{\bullet q,\alpha}^{\rm PSS}\subset O_{\bullet q}|_{V_{\alpha}^{\rm PSS}}$

and define stabilization maps

${\rm Stab}_{F_{\bullet q,\alpha}}(V_{\alpha}^{\rm
PSS})\to\partial^{\alpha}V_{\bullet q}^{\rm PSS}$

whose $G_{\bullet q}$-equivariantization is an open embedding. (We skip the
discussion of the companion bundle isomorphisms.) This bundle $F_{\bullet
q,\alpha}$ is indeed the orthogonal complement of $E_{\alpha}^{\rm PSS}$
inside $E_{\bullet q}^{\rm PSS}$; a good choice of an inner products can be
made inductively as in the proof of Proposition 5.64 which extend the inner
products we have already chosen on the obstruction bundles over the thickend
Floer moduli spaces. Then the stabilization map is still constructed using the
implicit function theorem which depends on choosing a family of local Banach
manifold charts, a family of approximate solutions, and a family of right
inverses. These objects can all be extended from the existing ones we have
chosen for the Floer moduli spaces. Therefore a compatible system of
stabilization maps can be constructed. Of course, the collar structure is
necessary for the construction.

####  7.1.5. Stable smoothing

We would like to construct a system of stable equivariant smoothings of
$V_{\bullet q}^{\rm PSS}$ resp. $V_{p\bullet}^{\rm SSP}$. Notice that we have
chosen stabilizations by $G_{pq}$-representations ${\bm{R}}_{pq}$ for all
$p<q$ in ${\mathcal{P}}^{\rm Floer}$. The method we used in Subsection 6.4 can
be used again here to construct stabilizations ${\bm{R}}_{\bullet q}^{\rm
PSS}$ and to construct smoothings on $\widehat{V}_{\bullet q}^{\rm
PSS}:=V_{\bullet q}^{\rm PSS}\times{\bm{R}}_{\bullet q}^{\rm PSS}$ whose
boundary is diffeomorphic to products of stabilized thickenings.

We can also require that the stabilizations satisfy the following property.
Namely, there exist smooth submersive evaluation maps

${\rm ev}_{\bullet}:\widehat{V}_{\bullet q}^{\rm PSS}\to M$

which coincides with the original evaluation map on $V_{\bullet q}^{\rm PSS}$.
Indeed, by observation of Abouzaid–McLean–Smith [AMS21, Lemma 4.5], for a
single smooth Kuranishi chart, on a further stabilization there exists a
smooth submersive extension of the evaluation map. Then when we inductively
construct stable smoothings, we can always achieve this extra condition.

It follows that by intersecting with the unstable manifolds of the given Morse
function $f:M\to{\mathbb{R}}$, one obtains smooth Kuranishi charts

$\widehat{K}_{xq}^{\rm PSS}=(G_{\bullet q}^{\rm PSS},\widehat{V}_{xq}^{\rm
PSS},\widehat{E}_{xq}^{\rm PSS},\widehat{S}_{xq}^{\rm PSS})$

where $\widehat{V}_{xq}^{\rm PSS}\subset\widehat{V}_{\bullet q}^{\rm PSS}$ is
the intersection with the (outer-collared) compactified unstable manifold
$\overline{W^{u}(\underline{\smash{x}})}\subset M$. It is straightforward to
check that we obtained a smooth K-chart lift of the flow bimodule $(M^{\rm
PSS})^{+}$.

Then after taking quotient, we obtained a D-chart lift of $(M^{\rm PSS})^{+}$
equipped with a collar structure and a scaffolding. The straightening that is
compatible with the collar structure and the scaffolding can be constructed in
a way similar to the case of Floer trajectories.

####  7.1.6. Normal complex structures and orientaions

The construction of normal complex structure and orientation on
$\widehat{K}_{xq}^{\rm PSS}$ essentially follows from the proof of Theorem
6.32. and the relevant discussions in [AB21, Section 11]. Two remarks should
be pointed out. First, the stabilization from [AMS21, Lemma 4.5] and the fiber
product with unstable submanifolds does not affect the _normal_ complex
structure, because the changes on the tangent bundles are all included in the
trivial summand of the corresponding representation of the isotropy group.
Second, when constructing orientation structures, the virtual vector bundle
associated with an object $x\in T^{\rm Morse}$ is defined by
$(TW^{u}(\underline{\smash{x}}),0)$.

###  7.2. Pearly bimodule

Consider the pearly bimodule defined in Section 4. Here we describe a K-chart
lift of it and show how to use the same argument to construct a stable
smoothing. First, for each $d$, consider an auxiliary moduli space which is
similar to $B_{d}$, $B_{d}^{\rm PSS}$, and $B_{d}^{\rm SSP}$ defined before.
Indeed, let ${\mathcal{F}}_{0,2}^{\rm pearl}(d)$ be the moduli space of
parametrized 2-marked holomorphic spheres into $\mathbb{CP}^{d}$ which have
degree $d$ and whose images are not entirely contained in a hyperplane. In
particular, each point of ${\mathcal{F}}_{0,2}^{\rm pearl}(d)$ is represented
by a stable map with 2 marked points that have a distinguished component whose
parametrization is fixed. ${\mathcal{F}}_{0,2}^{\rm pearl}(d)$ is a smooth
manifold (with no boundary or corners). Moreover there are holomorphic
submersions

${\rm ev}_{\pm}:{\mathcal{F}}_{0,2}^{\rm pearl}(d)\to\mathbb{CP}^{d}.$

Let

$B_{d}^{\rm pearl}\subset{\mathcal{F}}_{0,2}^{\rm pearl}(d)$

be the subset of elements whose evaluation at $z_{-}$ is
$[1,0,\ldots,0]\in\mathbb{CP}^{d}$. Then $B_{d}^{\rm pearl}$ is a smooth
manifold. There is a $G_{d}\cong U(d)$-action on $B_{d}^{\rm pearl}$.

Now consider the corresponding space
$\overline{\mathcal{M}}{}_{\bullet\bullet}^{\rm pearl}(d)$ of stable
parametrized holomorphic spheres in $M$ whose degree is $d$. Using multi-
layered thickening associated to a sequence of integers
$k_{1}<k_{2}<\cdots<k_{d}<\cdots$ one can produce a similar thickened moduli
space

$K_{\bullet\bullet}^{\rm pearl}(d)=(G_{d},V_{\bullet\bullet}^{\rm
pearl}(d),E_{\bullet\bullet}^{\rm pearl}(d),S_{\bullet\bullet}^{\rm pearl}).$

The domain $V_{\bullet\bullet}^{\rm pearl}(d)$ admits a $G_{d}$-action and an
equivariant projection map

$\pi_{\bullet\bullet}^{\rm pearl}:V_{\bullet\bullet}^{\rm pearl}(d)\to
B_{d}^{\rm pearl}.$

Moreover, by taking intersections with stable and unstable submanifolds of the
Morse function $f:M\to{\mathbb{R}}$, one obtains stratified charts

$K_{xy}^{\rm pearl}=(G_{xy},V_{xy}^{\rm pearl},E_{xy}^{\rm pearl},S_{xy}^{\rm
pearl})$

This collection of charts obviously defines a K-chart lift of the pearly
bimodule $M^{\rm pearl}$.

######  Theorem 7.7.

There exists a “single-layered” normally complex derived orbifold lift of the
outer-collared pearly bimodule $(M^{\rm pearl})^{+}$.

###### Sketch of Proof.

We can construct stable smoothings of $K_{\bullet\bullet}^{\rm pearl}(d)$
individualy for each $d\geq 1$ to obtain smooth Kuranishi charts. To save
notations, still denote them by $K_{\bullet\bullet}^{\rm pearl}(d)$. We can do
a further stabilization such that the evaluations at $z_{\pm}$ are smooth
submersions onto $M$. Hence each $K_{xy}^{\rm pearl}$ becomes smooth
stratified Kuranishi charts. Its outer-collaring is again smooth Kuranishi
charts. The scaffoldings are trivial as the Morse theory part is already
regular. Then taking quotient by $G_{xy}$ one obtains a D-chart lift of
$(M^{\rm pearl})^{+}$, denoted by $({\mathfrak{D}}^{\rm pearl})^{+}$. As the
scaffolding is trivial, this is a single-layered lift. Lastly, there is a
normal complex structure and an orientation on $({\mathfrak{D}}^{\rm
pearl})^{+}$ for the same reason as the case of [AMS21]. ∎

###  7.3. The homotopy

Lastly, we explain the collection of D-chart lifts of the system of moduli
spaces referred to as the “homotopy moduli spaces.” Although this ad hoc
collection lacks a formal package, the constructions are almost identical to
previous cases. The global chart construction, including the multi-layered
strategy, is the same as the case of Floer trajectories or PSS/SSP cases. Then
one obtains a weak K-chart lift of the collection of moduli spaces. By taking
outer-collaring and interpolating between the product Kuranishi section and
the boundary restriction of Kuranishi sections, one obtains a K-chart lift of
the outer-collared system. The construction of scaffolding (see Proposition
5.64) has completely the same induction procedure. We can also run the
smoothing procedure without essential differences.

##  Appendix A Product of canonical Whitney stratifications

###  A.1. Basics of Whitney stratifications

We review the basic notions of (Whitney) prestratifications and
stratification. Following Mather [Mat73], our convention is the same as our
previous work [BX22].

######  Definition A.1.

Let $S$ be a topological space. A prestratification on $S$ is a decomposition

$S=\bigsqcup_{\alpha\in{\mathfrak{A}}}S_{\alpha}$

of $S$ into the disjoint union of locally closed subsets satisfying the
following condition

  1. (1)

The decomposition is locally finite.

  2. (2)

(Axiom of frontier) If $\alpha,\beta\in{\mathfrak{A}}$,
$S_{\alpha}\cap\overline{S_{\beta}}\neq\emptyset$, then
$S_{\alpha}\subset\overline{S_{\beta}}$.

Each $S_{\alpha}$ is called a stratum of the prestratification. The axiom of
frontier induces a partial order among strata: $S_{\alpha}\leq S_{\beta}$ if
$S_{\alpha}\subset\overline{S_{\beta}}$. We use the symbol ${\mathfrak{A}}$ to
denote the prestratification as well as the partially ordered set of strata. A
space equipped with a prestratification ${\mathfrak{A}}$ is called an
${\mathfrak{A}}$-stratified space.

Now we consider the notion of stratifications. Two subsets $A,B\subseteq M$
are called equivalent at $x\in M$ if there exists an open neighborhood $U_{x}$
of $x$ such that $A\cap U_{x}=B\cap U_{x}$. An equivalence class is called a
setgerm at $x$. Given a prestratification ${\mathcal{A}}$ on $S\subseteq M$,
it assigns to each $x\in S$ a setgerm represented by the unique stratum that
contains $x$. A stratification of a subset $S\subset M$ is a rule which
assigns to each point $x\in S$ a set-germ $S_{x}$, such that for each $x\in
S$, there is an open neighborhood $U_{x}$ of $x$ and a prestratification of
$U_{x}\cap S$ such that the setgerm-valued function restricted to $U_{x}$ is
induced from this prestratification.

######  Definition A.2 (Whitney stratification).

Let $M$ be a smooth manifold.

  1. (1)

Given two disjoint smooth submanifolds $S$, $S^{\prime}$, we say that the pair
$(S,S^{\prime})$ satisfies Whitney’s condition (b) at $x\in
S^{\prime}\cap\overline{S}$ if the following is true: suppose $x_{i}\in S$,
$x_{i}^{\prime}\in S^{\prime}$ are two sequences converging to
$x\in\overline{S}\cap S^{\prime}$. Suppose the sequence of tangent spaces
$T_{x_{i}}S$ converges to a subspace $H\subset T_{x}M$ and the sequence of
secant lines $\overline{x_{i}x_{i}^{\prime}}$ converges to a line $L\subset
T_{x}M$, then $L\subset H$.

  2. (2)

Let $S\subset M$ be a subset. A prestratification on $S$ is called a Whitney
prestratification if each stratum $S_{\alpha}$ is a smooth submanifold and
each pair $(S_{\alpha},S_{\beta})$ of strata with
$S_{\beta}\subsetneq\overline{S_{\alpha}}$ satisfies Whitney’s condition (b)
at every point of $S_{\beta}$.

  3. (3)

A stratification on $S\subset M$ is called a Whitney stratification if for
each point $x\in S$ there exists an open neighborhood $U_{x}\subset S$ of $x$
and a Whitney prestratification on $U_{x}$ which induces this stratification
inside $U_{x}$. We use the symbol ${\mathcal{S}}$ to denote a Whitney
stratification on $S$, which is a setgerm-valued function on $S$.

On the same set there could be many different Whitney stratifications. One can