$f:A\to{\mathbb{R}}$ is called smooth if for each $a\in A$ there is an open
neighborhood $U_{a}\subset{\mathbb{R}}^{n}$ such that $f|_{A\cap U_{a}}$ can
be extended to a smooth function $f_{a}:U_{a}\to{\mathbb{R}}$. Then one can
define the notion of smooth maps $f:A\to B$ for subsets
$A\subseteq{\mathbb{R}}^{n}$ and $B\subseteq{\mathbb{R}}^{m}$. Then
compositions of smooth maps are still smooth. A smooth map $f:A\to B$ is a
diffeomorphism if it admits a smooth inverse.

The concept of smooth manifolds with corners is defined as follows. Let $M$ be
a topological space. A chart (of an $n$-dimensional smooth manifold with
corners) consists of an open subset $U\subset M$ and a homeomorphism
$\varphi:U\to[0,+\infty)^{n}$ onto an open subset. Two charts
$\varphi_{i}:U_{i}\to[0,+\infty)^{n}$, $i=1,2$, are compatible if
$\varphi_{1}\circ\varphi_{2}^{-1}:\varphi_{2}(U_{1}\cap
U_{2})\to\varphi_{1}(U_{1}\cap U_{2})$ is a diffeomorphism. A structure of
smooth $n$-dimensional manifold with corners is a maximal compatible atlas on
$M$. If $M$ is equipped with such a structure, we say that $M$ is a smooth
$n$-dimensional manifold with corners. Over smooth manifolds with corners, one
can define the notions of smooth vector bundles, smooth connections, and
smooth Riemannian metrics, etc.

The concept of manifold with faces is one which can exclude certain
pathological scenarios. Let $M$ be a smooth $n$-dimensional manifold with
corners. For each point $x\in M$ and each chart $\varphi:U\to[0,+\infty)^{n}$
defined over an open neighborhood of $x$, the number ${\rm depth}(x)$ of
coordinates of $\varphi(x)$ which are zero is independent of the choice of
charts. Given $k\leq n$, the subset

$\partial^{k}M:=\\{x\in M\ |\ {\rm depth}(x)=k\\}\subset M$

is naturally an $n-k$-dimensional smooth manifold. A connected face of $M$ is
a connected component of the set $\\{x\in M\ |\ {\rm depth}(x)=1\\}$. $M$ is
called a smooth manifold with faces if every $x\in M$ belongs to the closure
of exactly ${\rm depth}(x)$ different connected faces.

As we are treating various manifolds with faces which have more refined
structure of stratifications, we would like to introduce the following
concepts.

######  Definition 3.18.

Let ${\mathbb{A}}$ be a countable homogeneous poset.

  1. (1)

A smooth ${\mathbb{A}}$-manifold is a smooth manifold with faces $M$ which is
also a topological ${\mathbb{A}}$-manifold such that for each
$\alpha\in{\mathbb{A}}$, $M_{\alpha}$ is an open and closed subset of
$\partial^{{\rm depth}(\alpha)}M$ and such that each $x\in M$ has a smooth
corner chart.

  2. (2)

Let $G$ be a compact Lie group. A smooth $G$-action on a smooth
${\mathbb{A}}$-manifold $M$ is a $G$-action on the underlying topological
${\mathbb{A}}$-manifold of $M$ which acts on $M$ by smooth maps.

Note that any smooth ${\mathbb{A}}$-manifold $M$ has a natural structure of a
topological ${\mathbb{A}}$-manifold by forgetting the smooth structure.

######  Definition 3.19.

A smooth map $f$ from a smooth ${\mathbb{A}}$-manifold $M$ to a smooth
manifold without boundary $N$ is called _stratified submersive_ if for any
(open) stratum $M_{\alpha}$, the restriction

$f|_{M_{\alpha}}:M_{\alpha}\rightarrow N$

is a smooth submersion.

######  Lemma 3.20.

Suppose $f:M\to N$ is stratified submersive. Then for any $y\in N$, the space
$f^{-1}(y)$ is a smooth ${\mathbb{A}}$-manifold. Moreover, if $G$ is a compact
Lie group acting on $M$ and $f$ is $G$-invariant, then $f^{-1}(y)$ has a
smooth $G$-action.

###### Proof.

By Definition 3.19, the subsets $f^{-1}(y)\cap M_{\alpha}$ are smooth
manifolds. Using the collars near each stratum, we see that $f^{-1}(y)$ is a
smooth manifold with corners. We now verify that $f^{-1}(y)$ is a manifold
with faces, namely, each $x\in f^{-1}(y)$ is contained in exactly ${\rm
depth}(x)$ connected faces. Indeed, viewing $x$ as a point in $M$, by
definition, $x$ is contained in exactly ${\rm depth}(x)$ connected faces of
$M$. The implicit function theorem implies that $x$ is contained locally in
exactly ${\rm depth}(x)$ faces and these faces are still disconnected globally
in $f^{-1}(y)$. Hence $f^{-1}(y)$ is a smooth manifold with faces.

Next, we see that the decomposition

$f^{-1}(y)=\bigsqcup_{\alpha\in{\mathbb{A}}}f^{-1}(y)\cap M_{\alpha}$

induces an ${\mathbb{A}}$-space structure on $f^{-1}(y)$. Conditions of
Definition 3.15 can then be verified using the implicit function theorem.
Transversality implies that $f^{-1}(y)$ has smooth corner charts. Hence
$f^{-1}(y)$ is a smooth ${\mathbb{A}}$-manifold. The equivariant case follows
immediately from the definition. ∎

####  3.4.3. Stratified smooth orbifolds with corners and derived orbifold
charts

Smooth orbifolds with corners can be defined as in Section 2.1, with the
modification that for an $n$-dimensional orbifold with corners
${\mathcal{U}}$, each local chart $C=(U,{\mathit{\Gamma}},\psi)$ has the
following property: consider ${\mathbb{R}}^{n-k}\times{\mathbb{R}}_{\geq
0}^{k}$ where ${\mathit{\Gamma}}$ acts linearly on the first factor and
trivially on the second factor. Then
$U\subset{\mathbb{R}}^{n-k}\times{\mathbb{R}}_{\geq 0}^{k}$ is a
${\mathit{\Gamma}}$-invariant open subset and
$\psi:U/{\mathit{\Gamma}}\to{\mathcal{U}}$ is a homeomorphism onto an open
subset of ${\mathcal{U}}$. Just as the manifold case, one can define the
notion of smooth orbifold with faces and the notion of smooth
${\mathbb{A}}$-orbifolds for any homogeneous poset ${\mathbb{A}}$. We omit the
discussions of bundles, sections, and embeddings in the stratified sense. Note
that if $\alpha\in{\mathbb{A}}$ and ${\mathcal{U}}$ is an
${\mathbb{A}}$-orbifold, the $\partial^{\alpha}{\mathbb{A}}$-space
$\partial^{\alpha}{\mathcal{U}}$ is naturally a
$\partial^{\alpha}{\mathbb{A}}$-orbifold.

Now we can generalize the definition of derived orbifold charts (see
Definition 2.1) to the case when the domains have corners or faces.

######  Definition 3.21 (Stratified derived orbifold charts).

  1. (1)

An ${\mathbb{A}}$-stratified derived orbifold chart is a triple
$({\mathcal{U}},{\mathcal{E}},{\mathcal{S}})$, where ${\mathcal{U}}$ is a
smooth ${\mathbb{A}}$-orbifold, ${\mathcal{E}}\to{\mathcal{U}}$ is an orbifold
vector bundle, and ${\mathcal{S}}:{\mathcal{U}}\to{\mathcal{E}}$ is a
continuous section. We say that $({\mathcal{U}},{\mathcal{E}},{\mathcal{S}})$
is compact if ${\mathcal{S}}^{-1}(0)$ is compact.

  2. (2)

Given ${\mathbb{A}}_{i}$-stratified derived orbifold charts
$C_{i}=({\mathcal{U}}_{i},{\mathcal{E}}_{i},{\mathcal{S}}_{i})$, $i=1,2$, a
chart embedding ${\bm{\iota}}_{21}:C_{1}\hookrightarrow C_{2}$ consists of a
smooth stratified embedding

$\textstyle{{\mathcal{U}}_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\iota_{21}}$$\textstyle{{\mathcal{U}}_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathbb{A}}_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathbb{A}}_{2}}$

and a bundle embedding
$\widehat{\iota}_{21}:{\mathcal{E}}_{1}\to{\mathcal{E}}_{2}$ covering
$\iota_{21}$ which intertwines with the sections ${\mathcal{S}}_{1}$ and
${\mathcal{S}}_{2}$.

  3. (3)

Given an ${\mathbb{A}}$-space $X$, a derived orbifold presentation (or called
a D-chart presentation) of $X$ is a quadruple
$({\mathcal{U}},{\mathcal{E}},{\mathcal{S}},\psi)$ consisting of an
${\mathbb{A}}$-stratified derived orbifold chart
$({\mathcal{U}},{\mathcal{E}},{\mathcal{S}})$ with a homeomorphism of
${\mathbb{A}}$-spaces $\psi:{\mathcal{S}}^{-1}(0)\to X$.

The following set of notations are frequently used. If
$({\mathcal{U}},{\mathcal{E}},{\mathcal{S}},\psi)$ is a derived orbifold
presentation of the ${\mathbb{A}}$-space $X$, for any stratum $\alpha$, the
datum
$(\partial^{\alpha}{\mathcal{U}},{\mathcal{E}}|_{\partial^{\alpha}{\mathcal{U}}},{\mathcal{S}}|_{\partial^{\alpha}{\mathcal{U}}},\psi|_{{\mathcal{S}}^{-1}(0)\cap\partial^{\alpha}{\mathcal{U}}})$
is a derived orbifold presentation of $\partial^{\alpha}X$. In practice, we
might have a derived orbifold presentation of the boundary stratum
$\partial^{\alpha}X$ different from the restriction of a derived orbifold
presentation of $X$. To carry out our construction of perturbations, we would
like to relate the possibly different derived orbifold presentations of the
boundary strata. This motivates the following definition, which should be
thought of as a special case of the _Kuranishi diagram_ from [Abo22,
Definition 6.6]. Some notions appear in the following were defined in
Definition 2.3.

######  Definition 3.22.

A system of D-chart presentation of an ${\mathbb{A}}$-space $X$ consists of
the following objects.

  1. (1)

A collection of derived orbifold presentations

$\big{\\{}C_{\alpha}=({\mathcal{U}}_{\alpha},{\mathcal{E}}_{\alpha},{\mathcal{S}}_{\alpha},\psi_{\alpha})\big{\\}}_{\alpha\in{\mathbb{A}}}$

of the collection of boundary strata
$\\{\partial^{\alpha}X\\}_{\alpha\in{\mathbb{A}}}$ as stratified spaces.

  2. (2)

A collection of chart embeddings

$\big{\\{}{\bm{\iota}}_{\beta\alpha}:C_{\alpha}\to\partial^{\alpha}C_{\beta}\big{\\}}_{\alpha\leq\beta}.$

These objects need to satisfy the following conditions.

  1. (A)

The collection of chart embeddings satisfy the cocycle condition. More
precisely, for any triple of strata $\alpha\leq\beta\leq\gamma$, there holds

$\iota_{\gamma\beta}\circ\iota_{\beta\alpha}=\iota_{\gamma\alpha}.$

  2. (B)

Adjacent strata differ by a stabilization. More precisely, for any pair of
strata $\alpha\leq\beta$, there exist an orbifold vector bundle
${\mathcal{F}}_{\beta\alpha}\to{\mathcal{U}}_{\alpha}$ and a germ equivalence

${\rm
Stab}_{{\mathcal{F}}_{\beta\alpha}}(C_{\alpha})\simeq\partial^{\alpha}C_{\beta}$

In certain special cases the chart embeddings between different strata are
open. For example in the case of Morse flow category and the pearly bimodule.
We introduce the following notion.

######  Definition 3.23.

A system of derived orbifold presentations of an ${\mathbb{A}}$-space $X$ is
called single-layered if all chart embeddings
${\bm{\iota}}_{\beta\alpha}:C_{\alpha}\to\partial^{\alpha}C_{\beta}$ are open
embeddings, or equivalently, one can take ${\mathcal{F}}_{\beta\alpha}=0$ for
all $\alpha\leq\beta$.

####  3.4.4. Derived orbifold lifts of flow categories and bimodules

Now we can go back to the setup in Section 3.2. The following definition
imposes certain regular structures on morphism spaces of flow categories and
flow bimodules.

######  Definition 3.24.

Let $T^{\mathcal{P}}$ be a flow category over the poset ${\mathcal{P}}$. A
derived orbifold lift of $T^{\mathcal{P}}$, denoted by
${\mathfrak{D}}^{\mathcal{P}}$, consists of the following objects.

  1. (1)

A collection

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

of derived orbifold presentations of the
${\mathbb{A}}^{\mathcal{P}}_{pq}$-space $T_{pq}$ such that for each connected
component ${\mathcal{U}}_{pq,j}\subset{\mathcal{U}}_{pq}$, one has

${\rm dim}_{\mathbb{R}}{\mathcal{U}}_{pq,j}-{\rm
rank}_{\mathbb{R}}{\mathcal{E}}_{pq}|_{{\mathcal{U}}_{pq,j}}\equiv{\rm
ind}^{\mathcal{P}}(p)-{\rm ind}^{\mathcal{P}}(q)-1\ {\rm mod}\ 2N.$ (3.5)

  2. (2)

A collection of chart embeddings

$\big{\\{}{\bm{\iota}}_{prq}:C_{pr}\times
C_{rq}\hookrightarrow\partial^{prq}C_{pq}\big{\\}}_{p\leq r\leq q}$

(with the underlying poset identification
${\mathbb{A}}^{\mathcal{P}}_{pr}\times{\mathbb{A}}^{\mathcal{P}}_{rq}\cong\partial^{prq}{\mathbb{A}}^{\mathcal{P}}_{pq}$).
In particular, if
$\iota_{prq}:{\mathcal{U}}_{pr}\times{\mathcal{U}}_{rq}\hookrightarrow{\mathcal{U}}_{pq}$
is the associated domain embedding and
$\widehat{\iota}_{prq}:{\mathcal{E}}_{pr}\boxplus{\mathcal{E}}_{rq}\hookrightarrow{\mathcal{E}}_{pq}$
is the associated bundle embedding, then the following diagram commutes.

$\textstyle{{\mathcal{E}}_{pr}\boxplus{\mathcal{E}}_{rq}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\widehat{\iota}_{prq}}$$\textstyle{{\mathcal{E}}_{pq}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathcal{U}}_{pr}\times{\mathcal{U}}_{rq}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\iota_{prq}}$$\scriptstyle{{\mathcal{S}}_{pr}\times{\mathcal{S}}_{rq}}$$\textstyle{{\mathcal{U}}_{pq}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{\mathcal{S}}_{pq}}$
(3.6)

These objects need to satisfy the following conditions.

  1. (A)

For $p=q$, the space ${\mathcal{U}}_{pp}$ is a singleton with trivial isotropy
and ${\mathcal{E}}_{pp}=\\{0\\}$ is the trivial bundle.

  2. (B)

The chart embeddings satisfy the associativity. More precisely, whenever
$p<r<s<q$, the following diagram commutes.

$\textstyle{C_{pr}\times C_{rs}\times
C_{rq}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{\rm
id}\times{\bm{\iota}}_{rsq}}$$\scriptstyle{{\bm{\iota}}_{prs}\times{\rm
id}}$$\textstyle{\partial^{prs}C_{ps}\times
C_{rq}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{\bm{\iota}}_{psq}}$$\textstyle{C_{pr}\times\partial^{rsq}C_{rq}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{\bm{\iota}}_{prq}}$$\textstyle{\partial^{prsq}C_{pq}}$

  3. (C)

For each $pq$ and all $\alpha=pr_{1}\cdots
r_{l}q\in{\mathbb{A}}_{pq}^{\mathcal{P}}$, define

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

Then condition (B) implies that for each pair of elements $\alpha\leq\beta$ in
${\mathbb{A}}_{pq}^{\mathcal{P}}$, there is a well-defined chart embedding

${\bm{\iota}}_{\beta\alpha}:C_{\alpha}\hookrightarrow\partial^{\alpha}C_{\beta}.$

Then we require that the collection
$\\{C_{\alpha}\\}_{\alpha\in{\mathbb{A}}^{\mathcal{P}}_{pq}}$ of derived
orbifold presentations and the collection
$\\{{\bm{\iota}}_{\beta\alpha}\\}_{\alpha\leq\beta}$ of chart embeddings
constitute a system derived orbifold presentations of $T_{pq}$.

  4. (D)

The strict $\Pi$-equivariance condition: for any $a\in\Pi$ and $p<q$, there is
an isomorphism between derived orbifold charts (in the obvious sense)
$\tilde{\phi}_{a}:C_{pq}\to C_{a\cdot p\ a\cdot q}$ satisfying
$\tilde{\phi}_{a_{1}\cdot
a_{2}}=\tilde{\phi}_{a_{1}}\circ\tilde{\phi}_{a_{2}}$, and $\tilde{\phi}_{a}$
restricts to $\phi_{a}$ along the zero locus ${\mathcal{S}}_{pq}^{-1}(0)$ to
the map $\phi_{a}$ from Definition 3.7. Furthermore, $\tilde{\phi}_{0}={\rm
Id}$ for $a=0\in\Pi$ should be the identity map.

Now consider derived orbifold lifts of bimodules.

######  Definition 3.25.

Let $M$ be a flow bimodule from a flow category $T^{\mathcal{P}}$ to
$T^{{\mathcal{P}}^{\prime}}$ as in Definition 3.10. Suppose $T^{\mathcal{P}}$
resp. $T^{{\mathcal{P}}^{\prime}}$ is endowed with a derived orbifold lift

${\mathfrak{D}}^{\mathcal{P}}=\Big{(}\big{\\{}C^{\mathcal{P}}_{pq}=({\mathcal{U}}^{\mathcal{P}}_{pq},{\mathcal{E}}^{\mathcal{P}}_{pq},{\mathcal{S}}^{\mathcal{P}}_{pq},\psi^{\mathcal{P}}_{pq})\big{\\}}_{p<q},\big{\\{}{\bm{\iota}}_{\beta\alpha}^{\mathcal{P}}\big{\\}}_{\alpha\leq\beta}\Big{)}\text{
resp. }$
${\mathfrak{D}}^{{\mathcal{P}}^{\prime}}=\Big{(}\big{\\{}C^{{\mathcal{P}}^{\prime}}_{p^{\prime}q^{\prime}}=({\mathcal{U}}^{{\mathcal{P}}^{\prime}}_{p^{\prime}q^{\prime}},{\mathcal{E}}^{{\mathcal{P}}^{\prime}}_{p^{\prime}q^{\prime}},{\mathcal{S}}^{{\mathcal{P}}^{\prime}}_{p^{\prime}q^{\prime}},\psi^{{\mathcal{P}}^{\prime}}_{p^{\prime}q^{\prime}})\big{\\}}_{p^{\prime}<q^{\prime}},\big{\\{}{\bm{\iota}}_{\beta^{\prime}\alpha^{\prime}}^{{\mathcal{P}}^{\prime}}\big{\\}}_{\alpha^{\prime}\leq\beta^{\prime}}\Big{)}.$

A derived orbifold lift of $M$ compatible with (or extending)
${\mathfrak{D}}^{\mathcal{P}}$ and ${\mathfrak{D}}^{{\mathcal{P}}^{\prime}}$
consists of the following objects.

  1. (1)

A collection

$\big{\\{}C^{M}_{pp^{\prime}}=({\mathcal{U}}^{M}_{pp^{\prime}},{\mathcal{E}}^{M}_{pp^{\prime}},{\mathcal{S}}^{M}_{pp^{\prime}},\psi^{M}_{pp^{\prime}})\big{\\}}_{p\in{\mathcal{P}},p^{\prime}\in{\mathcal{P}}^{\prime}}$

of derived orbifold presentations of $M_{pp^{\prime}}$ for $M_{pp^{\prime}}$
as an ${\mathbb{A}}_{pp^{\prime}}$-space such that for each connected
component
${\mathcal{U}}_{pp^{\prime},j}^{M}\subset{\mathcal{U}}_{pp^{\prime}}^{M}$, one
has

${\rm dim}_{\mathbb{R}}{\mathcal{U}}_{pp^{\prime},j}^{M}-{\rm
rank}_{\mathbb{R}}{\mathcal{E}}_{pp^{\prime}}^{M}|_{{\mathcal{U}}_{pp^{\prime},j}^{M}}\equiv{\rm
ind}^{\mathcal{P}}(p)-{\rm ind}^{{\mathcal{P}}^{\prime}}(p^{\prime})\ {\rm
mod}\ 2N.$ (3.7)

  2. (2)

Given $p\in{\mathcal{P}}$ and $p^{\prime}\in{{\mathcal{P}}^{\prime}}$ for
$M_{pp^{\prime}}\neq\emptyset$, for any $p\leq q$, a chart embedding

${\bm{\iota}}^{M}_{pqp^{\prime}}:C^{\mathcal{P}}_{pq}\times
C^{M}_{qp^{\prime}}\hookrightarrow\partial^{pqp^{\prime}}C^{M}_{pp^{\prime}},$

and for any $q^{\prime}\leq p^{\prime}$, a chart embedding

${\bm{\iota}}^{M}_{pq^{\prime}p^{\prime}}:C^{M}_{pq^{\prime}}\times
C^{{\mathcal{P}}^{\prime}}_{q^{\prime}p^{\prime}}\hookrightarrow\partial^{pq^{\prime}p^{\prime}}C^{M}_{pp^{\prime}}.$

The precise meaning can be spelled out as in Equation (3.6).

These objects are required to satisfy the following conditions.

  1. (A)

For $p=q$, the chart embedding ${\bm{\iota}}_{pqp^{\prime}}^{M}$ is the
identity map after identifying $C^{\mathcal{P}}_{pq}$ with the trivial chart
for the singleton. Similarly, for $q^{\prime}=p^{\prime}$, the chart embedding
${\bm{\iota}}_{pq^{\prime}p^{\prime}}^{M}$ is also the identity map.

  2. (B)

The chart embeddings satisfy the associativity. Namely, given
$M_{pp^{\prime}}\neq\emptyset$, the following three diagrams commute if the
relevant topological spaces are nonempty.

For $p<q_{1}<q_{2}$, we have

$\textstyle{C^{\mathcal{P}}_{pq_{1}}\times C^{\mathcal{P}}_{q_{1}q_{2}}\times
C^{M}_{q_{2}p^{\prime}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{\rm
id}\times{\bm{\iota}}^{M}_{q_{1}q_{2}p^{\prime}}}$$\scriptstyle{{\bm{\iota}}^{\mathcal{P}}_{pq_{1}q_{2}}\times{\rm
id}}$$\textstyle{\partial^{pq_{1}q_{2}}C^{\mathcal{P}}_{pq_{2}}\times
C^{M}_{q_{2}p^{\prime}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{\bm{\iota}}^{M}_{pq_{2}p^{\prime}}}$$\textstyle{C^{\mathcal{P}}_{pq_{1}}\times\partial^{q_{1}q_{2}p^{\prime}}C^{M}_{q_{1}p^{\prime}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{\bm{\iota}}^{M}_{pq_{1}p^{\prime}}}$$\textstyle{\partial^{pq_{1}q_{2}p^{\prime}}C^{M}_{pp^{\prime}}.}$

For $q_{2}^{\prime}<q_{1}^{\prime}<p^{\prime}$, we have

$\textstyle{C^{M}_{pq_{2}^{\prime}}\times
C^{{\mathcal{P}}^{\prime}}_{q_{2}^{\prime}q_{1}^{\prime}}\times
C^{{\mathcal{P}}^{\prime}}_{q_{1}^{\prime}p^{\prime}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{\rm
id}\times{\bm{\iota}}^{{\mathcal{P}}^{\prime}}_{q_{1}^{\prime}q_{2}^{\prime}p^{\prime}}}$$\scriptstyle{{\bm{\iota}}^{M}_{pq_{2}^{\prime}q_{1}^{\prime}}\times{\rm
id}}$$\textstyle{\partial^{pq_{2}^{\prime}q_{1}^{\prime}}C^{M}_{pq_{1}^{\prime}}\times
C^{{\mathcal{P}}^{\prime}}_{q_{1}^{\prime}p^{\prime}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{\bm{\iota}}^{M}_{pq_{1}^{\prime}p^{\prime}}}$$\textstyle{C^{M}_{pq_{2}^{\prime}}\times\partial^{q_{2}^{\prime}q_{1}^{\prime}p^{\prime}}C^{{\mathcal{P}}^{\prime}}_{q_{2}^{\prime}p^{\prime}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{\bm{\iota}}^{M}_{pq_{2}^{\prime}p^{\prime}}}$$\textstyle{\partial^{pq_{2}^{\prime}q_{1}^{\prime}p^{\prime}}C^{M}_{pp^{\prime}}.}$

Finally, for $p<q$ and $q^{\prime}<p^{\prime}$, we have

$\textstyle{C^{{\mathcal{P}}}_{pq}\times C^{M}_{qq^{\prime}}\times
C^{{\mathcal{P}}^{\prime}}_{q^{\prime}p^{\prime}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{\rm
id}\times{\bm{\iota}}^{M}_{qq^{\prime}p^{\prime}}}$$\scriptstyle{{\bm{\iota}}^{M}_{pqq^{\prime}}\times{\rm
id}}$$\textstyle{\partial^{pqq^{\prime}}C^{M}_{pq^{\prime}}\times
C^{{\mathcal{P}}^{\prime}}_{q^{\prime}p^{\prime}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{\bm{\iota}}^{M}_{pq^{\prime}p^{\prime}}}$$\textstyle{C^{{\mathcal{P}}}_{pq}\times\partial^{qq^{\prime}p^{\prime}}C^{M}_{qp^{\prime}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{\bm{\iota}}^{M}_{pqp^{\prime}}}$$\textstyle{\partial^{pqq^{\prime}p^{\prime}}C^{M}_{pp^{\prime}}.}$

  3. (C)

For each $p,p^{\prime}$ and all $\alpha=pr_{1}\cdots r_{l}r_{l}^{\prime}\cdots
r_{1}^{\prime}p^{\prime}$, define

$C_{\alpha}^{M}=C^{\mathcal{P}}_{pr_{1}}\times\cdots\times
C^{M}_{r_{l}r_{l}^{\prime}}\times\cdots\times
C^{{\mathcal{P}}^{\prime}}_{r_{1}^{\prime}p^{\prime}}.$

As in Definition 3.24, for each pair $\alpha\leq\beta$ in
${\mathbb{A}}_{pp^{\prime}}$, there is a well-defined chart embedding

${\bm{\iota}}_{\beta\alpha}^{M}:C_{\alpha}^{M}\hookrightarrow C_{\beta}^{M}.$

It is required that the derived orbifold presentations
$\\{C_{\alpha}^{M}\\}_{\alpha\in{\mathbb{A}}_{pp^{\prime}}}$ and the
collection of chart embeddings
$\\{{\bm{\iota}}_{\beta\alpha}\\}_{\alpha\leq\beta}$ constitute a system of
derived orbifold presentations of $M_{pp^{\prime}}$.

  4. (D)

The strict $\Pi$-equivariance condition: for any $a\in\Pi$ and
$M_{pp^{\prime}}\neq\emptyset$, there is an isomorphism between derived
orbifold charts $\tilde{\phi}^{M}_{a}:C^{M}_{pp^{\prime}}\to C^{M}_{a\cdot p\
a\cdot p^{\prime}}$ satisfying $\tilde{\phi}^{M}_{a_{1}\cdot
a_{2}}=\tilde{\phi}^{M}_{a_{1}}\circ\tilde{\phi}^{M}_{a_{2}}$, and
$\tilde{\phi}^{M}_{a}$ restricts to the map $\phi^{M}_{a}$ from Definition
3.10 along the zero locus $({\mathcal{S}}^{M}_{pp^{\prime}})^{-1}(0)$.
Moreover, $\tilde{\phi}^{M}_{id}$ for $a=id\in\Pi$ is the identity map.
Moreover, the left and right actions of the charts from $T^{\mathcal{P}}$ and
$T^{\mathcal{P}^{\prime}}$ on the charts of $M$ should be $\Pi$-equivariant.

###  3.5. Additional structures

In order to construct a coherent system of FOP perturbations, we need more
structures on the derived orbifold lifts. They are the collar structure,
scaffolding, and straightening.

####  3.5.1. Collar structure

######  Definition 3.26.

  1. (1)

A collar structure on an ${\mathbb{A}}$-stratified derived orbifold chart
$({\mathcal{U}},{\mathcal{E}},{\mathcal{S}})$ is a collection of open chart
embeddings

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0.0pt\hbox{$\textstyle{\partial^{\alpha}{\mathcal{U}}\times[0,\epsilon)^{{\mathbb{F}}_{\alpha}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
50.8126pt\raise-48.46887pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-2.69444pt\hbox{$\scriptstyle{\theta_{\alpha}^{\rm
collar}}$}}}\kern 3.0pt}}}}}}\ignorespaces{\hbox{\kern
84.33809pt\raise-41.91333pt\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
54.33809pt\raise-41.91333pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern
84.33809pt\raise-41.91333pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{{\mathcal{U}}}$}}}}}}}\ignorespaces}}}}\ignorespaces}\
\ \forall\alpha\in{\mathbb{A}}$

(where ${\mathbb{F}}_{\alpha}\subset{\mathbb{A}}$ is the set of adjacent faces
to $\alpha$, see Definition 3.2 and Notation 3.14) covering the poset map

$\partial^{\alpha}{\mathbb{A}}\times{\mathbb{A}}^{(|{\mathbb{F}}_{\alpha}|)}\to{\mathbb{A}}$

satisfying the following conditions.

    1. (a)

For a pair of stratum $\alpha\leq\beta$,
$v\in{\mathcal{E}}|_{\partial^{\alpha}{\mathcal{U}}}$,
$t_{\beta}\in[0,\epsilon)^{{\mathbb{F}}_{\beta}}$ and
$t_{\beta\alpha}\in[0,\epsilon)^{{\mathbb{F}}_{\alpha}\setminus{\mathbb{F}}_{\beta}}$
(so that $(t_{\beta},t_{\beta\alpha})$ is identified with an element of
$[0,\epsilon)^{{\mathbb{F}}_{\alpha}}$), one has

$\widehat{\theta}_{\alpha}^{\rm
collar}(v,t_{\beta},t_{\beta\alpha})=\widehat{\theta}_{\beta}^{\rm
collar}(\widehat{\theta}_{\alpha}^{\rm
collar}(v,0,t_{\beta\alpha}),t_{\beta}).$

    2. (b)

For each $\alpha\in{\mathbb{A}}$, $x\in\partial^{\alpha}{\mathcal{U}}$ and
$t_{\alpha}\in[0,\epsilon)^{{\mathbb{F}}_{\alpha}}$, one has

$\widehat{\theta}_{\alpha}^{\rm
collar}({\mathcal{S}}(x),t_{\alpha})={\mathcal{S}}(\theta_{\alpha}^{\rm
collar}(x,t_{\alpha})).$

  2. (2)

A collar structure on a system of D-chart presentations
$((C_{\alpha})_{\alpha\in{\mathbb{A}}},({\bm{\iota}}_{\beta\alpha})_{\alpha\leq\beta})$
of an ${\mathbb{A}}$-stratified space $X$ (see Definition 3.22) consists of
collar structures on all derived orbifold charts
$C_{\alpha}=({\mathcal{U}}_{\alpha},{\mathcal{E}}_{\alpha},{\mathcal{S}}_{\alpha})$
for all $\alpha\in{\mathbb{A}}$. Namely, for each $\beta\in{\mathbb{A}}$, a
collection of open chart embedding

$\textstyle{{\mathcal{E}}_{\beta}|_{\partial^{\alpha}{\mathcal{U}}_{\beta}}\times[0,\epsilon)^{{\mathbb{F}}_{\alpha}\setminus{\mathbb{F}}_{\beta}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\widehat{\theta}_{\beta\alpha}^{\rm
collar}}$$\textstyle{{\mathcal{E}}_{\beta}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\partial^{\alpha}{\mathcal{U}}_{\beta}\times[0,\epsilon)^{{\mathbb{F}}_{\alpha}\setminus{\mathbb{F}}_{\beta}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\theta_{\beta\alpha}^{\rm
collar}}$$\textstyle{{\mathcal{U}}_{\beta}}$

for all $\alpha<\beta$ which satisfy conditions in item (1) above. We require
the following condition is satisfied. For each triple $\alpha<\beta<\gamma$

$\widehat{\iota}_{\gamma\beta}\Big{(}\widehat{\theta}_{\beta\alpha}^{\rm
collar}(v_{\beta},t_{\beta\alpha})\Big{)}=\widehat{\theta}_{\gamma\alpha}^{\rm
collar}\Big{(}\widehat{\iota}_{\gamma\beta}(v_{\beta}),t_{\beta\alpha},0_{\gamma\beta}\Big{)}$

for all
$v_{\beta}\in{\mathcal{E}}_{\beta}|_{\partial^{\alpha}{\mathcal{U}}_{\beta}}$
and
$t_{\beta\alpha}\in[0,\epsilon)^{{\mathbb{F}}_{\alpha}\setminus{\mathbb{F}}_{\beta}}$.
Here
$0_{\gamma\beta}=(0,\ldots,0)\in[0,\epsilon)^{{\mathbb{F}}_{\beta}\setminus{\mathbb{F}}_{\gamma}}$
and hence $(t_{\beta\alpha},0_{\gamma\beta})$ is regarded as a point of
$[0,\epsilon)^{{\mathbb{F}}_{\alpha}\setminus{\mathbb{F}}_{\gamma}}$.

  3. (3)

A collar structure on a derived orbifold lift of a flow category (using the
same notations as in Definition 3.24), is a collection of collar structures on
the chart $C_{pq}$ such that the following is satisfied.

    1. (a)

For each $\alpha=pr_{1}\cdots r_{l}q\in{\mathbb{A}}_{pq}^{\mathcal{P}}$, the
collar structures on $C_{pr_{1}},\ldots,C_{r_{l}q}$ define a product collar
structure on the product chart $C_{\alpha}$. Then the collection of collar
structures on $C_{\alpha}$ is a collar structure on the derived orbifold
presentation of $T_{pq}$ as defined in above item (2).

    2. (b)

The collar structure is strictly $\Pi$-equivariant.

  4. (4)

Suppose ${\mathfrak{D}}_{{\mathcal{P}}{\mathcal{P}}^{\prime}}$ is a derived
orbifold lift of a flow bimodule $M$ from $T^{\mathcal{P}}$ to
$T^{{\mathcal{P}}^{\prime}}$ which extends ${\mathfrak{D}}^{\mathcal{P}}$ and
${\mathfrak{D}}^{{\mathcal{P}}^{\prime}}$. Suppose
${\mathfrak{D}}^{\mathcal{P}}$ and ${\mathfrak{D}}^{{\mathcal{P}}^{\prime}}$
are equipped with collar structures. Then a collar structure on
${\mathfrak{D}}_{{\mathcal{P}}{\mathcal{P}}^{\prime}}$ which extends the
collar structure on ${\mathfrak{D}}^{\mathcal{P}}$ and
${\mathfrak{D}}^{{\mathcal{P}}^{\prime}}$ is a collection of collar structures
on all the derived orbifold charts
$({\mathcal{U}}_{pp^{\prime}}^{M},{\mathcal{E}}_{pp^{\prime}}^{M},{\mathcal{S}}_{pp^{\prime}}^{M},\psi_{pp^{\prime}}^{M})$
of the form

$\vbox{ \lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
35.08641pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{{\mathcal{E}}^{M}_{pp^{\prime}}|_{\partial^{\alpha}{\mathcal{U}}^{M}_{pp^{\prime}}}\times[0,\epsilon)^{{\mathbb{F}}_{\alpha}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
0.0pt\raise-30.43275pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
52.07687pt\raise 8.34723pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-3.2639pt\hbox{$\scriptstyle{\widehat{\theta}_{pp^{\prime},\alpha}^{\rm
collar}}$}}}\kern 3.0pt}}}}}}\ignorespaces{\hbox{\kern 89.1978pt\raise
0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
59.08641pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise 0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern 89.1978pt\raise
0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{{\mathcal{E}}_{pp^{\prime}}^{M}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
98.85808pt\raise-31.09944pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx<EMAIL_ADDRESS>0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\partial^{\alpha}{\mathcal{U}}_{pp^{\prime}}^{M}\times[0,\epsilon)^{{\mathbb{F}}_{\alpha}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
51.80748pt\raise-50.01274pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-2.08333pt\hbox{$\scriptstyle{\theta_{pp^{\prime},\alpha}^{\rm
collar}}$}}}\kern 3.0pt}}}}}}\ignorespaces{\hbox{\kern
89.08641pt\raise-42.84608pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
59.08641pt\raise-42.84608pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern
89.08641pt\raise-42.84608pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{{\mathcal{U}}^{M}_{pp^{\prime}}}$}}}}}}}\ignorespaces}}}}\ignorespaces}\
\ \forall p\in{\mathcal{P}},\ p^{\prime}\in{\mathcal{P}^{\prime}},\
\alpha\in{\mathbb{A}}_{pp^{\prime}}$

which satisfy similar compatibility conditions as the case of flow categories
using the factorizations of boundary strata. We omit the details.

######  Remark 3.27.

Although the description of collar structures seems to be very complicated,
they are constructed using the outer-collaring method which implies the
required conditions almost automatically.

####  3.5.2. Scaffolding

######  Definition 3.28.

A scaffolding of a system of D-chart presentations
$((C_{\alpha})_{\alpha\in{\mathbb{A}}},({\bm{\iota}}_{\beta\alpha})_{\alpha\leq\beta})$
of an ${\mathbb{A}}$-space $X$ is a collection of data

$\big{(}{\mathcal{F}}_{\beta\alpha},{\bm{\theta}}_{\beta\alpha}\big{)}_{\alpha\leq\beta}$

where for each pair $\alpha\leq\beta$

  1. (1)

the difference bundle ${\mathcal{F}}_{\beta\alpha}\to{\mathcal{U}}_{\alpha}$
is an orbifold vector bundle
${\mathcal{F}}_{\beta\alpha}\to{\mathcal{U}}_{\alpha}$. In notation, when
${\mathbb{A}}={\mathbb{A}}_{pq}^{\mathcal{P}}$ resp.
${\mathbb{A}}_{pp^{\prime}}$ and $\beta$ is the maximal element $pq$ resp.
$pp^{\prime}$, denote ${\mathcal{F}}_{\beta\alpha}$ by
${\mathcal{F}}_{pq,\alpha}$ resp. ${\mathcal{F}}_{pp^{\prime},\alpha}$.

  2. (2)

the stabilization map ${\bm{\theta}}_{\beta\alpha}$ is a germ equivalence

${\bm{\theta}}_{\beta\alpha}=(\theta_{\beta\alpha},\widehat{\theta}_{\beta\alpha}):{\rm
Stab}_{{\mathcal{F}}_{\beta\alpha}}(C_{\alpha})\simeq\partial^{\alpha}C_{\beta}$
(3.8)

which extends the chart embedding
${\bm{\iota}}_{\beta\alpha}:C_{\alpha}\hookrightarrow\partial^{\alpha}C_{\beta}$.
This germ equivalence induces a projection map

$\pi_{\beta\alpha}:\partial^{\alpha}C_{\beta}\to C_{\alpha}$

as well as a bundle splitting

${\mathcal{E}}_{\beta}|_{{\mathcal{U}}_{\alpha}}={\mathcal{E}}_{\alpha}\oplus{\mathcal{F}}_{\beta\alpha}$
(3.9)

where
${\mathcal{E}}_{\beta}|_{{\mathcal{U}}_{\alpha}}={\bm{\iota}}_{\beta\alpha}^{*}{\mathcal{E}}_{\beta}$
and a bundle isomorphism

$\vartheta_{\beta\alpha}:\pi_{\beta\alpha}^{*}({\mathcal{E}}_{\beta}|_{{\mathcal{U}}_{\alpha}})\cong{\mathcal{E}}_{\beta}|_{\partial^{\alpha}{\mathcal{U}}_{\beta}}.$
(3.10)

These objects need to satisfy the following conditions.

  1. (A)

For any triple of stratum $\alpha\leq\beta\leq\gamma$, as subbundles of
${\mathcal{E}}_{\gamma}|_{{\mathcal{U}}_{\alpha}}$ there holds

${\mathcal{F}}_{\gamma\alpha}={\mathcal{F}}_{\beta\alpha}\oplus{\mathcal{F}}_{\gamma\beta}|_{{\mathcal{U}}_{\alpha}}.$
(3.11)

  2. (B)

The stabilization map preserves stratum. More precisely, the following diagram
commutes

$\vbox{ \lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
26.36763pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
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0.0pt\hbox{$\textstyle{{\rm
Stab}_{{\mathcal{F}}_{\beta\alpha}}(C_{\alpha})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
31.31393pt\raise 6.51389pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-1.34723pt\hbox{$\scriptstyle{{\bm{\theta}}_{\beta\alpha}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 50.36763pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
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50.36763pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{\partial^{\alpha}C_{\beta}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
63.32831pt\raise-20.79276pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-0.42361pt\hbox{$\scriptstyle{{\bm{\iota}}_{\gamma\beta}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern
63.32831pt\raise-30.43553pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx<EMAIL_ADDRESS>0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{{\rm
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31.43846pt\raise-47.83551pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
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3.0pt}}}}}}\ignorespaces{\hbox{\kern
50.50174pt\raise-41.58553pt\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
50.50174pt\raise-41.58553pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\partial^{\alpha}C_{\gamma}}$}}}}}}}\ignorespaces}}}}\ignorespaces}.$

It follows that

$\pi_{\gamma\alpha}\circ(\iota_{\gamma\beta}|_{\partial^{\alpha}U_{\beta}})=\pi_{\beta\alpha}.$
(3.12)

  3. (C)

The bundle isomorphism (3.10) preserves stratum. More precisely, for any
triple of strata $\alpha\leq\beta\leq\gamma$, consider the following diagram

$\vbox{ \lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
37.31152pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
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0.0pt\hbox{$\textstyle{\pi_{\gamma\alpha}^{*}\left({\mathcal{E}}_{\gamma}|_{{\mathcal{U}}_{\alpha}}\right)|_{\partial^{\alpha}{\mathcal{U}}_{\beta}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
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1.0pt\raise 0.0pt\hbox{\lx<EMAIL_ADDRESS>0.0pt\hbox{\lx@xy@droprule}}}}\ignorespaces{}{\hbox{\hbox{\kern 1.0pt\raise
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1.0pt\raise 0.0pt\hbox{\lx<EMAIL_ADDRESS>0.0pt\hbox{\lx@xy@droprule}}}}\ignorespaces{}{\hbox{\hbox{\kern 1.0pt\raise
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1.0pt\raise 0.0pt\hbox{\lx<EMAIL_ADDRESS>0.0pt\hbox{\lx@xy@droprule}}}}\ignorespaces{}{\hbox{\hbox{\kern 1.0pt\raise
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0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
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0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-1.34723pt\hbox{$\scriptstyle{\vartheta_{\beta\alpha}\oplus\vartheta_{\gamma\beta\alpha}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern
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We explain the notations here. The first vertical equal arrow on the left is
due to (3.12) and the second one is due to (3.9). Here the requirement is
that, we require

$\vartheta_{\gamma\alpha}\left(\pi_{\beta\alpha}^{*}({\mathcal{F}}_{\gamma\beta}|_{{\mathcal{U}}_{\alpha}})\right)={\mathcal{F}}_{\gamma\beta}|_{\partial^{\alpha}{\mathcal{U}}_{\beta}}$

and the restriction to $\vartheta_{\gamma\alpha}$ to
$\pi_{\beta\alpha}^{*}({\mathcal{F}}_{\gamma\beta}|_{{\mathcal{U}}_{\alpha}})$
is equal to a linear isomorphism $\vartheta_{\gamma\beta\alpha}$ (which is in
the above commutative diagram). And we require that the above diagram
commutes.

  4. (D)

The stabilization maps satisfy the cocycle condition. Namely, for each triple
of strata $\alpha\leq\beta\leq\gamma$, the following diagram commutes.

$\vbox{ \lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
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50.291pt\raise-41.65831pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise 0.0pt\hbox{$\textstyle{{\rm
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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
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Here the “$=$” arrow is induced from the identity
${\mathcal{F}}_{\gamma\alpha}={\mathcal{F}}_{\gamma\beta}|_{{\mathcal{U}}_{\alpha}}\oplus{\mathcal{F}}_{\beta\alpha}$.

Now consider a derived orbifold lift of a flow category.

######  Definition 3.29.

A scaffolding of a derived orbifold lift of a flow category $T^{\mathcal{P}}$
consists of a collection of scaffoldings for the induced system of derived
orbifold chart presentations of $T_{pq}$

$\Big{(}\big{(}{\mathcal{F}}_{\beta\alpha},{\bm{\theta}}_{\beta\alpha}\big{)}_{\alpha\leq\beta}\Big{)}_{p<q}$

satisfying

  1. (1)

Suppose $pr_{1}\cdots r_{l}q=\alpha\leq\beta=ps_{1}\cdots s_{m}q$. Denote
$\alpha_{0}=pr_{1}\cdots s_{1}$, $\ldots$, $\alpha_{m}=s_{m}\cdots r_{l}q$.
Then as subbundles of ${\mathcal{E}}_{\beta}|_{{\mathcal{U}}_{\alpha}}$, in
view of the identification
${\mathcal{U}}_{\alpha}={\mathcal{U}}_{\alpha_{0}}\times\cdots\times{\mathcal{U}}_{\alpha_{m}}$,
one has

${\mathcal{F}}_{\beta\alpha}={\mathcal{F}}_{ps_{1},\alpha_{0}}\boxplus\cdots\boxplus{\mathcal{F}}_{s_{m}q,\alpha_{m}}.$

  2. (2)

With respect to the last identity, one has (as germs of maps)

${\bm{\theta}}_{\beta\alpha}={\bm{\theta}}_{ps_{1},\alpha_{0}}\times\cdots\times{\bm{\theta}}_{s_{m}q,\alpha_{m}}.$

######  Definition 3.30.

Given two flow categories $T^{\mathcal{P}}$ and $T^{{\mathcal{P}}^{\prime}}$
endowed with derived orbifold lifts ${\mathfrak{D}}^{\mathcal{P}}$,
${\mathfrak{D}}^{{\mathcal{P}}^{\prime}}$ respectively, let $M$ be a flow
bimodule from $T^{\mathcal{P}}$ to $T^{{\mathcal{P}}^{\prime}}$ endowed with a
compatible derived orbifold lift ${\mathfrak{D}}^{M}$. Suppose
${\mathfrak{D}}^{\mathcal{P}}$ and ${\mathfrak{D}}^{{\mathcal{P}}^{\prime}}$
come with scaffoldings

$\big{(}\big{\\{}{\mathcal{F}}^{\mathcal{P}}_{\beta\alpha},{\bm{\theta}}^{\mathcal{P}}_{\beta\alpha}\big{\\}}_{\alpha\leq\beta}\big{)}_{p<q},\text{
}\big{(}\big{\\{}{\mathcal{F}}^{{\mathcal{P}}^{\prime}}_{\beta^{\prime}\alpha^{\prime}},{\bm{\theta}}^{{\mathcal{P}}^{\prime}}_{\beta^{\prime}\alpha^{\prime}}\big{\\}}_{\alpha^{\prime}\leq\beta^{\prime}}\big{)}_{p^{\prime}<q^{\prime}}.$

A scaffolding of such a derived orbifold lift compatible with the given
scaffoldings is given by a collection of scaffoldings for the induced system
of derived orbifold chart presentations

$\big{(}\big{\\{}{\mathcal{F}}_{\tilde{\beta}\tilde{\alpha}},{\bm{\theta}}_{\tilde{\beta}\tilde{\alpha}}\big{\\}}_{\tilde{\alpha}\leq\tilde{\beta}}\big{)}_{\tilde{\alpha},\tilde{\beta}\in{\mathbb{A}}_{pp^{\prime}}^{M}}$

satisfying similar conditions as in Definition 3.29 using the factorization of
the boundary strata (see Definition 3.25 (C)).

####  3.5.3. Straightening

As we will perform FOP perturbations over stratified charts, we need to
consider straightenings in the stratified situation.

######  Definition 3.31 (Straightenings).

  1. (1)

Let $({\mathcal{U}},{\mathcal{E}},{\mathcal{S}})$ be an
${\mathbb{A}}$-stratified derived orbifold chart. A straightening consists of
a Riemannian metric on ${\mathcal{U}}$ and a connection on ${\mathcal{E}}$
such that for each $\alpha\in{\mathbb{A}}$, the restriction of the metric and
the connection onto ${\mathcal{U}}_{\alpha}$ (which is itself a smooth
orbifold) is a straightening of
$({\mathcal{U}}_{\alpha},{\mathcal{E}}|_{{\mathcal{U}}_{\alpha}},{\mathcal{S}}|_{{\mathcal{U}}_{\alpha}})$
(see Definition 2.1).

  2. (2)

Consider a system of derived orbifold presentations on an
${\mathbb{A}}$-stratified space $X$. A straightening consists of a collection
of straightenings of $C_{\alpha}$ for all $\alpha\in{\mathbb{A}}$ such that
for each pair $\alpha\leq\beta$ (with associated chart embedding
${\bm{\iota}}_{\beta\alpha}=(\iota_{\beta\alpha},\widehat{\iota}_{\beta\alpha}):C_{\alpha}\hookrightarrow\partial^{\alpha}C_{\beta}$),
the domain embedding $\iota_{\beta\alpha}$ is isometric and the bundle
embedding preserves the connection, i.e., the image of the bundle embedding is
preserved by the connection $\nabla^{{\mathcal{E}}_{\beta}}$ and the
restriction of $\nabla^{{\mathcal{E}}_{\alpha}}$ to the image of the bundle
embedding coincides with $\nabla^{{\mathcal{E}}_{\alpha}}$.

  3. (3)

Consider a derived orbifold lift ${\mathfrak{D}}^{\mathcal{P}}$ of a
topological flow category $T^{\mathcal{P}}$ over ${\mathcal{P}}$. A
straightening on ${\mathfrak{D}}^{\mathcal{P}}$ consists of a straightening on
all charts $C_{pq}$ satisfying the following condition. Fix $p<q$. For each
$\alpha=pr_{1}\cdots r_{l}q$, there is a product straightening on the
associated chart $C_{\alpha}=C_{pr_{1}}\times\cdots\times C_{r_{l}q}$. Then we
require that the collection of these straightenings for all
$\alpha\in{\mathbb{A}}_{pq}^{\mathcal{P}}$ is a straightening on the system of
derived orbifold presentations of $T_{pq}^{\mathcal{P}}$.

  4. (4)

It is straightforward to define the notion of straightenings on a derived
orbifold lift of a flow bimodule $M$ compatible with existing straightenings
on the lifts of the two flow categories $T^{\mathcal{P}}$ and
$T^{\mathcal{P}^{\prime}}$ with derived orbifold lifts
${\mathfrak{D}}^{\mathcal{P}}$ and ${\mathfrak{D}}^{\mathcal{P}^{\prime}}$.
Namely, it consists of a straightening on all charts $C^{M}_{pp^{\prime}}$
such that: for $\alpha=pq_{1}\cdots q_{k}q^{\prime}_{k^{\prime}}\cdots
q^{\prime}_{1}p^{\prime}\in{\mathbb{A}}_{pp^{\prime}}$ with
$M_{\alpha}\neq\emptyset$, there is a product straightening on the product
chart $C^{\mathcal{P}}_{pq_{1}}\times\cdots\times
C^{M}_{q_{k}q^{\prime}_{k^{\prime}}}\times\cdots\times
C^{\mathcal{P}^{\prime}}_{q^{\prime}_{1}p^{\prime}}$; we require that the
collection of these straightenings for all
$\alpha\in{\mathbb{A}}_{pp^{\prime}}$ is a straightening on the system of
derived orbifold presentations of $M_{pp^{\prime}}$.

####  3.5.4. Compatibility

We need the three kinds of additional structures (collar structure,
scaffolding, and straightening) to be compatible in a certain sense. These
compatibility conditions are necessary for the inductive construction of FOP
perturbations. For example, once a perturbation is chosen on all boundary
strata, it can be canonically extended to a neighborhood of the boundary using
the collar structure; meanwhile, there is another automatic extension using
the scaffolding. Hence a compatibility requirement between these two
structures is necessary. We also need the automatic extensions remain in the
class of FOP perturbations, which further requires that the straightening is
compatible with the collar structure and the scaffolding.

######  Remark 3.32.

In practice the three kinds of structures are constructed in different ways.
The collar structure is constructed by the outer-collaring construction, which
is very easy and straightforward. Then (see Subsection 5.8) we construct
scaffolding in the topological category. The scaffolding will play a role in
the smoothing process. After smoothing, the topological charts become smooth.
Then finally we construct the compatible straightenings.

######  Remark 3.33.

In the following, we just define the notions of compatibility of a derived
orbifold presentation. The relevant compatibility requirements for flow
categories and flow bimodules are then reduced to the corresponding
requirements for the induced derived orbifold presentations of all relevant
topological spaces.

######  Definition 3.34.

Consider a system of derived orbifold presentations
$\\{C_{\alpha}\\}_{\alpha\in{\mathbb{A}}}$ of an ${\mathbb{A}}$-stratified
space $X$. Suppose it is equipped with a collar structure and a scaffolding.
We say that they are compatible if the following conditions are satisfied.

  1. (1)

Given a triple $\alpha<\beta<\gamma$, we require that the following diagram
commutes.

$\textstyle{{\mathcal{E}}_{\gamma}|_{\partial^{\alpha}{\mathcal{U}}_{\gamma}}\times[0,\epsilon)^{{\mathbb{F}}_{\alpha}\setminus{\mathbb{F}}_{\beta}}\times\\{0\\}^{{\mathbb{F}}_{\beta}\setminus{\mathbb{F}}_{\gamma}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\widehat{\theta}_{\gamma\alpha}^{\rm
collar}}$$\textstyle{{\mathcal{E}}_{\gamma}|_{\partial^{\beta}{\mathcal{U}}_{\gamma}}}$$\textstyle{{\mathcal{E}}_{\gamma}|_{\partial^{\alpha}{\mathcal{U}}_{\beta}}\times[0,\epsilon)^{{\mathbb{F}}_{\alpha}\setminus{\mathbb{F}}_{\beta}}\times\\{0\\}^{{\mathbb{F}}_{\beta}\setminus{\mathbb{F}}_{\gamma}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\widehat{\theta}_{\gamma\alpha}^{\rm
collar}}$$\textstyle{{\mathcal{E}}_{\gamma}|_{{\mathcal{U}}_{\beta}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathcal{F}}_{\gamma\beta}|_{\partial^{\alpha}{\mathcal{U}}_{\beta}}\times[0,\epsilon)^{{\mathbb{F}}_{\alpha}\setminus{\mathbb{F}}_{\beta}}\times\\{0\\}^{{\mathbb{F}}_{\beta}\setminus{\mathbb{F}}_{\gamma}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\widehat{\theta}_{\gamma\alpha}^{\rm
collar}}$$\textstyle{{\mathcal{F}}_{\gamma\beta}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

Here the commutativity of the upper square is a requirement of the collar
structure. Moreover, the bottom horizontal arrow uses the splitting (3.9) and
$\widehat{\theta}_{\gamma\alpha}^{\rm collar}$ is the restriction to the
${\mathcal{F}}_{\gamma\beta}$ component.

  2. (2)

The stablization maps also respect the collars. More precisely, for
$\alpha<\beta<\gamma$, we require that

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Here the horizontal arrows are induced from the collar structure and the
vertical arrows are restrictions of the open chart embeddings associated to
the stabilization and scaffolding.

Finally we spell out the meaning of compatible straightenings.

######  Definition 3.35.

Consider a system of D-chart presentations equipped with a collar structure
and a scaffolding which are compatible. Then a straightening (see Definition
3.31) is compatible with the collar structure and the scaffolding if the
following conditions are satisfied.

  1. (1)

For each pair $\alpha<\beta$, near $\partial^{\alpha}{\mathcal{U}}_{\beta}$,
using the corner coordinates induced from the map $\theta_{\beta\alpha}^{\rm
collar}$, near $\partial^{\alpha}{\mathcal{U}}_{\beta}$, one has

$g_{\beta}=g_{\beta}|_{\partial^{\alpha}{\mathcal{U}}_{\beta}}+\sum_{i\in{\mathbb{F}}_{\alpha}\setminus{\mathbb{F}}_{\beta}}dt_{i}^{2}.$
(3.14)

Moreover, with respect to the bundle isomorphism
$\widehat{\theta}_{\beta\alpha}^{\rm collar}$, near
$\partial^{\alpha}{\mathcal{U}}_{\beta}$ one has

$\nabla^{{\mathcal{E}}_{\beta}}=\left(\widehat{\pi}_{\beta\alpha}^{\rm
collar}\right)^{*}\nabla^{{\mathcal{E}}_{\beta}}|_{\partial^{\alpha}{\mathcal{U}}_{\beta}}.$

  2. (2)

For each pair $\alpha\leq\beta$, the domain embedding

$\iota_{\beta\alpha}:{\mathcal{U}}_{\alpha}\hookrightarrow\partial^{\alpha}{\mathcal{U}}_{\beta}$

is isometric and totally geodesic. The normal bundle of this embedding has an
induced metric and metric connection. Via the stabilization map
${\bm{\theta}}_{\beta\alpha}$ which embeds ${\mathcal{F}}_{\beta\alpha}$ into
${\mathcal{E}}_{\beta}$, the bundle ${\mathcal{F}}_{\beta\alpha}$ then carries
a metric and metric connection.

  3. (3)

The bundle embedding
${\mathcal{F}}_{\beta\alpha}\hookrightarrow{\mathcal{E}}_{\beta}$ preserves
the connection.

  4. (4)

The stabilization map on the domain is isometric.

  5. (5)

Via the stabilization map (on the bundle), the connection on
${\mathcal{E}}_{\beta}|_{\partial^{\alpha}{\mathcal{U}}_{\beta}}$ near the
embedding image is the pullback connection, i.e., the direct sum of the
pullback connections on ${\mathcal{F}}_{\beta\alpha}$ and
${\mathcal{E}}_{\alpha}$.

###  3.6. Normal complex structure and orientations on flow categories

In order to carry out the FOP perturbations and define counts over
${\mathbb{Z}}$, we need to introduce normal complex structures on the derived
orbifold lifts and certain orientation structures.

######  Definition 3.36.

Let $T^{\mathcal{P}}$ be a flow category over ${\mathcal{P}}$ equipped with a
derived orbifold lift ${\mathfrak{D}}^{\mathcal{P}}$, with the collection of
derived orbifold presentations

$\big{\\{}C_{pq}=({\mathcal{U}}_{pq},{\mathcal{E}}_{pq},{\mathcal{S}}_{pq},\psi_{pq})\big{\\}}_{p\leq
q}.$

Moreover, assume that ${\mathfrak{D}}^{\mathcal{P}}$ has a scaffolding

${\mathfrak{F}}^{\mathcal{P}}=\Big{(}\big{(}{\mathcal{F}}_{\beta\alpha},{\bm{\theta}}_{\beta\alpha}\big{)}_{\alpha\leq\beta}\Big{)}_{p<q}.$

Then a normal complex structure (or normally complex lift) on the pair
${\mathfrak{D}}^{\mathcal{P}}$ and ${\mathfrak{F}}^{\mathcal{P}}$ consists of
the following data.

  1. (1)

A normal complex structure on each of the derived orbifold chart
$({\mathcal{U}}_{pq},{\mathcal{E}}_{pq},{\mathcal{S}}_{pq})$.

  2. (2)

A complex structure on each of the orbifold vector bundles
${\mathcal{F}}_{\beta\alpha}\to{\mathcal{U}}_{\alpha}$.

  3. (3)

For any pair $p,q\in{\mathcal{P}}$ with $T^{\mathcal{P}}_{pq}\neq\emptyset$
and any pair of strata $\alpha\leq\beta$ in ${\mathbb{A}}^{\mathcal{P}}_{pq}$,
the underlying map of the germ equivalence (3.8)

$\theta_{\beta\alpha}:{\rm
Stab}_{{\mathcal{F}}_{\beta\alpha}}({\mathcal{U}}_{\alpha})\simeq\partial^{\alpha}{\mathcal{U}}_{\beta}$

respects the normal complex structures.

  4. (4)

Under the same assumptions as the previous item, the map

$\widehat{\theta}_{\beta\alpha}:\pi^{*}_{\beta\alpha}{\mathcal{E}}_{\alpha}\oplus\pi^{*}_{\beta\alpha}{\mathcal{F}}_{\beta\alpha}\to{\mathcal{E}}_{\beta}|_{\partial^{\alpha}U_{\beta}}$

intertwines with the normal complex structures.

  5. (5)

The splitting of orbifold vector bundles (3.9) over ${\mathcal{U}}_{\alpha}$,
the bundle isomorphism (3.10), and the splitting (3.11) respect the normal
complex structures.

  6. (6)

These (normal) complex structures admit a strict $\Pi$-action in the obvious
sense.

######  Definition 3.37.

Let $T^{\mathcal{P}}$, ${\mathfrak{D}}^{\mathcal{P}}$, and
${\mathfrak{F}}^{\mathcal{P}}$ be as in Definition 3.36. Suppose we are given
a normal complex structure on the pair ${\mathfrak{D}}^{\mathcal{P}}$ and
${\mathfrak{F}}^{\mathcal{P}}$. Then a collar structure compatibile with
${\mathfrak{F}}^{\mathcal{P}}$ (see Definition 3.34) is said to be compatible
with the normal complex structure if the collar maps intertwine with the
normal complex structures.

######  Remark 3.38.

Note that the complex structure on ${\mathcal{F}}_{\beta\alpha}$ and the
normal complex structure on ${\mathcal{U}}_{\alpha}$ induce a normal complex
structure on ${\rm
Stab}_{{\mathcal{F}}_{\beta\alpha}}({\mathcal{U}}_{\alpha})$. Moreover, due to
the behavior of orbifold charts near the boundary and corners,
$\partial^{\alpha}{\mathcal{U}}_{\beta}$ indeed has a normal complex
structure.

######  Remark 3.39.

We could define the notion of normal complex structure in a more intrinsic way
which does not rely on the choice of scaffoldings.

######  Remark 3.40.

In our applications, the notion of normal complex structure is mostly useful
for considering the space ${\mathcal{U}}_{\alpha}$, as the orbifold vector
bundles ${\mathcal{E}}_{\alpha}$ and ${\mathcal{F}}_{\beta\alpha}$ would have
fiberwise complex structures in practice.

The normal complex structure is concerned with the tangential structure on the
normal directions to strata specified by the isomorphism classes of the
isotropy groups on an orbifold. In particular, an unorientable manifold has a
(trivial) normal complex structure when being viewed as an orbifold. To define
algebraic counts, we need to define orientation structures on the flow
categories and flow bimodules, which is an extra structure beyond the normal
complex structure.

######  Definition 3.41.

Let $V$ be a vector space over ${\mathbb{R}}$. The orientation line
$\mathfrak{o}_{V}$ of $V$ is defined to be the free ${\mathbb{Z}}$-module of
rank $1$ given by $H_{\dim V}(V,V\setminus\\{0\\};{\mathbb{Z}})$ with
${\mathbb{Z}}/2$-grading the mod $2$ reduction of $\dim V$. More generally, if
$(V^{+},V^{-})$ is a virtual vector space, its orientation line
$\mathfrak{o}_{(V^{+},V^{-})}$ is defined to be the tensor product of
orientation lines $\mathfrak{o}_{V^{+}}\otimes\mathfrak{o}_{V^{-}}^{\vee}$.

######  Definition 3.42.

An ${\mathbb{A}}$-stratified derived orbifold chart
$C=({\mathcal{U}},{\mathcal{E}},{\mathcal{S}})$ is called orientable if both
${\mathcal{U}}$ and ${\mathcal{E}}$ are orientable. Given a connected
component of an orientable derived orbifold chart, the orientation line
$\mathfrak{o}_{C}$ of $({\mathcal{U}},{\mathcal{E}},{\mathcal{S}})$ is defined
to be the orientation line of the virtual vector space

$(T_{x}{\mathcal{U}},{\mathcal{E}}_{x})$

for arbitrary $x\in{\mathcal{U}}$ lying in the top stratum of ${\mathcal{U}}$.

If $C=({\mathcal{U}},{\mathcal{E}},{\mathcal{S}})$ is an orientable
${\mathbb{A}}$-stratified derived orbifold chart, given an index
$\alpha\in{\mathbb{A}}$, the $\partial^{\alpha}{\mathbb{A}}$-stratified
derived orbifold chart
$\partial^{\alpha}C=(\partial^{\alpha}{\mathcal{U}},{\mathcal{E}}|_{\partial^{\alpha}{\mathcal{U}}},{\mathcal{S}}|_{\partial^{\alpha}{\mathcal{U}}})$
is also orientable. Moreover, it is easy to see that there is a natural
isomorphism

$\mathfrak{o}_{\partial^{\alpha}C}\xrightarrow{\sim}\mathfrak{o}_{C}\otimes(\mathfrak{o}_{\mathbb{R}}^{\vee})^{\otimes{\rm
depth}(\alpha)}.$

######  Definition 3.43.

Suppose $T^{\mathcal{P}}$, ${\mathfrak{D}}^{\mathcal{P}}$ and
${\mathfrak{F}}^{\mathcal{P}}$ are the same in Definition 3.36 An orientation
of such a normally complex derived orbifold lift is given by:

  1. (1)

A virtual vector space $(V_{p}^{+},V_{p}^{-})$ for each $p\in{\mathcal{P}}$.
Define $\mathfrak{o}_{p}:=\mathfrak{o}_{(V_{p}^{+},V_{p}^{-})}$.

  2. (2)

An isomorphism of orientation lines

$\mathfrak{o}_{C_{pq}}\xrightarrow{\sim}\mathfrak{o}_{p}^{\vee}\otimes\mathfrak{o}_{q}^{.}$
(3.15)

  3. (3)

For $p<r<q$, an isomorphism of orientation lines

$\mathfrak{o}_{C_{pr}}\otimes\mathfrak{o}_{C_{rq}}\otimes(\mathfrak{o}_{\mathcal{F}_{prq,pq}}\otimes\mathfrak{o}_{\mathcal{F}_{prq,pq}}^{\vee})\xrightarrow{\sim}\mathfrak{o}_{\partial^{prq}C_{pq}}.$
(3.16)

  4. (4)

The above isomorphisms are preserved by the $\Pi$-action.

Note that if the above derived orbifold lift is given an orientation, then for
any $\alpha\leq\beta$ in ${\mathbb{A}}^{\rm Floer}_{pq}$, there is an
isomorphism

$\mathfrak{o}_{C_{\alpha}}\xrightarrow{\sim}\mathfrak{o}_{\partial^{\alpha}C_{\beta}}.$

######  Definition 3.44.

Given a flow category $T^{\mathcal{P}}$ endowed with a derived orbifold lift
${\mathfrak{D}}^{\mathcal{P}}$, a scaffolding ${\mathfrak{F}}^{\mathcal{P}}$,
and a compatible collar structure, if they are further endowed with a normal
complex structure and orientation, we call such a datum an oriented and
normally complex derived orbifold lift and abbreviate it as
${\mathfrak{D}}^{\mathcal{P}}$.

If we have a flow bimodule $M$ between $T^{\mathcal{P}}$ and
$T^{\mathcal{P}^{\prime}}$ such that these three objects admit a compatible
derived orbifold lift, as well as a compatible scaffolding structure, one can
spell out the meaning of a normal complex structure and an orientation on $M$
similarly to Definition 3.36.

######  Definition 3.45.

Let $T^{\mathcal{P}}$ (resp. $T^{\mathcal{P}^{\prime}}$) be flow categories
endowed with a derived orbifold lift
${\mathfrak{D}}^{\mathcal{P}}=\\{C^{\mathcal{P}}_{pq}\\}_{p\leq q}$ (resp.
${\mathfrak{D}}^{\mathcal{P}^{\prime}}=\\{C^{\mathcal{P}^{\prime}}_{p^{\prime}q^{\prime}}\\}_{p^{\prime}\leq
q^{\prime}}$) and a scaffolding

${\mathfrak{F}}^{\mathcal{P}}=\Big{(}\big{(}{\mathcal{F}}_{\beta\alpha}^{\mathcal{P}},{\bm{\theta}}_{\beta\alpha}\big{)}_{\alpha\leq\beta}\Big{)}_{p<q}\text{
resp.
}{\mathfrak{F}}^{\mathcal{P}^{\prime}}=\Big{(}\big{(}{\mathcal{F}}_{\beta^{\prime}\alpha^{\prime}}^{\mathcal{P}^{\prime}},{\bm{\theta}}_{\beta^{\prime}\alpha^{\prime}}\big{)}_{\alpha^{\prime}\leq\beta^{\prime}}\Big{)}_{p^{\prime}<q^{\prime}}.$

Moreover, both the pair
$({\mathfrak{D}}^{\mathcal{P}},{\mathfrak{F}}^{\mathcal{P}})$ and the pair
$({\mathfrak{D}}^{\mathcal{P}^{\prime}},{\mathfrak{F}}^{\mathcal{P}^{\prime}})$
have a normal complex structure in the sense of Definition 3.36. Suppose $M$
is a flow bimodule from $T^{\mathcal{P}}$ to $T^{\mathcal{P}^{\prime}}$ with a
derived orbifold lift extending ${\mathfrak{D}}^{\mathcal{P}}$ and
${\mathfrak{D}}^{\mathcal{P}^{\prime}}$ (Definition 3.25)

${\mathfrak{D}}^{M}=\Big{(}\big{\\{}C^{M}_{pp^{\prime}}=({\mathcal{U}}^{M}_{pp^{\prime}},{\mathcal{E}}^{M}_{pp^{\prime}},{\mathcal{S}}^{M}_{pp^{\prime}},\psi^{M}_{pp^{\prime}})\big{\\}}_{p\in{\mathcal{P}},p^{\prime}\in{\mathcal{P}}^{\prime}},{\bm{\iota}}^{M}_{pp^{\prime}}\Big{)}$

and a scaffolding

${\mathfrak{F}}^{M}=\big{(}\big{\\{}{\mathcal{F}}_{\tilde{\beta}\tilde{\alpha}}^{M},{\bm{\theta}}_{\tilde{\beta}\tilde{\alpha}}^{M}\big{\\}}_{\tilde{\alpha}\leq\tilde{\beta}}\big{)}_{\tilde{\alpha},\tilde{\beta}\in{\mathbb{A}}_{pp^{\prime}}^{M}}$

compatible with ${\mathfrak{F}}^{\mathcal{P}}$ and
${\mathfrak{F}}^{\mathcal{P}^{\prime}}$ (Definition 3.30). Then a normal
complex structure (or normally complex lift) on this datum extending or
compatible with the given ones on
$({\mathfrak{D}}^{\mathcal{P}},{\mathfrak{F}}^{\mathcal{P}})$ and
$({\mathfrak{D}}^{\mathcal{P}^{\prime}},{\mathfrak{F}}^{\mathcal{P}^{\prime}})$
consists of the following data.

  1. (1)

A normal complex structure on each of the derived orbifold chart
$({\mathcal{U}}^{M}_{pp^{\prime}},{\mathcal{E}}^{M}_{pp^{\prime}},{\mathcal{S}}^{M}_{pp^{\prime}})$.

  2. (2)

A complex structure on the each of the orbifold vector bundles
${\mathcal{F}}_{\tilde{\beta}\tilde{\alpha}}^{M}\to{\mathcal{U}}^{M}_{\alpha}$.

  3. (3)

The germ equivalence $\theta^{M}_{\tilde{\alpha},\tilde{\beta}}$ and the
bundle map covering it $\widehat{\theta}^{M}_{\tilde{\alpha},\tilde{\beta}}$
all intertwine with the normal complex structures.

  4. (4)

Splittings or isomorphism of the form (3.9), (3.10), and (3.11) all respect
the (normal) complex structures.

  5. (5)

The respective (normal) complex structures are identified under the
$\Pi$-action.

######  Definition 3.46.

Under the same setting as in Definition 3.45, suppose we are given a
compatible normal complex structure on the pair ${\mathfrak{D}}^{M}$ and
${\mathfrak{F}}^{\mathcal{P}}$. Then a collar structure compatibile with
${\mathfrak{F}}^{M}$ (see Definition 3.34) is said to be compatible with the
normal complex structure if the collar maps intertwine with the normal complex
structures.

######  Definition 3.47.

Suppose
$(T^{\mathcal{P}},{\mathfrak{D}}^{\mathcal{P}},{\mathfrak{F}}^{\mathcal{P}})$,
$(T^{\mathcal{P}^{\prime}},{\mathfrak{D}}^{\mathcal{P}^{\prime}},{\mathfrak{F}}^{\mathcal{P}^{\prime}})$,
and $(M,{\mathfrak{D}}^{M},{\mathfrak{F}}^{M})$ be the same as in Definition
3.45. Moreover, assume that
$({\mathfrak{D}}^{\mathcal{P}},{\mathfrak{F}}^{\mathcal{P}})$ and
$({\mathfrak{D}}^{\mathcal{P}^{\prime}},{\mathfrak{F}}^{\mathcal{P}^{\prime}})$
are equipped with an orientation in the sense of Definition 3.43. Then an
orientation on the given normally complex lift of ${\mathfrak{D}}^{M}$ and
${\mathfrak{F}}^{M}$ is given by

  1. (1)

An isomorphism of orientation lines

$\mathfrak{o}_{C_{pp^{\prime}}^{M}}\xrightarrow{\sim}\mathfrak{o}_{p}^{\vee}\otimes\mathfrak{o}_{p^{\prime}}.$
(3.17)

  2. (2)

For $p<q$ and $p^{\prime}$, an isomorphism of orientation lines

$\mathfrak{o}_{C^{\mathcal{P}}_{pq}}\otimes\mathfrak{o}_{C^{M}_{qp^{\prime}}}\otimes(\mathfrak{o}_{\mathcal{F}^{M}_{pqp^{\prime},pq}}\otimes\mathfrak{o}_{\mathcal{F}^{M}_{pqp^{\prime},pq}}^{\vee})\xrightarrow{\sim}\mathfrak{o}_{\partial^{pqp^{\prime}}C_{pp^{\prime}}^{M}}.$
(3.18)

  3. (3)

For $p$ and $q^{\prime}<p^{\prime}$, an isomorphism of orientation lines

$\mathfrak{o}_{C^{M}_{pq^{\prime}}}\otimes\mathfrak{o}_{C^{\mathcal{P}^{\prime}}_{q^{\prime}p^{\prime}}}\otimes(\mathfrak{o}_{\mathcal{F}^{M}_{pq^{\prime}p^{\prime},pq}}\otimes\mathfrak{o}_{\mathcal{F}^{M}_{pq^{\prime}p^{\prime},pq}}^{\vee})\xrightarrow{\sim}\mathfrak{o}_{\partial^{pq^{\prime}p^{\prime}}C_{pp^{\prime}}^{M}}.$
(3.19)

  4. (4)

The above isomorphisms are preserved by the $\Pi$-action.

For a flow bimodule $M$ from $T^{\mathcal{P}}$ to $T^{\mathcal{P}^{\prime}}$
as in Definition 3.45 (with ${\mathfrak{D}}^{M}$, ${\mathfrak{F}}^{M}$ and a
compatible collar structure), if it is further endowed with a orientation
structure, we say it has an oriented and normally complex derived orbifold
lift and abbreviate as ${\mathfrak{D}}^{M}$.

###  3.7. Constructing FOP perturbations

We extend the notion of FOP perturbations from the case of a single derived
orbifold chart to the case of a derived orbifold lift of a flow category. The
case of a flow bimodule is similar.

######  Definition 3.48 (Perturbations on a derived orbifold lift).

Given a derived orbifold lift ${\mathfrak{D}}^{\mathcal{P}}$ of a flow
category $T^{\mathcal{P}}$ (see Definition 3.24).

  1. (1)

A perturbation on ${\mathfrak{D}}^{\mathcal{P}}$ consists of a system of
smooth perturbations

${\mathfrak{S}}^{\prime}=\Big{(}{\mathcal{S}}_{pq}^{\prime}:{\mathcal{U}}_{pq}\to{\mathcal{E}}_{pq}\Big{)}_{p<q}$

of the compact derived orbifold chart
$({\mathcal{U}}_{pq},{\mathcal{E}}_{pq},{\mathcal{S}}_{pq})$ (see Definition
2.1) such that the diagram (3.6) still commutes if we replace all
${\mathcal{S}}_{pq}$ by ${\mathcal{S}}_{pq}^{\prime}$.

  2. (2)

Suppose the derived orbifold lift is equipped with a scaffolding (see
Definition 3.28). Then we say that a perturbation ${\mathfrak{S}}^{\prime}$
respects the scaffolding if the following is true. For each pair of strata
$\alpha\leq\beta$ of ${\mathbb{A}}^{\mathcal{P}}_{pq}$, the datum
${\mathfrak{S}}^{\prime}$ induces a perturbation
${\mathcal{S}}_{\alpha}^{\prime}$ of $C_{\alpha}$ and a perturbation
${\mathcal{S}}_{\beta}^{\prime}$ of $C_{\beta}$ by taking the product. Then we
require

${\mathcal{S}}_{\beta}^{\prime}|_{\partial^{\alpha}{\mathcal{U}}_{\beta}}={\rm
Stab}_{{\mathcal{F}}_{\beta\alpha}}({\mathcal{S}}_{\alpha}^{\prime}).$

  3. (3)

Suppose the derived orbifold lift is equipped with a collar structure. Then we
say that a perturbation ${\mathfrak{S}}^{\prime}$ respects the collar
structure if for each $pq$ and each stratum
$\alpha\in{\mathbb{A}}_{pq}^{\mathcal{P}}$, the following diagram commutes.

$\textstyle{(\pi_{pq,\alpha}^{\rm
collar})^{*}({\mathcal{E}}_{pq}|_{\partial^{\alpha}{\mathcal{U}}_{pq}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\vartheta_{pq,\alpha}^{\rm
collar}}$$\textstyle{{\mathcal{E}}_{\partial^{\alpha}{\mathcal{U}}_{pq}}}$$\textstyle{\partial^{\alpha}{\mathcal{U}}_{pq}\times[0,\epsilon)^{{\mathbb{F}}_{\alpha}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(\pi_{pq,\alpha}^{\rm
collar})^{*}({\mathcal{S}}_{pq}^{\prime}|_{\partial^{\alpha}{\mathcal{U}}_{pq}})}$$\scriptstyle{\theta_{pq,\alpha}^{\rm
collar}}$$\textstyle{{\mathcal{U}}_{pq}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{\mathcal{S}}_{pq}^{\prime}}$

  4. (4)

Suppose the derived orbifold lift is normally complex and is equipped with a
package of compatible structures (scaffolding, collar structure, and
straightening). Then a perturbation ${\mathfrak{S}}^{\prime}$ is called an FOP
perturbation if respects the scaffolding, respects the collar structure, and
for each $pq$, with respect to the straightening, the restriction of
${\mathcal{S}}_{pq}^{\prime}$ to the interior of each stratum
$\partial^{\alpha}{\mathcal{U}}_{pq}$ is an FOP section of
${\mathcal{E}}_{pq}$.888Indeed being an FOP perturbation only needs to refer
to the straightening and the normal complex structure.

  5. (5)

Under the assumptions of the last item, an FOP perturbation is called strongly
transverse if the restriction of each ${\mathcal{S}}_{pq}^{\prime}$ to the
interior of each stratum of ${\mathcal{U}}_{pq}$ is strongly transverse (see
Definition 2.7).

######  Theorem 3.49.

Given a flow category $T^{\mathcal{P}}$, a derived orbifold lift
${\mathfrak{D}}^{\mathcal{P}}$ with a normally complex structure, together
with a package of compatible extra structures (scaffolding, collaring, and
straightening), there exists a strongly transverse FOP perturbation. Moreover,
we can make the perturbation $\Pi$-invariant, i.e.,

${\mathcal{S}}_{pq}^{\prime}={\mathcal{S}}_{a\cdot p\ a\cdot q}^{\prime}\
\forall a\in\Pi\ p,q\in{\mathcal{P}}$

after identifying $C_{pq}$ with $C_{a\cdot p\ a\cdot q}$.

###### Proof.

The construction is based on the same induction strategy as defining the
Hamiltonian Floer homology (as well as continuation maps etc.) using abstract
perturbations as in [FO99] and [LT98b]. The package of additional structures
is necessary because the FOP perturbation scheme is more rigid than the
traditional smooth or continuous multivalued perturbation scheme. We start
with a pair $p<q$ with minimal (nonzero) energy
${\mathcal{A}}^{\mathcal{P}}(q)-{\mathcal{A}}^{\mathcal{P}}(p)$. Then by
definition, ${\mathbb{A}}_{pq}^{\mathcal{P}}$ is a poset with a single element
and hence the chart
$C_{pq}=({\mathcal{U}}_{pq},{\mathcal{E}}_{pq},{\mathcal{S}}_{pq})$ is a
derived orbifold chart with no boundary or corners. By the absolute version of
Proposition 2.10, with respect to the normal complex structure and the
straightening, there exists a strongly transverse FOP perturbation
${\mathcal{S}}_{pq}^{\prime}:{\mathcal{U}}_{pq}\to{\mathcal{E}}_{pq}$ which
can be arbitrarily close (measured in $C^{0}$) to ${\mathcal{S}}_{pq}$. In
particular, ${\mathcal{S}}_{pq}^{\prime}$ has a compact zero locus. We can
also make such perturbations $\Pi$-invariant.

Now for a given pair $p<q$, we state our induction hypothesis: for pair $r<s$
with $p\leq r<s\leq q$ and $(p,q)\neq(r,s)$, we have constructed a strongly
transverse FOP section with respect to the straightening and normal complex
structure

${\mathcal{S}}_{rs}^{\prime}:{\mathcal{U}}_{rs}\to{\mathcal{E}}_{rs}$

satisfying the compatibility condition for a perturbation (with respect to the
collar structure and scaffolding, see Definition 3.48). We would like to
construct a strongly transverse FOP perturbation ${\mathcal{S}}_{pq}^{\prime}$
which extends the existing ones and which still satisfies the compatibility
conditions.

For each proper stratum $\alpha=pr_{1}\cdots r_{l}q$ of $pq$, the product of
${\mathcal{S}}_{pr_{1}}^{\prime}$, $\ldots$, ${\mathcal{S}}_{r_{l}q}^{\prime}$
provides a section

${\mathcal{S}}_{\alpha}^{\prime}:={\mathcal{S}}_{pr_{1}}^{\prime}\times\cdots\times{\mathcal{S}}_{r_{l}q}^{\prime}:{\mathcal{U}}_{\alpha}\to{\mathcal{E}}_{\alpha}.$

Then via the stabilization map

${\bm{\theta}}_{pq,\alpha}:{\rm
Stab}_{{\mathcal{F}}_{pq,\alpha}}(C_{\alpha})\simeq\partial^{\alpha}C_{pq}$

one obtains a section

${\mathcal{S}}_{pq,\alpha}^{\prime}:\partial^{\alpha}{\mathcal{U}}_{pq}\to{\mathcal{E}}_{pq}|_{\partial^{\alpha}{\mathcal{U}}_{pq}}$

which is the stabilization of ${\mathcal{S}}_{\alpha}^{\prime}$. We check the
following conditions.

  1. (1)

The collection of ${\mathcal{S}}_{pq,\alpha}^{\prime}$ agree on overlaps.
Indeed, for each pair of stratum $\alpha\leq\beta$, by the compatibility
condition (see item (2) of Definition 3.48) satisfied by the existing
perturbations (which is assumed as induction hypothesis), one has

${\mathcal{S}}_{\beta}^{\prime}|_{\partial^{\alpha}{\mathcal{U}}_{\beta}}={\rm
Stab}_{{\mathcal{F}}_{\beta\alpha}}({\mathcal{S}}_{\alpha}^{\prime}).$

Then by the definition of scaffolding (see Definition 3.28 and (3.13)), one
has

${\mathcal{S}}_{pq,\beta}^{\prime}|_{\partial^{\alpha}{\mathcal{U}}_{pq}}={\rm
Stab}_{{\mathcal{F}}_{pq,\beta}}({\mathcal{S}}_{\beta}^{\prime}|_{\partial^{\alpha}{\mathcal{U}}_{\beta}})={\rm
Stab}_{{\mathcal{F}}_{pq,\beta}}({\rm
Stab}_{{\mathcal{F}}_{\beta\alpha}}({\mathcal{S}}_{\alpha}^{\prime}))\\\ ={\rm
Stab}_{{\mathcal{F}}_{pq,\alpha}}({\mathcal{S}}_{\alpha}^{\prime})={\mathcal{S}}_{pq,\alpha}^{\prime}.$

  2. (2)

The collection of ${\mathcal{S}}_{pq,\alpha}^{\prime}$ respect the collar
structure. This is a consequence of the induction hypothesis and the fact that
the scaffolding respects the collar structure (see Definition 3.34).

  3. (3)

Each ${\mathcal{S}}_{pq,\alpha}^{\prime}$ is an FOP perturbation. This is a
consequence of the compatibility condition on the additional structures (see
Definition 3.35) which guarantees that the stabilization of an FOP section is
still an FOP section.

  4. (4)

Each ${\mathcal{S}}_{pq,\alpha}^{\prime}$ is strongly transverse within the
interior of $\partial^{\alpha}{\mathcal{U}}_{pq}$. Notice that

${\rm Int}{\mathcal{U}}_{\alpha}={\rm
Int}{\mathcal{U}}_{pr_{1}}\times\cdots\times{\rm Int}{\mathcal{U}}_{r_{l}q}.$

Corollary 2.14 implies that the restriction of
${\mathcal{S}}_{\alpha}^{\prime}$ to the interior of ${\mathcal{U}}_{\alpha}$
is an FOP section. Moreover, as strong transversality is preserved under
stabilization, we know that ${\mathcal{S}}_{pq,\alpha}^{\prime}$ is strongly
transverse within ${\rm Int}\partial^{\alpha}{\mathcal{U}}_{pq}={\rm
Stab}_{{\mathcal{F}}_{pq,\alpha}}({\rm Int}{\mathcal{U}}_{\alpha})$.

Therefore, we can extend the collection of
${\mathcal{S}}_{pq,\alpha}^{\prime}$ to a neighborhood of
$\partial{\mathcal{U}}_{pq}$ using the collar structure, simply by pulling
back the existing perturbations using the projection maps. Hence we have
obtained a section ${\mathcal{S}}_{pq}^{\prime}$ of ${\mathcal{E}}_{pq}$
defined in an open neighborhood of $\partial{\mathcal{U}}_{pq}$. The
conditions on the collar structure imply that the extension is well-defined.
The compatibility between straightenings and the collar structure implies that
the extension is still an FOP section. As we can extend strongly transverse
FOP sections in the standard “CUDV” fashion (see the relative version of
Proposition 2.10), one can construct a strongly FOP perturbation
${\mathcal{S}}_{pq}^{\prime}$ which extends the existing ones near the
boundary. The compactness assumption on ${\mathcal{S}}_{pq}^{-1}(0)$ implies
that one can make the perturbed zero locus
$({\mathcal{S}}_{pq}^{\prime})^{-1}(0)$ compact. The inductive construction
can then be carried on. Moreover, the $\Pi$-equivariance of the perturbation
can be maintained in the induction process. ∎

###  3.8. Chain complexes and maps over the integers

We recall the notion of Novikov coefficient ring.

######  Definition 3.50.

  1. (1)

The integral Novikov ring is the ring of formal Laurent series in a single
variable $T$ with integer coefficients, i.e.

$\Lambda:={\mathbb{Z}}[[T]][T^{-1}]=\Big{\\{}{\mathfrak{x}}=\sum_{i=-m}^{\infty}a_{i}T^{i},\
{\rm where}\ m\in{\mathbb{Z}}\ {\rm and}\ a_{i}\in{\mathbb{Z}}\Big{\\}}.$

  2. (2)

The valuation on $\Lambda$ is the map

${\rm val}:\Lambda\to{\mathbb{Z}},\ {\rm val}(\sum a_{i}T^{i})={\rm
min}\left\\{i\ |\ a_{i}\neq 0\right\\}.$

Denote

$\Lambda_{0}:=\\{{\mathfrak{x}}\in\Lambda\ |\ {\rm val}({\mathfrak{x}})=0\\}.$

and

$\Lambda_{+}:=\\{{\mathfrak{x}}\in\Lambda\ |\ {\rm val}({\mathfrak{x}})>0\\}.$

####  3.8.1. The chain complexes

Now we start to build the chain complexes and chain maps. First, associated to
a poset ${\mathcal{P}}$ as in Setup 3.5, one can define a
${\mathbb{Z}}/2N$-graded free $\Lambda$-module as follows. Define

$C_{*}^{\mathcal{P}}:=\left\\{\sum_{p\in{\mathcal{P}}}m_{p}p\ |\
m_{p}\in{\mathbb{Z}},\ \forall c>0,\\#\\{p\ |\ m_{p}\neq
0,{\mathcal{A}}^{\mathcal{P}}(p)<c\\}<\infty\right\\}.$

This is a free abelian group graded by the index function ${\rm
ind}^{\mathcal{P}}:{\mathcal{P}}\to{\mathbb{Z}}/2N$. Define a $\Lambda$-module
structure on $C_{*}$ by

$T^{a}\Big{(}\sum_{p\in{\mathcal{P}}}m_{p}p\Big{)}:=\sum_{p\in{\mathcal{P}}}m_{p}(a\cdot
p).$

Conditions of Setup 3.5 implies that $C_{*}$ is a ${\mathbb{Z}}/2N$-graded
free $\Lambda$-module whose rank is equal to the cardinality of
${\mathcal{P}}/\Pi$.

Isolated zeroes in the free locus of each oriented derived orbifold chart
induce integer counts. More precisely, fix a strongly transverse FOP
perturbation

${\mathfrak{S}}^{\prime}:=\Big{(}{\mathcal{S}}_{pq}^{\prime}:{\mathcal{U}}_{pq}\to{\mathcal{E}}_{pq}\Big{)}_{p<q}.$

For each pair $p<q$, we can write

${\mathcal{U}}_{pq}=\bigsqcup_{i=-\infty}^{+\infty}{\mathcal{U}}_{pq}^{[i]}$

where ${\mathcal{U}}_{pq}^{[i]}\subset{\mathcal{U}}_{pq}$ is the open and
closed subset of points whose local virtual dimension is $i$. It follows from
(3.5) that

${\mathcal{U}}_{pq}^{[i]}\neq\emptyset\Longrightarrow i\equiv{\rm
ind}^{\mathcal{P}}(p)-{\rm ind}^{\mathcal{P}}(q)-1\ {\rm mod}\ 2N.$

Then let $n_{pq}$ be the count (with signs) of zeroes of
${\mathcal{S}}_{pq}^{\prime}$ in the free locus of the zero-dimensional
component ${\mathcal{U}}_{pq}^{[0]}$. The sign of an isolated zero point of
${\mathcal{S}}_{pq}^{\prime}$ is uniquely determined by the sign read off from
the isomorphism (3.15). The compactness assumption (see item (3) of Definition
3.7) implies that $n_{pq}$ is finite. Moreover, by the finiteness of
${\mathcal{P}}/\Pi$ and the compactness condition, these counts define a
${\mathbb{Z}}$-linear map

$d^{\mathcal{P}}:C_{*}^{\mathcal{P}}\to C_{*-1}^{\mathcal{P}}.$

The $\Pi$-equivariance of the perturbation implies that

$n_{pq}=n_{a\cdot p\ a\cdot q},\ \forall a\in{\mathbb{Z}}\ and\
p,q\in{\mathcal{P}}$

which further implies that $d^{\mathcal{P}}$ is $\Lambda$-linear. By looking
at 1-dimensional components of all derived orbifold charts, using the
coherence of the orientations (3.16), one can see that $d^{\mathcal{P}}$ is a
differential map, i.e., $d^{\mathcal{P}}\circ d^{\mathcal{P}}=0$. Therefore,
one obtains a chain complex

$(C_{*}^{\mathcal{P}},d^{\mathcal{P}})$

of $\Lambda$-modules. One hence obtains the homology

$H_{*}^{\mathcal{P}}=\bigoplus_{i\in{\mathbb{Z}}/2N}H_{i}^{\mathcal{P}}$

which has a natural $\Lambda$-module structure. One can show that the chain
homotopy equivalence class of the chain complex does not depend on the choice
of the perturbations. We omit the details because such a fact is not needed in
our application.

####  3.8.2. The chain maps

Now suppose we have two topological flow categories, $T^{\mathcal{P}}$ over
${\mathcal{P}}$ and $T^{{\mathcal{P}}^{\prime}}$ over ${\mathcal{P}}^{\prime}$
and a flow bimodule $M$ from $T^{\mathcal{P}}$ to
$T^{{\mathcal{P}}^{\prime}}$.

######  Theorem 3.51.

Assume the following conditions.

  1. (1)

There is an oriented and normally complex derived orbifold lift
${\mathfrak{D}}^{\mathcal{P}}$ resp. ${\mathfrak{D}}^{{\mathcal{P}}^{\prime}}$
of $T^{\mathcal{P}}$ resp. $T^{{\mathcal{P}}^{\prime}}$.

  2. (2)

There is an oriented and normally complex derived orbifold lift
${\mathfrak{D}}_{{\mathcal{P}}{\mathcal{P}}^{\prime}}$ of $M$ which extends
${\mathfrak{D}}^{\mathcal{P}}$ and ${\mathfrak{D}}^{{\mathcal{P}}^{\prime}}$.

  3. (3)

On these derived orbifold lifts there exist compatible collar structures,
scaffoldings, and straightenings.

Suppose we are given a strongly transverse FOP perturbation on
${\mathfrak{D}}^{\mathcal{P}}$ and a strongly transverse FOP perturbation on
${\mathfrak{D}}^{{\mathcal{P}}^{\prime}}$. Then there exists a strongly
transverse FOP perturbation on
${\mathfrak{D}}_{{\mathcal{P}}{\mathcal{P}}^{\prime}}$ which extends the
existing ones which is $\Pi$-invariant.

###### Proof.

This theorem is essentially a relative version of Theorem 3.49 and the proof
is the same as we can always extend strongly transverse FOP perturbations from
local to global using the standard “CUDV” fashion (see the relative version of
Proposition 2.10). ∎

Now under the assumptions of Theorem 3.51, the existing strongly transverse
FOP perturbations induce chain complexes

$\displaystyle\ (C_{*}^{\mathcal{P}},d^{\mathcal{P}}),\ $ $\displaystyle\
(C_{*}^{{\mathcal{P}}^{\prime}},d^{{\mathcal{P}}^{\prime}}).$

We would like to define a chain map using the extended FOP perturbations on
${\mathfrak{D}}_{{\mathcal{P}}{\mathcal{P}}^{\prime}}$. Indeed, for
$p\in{\mathcal{P}}$ and $p^{\prime}\in{\mathcal{P}}^{\prime}$, consider the
perturbation

${\mathcal{S}}_{pp^{\prime}}^{\prime}:{\mathcal{U}}_{pp^{\prime}}\to{\mathcal{E}}_{pp^{\prime}}.$

Let $n_{pp^{\prime}}$ be the count (with signs read off from the isomorphism
between orientation lines (3.17)) of zeros of
${\mathcal{S}}_{pp^{\prime}}^{\prime}$ in the free locus of all components of
${\mathcal{U}}_{pp^{\prime}}$ with local virtual dimension zero. The
compactness condition on $M_{pp^{\prime}}$ (see Definition 3.10) implies that
$n_{pp^{\prime}}$ is finite. Moreover, (3.7) implies that

$n_{pp^{\prime}}\neq 0\Longrightarrow{\rm ind}^{\mathcal{P}}(p)\equiv{\rm
ind}^{{\mathcal{P}}^{\prime}}(p^{\prime})\ {\rm mod}\ 2N.$

Then we formally define

$\Psi^{M}:C_{*}^{\mathcal{P}}\to C_{*}^{{\mathcal{P}^{\prime}}}$

by linearly extending

$\Psi^{M}(p)=\sum_{p^{\prime}\in{\mathcal{P}}^{\prime}}n_{pp^{\prime}}p^{\prime}.$

We claim that the above is a legitimate element of
$C_{*}^{{\mathcal{P}}^{\prime}}$. Indeed, this is a consequence of the
condition (3.4) of Definition 3.10 and the finiteness of
${\mathcal{P}}^{\prime}/\Pi$. Hence $\Psi^{M}$ is a well-defined morphisms of
graded abelian group. Moreover, by the $\Pi$-equivariance of all structures,
one has

$n_{pp^{\prime}}=n_{a\cdot p\ n\cdot p^{\prime}},\ \forall a\in\Pi.$

Hence $\Psi^{M}$ is $\Lambda$-linear. Lastly, by the description of
codimension one stratum of all bimodule moduli spaces and the coherence of
orientations (3.18) and (3.19), similar to the case of $d^{\mathcal{P}}\circ
d^{\mathcal{P}}=0$, one has that $\Psi^{M}$ is a chain map, i.e.,

$\Psi^{M}\circ d^{\mathcal{P}}=d^{{\mathcal{P}}^{\prime}}\circ\Psi^{M}.$

######  Remark 3.52.

We would like to remark that the definitions of $d^{\mathcal{P}}$ and
$\Psi^{M}$ from counting points in moduli spaces of virtual dimension $0$, and
the usual proof for $(d^{\mathcal{P}})^{2}=0$ and
$d^{\mathcal{P}^{\prime}}\circ\Psi^{M}=\Psi^{M}\circ d^{\mathcal{P}}$ from
counting the boundary points of moduli spaces of virtual dimension $1$ works
in our setting, because the pseudocycle condition on the free locus of
$({\mathcal{S}}_{pq}^{\prime})^{-1}(0)$ guarantees that the topological
boundary $\partial(({\mathcal{S}}_{pq}^{\prime})^{-1}(0))^{\rm free}$ has
codimension at least $2$ so the desired compactness property, therefore the
finiteness of algebraic counts holds.

##  4\. Proof of the integral Arnold conjecture

In this section, we define the topological flow category associated with a
non-degenerate Hamiltonian and the topological flow category constructed from
a Morse–Smale function. Then we define two bimodules between these two flow
categories, which are respectively the space-level lifts of the well-known
Piunikhin–Salamon–Schwarz map (the PSS map) and its inverse (the SSP map). We
state in Theorem 4.7 the most important technical result in this paper,
namely, the existence of derived orbifold lifts of the aforementioned objects,
whose proof is contained in later sections. Finally, we explain how to use the
perturbation scheme developed in Section 2 and Section 3 to prove Theorem A.

The following is the standing assumption of our discussions later on.

######  Hypothesis 4.1.

  1. (1)

The symplectic manifold $(M,\omega)$ is integral, i.e., the de Rham cohomology
class $[\omega]\in H_{dR}^{2}(M)\cong H^{2}(M;{\mathbb{R}})$ lies in the image
of $H^{2}(M;{\mathbb{Z}})\to H^{2}(M;{\mathbb{R}})$.

  2. (2)

$H$ is a nondegenerate Hamiltonian on $(M,\omega)$.

  3. (3)

All 1-periodic orbits of $H$ are embedded and any two of them are disjoint.

  4. (4)

The symplectic actions (4.1) of all capped 1-periodic orbits are integral.

  5. (5)

$J$ is a time-independent $\omega$-compatible almost complex structure such
that for all Floer trajectories with smooth domains (i.e., no sphere bubbles
are attached and the trajectories are not broken) $u:\Theta\to X$, the
linearized operator is surjective.

We explain why it suffices to prove the Arnold conjecture under these
assumptions in order to establish it in full generality.

  1. (1)

As we explained in the introduction of this paper, the case of Arnold
conjecture for which $[\omega]$ is rational implies the general case, and this
is equivalent to the case when $[\omega]\in H^{2}(M;\mathbb{Z})$ by suitably
rescaling $\omega$ and $H$ using a common integer-valued factor.

  2. (2)

Because any $C^{2}$-small perturbation of a nondegenerate Hamiltonian $H$ will
not change the number of periodic orbits, we can freely perturb $H$ to
guarantee that all $2\pi$-periodic orbits are nondegenerate.

  3. (3)

As the 2-dimensional case of the integral Arnold conjecture is known, we can
restrict ourselves to the case when ${\rm dim}M\geq 4$. Then one can slightly
perturb $H$ so that distinct periodic orbits do not intersect and all periodic
orbits are embedded.

  4. (4)

Given a nondegenerate Hamiltonian $H$ (whose capped 1-periodic orbits are all
discrete), by adding a $C^{2}$-small, $t$-independent function
$f:M\to{\mathbb{R}}$ whose restriction to a neighborhood of the image of each
periodic orbit is a constant does not change the set of capped 1-periodic
orbits but will shift their symplectic actions by constants. Then we can add a
sufficiently small function $f$ which makes the symplectic actions of all
capped periodic orbits rational. Then the symplectic actions can be made
integral by a further rescaling.

  5. (5)

For any given $J$, one can slightly perturb $H$ to achieve transversality for
all smooth Floer trajectories. Moreover, the perturbed Hamiltonian can be
chosen to agree with $H$ up to second order on each 1-periodic orbits of $H$
(see [FHS95, Theorem 5.1]).

###  4.1. Hamiltonian Floer flow categories

####  4.1.1. The moduli spaces of Floer trajectories

We first review the basics about the moduli spaces of stable Floer
trajectories. Let $(M,\omega)$ be a compact symplectic manifold and
$H=(H_{t})_{t\in S^{1}}$ be a smooth 1-periodic Hamiltonian on $M$. A capped
1-periodic orbit is a pair $p=([u],\underline{\smash{p}})$ where
$\underline{\smash{p}}:S^{1}\to M$ is a 1-periodic orbit of the Hamiltonian
vector field $X_{H_{t}}$ and $[u]$ is an equivalence class of maps
$u:{\mathbb{D}}\to M$ such that (we view ${\mathbb{D}}$ as the unit disk in
${\mathbb{C}}$)

$u(e^{2\pi{\bm{i}}t})=\underline{\smash{p}}(t);$

two such maps $u_{1},u_{2}:{\mathbb{D}}\to M$ are equivalent if

$\int_{\mathbb{D}}u_{1}^{*}\omega=\int_{{\mathbb{D}}}u_{2}^{*}\omega.$

Define the symplectic action of a capped orbit $p=([u],\underline{\smash{p}})$
to be

${\mathcal{A}}_{H}(p)=\int_{\mathbb{D}}u^{*}\omega+\int_{S^{1}}H_{t}(\underline{\smash{p}}(t))dt.$
(4.1)

We denote by ${\rm Per}(H)$ the set of contractible 1-periodic orbits of $H$,
whose elements are denoted by $\underline{\smash{p}},\underline{\smash{q}}$,
etc. Denote by $\widetilde{{\rm Per}}(H)$ the set of capped 1-periodic orbits
of $H$, whose elements are denoted by $p,q$, etc. It follows from the
integrality assumption on the symplectic class that $\widetilde{{\rm Per}}(H)$
is a ${\mathbb{Z}}$-covering of ${\rm Per}(H)$.

Choose an $\omega$-compatible almost complex structure $J$ on $M$. Let
$\Theta={\mathbb{R}}\times S^{1}$ be the infinite cylinder with standard
coordinates $z=s+{\bm{i}}t$. The Floer equation is the first-order equation
for smooth maps $u:\Theta\to M$

$\frac{\partial u}{\partial s}+J\left(\frac{\partial u}{\partial
t}-X_{H_{t}}(u)\right)=0.$ (4.2)

The associated energy for a map $u:\Theta\to M$ is defined to be

$E_{H}(u):=\frac{1}{2}\int_{\Theta}\left(\left|\frac{\partial u}{\partial
s}\right|^{2}+\left|\frac{\partial u}{\partial
t}-X_{H_{t}}(u)\right|^{2}\right)dsdt.$

Any solution to (4.2) with finite energy necessarily converges as
$s\to\pm\infty$ to periodic orbits of $H$. Then one can use a pair of (capped)
orbits to label solutions. Let
${\mathcal{M}}_{\underline{\smash{p}}\underline{\smash{q}}}^{\rm Floer}$ be
the set of solutions to (4.2) which converge to $\underline{\smash{p}}$ resp.
$\underline{\smash{q}}$ as $s\to-\infty$ resp. $s\to+\infty$, modulo the
obvious time translation. Denote by

${\mathcal{M}}_{pq}^{\rm
Floer}\subset{\mathcal{M}}_{\underline{\smash{p}}\underline{\smash{q}}}^{\rm
Floer}$

the subset of equivalence classes of solutions $u$ such that the concatenation
of $p$ and $u$ is equivalent to $q$ as capped 1-periodic orbits. Then for any
solution $u$ to (4.2) representing an element of ${\mathcal{M}}_{pq}^{\rm
Floer}$, it is standard that

$E_{H}(u)={\mathcal{A}}_{H}(q)-{\mathcal{A}}_{H}(p).$ (4.3)

As a result the Floer differential increases the symplectic action.

The moduli spaces in general are not compact with respect to the
$C^{\infty}_{\rm loc}$-topology. Indeed, the space ${\mathcal{M}}_{pq}^{\rm
Floer}$ admits a natural compactification called the Gromov–Floer
compactification, denoted by $\overline{\mathcal{M}}{}_{pq}^{\rm Floer}$,
incorporating both bubbling of holomorphic spheres and breaking of Floer
trajectories, see e.g. [FO99, Section 18]. It is standard knowledge that
$\overline{\mathcal{M}}{}_{pq}^{\rm Floer}$ is a compact Hausdorff topological
space.999It is possible that ${\mathcal{M}}_{pq}^{\rm Floer}=\emptyset$ while
$\overline{\mathcal{M}}{}_{pq}^{\rm Floer}\neq\emptyset$.

####  4.1.2. The flow category

We now package the collection of Floer moduli spaces into a flow category.
First we see how this system of moduli spaces fits into Setup 3.5. Let
$N\in{\mathbb{Z}}_{\geq 0}$ be the minimal Chern number of $(M,\omega)$. The
integral symplectic form $\omega$ defines a homomorphism

$\omega:\pi_{2}(M)\to{\mathbb{Z}}.$

Define

$\Pi:=\pi_{2}(M)/{\rm ker}\omega.$

Then $\Pi$ is an infinite cyclic group and $\omega$ induces an injection

$\omega:\Pi\to{\mathbb{Z}}$

Then we define

${\mathcal{P}}^{\rm Floer}:=\widetilde{\rm Per}(H).$

Define the partial order by the existence of nonempty Floer moduli spaces,
i.e.,

$p\leq q\Longrightarrow\overline{\mathcal{M}}{}_{pq}^{\rm
Floer}\neq\emptyset.$

It has a free $\Pi$-action defined by taking the connected sum between a
representative of a capped 1-periodic orbit and a representative of an element
$a\in\Pi$. Define

$\displaystyle\ {\mathcal{A}}^{{\mathcal{P}}^{\rm Floer}}:={\mathcal{A}}_{H},\
$ $\displaystyle\ {\rm ind}^{{\mathcal{P}}^{\rm Floer}}:=\text{ Conley--
Zehnder\ index}.$

Then ${\mathcal{P}}^{\rm Floer}$ satisfies conditions of Setup 3.5.

######  Notation 4.2.

As a convention, objects labelled by ${\mathcal{P}}^{\rm Floer}$ are often
also labelled by ${\rm Floer}$. For example, we abbreviate

$\displaystyle\ {\mathcal{A}}^{\rm Floer}:={\mathcal{A}}^{{\mathcal{P}}^{\rm
Floer}},\ $ $\displaystyle\ {\rm ind}^{\rm Floer}:={\rm
ind}^{{\mathcal{P}}^{\rm Floer}}.$

One can then describe the stratifications on Floer moduli spaces. As in
Notation 3.6, for each pair $p,q\in{\mathcal{P}}^{\rm Floer}$, one has the
homogeneous poset

${\mathbb{A}}_{pq}^{\rm Floer}:=\big{\\{}\alpha=pr_{1}\cdots r_{l}q\ |\
p<r_{1}<\cdots<r_{l}<q\big{\\}}.$

Then the moduli space $\overline{\mathcal{M}}{}_{pq}^{\rm Floer}$ an
${\mathbb{A}}_{pq}^{\rm Floer}$-stratified topological space. Indeed, for each
$\alpha=pr_{1}\cdots r_{l}q$, one has

$\partial^{\alpha}\overline{\mathcal{M}}{}_{pq}^{\rm
Floer}\cong\overline{\mathcal{M}}{}_{pr_{1}}^{\rm
Floer}\times\cdots\times\overline{\mathcal{M}}{}_{r_{l}q}^{\rm Floer}$

as the $\alpha$-stratum is the subset of “broken trajectories” of a type
described by $\alpha$.

######  Definition 4.3.

The Hamiltonian Floer flow category (associated to $H$ and $J$), denoted by
$T^{\rm Floer}$, is the topological flow category over ${\mathcal{P}}^{\rm
Floer}$ whose morphism space between $p,q\in\widetilde{{\rm Per}}(H)$ is the
moduli space $\overline{\mathcal{M}}{}_{pq}^{\rm Floer}$, and whose
composition maps are the natural inclusions

$\overline{\mathcal{M}}{}_{pr}^{\rm
Floer}\times\overline{\mathcal{M}}{}_{rq}^{\rm
Floer}\to\overline{\mathcal{M}}{}_{pq}^{\rm Floer},\ \forall p\leq r\leq q.$

###  4.2. Morse flow category

We describe the definition of the more classical Morse flow category and
explain how it fits into the general framework of this paper. We declare that
in this paper Morse flows are always the ascending flow, i.e., the flow
generated by the gradient vector field. Suppose $(f,g)$ is a Morse–Smale pair
on $M$, namely, $f$ is a Morse function and $g$ is a Riemannian metric such
that the unstable manifold of any critical point intersects transversely with
any stable manifold. We use $\underline{\smash{x}},\underline{\smash{y}}$ to
denote the critical points of $f$. To ensure the moduli spaces of
(unparametrized) gradient flow lines to have smooth structures, we assume that
near each $\underline{\smash{x}}\in{\rm crit}(f)$, there exists a coordinate
chart $(x_{1},\dots,x_{2n})$ such that $f=\pm x_{1}^{2}\pm\cdots\pm
x_{2n}^{2}$ and $g=dx_{1}\otimes dx_{1}+\cdots dx_{2n}\otimes dx_{2n}$. Then
by [Weh12], for any pair of cricial points
$\underline{\smash{x}},\underline{\smash{y}}\in{\rm crit}(f)$, the moduli
space

$\overline{\mathcal{M}}{}^{\rm
Morse}_{\underline{\smash{x}}\underline{\smash{y}}}$

of unparametrized broken flow lines connecting $x$ and $y$ is a smooth
manifold with faces. It is standard knowledge that upon choosing orientations
on the unstable manifolds of all critical points, one can count rigid Morse
flow lines (with signs) and define a ${\mathbb{Z}}$-graded chain complex over
${\mathbb{Z}}$ and its homology coincides with $H_{*}(M;{\mathbb{Z}})$.

We want to fit the Morse flow category into the abstract framework. Define

${\mathcal{P}}^{\rm Morse}:=\Pi\times{\rm crit}(f)$

with a partial order defined by

$(a,\underline{\smash{x}})\leq(b,\underline{\smash{y}})\Longleftrightarrow
a=b\ {\rm and}\
\overline{\mathcal{M}}{}_{\underline{\smash{x}}\underline{\smash{y}}}^{\rm
Morse}\neq\emptyset.$

$\Pi$ naturally acts freely on ${\mathcal{P}}^{\rm Morse}$. The action
function is defined by

${\mathcal{A}}^{\rm Morse}(a,\underline{\smash{x}})=\omega(a)+\epsilon
f(\underline{\smash{x}})$

where $\epsilon$ is a sufficiently small positive number. The index function
is defined by

${\rm ind}^{\rm Morse}(a,\underline{\smash{x}})\equiv\frac{1}{2}{\rm
dim}M-{\rm Morse\ index\ of\ }\underline{\smash{x}}\ {\rm mod}\ 2N.$

One can easily check that the triple $({\mathcal{P}}^{\rm
Morse},{\mathcal{A}}^{\rm Morse},{\rm ind}^{\rm Morse})$ satisfies conditions
of Setup 3.5. Then following Notation 3.6 one obtains a collection of
homogeneous posets ${\mathbb{A}}_{xy}^{\rm Morse}$ for all
$x,y\in{\mathcal{P}}^{\rm Morse}$. Define

$\overline{\mathcal{M}}{}_{xy}^{\rm
Morse}=\left\\{\begin{array}[]{cc}\overline{\mathcal{M}}{}_{\underline{\smash{x}}\underline{\smash{y}}}^{\rm
Morse},&\ {\rm if}\ x=(a,\underline{\smash{x}})\ {\rm and}\
y=(a,\underline{\smash{y}}),\\\ \emptyset,&\ {\rm
otherwise}.\end{array}\right.$

Then each $\overline{\mathcal{M}}{}_{xy}^{\rm Morse}$ is stratified by the
poset ${\mathbb{A}}_{xy}^{\rm Morse}$. We can then define the flow category
$T^{\rm Morse}:=T^{\rm Morse}(f,g)$ associated with $(f,g)$ as follows.

######  Definition 4.4.

The Morse flow category $T^{\rm Morse}$ is the topological flow category over
${\mathcal{P}}^{\rm Morse}$ whose morphism spaces are $T_{xy}^{\rm
Morse}=\overline{\mathcal{M}}{}_{xy}^{\rm Morse}$ and whose composition maps
are the natural inclusions

$\overline{\mathcal{M}}{}_{xz}^{\rm
Morse}\times\overline{\mathcal{M}}{}_{zy}^{\rm
Morse}\cong\partial^{xzy}\overline{\mathcal{M}}{}_{xy}^{\rm
Morse}\hookrightarrow\overline{\mathcal{M}}{}_{xy}^{\rm Morse}.$

It is obvious that $T^{\rm Morse}$ is a strict $\Pi$-equivariant flow
category.

As transversality is already achieved for the Morse flow category and there is
no orbifold behavior, one does not need to use derived orbifold lift nor
normal complex structures to define the resulting chain complex. However we
would like to formally put this case into the general framework of the
previous section as it will be necessary when we connect the Morse flow
category to objects where transversality fails and orbifold behavior appears.

First, following the abstract outer-collaring recipe, one can construct an
outer-collaring of the Morse flow category. Fix the width $r=1$. The outer-
collared Morse flow category $(T^{{\rm Morse}})^{+}$ has morphism spaces being
$(\overline{\mathcal{M}}{}_{xy}^{{\rm Morse}})^{+}$. A priori this is only a
topological flow category. However, one can equip the morphism spaces and
composition maps with smooth structure. Indeed, by [Weh12, Theorem 1.4] which
proves the associativity of gluing maps for the special kind of Morse–Smale
pair $(f,g)$ (see also [Qin18]), the original Morse flow category has
compatible “interior” collars. Then the outer-collaring construction does not
alter the feature and put a smooth structure on each space
$(\overline{\mathcal{M}}{}_{xy}^{\rm Morse})^{+}$ such that the composition
maps are smooth.

Second, as transversality is already achieved and the moduli spaces are
manifolds but not orbifolds, the collection of outer-collared moduli spaces
together with the zero obstruction bundle and the zero Kuranishi map form a
derived orbifold lift of $(T^{\rm Morse})^{+}$, denoted by
${\mathfrak{D}}^{\rm Morse}$. The outer-collaring provides a collar structure
on this lift. There is also the trivial scaffolding and the trivial
straightening, which are obviously compatible with the collar structure. In
the manifold case, there is only the trivial normal complex structure. Lastly,
it is a classical knowledge that upon choosing orientations on all unstable
manifolds the Morse moduli spaces inherit coherent orientations. Here the
orientation line ${\mathfrak{o}}_{x}$ associated with a capped critical point
$x=(a,\underline{\smash{x}})$ is defined to be the orientation line of the
tangent space of the stable submanifold of $\nabla_{g}f$ at
$\underline{\smash{x}}$. We summarize these observations as follows.

######  Lemma 4.5.

A choice of orientations on all unstable manifolds of $\nabla^{g}f$ makes
${\mathfrak{D}}^{\rm Morse}$ an oriented and normally complex drived orbifold
lift of $(T^{\rm Morse})^{+}$ equipped with a compatible package of additional
structures. Moreover, the $0$-perturbation is a strongly transverse FOP
perturbation on ${\mathfrak{D}}^{\rm Morse}$ which defines the
${\mathbb{Z}}/2N$-graded chain complex $(C_{*}(f;\Lambda),d^{\rm Morse})$ of
$\Lambda$-modules, whose homology is isomorphic to the $\Lambda$-module

$H^{(2N)}_{n-*}(M;\Lambda)=\bigoplus_{i\in{\mathbb{Z}}/2N}H^{(2N)}_{n-i}(M;{\mathbb{Z}})\otimes_{\mathbb{Z}}\Lambda.$

Here

$H_{i}^{(2N)}(M;{\mathbb{Z}}):=\bigoplus_{j\equiv i\ {\rm mod}\
2N}H_{j}(M;{\mathbb{Z}}).$

###  4.3. PSS and SSP bimodules

####  4.3.1. Moduli spaces

Now we describe the moduli spaces which allow us to interpolate between Floer
theory and Morse theory. We first set up a convention: PSS moduli spaces are
defined by objects with input from Morse critical points and output from
1-periodic (capped) Hamiltonian orbits and SSP moduli spaces parametrizes
objects in the reversed direction.

We only describe PSS moduli spaces in detail. The case of SSP moduli spaces is
similar. To define the equation and hence the moduli space, we make the
following choices.

  1. (1)

An integer $C^{\rm PSS}\in{\mathbb{Z}}$ satisfying

$\inf_{(t,x)\in S^{1}\times M}H_{t}(x)>C^{\rm PSS}.$ (4.4)

  2. (2)

A smooth, non-decreasing cut-off function $\rho:{\mathbb{R}}\to[0,1]$ such
that

$\rho(s)=\left\\{\begin{array}[]{cc}0,&\ s\leq 0,\\\ 1,&\ s\geq
1.\end{array}\right.$ (4.5)

Define a $2$-parameter family of functions

$H_{s,t}^{{\rm PSS}}(x)=(1-\rho(s))C^{\rm PSS}+\rho(s)H_{t}(x).$

Then over the cylinder $\Theta$ one has the vector-field-valued $1$-form

$X_{H}^{\rm PSS}\otimes dt$

which is defined by the Hamiltonian vector field of $H_{s,t}^{{\rm PSS}}(x)$.
Note that it vanishes for $s\leq 0$. The PSS equation is

$\frac{\partial u}{\partial s}+J\left(\frac{\partial u}{\partial t}-X_{H}^{\rm
PSS}(u)\right)=0,\ {\rm where}\ u\in C^{\infty}(\Theta,M).$ (4.6)

The energy of a solution is defined to be and is computed as

$\begin{split}E^{\rm PSS}(u):=&\ \int_{\Theta}\|\partial_{s}u\|^{2}dsdt\\\ =&\
\int_{\Theta}\omega(\partial_{s}u,\partial_{t}u-X_{H}^{\rm PSS}(u))dsdt\\\ =&\
\int_{\Theta}u^{*}\omega+\int_{\Theta}dH_{s,t}^{\rm PSS}(\partial_{s}u)dsdt\\\
=&\ \int_{\Theta}u^{*}\omega+\int_{\Theta}\frac{\partial}{\partial
s}(H_{s,t}^{\rm PSS}(u))dsdt-\int_{\Theta}\frac{\partial H^{\rm
PSS}_{s,t}}{\partial s}dsdt.\end{split}$

If the energy is finite, then by Gromov’s removal of singularity theorem, $u$
extends to a $J$-holomorphic map near $-\infty$ and hence automatically
determines a cap for the periodic orbit at $+\infty$. Hence one can use a
capped 1-periodic orbit $p\in\widetilde{{\rm Per}}(H)$ to label the moduli
space of PSS solutions. It is similar to the case of Floer trajectories that
${\mathcal{M}}{}_{\bullet p}^{\rm PSS}$ admits a natural compactification
consisting of “stable PSS trajectories,” i.e. configurations with both
cylindrical and spherical components. Denote the compactification by

$\overline{\mathcal{M}}{}_{\bullet p}^{\rm PSS}.$ (4.7)

For each map $u$ representing an element of $\overline{\mathcal{M}}{}_{\bullet
p}^{\rm PSS}$, one has the following energy identity

$E^{\rm PSS}(u)={\mathcal{A}}_{H}(p)-C^{\rm
PSS}-\int_{\Theta}\rho^{\prime}(s)(H(u)-C^{\rm PSS})dsdt.$

An important consequence of (4.4) is that

$\overline{\mathcal{M}}{}_{\bullet p}^{\rm
PSS}\neq\emptyset\Longrightarrow{\mathcal{A}}_{H}(p)-C^{\rm PSS}>0.$ (4.8)

We introduce

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

and call it the topological energy of a solution $u$.

Now we bring in the Morse theory of a Morse–Smale pair $(f,g)$. Given
$\underline{\smash{x}}\in{\rm crit}(f)$, consider the unstable manifold
$W^{u}(\underline{\smash{x}})\subset M$ of the flow of $\nabla^{g}f$ and its
compactification $\overline{W^{u}(\underline{\smash{x}})}$. Then define

$\overline{\mathcal{M}}{}_{\underline{\smash{x}}p}^{\rm PSS}:={\rm
ev}_{\bullet}^{-1}(\overline{W^{u}(\underline{\smash{x}})}),$

where ${\rm ev}_{\bullet}:\overline{\mathcal{M}}{}_{\bullet p}^{\rm PSS}\to M$
is the evaluation map at $-\infty$. Moreover, if
$x=(a,\underline{\smash{x}})\in{\mathcal{P}}^{\rm Morse}$ and
$p\in{\mathcal{P}}^{\rm Floer}$, then define

$\overline{\mathcal{M}}{}_{xp}^{\rm
PSS}:=\overline{\mathcal{M}}{}_{\underline{\smash{x}}\ (-a)\cdot p}^{\rm
PSS}.$

The stratifications on the PSS moduli spaces fit into the general framework of
bimodules described in the previous section. Recall that to
$x\in{\mathcal{P}}={\mathcal{P}}^{\rm Morse}$ and
$p\in{\mathcal{P}}^{\prime}={\mathcal{P}}^{\rm Floer}$ one associates a
homogeneous poset

${\mathbb{A}}_{xp}:={\mathbb{A}}_{xp}^{\rm PSS}:=\\{xy_{1}\cdots
y_{k}q_{l}\cdots q_{1}p\ |\ x<y_{1}<\cdots<y_{k},q_{l}<\cdots<q_{1}<p\\}$

which has the natural partial order induced from inclusion and the depth
function

${\rm depth}(xy_{1}\cdots y_{k}q_{l}\cdots q_{1}p)=k+l.$

The PSS moduli space $\overline{\mathcal{M}}{}_{xp}^{\rm PSS}$ is stratified
by ${\mathbb{A}}_{xp}^{\rm PSS}$ with corresponding strata

$\left(\overline{\mathcal{M}}{}_{xp}^{\rm PSS}\right)_{xy_{1}\cdots
y_{k}q_{l}\cdots q_{1}}\cong\overline{\mathcal{M}}{}_{xy_{1}}^{\rm
Morse}\times\cdots\times\overline{\mathcal{M}}{}_{y_{k-1}y_{k}}^{\rm
Morse}\times\overline{\mathcal{M}}{}_{y_{k}q_{l}}^{\rm
PSS}\times\overline{\mathcal{M}}{}_{q_{l}q_{l-1}}^{\rm
Floer}\times\cdots\times\overline{\mathcal{M}}{}_{q_{1}p}^{\rm Floer}$

i.e., the subset of configurations with breakings described by the word
$xy_{1}\cdots y_{k}q_{l}\cdots q_{1}p$. Gromov compactness shows that there
are at most finitely many nonempty stratum.

To define the SSP moduli spaces, let us choose an integer $C^{\rm
SSP}\in{\mathbb{Z}}$ such that

$\sup_{(t,x)\in S^{1}\times M}H_{t}(x)<C^{\rm SSP}.$ (4.9)

Using the same cut-off function as in (4.5), we can write down another
$2$-parameter family of functions

$H_{s,t}^{{\rm SSP}}(x)=(1-\rho(s))H_{t}(x)+\rho(s)C^{{\rm SSP}}.$ (4.10)

Denote the associated vector-field-valued $1$-form on $\Theta$ by $X_{H}^{{\rm
SSP}}\otimes dt$, then the SSP equation is written as

$\frac{\partial u}{\partial s}+J\left(\frac{\partial u}{\partial t}-X_{H}^{\rm
SSP}(u)\right)=0,\ {\rm where}\ u\in C^{\infty}(\Theta,M).$

Because $X_{H}^{{\rm SSP}}dt$ vanishes for $s\geq 0$, any solution $u$ extends
to a $J$-holomorphic map near $+\infty$, which determines a cap for the
periodic orbit at $-\infty$. Given $p\in\widetilde{{\rm Per}}(H)$, we cam
similarly consider “stable SSP trajectories” which constitute a compact moduli
space

$\overline{\mathcal{M}}{}_{p\bullet}^{{\rm SSP}}.$

For any representative $u$ of a point in
$\overline{\mathcal{M}}_{p\bullet}^{{\rm SSP}}$, its topological energy is
defined to be

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

Due to the choice (4.9), we see that $d_{p\bullet}>0$ as long as
$\overline{\mathcal{M}}_{p\bullet}^{{\rm SSP}}\neq\emptyset$. Now suppose
$\underline{\smash{x}}\in{\rm crit}(f)$, let
$W^{s}(\underline{\smash{x}})\subset M$ be the stable submanifold of
$\nabla^{g}f$ and denote by $\overline{W^{s}(\underline{\smash{x}})}$ its
compactification. Introduce the moduli space

$\overline{\mathcal{M}}^{{\rm SSP}}_{p\underline{\smash{x}}}:={\rm
ev}_{\bullet}^{-1}(\overline{W^{s}(\underline{\smash{x}})}),$

where ${\rm ev}_{\bullet}:\overline{\mathcal{M}}_{p\bullet}^{{\rm SSP}}\to M$
is the evaluation map at $+\infty$. If
$x=(a,\underline{\smash{x}})\in{\mathcal{P}}^{\rm Morse}$ and
$p\in{\mathcal{P}}^{\rm Floer}$, then define

$\overline{\mathcal{M}}{}_{px}^{\rm SSP}:=\overline{\mathcal{M}}{}_{a\cdot p\
\underline{\smash{x}}}^{\rm SSP}.$

Similar to the PSS case, the homogeneous posets associated with SSP spaces are
denoted by

${\mathbb{A}}_{px}^{\rm SSP}:=\\{pq_{1}\cdots q_{l}y_{k}\cdots y_{1}x|\
p<q_{1}<\cdots<q_{l},y_{k}<\cdots<y_{1}<x\\}$

for $p\in{\mathcal{P}}^{\rm Floer}$ and $x\in{\mathcal{P}}^{\rm Morse}$,
endowed with depth function

${\rm depth}(pq_{1}\cdots q_{l}y_{k}\cdots y_{1}x)=k+l.$

$\overline{\mathcal{M}}{}_{px}^{\rm SSP}$ is stratified by
${\mathbb{A}}_{px}^{\rm SSP}$ with corresponding strata

$\left(\overline{\mathcal{M}}{}_{px}^{\rm SSP}\right)_{pq_{1}\cdots
q_{l}y_{k}\cdots y_{1}x}\cong\overline{\mathcal{M}}{}_{pq_{1}}^{\rm
Floer}\times\cdots\times\overline{\mathcal{M}}{}_{q_{l-1}q_{l}}^{\rm
Floer}\times\overline{\mathcal{M}}{}_{q_{l}y_{k}}^{\rm
SSP}\times\overline{\mathcal{M}}{}_{y_{k}y_{k-1}}^{\rm
Morse}\times\cdots\times\overline{\mathcal{M}}{}_{y_{1}x}^{\rm Morse}.$

####  4.3.2. The bimodule structure

The PSS and SSP moduli spaces can be packaged into two flow bimodules (see
Definition 3.10). Recall that one has the Floer flow category $T^{\rm Floer}$
and the Morse flow category $T^{\rm Morse}$. Essentially by the way we
compactify the PSS resp. SSP moduli spaces we see that one can define a flow
bimodule $M^{\rm PSS}$ from $T^{\rm Morse}$ to $T^{\rm Floer}$ and a flow
bimodule $M^{\rm SSP}$ from $T^{\rm Floer}$ to $T^{\rm Morse}$. We explain the
specific terms for the PSS bimodule; the SSP case is completely symmetric.

######  Proposition 4.6.

For $x\in{\mathcal{P}}^{{\rm Morse}}$ and $p\in{\mathcal{P}}^{\rm Floer}$,
define $M^{{\rm PSS}}_{xp}:=\overline{\mathcal{M}}{}_{xp}^{\rm PSS}$. Then
together with the natural inclusion of the boundary strata

$\overline{\mathcal{M}}{}^{{\rm
Morse}}_{xy}\times\overline{\mathcal{M}}{}_{yp}^{\rm
PSS}\to\overline{\mathcal{M}}{}_{xp}^{\rm PSS},$
$\overline{\mathcal{M}}{}_{xq}^{\rm
PSS}\times\overline{\mathcal{M}}{}_{qp}^{\rm
Floer}\to\overline{\mathcal{M}}{}_{xp}^{\rm PSS},$

the spaces $M^{{\rm PSS}}_{xp}$ define a flow bimodule from $T^{\rm Morse}$ to
$T^{\rm Floer}$.

###### Proof.

For any $x\in{\mathcal{P}}^{{\rm Morse}}$ and $p\in{\mathcal{P}}^{\rm Morse}$
the moduli space $M^{{\rm PSS}}_{xp}$ is compact due to Gromov compactness and
the bound on the topological energy. For such $x$ and $p$, indeed the
difference of energies ${\mathcal{A}}^{\rm Morse}(x)-{\mathcal{A}}^{\rm
Floer}(p)$ is uniformly bounded. The associaticity of inclusions of boundary
strata follows from the construction, so is the strict $\Pi$-equivariance
property. ∎

We refer to such a bimodule as the PSS bimodule, and the version for the SSP
spaces as SSP bimodule, denoted by $M^{\rm PSS}$ and $M^{\rm SSP}$
respectively.

###  4.4. The pearly bimodule

We consider the moduli space of (parametrized) $J$-holomorphic maps
$u:\mathbb{CP}^{1}\to M$ with two marked points $z_{-}=0$ and $z_{+}=\infty$.
Given a homology class $A\in H_{2}(M;{\mathbb{Z}})$, let
${\mathcal{M}}{}_{0,2}^{\rm pearl}(M,J;A)$ be the moduli space of
(parametrized) $J$-holomorphic maps representing the class $A$ whose domains
are smooth. It has a Gromov compactification
$\overline{\mathcal{M}}{}_{0,2}^{\rm pearl}(M,J;A)$. Each element of this
compactification is represented by a stable map whose domain has a
distinguished component whose parametrization is fixed. Moreover, by
intersecting with the unstable manifold of $\underline{\smash{x}}\in{\rm
crit}f$ at $z_{-}$ and with the stable manifold of
$\underline{\smash{y}}\in{\rm crit}f$ at $z_{+}$, we have a moduli space

$\overline{\mathcal{M}}{}_{\underline{\smash{x}}\underline{\smash{y}}}^{\rm
pearl}(M,J;A)$

For $a\in\Pi$, define

$\overline{\mathcal{M}}{}_{\underline{\smash{x}}\underline{\smash{y}}}^{\rm
pearl}(a):=\bigcup_{\omega(A)=\omega(a)}\overline{\mathcal{M}}{}_{\underline{\smash{x}}\underline{\smash{y}}}^{\rm
pearl}(M,J;A).$

Then given $x,y\in{\mathcal{P}}^{\rm Morse}$ which can be written as
$(a,\underline{\smash{x}})$ and $(b,\underline{\smash{y}})$ where $a,b\in\Pi$
and $\underline{\smash{x}},\underline{\smash{y}}\in{\rm crit}f$, define

$\overline{\mathcal{M}}{}_{xy}^{\rm
pearl}:=\overline{\mathcal{M}}{}_{\underline{\smash{x}}\underline{\smash{y}}}^{\rm
pearl}(b-a).$

The pearly moduli spaces are stratified by configurations which have broken
Morse trajectories either on the incoming edge or on the out-going edge.
Indeed, for ${\mathcal{P}}={\mathcal{P}}^{\prime}={\mathcal{P}}^{\rm Morse}$,
as in Notation 3.6 there is a system of homogeneous posets indexed by pairs of
$x,y\in{\mathcal{P}}^{\rm Morse}$. More explicitly, given
$x,y\in{\mathcal{P}}^{\rm Morse}$, define

${\mathbb{A}}_{xy}^{\rm pearl}:=\left\\{xx_{1}\cdots x_{k}y_{l}\cdots y_{1}y\
\left|\ \begin{array}[]{c}x_{i}=(a,\underline{\smash{x}}_{i}),\
y_{j}=(b,\underline{\smash{y}}_{j}),\\\
\underline{\smash{x}}<\underline{\smash{x}}_{1}<\cdots<\underline{\smash{x}}_{k},\
\underline{\smash{y}}_{l}<\cdots<\underline{\smash{y}}_{1}<\underline{\smash{y}}\end{array}\right.\right\\}.$

Note that all the $x_{i}$’s resp. $y_{j}$’s are marked with the same class
$a\in\Pi$ resp. $b\in\Pi$, because otherwise, the morphism space $T^{\rm
Morse}_{x_{i}x_{i+1}}$ or $T^{\rm Morse}_{y_{j}y_{j+1}}$ is empty by the
construction in Section 4.2. Given $\alpha=xx_{1}\cdots x_{k}y_{l}\cdots
y_{1}y\in{\mathbb{A}}_{xy}^{\rm pearl}$, the corresponding stratum in the
pearly moduli space is

$\Big{(}\overline{\mathcal{M}}{}_{xy}^{\rm
pearl}\Big{)}_{\alpha}\cong\overline{\mathcal{M}}{}_{xx_{1}}^{\rm
Morse}\times\cdots\times\overline{\mathcal{M}}{}_{x_{k-1}x_{k}}^{\rm
Morse}\times\overline{\mathcal{M}}{}_{x_{k}y_{l}}^{\rm
pearl}\times\overline{\mathcal{M}}_{y_{l}y_{l-1}}^{\rm
Morse}\times\cdots\times\overline{\mathcal{M}}{}_{y_{1}y}^{\rm Morse}.$

There are again only finitely many nonempty strata, thanks to Gromov
compactness. Using the formulation of flow bimodules, we can see that the
collection of pearly moduli spaces and the product structures of various
strata provide a flow bimodule from $T^{\rm Morse}$ to $T^{\rm Morse}$, which
we call the pearly bimodule and denote it by $M^{\rm pearl}$. The strict
$\Pi$-action on $M^{\rm pearl}$ follows from the construction. The outer-
collaring (of width 1) of $M^{\rm pearl}$ provides a bimodule $(M^{\rm
pearl})^{+}$ from $(T^{\rm Morse})^{+}$ to $(T^{\rm Morse})^{+}$.

###  4.5. Main Theorems

Using the notations introduced above, we can state the output of Sections 5,
6, and 7.

######  Theorem 4.7.

Denote by $(T^{\rm Floer})^{+}$ the outer-collaring (see Section 3.3.2) of the
Hamiltonian Floer flow category $T^{\rm Floer}$. Then $(T^{\rm Floer})^{+}$
admits a derived orbifold lift (Definition 3.24)

${\mathfrak{D}}^{\rm Floer}=\Big{(}\big{\\{}C^{\rm
Floer}_{pq}=({\mathcal{U}}^{\rm Floer}_{pq},{\mathcal{E}}^{\rm
Floer}_{pq},{\mathcal{S}}^{\rm Floer}_{pq},\psi^{\rm
Floer}_{pq})\big{\\}}_{p<q},\big{\\{}{\bm{\iota}}_{\beta\alpha}^{\rm
Floer}\big{\\}}_{\alpha\leq\beta}\Big{)},$

with a collar structure (Definition 3.26)

$\\{\widehat{\theta}_{\beta\alpha}^{{\rm collar},{\rm
Floer}}\\}_{\alpha\leq\beta},$

and a scaffolding (Definition 3.28)

$\big{(}{\mathcal{F}}_{\beta\alpha}^{\rm
Floer},{\bm{\theta}}_{\beta\alpha}^{\rm Floer}\big{)}_{\alpha\leq\beta}$

such that they are compatible (Definition 3.34). Moreover, such a lift can be
upgraded to an oriented and normally complex derived orbifold lift (Definition
3.44). Further, one can equip the D-chart lift a straightening which is
compatible with the collar and scaffoldings.

The proof is completed in Theorem 6.31.

###### Proof of Theorem B.

From Theorem 4.7 this is a corollary of Theorem 3.49 and the discussion in
Section 3.8.1. ∎

######  Theorem 4.8.

Let $(M^{\rm PSS})^{+}$ be the outer-collaring of the flow bimodule $M^{\rm
PSS}$ from $(T^{\rm Morse})^{+}$ to $(T^{\rm Floer})^{+}$. Then $(M^{\rm
PSS})^{+}$ has an oriented and normally complex derived orbifold lift

${\mathfrak{D}}^{\rm PSS}=\Big{(}\big{\\{}C^{\rm PSS}_{xp}=({\mathcal{U}}^{\rm
PSS}_{xp},{\mathcal{E}}^{\rm PSS}_{xp},{\mathcal{S}}^{\rm PSS}_{xp},\psi^{\rm
PSS}_{xp})\big{\\}}_{x,p},\big{\\{}{\bm{\iota}}_{\beta\alpha}^{\rm
PSS}\big{\\}}_{\alpha\leq\beta}\Big{)},$

with a collar structure

$\\{\widehat{\theta}_{\beta\alpha}^{{\rm collar},{\rm
PSS}}\\}_{\alpha\leq\beta},$

and a scaffolding (Definition 3.28)

$\big{(}{\mathcal{F}}_{\beta\alpha}^{\rm PSS},{\bm{\theta}}_{\beta\alpha}^{\rm
PSS}\big{)}_{\alpha\leq\beta}$

such that they are compatible, and they extend the given structures on
${\mathfrak{D}}^{\rm Floer}$ and ${\mathfrak{D}}^{\rm Morse}$. The same
statement holds by reversing the role of $(T^{\rm Floer})^{+}$ to $(T^{\rm
Morse})^{+}$ and replace ${\rm PSS}$ by ${\rm SSP}$.

The proof is provided in Section 7. The following assertion is proved
similarly as the arguments in Section 6.3.1.

######  Lemma 4.9.

Both ${\mathfrak{D}}^{\rm PSS}$ and ${\mathfrak{D}}^{\rm SSP}$ have a
straightening extending the straightening on ${\mathfrak{D}}^{\rm Floer}$, and
they are compatible with the respective collar structure and scaffolding. ∎

As a consequence, Theorem 3.51 and the discussion in Section 3.8.1 define the
$\Lambda$-linear chain maps

$\Psi^{\rm PSS}:CM_{*}(f;\Lambda)\to CF_{*}(H;\Lambda),$ $\Psi^{\rm
SSP}:CF_{*}(H;\Lambda)\to CM_{*}(f;\Lambda).$

######  Theorem 4.10.

Let $(M^{\rm pearl})^{+}$ be the flow bimodule from $(T^{\rm Morse})^{+}$ to
$(T^{\rm Morse})^{+}$ obtained from the outer-collaring of the pearly bimodule
$M^{\rm pearl}$. Then $(M^{\rm pearl})^{+}$ has an oriented and normally
complex derived orbifold lift

${\mathfrak{D}}^{\rm pearl}=\Big{(}\big{\\{}C^{\rm
pearl}_{xy}=({\mathcal{U}}^{\rm pearl}_{xy},{\mathcal{E}}^{\rm
pearl}_{xy},{\mathcal{S}}^{\rm pearl}_{xy},\psi^{\rm
pearl}_{xy})\big{\\}}_{x,y},\big{\\{}{\bm{\iota}}_{\beta\alpha}^{\rm
pearl}\big{\\}}_{\alpha\leq\beta}\Big{)},$

with a collar structure

$\\{\widehat{\theta}_{\beta\alpha}^{{\rm collar},{\rm
pearl}}\\}_{\alpha\leq\beta},$

such that the induced derived orbifold presentation on each
$\overline{\mathcal{M}}{}^{\rm pearl}_{xy}$ is single-layered (Definition
3.23), i.e., ${\mathfrak{D}}^{\rm pearl}$ has a trivial scaffolding. Moreover,
the orientation structure, normally complex structure, and the collar
structure extend the existing ones on ${\mathfrak{D}}^{\rm Morse}$.

The proof is provided in Section 7. By the argument in Section 6.3.1 (but
simpler because the scaffolding is absent in this case), we can find a
straightening of ${\mathfrak{D}}^{\rm pearl}$ which is compatible with the
collar structure. We fix it once for all.

######  Corollary 4.11.

The FOP countings associated to the oriented and normally complex derived
orbifold lift ${\mathfrak{D}}^{\rm pearl}$ and the chosen straightening
thereon defines a chain map

$\Psi^{\rm pearl}:CM_{*}(f;\Lambda)\to CM_{*}(f;\Lambda)$

which is unitriangular, i.e.,

$\Psi^{\rm pearl}-{\rm Id}\in({\rm End}_{\Lambda}(CM_{*}(f;\Lambda)))_{+}.$

In particular, the induced map on homology

$\Psi^{\rm pearl}:H_{*}(M;\Lambda)\to H_{*}(M;\Lambda)$

is invertible.

###### Proof.

The construction of $\Psi^{\rm pearl}$ follows from Theorem 3.51 and
discussions in Section 3.8. To prove that $\Psi^{\rm pearl}$ is unitriangular,