each manifold with $\mathbb{R}$ and use the extra $\mathbb{R}$ coordinate
together with a normal coordinate vector field to insert a rotation in the
framing such that we can glue these two manifolds along their common boundary
stratum to a new framed manifold. This is the construction used in Lemma 21.

######  Lemma 49.

Suppose we are in the situation of Proposition 24: we have perturbation data
$\mathfrak{P}_{k}$ for $\mathcal{C}(H^{\\#k}_{h},J^{\\#k}_{h})$ and
perturbation data $\mathfrak{P}$ for $\mathcal{C}(H_{h},J_{h})$ such there is
a $C_{k\ell/C_{k}}\simeq C_{\ell}$-equivariant isomorphism virtually smooth
flow categories between a restriction of
$\mathcal{C}(H^{\\#k}_{h},J^{\\#k}_{h})^{C_{k}}$ and a restriction of
$\mathcal{C}^{\prime}(H_{h},J_{h})_{r,\mathcal{E}_{k}}$ where
$\mathcal{E}_{k}$ is some $E^{s}$-parameterization and
$\mathcal{C}^{\prime}(H_{h},J_{h})_{r}$ is a restratification of
$\mathcal{C}^{\prime}(H_{h},J_{h})$ associated to the extension of
perturbation data as in Lemma 20.

Suppose further that $\mathcal{C}^{\prime}(H_{h},J_{h})_{\mathcal{E}^{f}_{h}}$
has been given a framing as in Lemma 48. Write $\mathcal{E}^{f}_{h,0,ab}$ for
the direct sum of all the terms in (178) not contributing to
$\mathfrak{E}^{f}_{ab}$. Then, fixing an extension of $\mathcal{E}^{f}_{h}$ to
$\mathcal{C}^{\prime}(H^{\\#k}_{h},J^{\\#k}_{h})$, there is an
$E^{s}$-parameterization $\mathcal{E}^{f}_{h,k,0}$ of
$\mathcal{C}^{\prime}(H^{\\#k},J^{\\#k})$ such that, writing
$\mathcal{E}^{f}_{h,k}=\mathcal{E}^{f}_{h,k,0}\oplus\mathcal{E}^{f}_{h}$,
there is a framing of
$\mathcal{C}^{\prime}(H^{\\#k}_{h},J^{\\#k}_{h})_{\mathcal{E}^{f}_{h,k}}$ such
that the induced framing of
$\mathcal{C}^{\prime}(H^{\\#k}_{h},J^{\\#k}_{h})_{\mathcal{E}^{f}_{h,k}}^{C_{k}}$
agrees under the previous isomorphism with the framing of
$\mathcal{C}^{\prime}(H_{h},J_{h})_{r,\mathcal{E}_{k}\oplus\mathcal{E}^{f}_{h,k,0}\oplus\mathcal{E}^{f}_{h}}$
induced by applying Lemma 45 to the framing of
$\mathcal{C}^{\prime}(H_{h},J_{h})_{\mathcal{E}^{f}_{h}}$ to get a framing of
$\mathcal{C}^{\prime}(H_{h},J_{h})_{r,\mathcal{E}^{f}_{h}}$ and subsequently
stabilizing. Moreover, when restricting to the objects of
$\mathcal{C}(H_{ab},J_{ab})$, the framing of
$\mathcal{C}^{\prime}(H^{\\#k}_{h},J^{\\#k}_{h})_{\mathcal{E}^{f}_{h,k}}$ is
simply a the stabilization of the framing of
$\mathcal{C}^{\prime}(H^{\\#k}_{ab},J^{\\#k}_{ab})_{\mathcal{E}^{f}_{h,k}}$
produced by applying Lemma 47 with $\mathcal{E}^{f}_{k,0}$ in that lemma given
by $\mathcal{E}^{f}_{h,0,ab}\oplus\mathcal{E}^{f}_{h,k,0}$.

###### Proof.

This follows completely analogously to Lemma 47: we first choose a
sufficiently large initial $\mathcal{E}^{f}_{h,k,0}$ such that all the
$E^{s}$-parameterizations of $\mathcal{V}^{1}_{h,k,ab}$ underlying the
parameterizations of the framings of the obstruction bundles of
$\mathcal{C}(H^{\\#k},J^{\\#k}_{ab})_{r,\mathcal{E}^{f}_{h,k}\oplus\mathcal{E}^{f}_{h,0,ab}\oplus\mathcal{E}^{f}_{h,k,0}}$
are sub-$E^{s}$-parameterizations of this; we then do the induction of Lemma
168 arranging that the vector space assigned to an object always contains the
vector space assigned to that object by $\mathcal{E}^{f}_{h,k,0}$, and
otherwise requiring that the data $\lambda_{V(x,y)}$ and
$\lambda_{V^{1}(x,y)}$ satisfy the conditions of the lemma whenever the lemma
statement in fact fixes these data. ∎

##  9 Background on Homotopy Theory

###  9.1 Conventions

For the entire paper we fix a Grothendieck Universe $\overline{\mathcal{U}}$.
All model categories will consider will have all objects elements of
$\overline{\mathcal{U}}$.

All groups denoted $G$ will be compact Lie groups; in most cases they are
finite groups. In Section 10.6 we will consider the group
$C_{p^{\infty}}=(\varprojlim C_{p^{n}})\subset S^{1}$ in a general sense, but
we will never explicitly consider the action of this group on any spaces.

We will write $Top$ for the category of compactly generated weak-Hausdorff
(CGWH) spaces, $G-Top$ for the category of (CGWH) spaces with continuous
$G$-action, and $Top_{*},G-Top_{*}$ for their based variants. All of these
categories are enriched over themselves, are symmetric monoidal, and admit all
small limits and colimits.

###  9.2 Model Categories

In this section, we briefly review what we need to know about model
categories. A reference for most of what we need about model categories is
[56]; note that we can always replace conditions on existence of all colimits
with existence of all $\overline{\mathcal{U}}$-small colimits.

A model category $\mathcal{C}$ comes with three classes of distinguished
morphisms: the _cofibrations_ , _fibrations_ , and _weak equivalences_. These
are each closed under composition. Model categories always admit initial
objects $0$ and final objects $1$; objects admitting a cofibration from $0$
are called _cofibrant_ , and objects admitting a fibration to $1$ are called
_fibrant_. A _trivial (co)fibration_ is a morphism which is a (co)fibration
and also a weak equivalence.

There is a functor from a model category $\mathcal{C}$ to its homotopy
category $Ho(\mathcal{C})$; this functor exhibits $Ho(\mathcal{C})$ as a
localization of $\mathcal{C}$ at the weak equivalences. In particular, there
is a functor $\mathcal{C}\to Ho(\mathcal{C})$, and given a weak equivalence
$f:X\to Y$ in $\mathcal{C}$, its image in $Ho(\mathcal{C})$ has a canonical
inverse.

Model categories $\mathcal{C}$ admit _weak factorization systems_ : given any
morphism $f:X\to Y$, there is a factorization $X\to Z_{1}\to Y$ where $X\to
Z_{1}$ is a cofibration and $Z_{1}\to Y$ is a trivial fibration, and also a
factorization $X\to Z_{2}\to Y$ where $X\to Z_{2}$ is a trivial cofibration
and $Z_{2}\to Y$ is a fibration. Moreover, these weak factorization systems
are _functorial_ in the sense that if we think of $X\xrightarrow{f}Y$ as a
diagram, then a map of diagrams extends to a map of factorization diagrams
$X\to Y\to Z$.

In particular, applying the weak factorization to the canonical maps $0\to X$
gives diagrams $0\to cX\to X$, functorial in $X$, where the first map is a
cofibration and the second is a trivial fibration. Thus there is a cofibrant
replacement functor $c:\mathcal{C}\to\mathcal{C}$ sending objects to cofibrant
objects, and a natural transformation $c\to id$ with all defining maps being
trivial fibrations. Dually there is fibrant replacement functor
$f:\mathcal{C}\to\mathcal{C}$ and a natural transformation $id\to f$ with all
defining maps trivial cofibrations.

We will repatedly use the following

######  Lemma 50.

Cofibrant replacement preserves fibrancy of objects, and vice versa.

###### Proof.

The object $X$ is fibrant if $X\to 1$ is a fibration; but then $cX\to X$ is a
trivial fibration, so $cX\to 1$ is a fibration because fibrations are closed
under composition. The proof for the other statement is similar. ∎

There is a notion of _left homotopy_ and _right homotopy_ on morphisms from
$X$ to $Y$ in $\mathcal{C}$. It is a fundamental theorem (e.g. [56, Theorem
1.2.10]) that if $X$ is cofibrant and $Y$ is fibrant then these notions
coincide, homotopic maps are sent to the same morphism in the homotopy
category, and the map $\mathcal{C}(X,Y)/\sim\to Ho(X,Y)$ is an isomorphism,
where the equivalence relation on the domain is homotopy of maps.

###  9.3 Orthogonal Spectra

In this section, we give a more detailed review of the properties of
orthogonal spectra that we need. Throughout, $G$ will be a finite group.
Helpful references include [87, 88] and the foundational [73], as well as
[53]. We will use the model structure on orthogonal spectra defined in [73],
although we will largely avoid any discussion of universes, as explained
below.

#### Universes.

A $G$-universe is a direct sum of orthogonal $G$-representations which, if it
contains a copy of an irreducible representation $V$ of $G$, contains
infinitely many copies of $V$. A complete $G$-universe is one that contains
the regular representation.

Throughout this paper, we will fix a single complete $S^{1}$-universe
$\mathcal{U}\supset\mathbb{R}^{\infty}$ (where the inclusion is equivariant
and $G$ acts trivially on $\mathbb{R}^{\infty}$). Whenever we consider
$C_{p^{k}}$-equivariant stable homotopy theory, we will use the complete
$C_{p^{k}}$-universe obtained from $\mathcal{U}$ by restricting along the
inclusion map $C_{p^{k}}\subset S^{1}$.

For every closed subgroup $G\subset S^{1}$ we fix a copy of an embedding of
the regular representation $\rho_{G}\subset\mathcal{U}$.

#### Orthogonal spectra.

Given pair of finite-dimensional dimensional sub-$G$-representations
$V,W\subset\mathcal{U}$, we have an associated $G$-space $O_{G}(V,W)$, which
is the Thom space of the bundle $\xi(V,W)\to L(V,W)$, where $L(V,W)$ is the
space of linear isometric inclusions from $V$ to $W$ and the fiber of
$\xi(V,W)$ of over $\phi$ is $(Im\phi)^{\perp}$. These assemble into into a
topological category $O_{G}$; one definition of an orthogonal $G$-spectrum
[73, Definition 4.1] is an functor $X:O_{G}\to G-Top_{*}$ enriched over
$G-Top_{*}$, the structure maps

$O_{G}(V,W)\wedge X(V)\to X(W)$

are $G$-equivariant. A map of $G$-spectra (with respect to the universe
$\mathcal{U}$) is natural transformation of enriched functors.

Let $O$ be the category of $O_{G}$ consisting only of the representations
$\mathbb{R}^{n}=\mathbb{R}^{n}\oplus
0^{\infty}\subset\mathbb{R}^{\infty}\subset\mathcal{U}$. The functor $X$ can
functorially and canonically recovered from the functor $X|_{O}:O\to
G-Top_{*}$ up to a canonical isomorphism, via the map

$L(\mathbb{R}^{n},W)_{*}\wedge_{O^{\mathbb{R}^{n}}}X(\mathbb{R}^{n})\simeq
X(W);$

which arises from a factorization of the structure map
$O_{G}(\mathbb{R}^{n},W)\wedge X(\mathbb{R}^{n})\to X(W)$ using the fact that
$O_{G}(\mathbb{R}^{n},W)=L(\mathbb{R}^{n},W)$ [88, Remark 2.7].

We will _always_ use the latter definition of an orthogonal $G$-spectrum as an
enriched functor $O\to G-Top$. The category of orthogonal $G$-spectra in this
sense will be denoted $G-OSp$. This category is a symmetric monoidal category.

However, when computing various quantities like fixed points we will define
the associated functor $O_{G}\to G-Top_{*}$ (where we _always use the one
$G$-universe $U$_) to do the computation, and then if the final object is an
$H$-spectrum, we will restrict the resulting object back to $O$ to produce a
functor $O\to H-Top$. This convention will allow us to make clean statements
about the interaction of various functors and change of groups.

In particular, given a closed inclusion of Lie groups $H\subset G$, an
$H$-spectrum $X$, and a $G$-spectrum $Y$, it makes sense to talk about an
$H$-equivariant map $f:X\to Y$ by implicitly viewing $Y$ as an $H$-spectrum
via the forgetful functor $G-Top\to H-Top$. The associated map on spectra,
called ‘group restriction’, is denoted

$R^{G}_{H}:G-OSp\to H-OSp.$

Thus, in the situation above, when we wish to be more precise we will write
$f$ as a map

$f:X\to R^{G}_{H}Y.$

However, due to a number of helpful properties of the group restriction
functor will allow us to drop many instances of $R^{G}_{H}$ from our notation,
which will help both with typesetting and conceptual understanding.

#### Equivariant Homotopy Groups.

For $j\geq 0$ have by definition that

$\pi^{G}_{j}(X):=\operatorname{colim}_{n}[\SS(\rho_{G}^{\oplus
n}\oplus\mathbb{R}^{j}),X(\rho_{G}^{\oplus n})]_{G}$

where $\rho_{G}$ is the regular representation of $G$ and $[\cdot,\cdot]_{G}$
denotes equivariant homotopy classes of maps. Here $\SS$ is the _sphere
spectrum_ , reviewed below in the paragraph on free spectra.

For $j<0$ we have that

$\pi^{G}_{j}(X):=\operatorname{colim}_{n}[S(\rho_{G}^{\oplus
n}),X(\rho_{G}^{\oplus n}\oplus\mathbb{R}^{j})]_{G},$

#### Equivalences and model structures.

A _level equivalence_ of orthogonal $G$-spectra is a map $X\to Y$ such that
the associated maps $X(\mathbb{R}^{n})\to Y(\mathbb{R}^{n})$ are weak
$G$-homotopy equivalences.

A _stable equivalence_ of orthogonal $G$-spectra is a map of orthogonal
$G$-spectra such that the associaed maps on $H$-stable homotopy groups
$\pi^{H}_{k}$ are isomorphisms for all closed subgroups $H\subset G$ and all
$k\in\mathbb{Z}$.

The stable equivalences of orthogonal $G$-spectra form the the weak
equivalences of several model structures on $G-OSp$. We will use the one
defined in [73, Theorem 2.4]. This model structure has the property that the
objects $F_{V}X$ for all $G$-CW complexes $X$ are always cofibrant, where
$F_{V}$ the free spectrum on the $G$-representation $V\subset\mathcal{U}$, a
notion reviewed below. More generally, the cofibrations (called
$q$-cofibrations in [73]) are generated by $F_{V}i$, where $i:(G/H\times
S^{n-1})_{*}\to(G/H\times D^{n})_{*}$ are the cell cofibrations.

Given a functor $F$ on $G-OSp$ with values in a category with weak
equivalences, we will say that it is _homotopical on $X$_ if we would get a
diagram of weak equivalences relating $F(X)$ and $F(cfX)$ from the
factorization diagram. Similar language will be used for bifunctors; and if
the target is not a category with an obvious model-category structure, we
simply mean that the weak equivalences of the codomain of $F$ are exactly the
isomorphisms. A _derived_ functor for $F$ is a functor on $Ho(F)$ that is
naturally isomorphic to $Ho(Fcf)$.

#### Basics.

Given $X\in G-OSp$ and $K\in G-Top_{*}$ one can form $X\wedge K$ by smashing
$X$ with $K$ levelwise, i.e.

$(X\wedge K)(V):=X(V)\wedge K,$

and also form the mapping spectrum [88, Example 5.2]

$X^{K}(V):=G-Top_{*}(K,X(V)).$

These form an adjoint pair:

$G-OSp(X,Y^{K})\simeq G-OSp(A\wedge X,Y).$

Similarly, given $Z\in G-OSp$, there is a mapping space

$map(X,Z)\in G-Top_{*}$

given by the set of natural transformations topologized as a subspace of
$\prod_{n}G-Top_{*}(X(\mathbb{R}^{n}),Y^{(}\mathbb{R}^{n}))$.

We write $\Omega^{V}X=X^{S^{V}}$.

For the model structure of [73, Theorem 2.4], given a cofibrant $X\in G-OSp$
and a fibrant $Y\in G-OSp$, a pair of maps $f_{0},f_{1}:X\to Y$ are homotopic
in the sense of model categories if and only if there is a map

$h:f:X\wedge[0,1]_{*}\to Y$

such that its restrictions to $X\wedge\\{i\\}_{*}$ agree with $f_{i}$ for
$i=0,1$.

#### Free Spectra.

For $V\in O_{G}$, one defines _free spectrum_

$F_{V}(W)=O_{G}(V,W),\text{ i.e. }F_{V}=O_{G}(V,\cdot).$

(Note that the kinds of free spectra allowed _depend_ on the universe being
considered; but for each $G\subset S^{1}$ we have fixed the universe that we
are using, so this notation is unambiguous.)

There are canonical isomorphisms of orthogonal spectra [88, Eq. 5.8]

$F_{V}\wedge F_{W}\simeq F_{V\oplus W}.$

More generally, for $K\in G-Top_{*}$, one sets

$F_{V}K:=F_{V}\wedge K,$

and we have isomorphisms of orthogonal spectra

$F_{V}K_{1}\wedge F_{V}K_{2}\simeq F_{V\oplus W}K_{1}\wedge K_{2}.$

The functor

$G-Top_{*}\to G-OSp,K\mapsto F_{V}K$

is left adjoint to the evaluation functor at $V$; thus there is a natural map

$F_{V}A\to\Omega^{V}(\Sigma^{\infty}A)$

adjoint to the adjunction unit (in $G-Top_{*}$) $A\to\Omega^{V}(A\wedge
S^{V})=\Omega^{V}(\Sigma^{\infty}A)(V)$. This map turns out to be a stable
equivalence [88, Proposition 5.12].

The sphere spectrum is $\SS:=F_{0}$. This makes sense as an object in $G-OSp$
for any $G$, and for any closed Lie subgroup $H\subset G$ we have
$R^{G}_{H}\SS=\SS$. We write

$\Sigma^{\infty}X=F_{0}X\in G-OSp$

for a $G$-space $X$.

There are also canonical maps of orthogonal $G$-spectra

$F_{W\oplus V}\Sigma^{V}X\to F_{W}X.$ (179)

which are weak equivalences when $X$ and $Y$ are $G$-CW complexes.

#### Mapping Spectra.

Given $V\in O_{G}$ and $X\in G-OSp$, there is a _shift spectrum_

$sh^{V}X(W)=X(V\oplus W)$

with structure maps induced from the inclusions $O_{G}(W)\subset O_{G}(V\oplus
W)$ which are natural in $W$.

Given $X,Y\in G-OSp$, there is an internal mapping spectrum $F(X,Y)$ [88,
5.11] with

$F(X,Y)(V)=map(X,sh^{V}Y).$

The internal function spectrum makes $G-OSp$ into a closed symmetric monoidal
category with unit $\SS$ [73, Theorem 3.1]. As usual, the composition maps

$F(X,Y)\wedge X\to Y$

are the counit of the adjunction

$F(Y\wedge X,Z)\simeq F(Y,F(X,Z))$

(valid for each $X$).

The mapping spectrum $F(X,Y)$ is homotopical whenever $X$ is cofibrant and $Y$
is fibrant. This follows from [88, Proposition 5.17] and the characterization
of the fibrant objects as $G-\Omega$ spectra [73, Theorem 2.4], namely those
orthogonal $G$-spectra $X$ such that for all closed subgroups $H\subset G$ and
all pairs of $H$-representations in the universe $\mathcal{U}$, the map
$X(V)\to\Omega^{W}X(VoplusW)$ adjoint to the structure map $X(V)\wedge
S^{W}\to X(V\oplus W)$ is a weak $H$-equivalence.

#### Geometric Fixed Points.

Given a closed _normal_ subgroup $H\subset G$, there is a monoidal geometric
fixed point functor

$\Phi^{H}:G-OSp\to H-OSp.$

There are several differing definitions which all give isomorphic functors on
homotopy categories. We always use variant of the definition [73] which gives
a monoidal functor; this definition agrees with the definition in [53] under a
nontrivial isomorphism, and differs from the definition in [88].

We have the helpful propositions:

######  Proposition 29 (Proposition 4.5 [73] ).

For $X\in G-Top_{*}$, there is a natural isomorphism in $H-OSp$

$\Phi^{H}(F_{V}X)\simeq F_{V^{H}}X^{H}.$ (180)

which preserves ($q$-)cofibrations and acyclic ($q$-)cofibrations.

The argument of [73, Proposition 4.5] shows also that

######  Lemma 51.

The functor $\Phi^{H}$ commutes with the formation of mapping telescopes (see
below.)

######  Proposition 30 (Proposition 4.5[73]).

For $X,Y\in G-OSp$, there is a natural map in $G/H-OSp$

$\Phi^{H}X\wedge\Phi^{H}Y\to\Phi^{H}(X\wedge Y)$

which is an isomorphism if one of $X$ and $Y$ are cofibrant.

The functor $\Phi^{H}X$ is homotopical whenever $X$ is cofibrant [73, Prop.
4.12, Corr. 4.13].

#### Interactions with change of group.

We now compile several results about change of group.

######  Lemma 52 (Theorem IV.1.2 [73]).

Let $H\subset G$ be a closed subgroup of a compact Lie group $G$, and let
$R^{G}_{H}$ be the restriction functor from orthogonal $G$-spectra to
orthogonal $H$-spetra. Then $R^{G}_{H}$ preserves stable equivalences,
$q$-fibrations, and $q$-cofibrations. In particular, they preserve fibrant and
cofibrant objects.

It follows immediately from the definitions above there is an equality of
bifunctors

$F(R^{G}_{H}X,R^{G}_{H}Y)=R^{G}_{H}F(X,Y).$ (181)

It follows immediately from [73, Definition 4.1] that we have an equality of
functors

$\Phi^{H}R^{G}_{G^{\prime}}=R^{G}_{G^{\prime}}\Phi^{H}.$ (182)

#### Mapping Telescopes.

We will collate some results about mapping telescopes below. First, a mapping
telescope of a family of spaces is

$X_{0}\xrightarrow{f_{0}}X_{1}\xrightarrow{f_{1}}\cdots\xrightarrow{f_{n-1}}X_{n}$

is the pushout

$Tel(X_{0}\to\cdots\to
X_{n}):=(X_{0}(V)\times[0,1])\cup_{f^{\prime}_{1}}(X_{1}(V)\times[0,1])\cup_{f^{\prime}_{2}}\cdots\cup_{f^{\prime}_{n-1}}(X_{n}\times\\{0\\})$

where $f^{\prime}_{k}:X_{k}\times\\{1\\}\to X_{k+1}\times\\{0\\}$ is is just
the action of $f_{k}$ on the first factor. The mapping telescope of an
infinite sequence

$X_{0}\to X_{1}\to\cdots$ (183)

is the colimit of the mapping telescope of the finite subsequences, i.e.

$Tel(X_{0}\to X_{1}\to\cdots)=\operatorname{colim}_{n}(X_{0}\to
X_{1}\to\cdots\to X_{n})$

under the natural closed embeddings of these spaces. When all the $X_{i}$ are
$G$-spaces and $f_{i}$ are $G$-maps then mapping telescopes inherit a
$G$-action, and if all spaces involved are $G$-CW complexes and $G$-CW maps
then the mapping telescope is a $G$-CW complex.

There are collapse maps

$coll_{0,1}:Tel(X_{0}\to\cdots\to X_{n})\to Tel(X_{1}\to\cdots\to X_{n})$

induced by the collapse map $X_{0}\times[0,1]\to X_{1}$, and they are
(equivariant) homotopy equivalences, with explicit homotopy-inverse the closed
inclusion

$i_{1,0}:Tel(X_{1}\to\cdots\to X_{n})\subset Tel(X_{0}\to\cdots\to X_{n})$
(184)

Taking colimits in $n$ shows that the analogous collapse and inclusion maps
(denoted with in the same way) on infinite mapping telescopes are
$(G-)$-homotopy equivalences as well. One defines
$coll_{a,b}=coll_{a,a+1}coll_{a+1,a+2}\cdots coll_{b-1,b}$ and similarly for
$i_{b,a}$.

Given a diagram like (183) of orthogonal $G$-spectra, the mapping telescope is
defined by applying the above constructions level-wise. One thus immediately
sees that if all $X_{i}$ are cofibrant $G$-spectra then so is the mapping
telescope. Taking mapping telescopes manifestly commutes with group
restriction, and the collapse maps and their canonical homotopy inverses when
applied level-wise define corresponding maps of mapping telescopes of
orthogonal $G$-spectra which are also level equivalences.

The following lemma is standard:

######  Lemma 53.

Consider a diagram of $X_{i}\in G-Top_{*}$

$X_{0}\subset X_{1}\subset X_{2}\subset\ldots\subset
X=\operatorname{colim}_{i}X_{i}.$

Then if all inclusions above are closed inclusions then the canonical map from
the mapping telescope $Tel(X_{0}\to\ldots)\to X$ is a weak equivalence.

We have the following useful standard result:

######  Lemma 54.

Given a diagram of orthogonal $G$-spectra equipped with levelwise
nondegenerate basepoints

$Y_{0}\to Y_{1}\to\cdots$

(with either finitely or infinitely many terms) where all maps are level-wise
closed embeddings, we have a functorial isomorphism

$\operatorname{colim}\pi_{G}^{j}(Y_{i})\simeq\pi_{G}^{j}(\operatorname{colim}Y_{i})$
(185)

of equivariant homotopy groups.

###### Proof.

This is proven exactly like its analog for $G=1$, which is explained in [87,
Proposition 1.24]. For $j\geq 0$, by [56, Prop 2.4.2], any map from
$S(\rho_{G}^{\oplus n}\oplus\mathbb{R}^{j})$ to
$\operatorname{colim}_{i}Y_{i}(\rho_{G}^{\oplus n})]_{G}$ factors through some
$Y_{i}(\rho_{G}^{\oplus n})]_{G}$, as does any homotopy between a pair of such
maps. Thus the canonical map

$\operatorname{colim}_{i}[S(\rho_{G}^{\oplus
n}\oplus\mathbb{R}^{j}),Y_{i}(\rho_{G}^{\oplus n})]_{G}\to[S(\rho_{G}^{\oplus
n}\oplus\mathbb{R}^{j}),\operatorname{colim}_{i}Y_{i}(\rho_{G}^{\oplus
n})]_{G}$

is an isomorphism. Passing to colimits in $n$ we conclude that the canonical
map (185) is an isomorphism.

For $j<0$, and a directly analogous argument goes through.

∎

This result easily implies the following helpful lemmata on mapping telescopes
of equivariant orthogonal spectra:

######  Lemma 55.

Given a (strictly) commutative diagram

${X_{0}}$${X_{1}}$${X_{2}}$${\cdots}$${X^{\prime}_{0}}$${X^{\prime}_{1}}$${X^{\prime}_{2}}$${\cdots}$
(186)

of orthogonal $G$-spectra with all the vertical arrows stable equivalences,
the induced map on mapping telescopes

$Tel(X_{0}\to X_{1}\to\cdots)\to Tel(X^{\prime}_{0})\to Tel(X^{\prime}_{0}\to
X^{\prime}_{1}\to\cdots)$

is a stable equivalence.

###### Proof.

A stable equivalence of orthogonal $G$-spectra is one for which the induced
maps on $\pi^{H}_{k}$ are isomorphisms for all subgroups $H\subset G$ and all
$k\in\mathbb{Z}$. If the diagram (186) has finitely many entries and
terminates with a morphism $X_{n}\to X^{\prime}_{n}$ for some $n$, then there
is a commutative diagram

${Tel(X_{0}\to X_{1}\to\cdots)}$${X_{n}}$${Tel(X^{\prime}_{0}\to
X^{\prime}_{1}\to\cdots)}$${X^{\prime}_{n}}$ (187)

where the horizontal arrows are the inclusion maps (and are thus level
equivalences), the left vertical arrow is the induced map on mapping
telescopes, and the right vertical arrow is the map in the diagram (186). In
the case where (186) has infinitely many entries, the statement for finite-
size diagrams implies that the map

$\operatorname{colim}_{n}\pi^{H}_{j}(Tel(X_{0}\to\cdots\to
X_{n}))\to\operatorname{colim}_{n}\pi^{H}_{j}(Tel(X^{\prime}_{0}\to\cdots\to
X^{\prime}_{n}))$

is an isomorphism for all $H,j$; the previous lemma then implies that the map
between mapping telescopes is a stable equivalence. ∎

######  Lemma 56.

Given a diagram

${X_{0}}$${X_{1}}$${X_{2}}$${\cdots}$${X^{\prime}_{0}}$${X^{\prime}_{1}}$${X^{\prime}_{2}}$${\cdots}$$\scriptstyle{a_{0}}$$\scriptstyle{f_{0}}$$\scriptstyle{a_{1}}$$\scriptstyle{f_{1}}$$\scriptstyle{f_{2}}$$\scriptstyle{a_{2}}$$\scriptstyle{b_{0}}$$\scriptstyle{b_{1}}$$\scriptstyle{b_{2}}$
(188)

of orthogonal $G$-spectra which is _not_ commutative, but such that one has
chosen homotopies $h_{i}:X_{i}\wedge[0,1]_{*}\to X^{\prime}_{i+1}$ from
$f_{i+1}a_{i}$ to $b_{i}f_{i}$, there is a map

$Tel(X_{0}\to X_{1}\to\cdots)\to Tel(X^{\prime}_{0})\to Tel(X^{\prime}_{0}\to
X^{\prime}_{1}\to\cdots)$ (189)

defined by as the induced map on colimits of finite mapping telescopes by maps
which, when evaluated on the representation $V$, map $X_{i}(V)\times\\{0\\}$
to $X^{\prime}_{i}(V)\times\\{0\\}$ via $f_{i}(V)$, and more generally map
$X_{i}(V)\times[0,1]$ to
$X^{\prime}_{i}(V)\times[0,1]\cup_{b_{i}(V)}X^{\prime}_{i+1}\times\\{0\\}$ via
by sending $(x,t)\mapsto(h_{i}(x,2t-1),0)$ for $t\geq 1/2$, and otherwise
$(x,t)\mapsto(f_{i}(x),t/2)$ for $t\leq 1/2$.

If all the vertical arrows of (188) are stable equivalences then the map (189)
is a stable equivalence.

###### Proof.

The proof of this lemma is _identical_ to the proof of the previous lemma. ∎

##  10 Floer Homotopy and its Properties

In this section, we apply the tools that we have developed in this paper to
construct the genuine equivariant Floer homotopy type, as well as the
cyclotomic structure on symplectic cohomology. First, in Section 10.1, we will
combine the results we have established so far and use these to define the
(genuine equivariant) Floer homotopy and cohomotopy types
$CF_{\bullet}(H,J;\SS)$ and $CF^{\bullet}(H,J;\SS)$ for admissible Floer data
$(H,J)$. A-priori these depend on many choices, which we will refer to as the
_homotopy definition data_ ; in this same section, we describe a stabilization
of homotopy definition data, and explain how the geometric results we have
established allow us to establish maps and relations between maps between
these Floer homotopy types. Then, in Section 10.2, we describe some general
results about flow categories (which in fact are used in Section 10.1), and
use these and other arguments regarding the framings of our virtually smooth
flow categories $\mathcal{C}^{\prime}(H,J)$ to show that on homology, the
spectra and maps of spectra defined in Section 10.2 define, when the homotopy
definition data are chosen appropriately (and in particular when the Floer
data are sufficiently regular), spectra and maps of spectra which correspond
to the Floer chain complex and the Floer continuation maps on Floer homology.
Subsequently, in Section 10.4, we use these aforementioned results to prove
invariance of the full genuine equivariant Floer homotopy types
$CF_{\bullet}(H,J;\SS)$ from all choices made.

We then proceed with the discussion of the cyclotomic structure on symplectic
cohomology. In order to construct symplectic cohomology, we imitate the usual
construction of symplectic cohomology as a colimit of Hamiltonian Floer
homology groups.

Let us recall the construction of symplectic cohomology. Suppose that $(H,J)$
is a _strongly nondegenerate_ convex linear Floer datum on
$(M,\omega,\lambda)$. Thus, the Floer data $(H^{\\#k},J^{\\#k})$ are also
convex linear nondegenerate Floer data, with the slope of $H^{\\#k}$ being $k$
times the slope of $H$. Fix a prime $p$. If we choose a convex continuation
datum $(H_{s},J_{s})$ from $H_{1}$ to $H^{\\#p}$, then
$(H^{\\#p^{k}}_{s},J^{\\#p^{k}}_{s})$ is a convex continuation datum from
$(H^{\\#p^{k}},J^{\\#p^{k}})$ to $(H^{\\#p^{k+1}},J^{\\#p^{k+1}})$. If all the
Floer data and continuation data are also regular, then we have a diagram of
Floer cohomology groups

$CF^{*}(H,J)\to CF^{*}(H^{\\#p},J^{\\#p})\to\cdots.$

The symplectic cohomology $SH^{*}(M)$ is the colimit of the corresponding
cohomology groups

$HF^{*}(H,J)\to HF^{*}(H^{\\#p},J^{\\#p})\to\cdots.$ (190)

It turns out to be independent of all the data chosen above up to canonical
isomorphism [4].

To construct spectral symplectic cohomology $SH^{\bullet}(M;\SS)$ one imitates
this construction. The geometric constructions of the paper (reviewed in
Section 10.1 below) allow us to define Floer cohomotopy types

$CF^{\bullet}(H^{\\#k},J^{\\#k};\SS)_{\mathfrak{H}^{k}}:=F(|\mathcal{C}(H^{\\#k},J^{\\#k})|_{\mathfrak{H}^{k}},\SS).$

where $|\mathcal{C}(H^{\\#k},J^{\\#k})|_{\mathfrak{H}^{k}}$ is the orthogonal
spectrum associated to a virtual smoothing of the proper flow category
$\mathcal{C}(H^{\\#k},J^{\\#k})$, together with a framing, an embedding of the
framing, and so forth, as needed in Sections 4 \- 8; all the choices made in
the construction are denoted by $\mathfrak{H}^{k}$, and are referred to as the
_homotopy definition data_. Here, the function spectrum $F(\cdot,\SS)$ is
taken in the derived sense. The constructions of this paper allow us to define
a diagram of orthogonal spectra

$CF^{\bullet}(H,J;\SS)_{\mathfrak{H}^{1}}\to
CF^{\bullet}(H^{\\#p},J^{\\#p};\SS)_{\mathfrak{H}^{p}}\to
CF^{\bullet}(H^{\\#p^{2}},J^{\\#p^{2}};\SS)_{\mathfrak{H}^{p^{2}}}\to\cdots$
(191)

These maps lift the maps (190) in the sense that upon taking (derived) smash
product with $H\mathbb{Z}$ and subsequently taking stable homotopy groups one
recovers the maps (190). Thus, the spectral symplectic cohomology reduces to
the usual symplectic cohomology upon smashing with $H\mathbb{Z}$ and taking
homotopy groups as well.

Constructing the cyclotomic structure is somewhat more challenging due to the
potentially large numbers of compatibility conditions involved. As stated in
the introduction, we use the definition of a _genuine $p$-cyclotomic
spectrum_, originally defined by Hesselholt-Madsen [51]; we use closely
related variant of this definition introduced by Nikolaus-Scholze [77] which
still uses genuine equivariant spectra, but uses $\infty$-categories to keep
track of some of the coherence data required. We wish to show that
$SH^{\bullet}(M;\SS)$ is the underlying spectrum of a genuine $p$-cyclotomic
spectrum for every $p$. (We do not work out the compatibility conditions
between different primes $p$ in this paper, although it is essentially a
tedious exercise given the tools developed in this paper.) The notion of a
genuine $p$-cyclotomic spectrum, reviewed in Section 10.6, is purely
$\infty$-categorical, while our constructions are concrete and geoometric.
Now, the symmetries of the geometric situation translate furnish part of the
data needed for the $\infty$-categorical notion. Indeed, a a genuine
$p$-cyclotomic spectrum $X$ is, for each $\ell$, a $C_{p^{\ell}}$-spectrum
$X_{\ell}$, such that $X_{\ell-1}$ is weakly equivalent to
$F^{C_{p^{\ell}}}_{C_{p^{\ell-1}}}X_{\ell}$, together with cyclotomic
structure equivalences

$X_{\ell-1}\to\Phi^{C_{p}}X_{\ell}$ (192)

such that all this data is compatible up to homotopy as $k$ varies. The
geometry of the virtual smoothings of the flow categories of Floer moduli
spaces that we have established in the earlier portions of this paper gives us
each piece of this structure. To define $X_{\ell}$, we consider the diagram
(191) over all $k\geq\ell$, and note that the relevant flow categories are
naturally $C_{p^{\ell}}$-equivariant, and all the smoothings, framings, etc.
can be chosen to be so as well. As such, the construction of this paper give
us a lift of that part of the diagram (191) thought of as a diagram in the
homotopy category of spectra to a diagram of $C_{p^{\ell}}$-spectra; taking
the homotopy colimit in this latter category defines $X_{\ell}$. The
equivalences (192) are then defined by taking the map on homotopy colimits
induced by the weak equivalences

$CF^{\bullet}(H^{\\#p^{k-1}},J^{\\#p^{k-1}};\SS)_{\mathfrak{H}^{p^{k-1}}}\simeq\Phi^{C_{p}}CF^{\bullet}(H^{\\#p^{k}},J^{\\#p^{k}};\SS)_{\mathfrak{H}^{p^{k}}}$

which arise essentially for free from Theorem 8 once one has done all
constructions equivariantly, as detailed earlier in this paper.

Thus the geometric constructions give the input to the data of a genuine
$p$-cyclotomic spectrum. The challenge is to verify that the compatibility
conditions betwen these various maps required by the definition of a genuine
$p$-cyclotomic structure hold. All geometric choices to be made are made in
the straightforward Section 10.3 (which combines many geometric results in the
paper). This data is then carefully repackaged in Sections 10.5 and Section
10.6. The first basic issue is that one is working with many different
categories of genuine equivariant spectra (specifically, orthogonal
$C_{p^{k}}$ spectra for varying $k$) _for which the notions of cofibrant and
fibrant spectra are all distinct_. Using summary of the geometric results of
Section 10.1, we explain, in Section 10.5, how to take a sequence of
cofibrant-fibrant replacements of the resulting objects and maps in the
varying model categories of orthogonal spectra, as well as explicit models of
homotopy colimits of the resulting diagrams, in order to define certain
relatively concrete representatives of the objects $X_{\ell}$ in the
categories of $C_{p^{\ell}}$-orthogonal spectra. Subsequently, using the
explicit nature of the homotopy colimits and the sequences of fibrant-
cofibrant replacements taken, we show in Section 10.6 that this process also
gives the compatibility data needed for these maps to define a lift of
$SH^{\bullet}(M;\SS)$ to the category of genuine $p$-cyclotomic spectra.

We now proceed with the construction.

###  10.1 Basic Results on Floer Homotopy

In this section we use the earlier geometric geometric constructions of this
paper in order to define Floer homotopy types and define relationships between
them. We will work with two kinds of homotopy types: _homological_ and
_cohomological_. One will be the Spanier-Whitehead dual of the other, and the
relationship between them be analogous to the relationship between Floer
_homology_ and Floer _cohomology_. The dualization from homological Floer
homotopy types to cohomological Floer homotopy types will be introduced to
reverse the directions of various maps defined by virtually smooth flow
categories, which is important when defining symplectic cohomology since in
that setting the continuation maps used in the definition of symplectic
cohomology only go one way, in the direction of increasing slope of the
Hamiltonian.

Throughout, we are working with an admissible symplectic manifold.

######  Remark 63.

Below, whenever we consider free spectra $F_{V}$ for $V$ a
$C_{p^{k}}$-representation, we find an $S^{1}$-representation $V^{\prime}$
which gives $V$ upon restriction, and fix an embedding of $V^{\prime}$ into
$\mathcal{U}$. Often we will have pairs of representations $V_{1}\subset
V_{2}$ where $V_{1}$ is a $C_{p^{k}}$-representation, $V_{2}$ is a
$C_{p^{\ell}}$ representation for $\ell\geq k$, and the embedding is
$C_{p^{\ell}}$-equivariant. In that case, if we have already picked a lift of
$V_{1}$ to an $S^{1}$-represetation $V^{\prime}_{1}$ and an embedding
$V^{\prime}_{1}\subset\mathcal{U}$, we proceed by choosing a lift of $V_{2}$
an $S^{1}$-representation $V^{\prime}_{2}$ which contains $V^{\prime}_{1}$ as
a sub-representation, and extend of $V^{\prime}_{1}$ to an embedding of
$V^{\prime}_{2}$. In the cases we consider it will be straightforward to see
that such embeddings exist and that their choices of embeddings do not affect
the results up to isomorphism.

######  Proposition 31.

Let $(H,J)$ be an admissible Floer datum on $M$. Then, given a choice of
integralization data $\mathfrak{A}$, there is a proper flow category
$\mathcal{C}(H,J)$. There exists a compatible regular system of perturbation
data $\mathfrak{P}$ with underlying integralization data $\mathfrak{A}$ and a
smooth structure on the virtual smoothing $\mathcal{C}^{\prime}(H,J)$ of
$\mathcal{C}(H,J)$ associated to $\mathfrak{P}$. If the polarization class
$\rho\in KO^{1}(LM)$ of Section 8.6 vanishes, then, by choosing a
trivialization of the polarization map $\tilde{\rho}$, the virtual smoothing
can be framed, and admits extension $\bar{\mathbf{E}}$ of an embedding
$\mathbf{E}$. If $(H,J)$ are $C_{k}$ equivariant, and the polarization class
$\rho\in KO^{1}_{C_{k}}(LM)$ of Section 8.6 vanishes, then all of the above
data can be chosen to be equivariant. The associated stable homotopy type is
the orthogonal spectrum

$|\mathcal{C}^{\prime}(H,J)|=F_{\tilde{V}^{1}}\\{\mathcal{C}^{\prime}(H,J)\\}$
(193)

where $\tilde{V}^{1}$ is the _shift space_ associated to
$\mathcal{C}^{\prime}$ (Remark 38 and Definition 63) and
$\\{\mathcal{C}^{\prime}(H,J)\\}$ is a $C_{k}$-space with the equivariant
homotopy type of a $C_{k}$-CW complex. This orthogonal spectrum depends on all
of the data described above; we call the data
$\mathfrak{H}=(\mathfrak{A},\mathfrak{P},\tilde{\rho},\mathbf{E},\bar{\mathbf{E}})$
the _homotopy definition data_ , and we will refer to
$\tilde{V}^{1}=\tilde{V}^{1}_{\mathfrak{H}}$ and
$\\{\mathcal{C}^{\prime}(H,J)\\}_{\mathfrak{H}}$ when we wish to keep track of
the dependence on the homotopy definition data.

We will define the _Floer homotopy type_ associated to $\mathfrak{H}$ to be

$CF_{\bullet}(H,J)_{\mathfrak{H}}:=|\mathcal{C}^{\prime}(H,J)|_{\mathfrak{H}}:=F_{\tilde{V}^{1}_{\mathfrak{H}}}\\{\mathcal{C}^{\prime}(H,J)\\}_{\mathfrak{H}}.$

The _Floer cohomotopy type_ associated to $\mathfrak{H}$ is the (equivariant)
Spanier-Whitehead dual of the above object in the (equivariant) stable
homotopy category, and is represented in the equivariant stable homotopy
category by the function spectrum

$CF^{\bullet}(H,J)_{\mathfrak{H}}:=F(|\mathcal{C}^{\prime}(H,J)_{\mathfrak{H}}|,f\SS),$

where $f$ denotes fibrant replacement.

###### Proof of Proposition 31.

This follows from a combination of the results we have established in this
paper. Namely, we invoke Propositions 7, 16, and 28 to produce the virtually
smooth framed flow category $\mathcal{C}^{\prime}(H,J)$. To produce the
embedding and its extension we invoke Propositions 3 and Lemma 19. The
definition of $\\{\mathcal{C}^{\prime}(H,J)\\}$ is given in Definitions 33 and
46, and the statement that it has the homotopy type of a $G$-CW complex is
Proposition 2. ∎

In order to relate these homotopy types, we will use the continuation map
construction, for which virtual smoothings are constructed in Section 7.8. We
first need to introduce the notion of a stabilization of a homotopy definition
datum:

######  Definition 90.

Let $\mathfrak{H}^{\prime}$ and $\mathfrak{H}$ be homotopy definition data for
$\mathcal{C}(H,J)$. We say that $\mathfrak{H}^{\prime}$ is a stabilization of
$\mathfrak{H}$ if the following conditions hold:

  1. 1.

The integralization datum $\bar{\mathfrak{A}}^{\prime}$ underlying
$\mathfrak{H}^{\prime}$ is an extension of the integralization datum
$\bar{\mathfrak{A}}$ underlying $\mathfrak{H}$ in the sense of Lemma 32, and

  2. 2.

Writing $\mathfrak{P}$ and $\mathfrak{P}^{\prime}$ for the perturbation data
underlying $\mathfrak{H}$ and $\mathfrak{H}^{\prime}$, respectively, we have
that there is an isomorphism $f$ of virtually smooth equivariant flow
categories from $\mathcal{C}^{\prime}(H,J)_{\mathfrak{P}^{\prime}}$ to (a
restriction of) $\mathcal{C}^{\prime}(H,J)_{\mathfrak{P},r,\mathcal{E}}$,
which is the stabilization by an $E^{s}$-parameterization $\mathcal{E}$ of the
restratification $\mathcal{C}^{\prime}(H,J)_{\mathfrak{P},r}$ of
$\mathcal{C}^{\prime}(H,J)_{\mathfrak{P}}$ associated to the fact that
$\bar{\mathfrak{A}}^{\prime}$ is an extension of $\bar{\mathfrak{A}}$

  3. 3.

There is an isomorphism $f^{E}:\mathcal{E}^{\prime}\to\mathcal{E}^{s}$, of
$E^{s}$-parameterizations, where $\mathcal{E}^{\prime}$ is the
$E^{s}_{2}$-parameterization underlying
$\mathcal{C}^{\prime}(H,J)_{\mathfrak{P}^{\prime}}$, and $\mathcal{E}^{s}$ is
the $E^{s}_{2}$-parameterization underlying the framing of
$\mathcal{C}^{\prime}(H,J)_{\mathfrak{P},r,\mathcal{E}}$ induced from that of
$\mathcal{C}(H,J)^{\prime}_{\mathfrak{P}}$ by Lemmas 45 and 8.7. Moreover,
under the identifications $f$ and $f^{E}$, the framing of
$\mathcal{C}^{\prime}(H,J)_{\mathfrak{P}^{\prime}}$ agrees with the framing of
$\mathcal{C}^{\prime}(H,J)_{\mathfrak{P},r,\mathcal{E}}$ induced by these same
Lemmas.

  4. 4.

Furthermore,the embeddings underlying $\mathfrak{H}^{\prime}$ agree (under $f$
and the map $f^{F}$ on $F^{s}_{2}$-parameterizations induced by $f^{E}$) with
the embeddings induced from the embeddings underlying $\mathfrak{H}$ via Lemma
20 and 17, and similarly the extension of the embedding underlying
$\mathfrak{H}^{\prime}$ is given (under $f^{F}$) by applying Lemmas 20 and 17.

The same definition applies to make sense of stabilizations of homotopy
definition data for $\mathcal{C}(H_{s},J_{s})$ or $\mathcal{C}(H_{h},J_{h})$,
where $(H_{s},J_{s})$ are convex continuation data and $(H_{h},J_{h})$ are
convex continuation homotopy data.

######  Proposition 32.

Say $\mathfrak{H}^{\prime}$ is a stabilization of of the homotopy definition
datum $\mathfrak{H}$. Using the notation of Definition 90, write
$V^{1}_{\mathcal{E}}$ and $\mathcal{V}^{1}_{\mathcal{E}^{s}}$ for the shift
spaces underlying $\mathcal{E}$ and $\mathcal{E}^{s}$, and write

$\Delta\bar{A}_{max}=|\mathfrak{A}^{\prime}|\setminus|\mathfrak{A}|$

where $(\mathfrak{A},\mathfrak{A}^{\prime})$ are the marked actions underlying
the integralization data $(\bar{\mathfrak{A}},\bar{\mathfrak{A}}^{\prime})$,
respectively. Then there is an equivariant homeomorphism

$\\{\mathcal{C}(H,J)\\}_{\mathfrak{H}^{\prime}}=\Sigma^{V^{1}_{\mathcal{E}}\oplus\mathbb{R}^{\Delta\bar{A}_{max}}+M}\\{\mathcal{C}(H,J)\\}_{\mathfrak{H}}$
(194)

and an isomorphism of orthogonal spectra

$|\mathcal{C}(H,J)|_{\mathfrak{H}^{\prime}}\simeq
F_{V^{1}_{\mathcal{E}}\oplus\mathbb{R}^{\Delta\bar{A}_{max}}}\Sigma^{V^{1}_{\mathcal{E}}\oplus\mathbb{R}^{\Delta\bar{A}_{max}}+M}|\mathcal{C}(H,J)|_{\mathfrak{H}},$
(195)

where $M$ is the restratification embedding shift of Lemma 20. Moreover we can
replace $(H,J)$ with $(H_{s},J_{s})$ or with $(H_{h},J_{h})$ in any of the
statements above.

###### Proof.

The homeomorphism (194) follows immediately from Property (2) of Definition 90
together with Proposition 6 and Lemma 18.

The isomorphism (195) follows from (194) together with the statement that,
writing $V^{1}_{\mathcal{E}_{0}}$ for the shift space associated to
$\mathcal{C}^{\prime}(H,J)_{\mathfrak{P}}$, we have an equivariant isomorphism

$V^{1}_{\mathcal{E}_{0}}\oplus
V^{1}_{\mathcal{E}}\oplus\mathbb{R}^{\Delta\bar{A}_{max}}\simeq
V^{1}_{\mathcal{E}^{s}}.$ (196)

But this follows from Property (3) of Definition 90 together with the fact
that in the context of Lemma 45 , the shift space associated to the part of
the $E^{s}$-parameterization $V^{1}_{can}$ of that lemma changes exactly by
$\mathbb{R}^{\Delta\bar{A}_{max}+M}\oplus V^{1}_{\mathcal{E}_{1}}$ for some
semi-free parameterization $\mathcal{E}_{1}$ (see Remark 50, in the context of
which $M=\Delta\mathfrak{A}_{lower}$) while in the context of Lemma 8.7, we
simply add $\mathcal{V}^{1}_{\mathcal{E}_{2}}$ to the shift space for some
semi-free parameterization $\mathcal{E}_{2}$; and in the end we have
$\mathcal{E}=\mathcal{E}_{1}\oplus\mathcal{E}_{2}$. Combining these statements
gives exactly (196). The proofs with $(H,J)$ replaced by $(H_{s},J_{s})$ or
$(H_{h},J_{h})$ are identical. ∎

Below we summarize the geometric data associated to continuation maps:

######  Proposition 33.

Let $(H_{\pm},J_{\pm})$ be a pair of admissible floer data for $M$, and let
$(H_{s},J_{s})$ be admissible continuation data from $(H_{-},J_{-})$ to
$(H_{+},J_{+})$. If the polarization class $\rho\in KO^{1}(LM)$ vanishes,
then, given homotopy definition data $\mathfrak{H}_{\pm}$ for
$|\mathcal{C}^{\prime}(H_{\pm},J_{\pm})|$, there exists a homotopy definition
datum $\mathfrak{H}$ defining a virtual smoothing
$\mathcal{C}^{\prime}(H_{s},J_{s})$ of $\mathcal{C}(H,J)$, such that the
restrictions of $\mathfrak{H}$ to $\mathcal{C}(H_{\pm},J_{\pm})$ are
stabilizations of $\mathfrak{H}_{\pm}$, and such that the integral actions are
given by shifts of the original integral actions for
$\mathcal{C}(H_{\pm},J_{\pm})$ as in (102). Moreover, if either of
$\mathfrak{H}_{\pm}$ are $C_{k}$-equivariant, and the polarization class
$\rho\in KO^{1}_{C_{k}}(LM)$ vanishes then the restriction of $\mathfrak{H}$
to the corresponding flow subcategory $\mathcal{C}(H_{\pm},J_{\pm})$ can be
chosen to carry a $C_{k}$ equivariant structure, with the stabilization from
$\mathfrak{H}_{\pm}$ to $\mathfrak{H}|_{\mathcal{C}(H_{\pm},J_{\pm})}$ an
equivariant stabilization.In particular, the shift spaces
$(\tilde{V}^{1}_{+},\tilde{V}^{1}_{-},\tilde{V}^{1})$ associated to
$(\mathfrak{H}_{-},\mathfrak{H}_{+},\mathfrak{H})$ satisfy
$\tilde{V}^{1}=\tilde{V}^{1}_{\pm}\oplus\tilde{V}^{1}_{\pm,s}$, where this
decomposition is a decomposition into $C_{k}$-representations if
$(H_{\pm},J_{\pm})$ is $C_{k}$-equivariant.

We have an inclusion

$\Sigma^{V^{1}_{+,s}}\\{\mathcal{C}(H_{+},J_{+})\\}_{\mathfrak{H}_{+}}\subset\\{\mathcal{C}(H_{s},J_{s})\\}_{\mathfrak{H}}$
(197)

which is an equivariant homeomorphism onto its image, and the quotient by the
inclusion is equivariantly homeomorphic to a shift of the space underlying
$|\mathcal{C}(H_{-},J_{+})|$:

$\\{\mathcal{C}(H_{s},J_{s})\\}_{\mathfrak{H}}/\left(\Sigma^{V^{1}_{+,s}}\\{\mathcal{C}(H_{+},J_{+})\\}_{\mathfrak{H}_{+}}\right)\simeq\Sigma^{\mathbb{R}\oplus
V^{1}_{-,s}}\\{\mathcal{C}(H_{-},J_{-})\\}_{\mathfrak{H}_{-}}\simeq\\{\mathcal{C}(H_{-},J_{-})\\}_{\mathfrak{H}}.$
(198)

###### Proof.

Write $\mathfrak{P}_{\pm}$ for the perturbation data underlying
$\mathfrak{H}_{\pm}$. We choose integralization data via Lemma 30. We then
initially choose perturbation data via Lemma 19, and then subsequently a
trivial stabilization of these perturbation data which gives the associated
topological smoothing a smooth structure via Lemma 20 (see Definition 83).
This virtual smoothing is framed via Lemmas 39 and 46. Choosing
$\mathcal{C}_{1}=\mathcal{C}(H_{-},J_{-})$ and
$\mathcal{C}_{2}=\mathcal{C}(H_{+},J_{+})$ in Proposition 4 and using Lemma 17
to define $\mathbf{E}_{i}$ and $\bar{\mathbf{E}}_{i}$ in this Proposition
defines the embedding and the extension of the embedding.

We now apply Lemma 59 below; the inclusion (197) is simply (208) applied to
this setting. Similarly, the homeomorphism (198) is simply (210) combined with
Proposition 32 (here $M=1$, as per Remark 50; see also (196)).

∎

We can use this geometric construction to produce a map of orthogonal spectra:

######  Proposition 34.

The data $\mathfrak{H}$ as in Proposition 33 determines a map $f$, unique up
to (equivariant) homotopy, of the form

$\Sigma^{\mathbb{R}\oplus
V^{1}_{-},s}\\{\mathcal{C}(H_{-},J_{-})\\}_{\mathfrak{H}_{-}}\xrightarrow{f}\Sigma^{\mathbb{R}\oplus
V^{1}_{+},s}\\{\mathcal{C}(H_{+},J_{+})\\}_{\mathfrak{H}_{+}},$ (199)

which in turns defines the map $\bar{f}=F_{\mathbb{R}\oplus
V^{1}_{\mathfrak{H}}}f$ on orthogonal spectra sitting in the diagram below:

$\begin{gathered}|\mathcal{C}(H_{-},J_{-})|_{\mathfrak{H}_{-}}\leftarrow
F_{\mathbb{R}\oplus V^{1}_{-,s}}\Sigma^{\mathbb{R}\oplus
V^{1}_{-,s}}|\mathcal{C}(H-,J_{-})|_{\mathfrak{H}_{-}}\simeq\\\
F_{\mathbb{R}\oplus V^{1}}\Sigma^{\mathbb{R}\oplus
V^{1}_{-},s}\\{\mathcal{C}(H_{-},J_{-})\\}_{\mathfrak{H}_{-}}\xrightarrow{\bar{f}}F_{\mathbb{R}\oplus
V^{1}}\Sigma^{\mathbb{R}\oplus
V^{1}_{+},s}\\{\mathcal{C}(H_{+},J_{+})\\}_{\mathfrak{H}_{+}}\\\ \simeq
F_{\mathbb{R}\oplus V^{1}_{+,s}}\Sigma^{\mathbb{R}\oplus
V^{1}_{+,s}}|\mathcal{C}(H_{+},J_{+})|_{\mathfrak{H}_{+}}\to|\mathcal{C}(H_{+},J_{+})|_{\mathfrak{H}_{+}}.\end{gathered}$
(200)

Here the top arrow pointing left and the bottom arrow pointing right denote
the canonical weak equivalences (179). In particular, the above diagram
defines a map in $[f]$ in the homotopy category from
$|\mathcal{C}(H_{-},J_{-})|_{\mathfrak{H}_{-}}\to|\mathcal{C}(H_{+},J_{+})|_{\mathfrak{H}_{+}}$.

###### Proof.

The existence of the map $f$ follows from Lemma 60 stated and proven below. To
produce $f$, we apply that Lemma with
$\\{\mathcal{C}\\}=\\{\mathcal{C}(H_{s},J_{s})\\}_{\mathfrak{H}}$, so
$\\{\mathcal{C}_{2}\\}=\\{\mathcal{C}(H_{+},J_{+}\\}_{\mathfrak{H}}$, and thus
$\Sigma^{V_{2}}\\{\mathcal{C}_{2}\\}=\Sigma^{V^{1}_{+},s}\\{\mathcal{C}(H_{+},J_{+})\\}_{\mathfrak{H}_{+}}$.
Meanwhile, one gets that
$\\{\mathcal{C}_{1}\\}=\\{\mathcal{C}(H_{-},J_{-})\\}_{\mathfrak{H}}\simeq\Sigma^{\mathbb{R}\oplus
V^{1}_{-,s}}\\{\mathcal{C}(H_{-},J_{-})\\}_{\mathfrak{H}_{-}}$ by (198). Thus
the map $f:\\{\mathcal{C}_{1}\\}\to\Sigma^{\mathbb{R}\oplus
V_{2}}\\{\mathcal{C}_{2}\\}$ produced by that lemma takes the the form of
(199).

To conclude it suffices to explain the isomorphisms on the top right and the
bottom left of (200). The top right follows from the fact that
$|\mathcal{C}(H_{-},J_{-})|_{\mathfrak{H}_{-}}=F_{V^{1}_{-}}\\{\mathcal{C}(H_{-},J_{-})\\}_{\mathfrak{H}_{-}}$
and the fact that $V^{1}_{-,s}\oplus V^{1}_{-}=V^{1}$; and the one on the
bottom left follows by a similar argument. ∎

Finally, we can use the constructions regarding continuation homotopy data of
Section 7.10 to produce commutative diagrams associated to maps of Floer
homotopy types. Below we summarize the geometric results proven in the paper.

######  Proposition 35.

Suppose that the polarization class $\rho\in KO^{1}_{C_{k}}(LM)$ vanishes, and
we are given $C_{k}$-equivariant homotopy continuation data $(H_{h},J_{h})$ as
well as

  * •

Homotopy definition data $\mathfrak{H}_{a}$, $a\in\\{1,2,3,4\\}$ for the
associated floer data $(H_{a},J_{a})$, and

  * •

homotopy definition data $\mathfrak{H}_{ab}$,
$(a,b)\in\\{(1,2),(1,3),(2,4),(3,4)\\}$ for the associated continuation data
$(H_{ab},J_{ab})$, such that for every such $(a,b)$ the homotopy definition
data $\mathfrak{H}_{ab,a}$ obtained from $\mathfrak{H}_{ab}$ by restriction to
$\mathcal{C}(H_{a},J_{a})$ are stabilizations of $\mathfrak{H}_{a}$, and
similarly the homotopy definition data $\mathfrak{H}_{ab,b}$ obtained from
$\mathfrak{H}_{ab}$ by restriction to $\mathcal{C}(H_{b},J_{b})$ are
stabilizations of $\mathfrak{H}_{b}$, and

  * •

The integralization data underlying the $\mathfrak{H}_{ab}$ satisfy the
conditions of Lemma 37.

Moreover, all of the above are assumed to be $C_{k}$-equivariant. Then there
exists a homotopy definition datum $\mathfrak{H}_{h}$ for
$\mathcal{C}(H_{h},J_{h})$ which such that the homotopy definition data
$\mathfrak{H}_{h,ab}$ for $\mathcal{C}(H_{ab},J_{ab})$ obtained from
$\mathfrak{H}_{h}$ by restriction are all stabilizations of
$\mathfrak{H}_{ab}$, for $(a,b)\in\\{(1,2),(1,3),(2,4),(3,4)\\}$. Write
$(V^{1}_{h},\\{V^{1}_{ab}\\},\\{V^{1}_{a}\\})$ for the shift spaces assoicated
to $(\mathfrak{H}_{h},\\{\mathfrak{H}_{ab}\\},\\{\mathfrak{H}_{a}\\})$. Then
there are $G$-vector spaces
$(\\{V^{1}_{ab,s}\\},\\{V^{1}_{a,\tilde{s}}\\},\\{V^{1}_{a,s_{ab}}\\})$ and
decompositions

$V^{1}_{h}=V^{1}_{ab}\oplus V^{1}_{ab,s}=V^{1}_{a}\oplus
V^{1}_{a,\tilde{s}},V^{1}_{ab}=V^{1}_{a}\oplus V^{1}_{a,s_{ab}}$

(the last decomposition are the decompositions of shift spaces described in
Proposition 33). This forces additional isomorphisms

$V^{1}_{a,\tilde{s}}=V^{1}_{a,s_{ab}}\oplus
V^{1}_{ab,s}=V^{1}_{a,s_{ab^{\prime}}}\oplus V^{1}_{ab^{\prime},s}$
$V^{1}_{b,\tilde{s}}=V^{1}_{b,s_{ab}}\oplus
V^{1}_{ab,s}=V^{1}_{a^{\prime},s_{a^{\prime}b}}\oplus V^{1}_{a^{\prime}b,s}.$

Writing $\mathcal{C}_{i}$ for $\mathcal{C}(H_{i},J_{i})$, and writing
$\mathcal{C}_{abc}$ for the subcategory on the objects of
$\mathcal{C}(H_{i},J_{i})$ as $i=a,b,c$, we have an inclusion

$\Sigma^{V^{1}_{4,\tilde{s}}}\\{\mathcal{C}_{4}\\}_{\mathfrak{H}_{4}}\to\\{\mathcal{C}_{234}\\}_{\mathfrak{H}},\\{\mathcal{C}_{234}\\}_{\mathfrak{H}}/\Sigma^{V^{1}_{4,\tilde{s}}}\\{\mathcal{C}_{4}\\}_{\mathfrak{H}_{4}}\simeq\Sigma^{\mathbb{R}\oplus
V^{1}_{2,\tilde{s}}}\\{\mathcal{C}_{2}\\}_{\mathfrak{H}_{2}}\vee\Sigma^{\mathbb{R}\oplus
V^{1}_{3,\tilde{s}}}\\{\mathcal{C}_{3}\\}_{\mathfrak{H}_{3}}.$ (201)

Moreover there is an inclusion
$\\{\mathcal{C}_{234}\\}_{\mathfrak{H}}\to\\{\mathcal{C}(H_{h},J_{h})\\}_{\mathfrak{H}}$
and composing the previous inclusion with this one gives an inclusion

$\Sigma^{V^{1}_{4,\tilde{s}}}\\{\mathcal{C}_{4}\\}_{\mathfrak{H}_{4}}\to\\{\mathcal{C}(H_{h},J_{h})\\}_{\mathfrak{H}},\\{\mathcal{C}(H_{h},J_{h})\\}_{\mathfrak{H}}/\Sigma^{V^{1}_{4,\tilde{s}}}\\{\mathcal{C}_{4}\\}_{\mathfrak{H}_{4}}\simeq\\{\mathcal{C}_{123}\\}_{\mathfrak{H}};$
(202)

In particular by the previous two claims there is an inclusion

$\Sigma^{\mathbb{R}\oplus
V^{1}_{2,\tilde{s}}}\\{\mathcal{C}_{2}\\}_{\mathfrak{H}_{2}}\vee\Sigma^{\mathbb{R}\oplus
V^{1}_{3,\tilde{s}}}\\{\mathcal{C}_{3}\\}_{\mathfrak{H}_{3}}\xrightarrow{i_{23}}\\{\mathcal{C}_{123}\\}_{\mathfrak{H}},\\{\mathcal{C}_{123}\\}_{\mathfrak{H}}/Imi_{23}\simeq\Sigma^{\mathbb{R}\oplus\mathbb{R}\oplus
V^{1}_{1,\tilde{s}}}\\{\mathcal{C}_{1}\\}_{\mathfrak{H}_{1}}.$ (203)

###### Proof.

We choose the integralization data via Lemma 37, the perturbation data via
Proposition 22, the smoothings via Proposition 23, and the framings via Lemma
39 and Lemma 48.

Choosing and $\mathcal{C}_{1}=\mathcal{C}(H_{1},J_{2})$ and
$\mathcal{C}_{2}=\mathcal{C}(H_{24},J_{24})\cup\mathcal{C}(H_{34},J_{34})$ in
Proposition 4 and using Lemma 17 to define $\mathbf{E}_{i}$ and
$\bar{\mathbf{E}}_{i}$ in this Proposition defines the embedding and the
extension of the embedding.

All homeomorphisms are produced (as above in the proof of Proposition 35) via
Lemma 59, Proposition 32, and Lemma 18, with the restratification embedding
shifts in Lemma 18 giving rise to the $\mathbb{R}$ factors in these
homeomorphisms above. The computation of the restratification embedding shifts
comes coming from the fact that $V^{1}_{can}$ (see (104); the definition is
the same in this case) does not contain coordinates corresponding to the
marked continuation map points $s_{A^{1}_{c}}$ and $s_{A^{2}_{c}}$. ∎

######  Lemma 57.

Given two maps of pointed Hausdorff spaces $g_{1},g_{2}:X\to Y$, when
$\overline{g_{1}^{-1}(*)^{c}}\cap\overline{g_{2}^{-1}(*)^{c}}=\emptyset$,
write $g_{1}\underline{\vee}g_{2}X\to Y_{*}$ for the map defined by
$g_{1}\underline{\vee}g_{2}(x)=g_{i}(x)$ if $x\in g_{i}^{-1}(*)^{c}$ for
$i=1,2$, and otherwise $g_{1}\underline{\vee}g_{2}(x)=*$; by the non-
intersection condition this is a well-defined continuous map. When $X$ and $Y$
are pointed CW complexes, this map represents the class of $g_{1}+g_{2}$ in
the stable homotopy category.

###### Proof.

This occurs because of a canonical homotopy. Indeed, given any two maps
$g_{1},g_{2}:X\to Y$, there one can define the class of the map $g_{1}+g_{2}$
concatenating the maps $\Sigma g_{1}$ and $\Sigma g_{2}$ along the suspension
coordinate, i.e. by composing $\Sigma g_{1}\wedge\Sigma g_{2}$ with the
natural map $\Sigma X\to\Sigma X\wedge\Sigma X$; and then formally
desuspending this map. But this latter map is homotopic to $\Sigma
g_{1}\underline{\vee}\Sigma g_{2}$ whenever the latter makes sense. ∎

######  Lemma 58.

We can choose the maps associated by Proposition 34 to the data (201) to be
$\Sigma^{V^{1}_{24,s}}f_{24}\vee(-\Sigma^{V^{1}_{34,s}}f_{34})$, where the
$(-\Sigma^{V^{1}_{34,s}}f_{34})$ denotes the composition of reflection in
$\mathbb{R}$ shift coordinate of $\Sigma^{R\oplus
V^{1}_{3,\tilde{s}}}\\{\mathcal{C}_{3}\\}_{\mathfrak{H}_{3}}$ with
$(\Sigma^{V^{1}_{34,s}}f_{34})$. Moreover,
$\\{\mathcal{C}_{123}\\}_{\mathfrak{H}}$ is homotopy equivalent to the cone on
$\Sigma^{V^{1}_{12,s}}f_{12}\underline{\vee}\Sigma^{1}_{13,s}f_{13}$, and the
map
$\\{\mathcal{C}_{123}\\}_{\mathfrak{H}}\to\Sigma^{V^{1}_{4,\tilde{s}}\\{\mathcal{C}_{4}\\}_{\mathfrak{H}_{4}}}$
associated by Proposition 34 to (202) can be taken to agree with
$\Sigma^{V^{1}_{24,s}}f_{24}\vee\Sigma^{V^{1}_{34,s}}f_{34}$ on the images of
$\Sigma^{\mathbb{R}\oplus
V^{1}_{2,\tilde{s}}}\\{\mathcal{C}_{2}\\}_{\mathfrak{H}_{2}}\vee\Sigma^{\mathbb{R}\oplus
V^{1}_{3,\tilde{s}}}\\{\mathcal{C}_{3}\\}_{\mathfrak{H}_{3}}$ in
$\\{\mathcal{C}_{123}\\}_{\mathfrak{H}}$.

###### Proof.

The first claim follows from the following reasoning. Given a pushout diagram

${X}$${Y_{1}}$${Y_{2}}$${Z}$$\scriptstyle{a}$$\scriptstyle{b}$

with the map to $X\to Z$ called $C$ and both $a$ and $b$ being cofibrations,
if we have chosen homotopy equivalences $h_{1}:Y_{1}/X\to Cone(a)$ and
$h_{2}:Y_{2}/X\to Cone(b)$ then the wedge of these defines an equivalence
$h_{3}:Y_{1}/X\wedge Y_{2}/X\to Cone(c)$. Moreover, using this definition, the
map give by the compositio of $h_{3}$ with the Puppe map is manifestly the
wedge products associated to the compositions of the maps associated to
$h_{i}$ and the corresponding Puppe maps for $i=1,2$.

The next claim goes as follow. If we consider
$\\{\mathcal{C}_{123}\\}_{\mathfrak{H}}/\Sigma^{\mathbb{R}\oplus
V^{1}_{2,\tilde{s}}}\\{\mathcal{C}_{2}\\}_{\mathfrak{H_{2}}}$ then this is
homeomorphic to $\Sigma^{\mathbb{R}\oplus
V^{1}_{13,s}}\\{\mathcal{C}_{13}\\}_{\mathfrak{H}_{13}}$; by the fact that
every triple of maps in a long exact puppe sequence is homotopy equivalent to
$X\xrightarrow{f}Y\to C(f)$, we have that $\Sigma^{\mathbb{R}\oplus
V^{1}_{13,s}}\\{\mathcal{C}_{13}\\}_{\mathfrak{H}_{13}}$ is homotopy
equivalent to $\Sigma^{V^{1}_{13,s}}f_{13}$. This statement together with the
corresponding statemtn with $2$ and $3$ exchanged implies the claim.

The last claim goes as follows: if we extend the earlier diagram as

${X}$${Y_{1}}$${Y_{2}}$${Z}$${W}$$\scriptstyle{a}$$\scriptstyle{b}$

where the map $Z\to W$ is a cofibration, and we call the map $X\to W$ as $d$,
then there is an inclusion $Cone(c)\to Cone(d)$, and we can choose the
equivalence $W/X\to Cone(d)$ to restrict to the previously chosen equivalence
$Z/X=Y_{1}/X\wedge Y_{2}/X\to Cone(c)$. ∎

######  Proposition 36.

Using the notation of Proposition 35, let $f_{ab}:\Sigma^{\mathbb{R}\oplus
V^{1}_{a,s_{ab}}}\\{\mathcal{C}_{a}\\}_{\mathfrak{H}_{a}}\to\Sigma^{\mathbb{R}\oplus
V^{1}_{b,s_{ab}}}\\{\mathcal{C}_{b}\\}_{\mathfrak{H}_{b}}$ be the maps
associated by Proposition 34 to the homotopy definition data
$\mathfrak{H}_{ab}$. Then we have a homotopy

$\Sigma^{V^{1}_{24},s}f_{24}\Sigma^{V^{1}_{12},s}f_{12}\underline{\vee}(-\Sigma^{V^{1}_{34},s}f_{34})\Sigma^{V^{1}_{13},s}f_{13}.$
(204)

With notation as in Proposition 34, we have equality in the stable homotopy
category

$[f_{24}][f_{12}]=-[f_{34}][f_{13}].$ (205)

###### Proof.

First, we will explain why the domains and codomains of the maps in (204)
match appropriately so that this equation make sense. Indeed, the map
$\Sigma^{V^{1}_{ab},s}f_{ab}$ defines a map

$\Sigma^{\mathbb{R}\oplus
V^{1}_{a,\hat{s}}}\\{C_{a}\\}_{\mathfrak{H}_{a}}\simeq\Sigma^{V^{1}_{ab,s}}\Sigma^{\mathbb{R}\oplus
V^{1}_{a,s_{ab}}}\\{C_{a}\\}_{\mathfrak{H}_{a}}\xrightarrow{\Sigma^{V^{1}_{ab},s}f_{ab}}\Sigma^{V^{1}_{ab,s}}\Sigma^{\mathbb{R}\oplus
V^{1}_{b,s_{ab}}}\\{C_{b}\\}_{\mathfrak{H}_{b}}\simeq\Sigma^{\mathbb{R}\oplus
V^{1}_{b,\hat{s}}}\\{C_{b}\\}_{\mathfrak{H}_{b}},$ (206)

using the identities $V^{1}_{a,s_{ab}}\oplus V^{1}_{ab,s}=V^{1}_{a,\hat{s}}$;
we use these isomorphisms to make sense of the compositions in (204).

Then, we will explain how (204) implies (205). Indeed, the identity (205)
holds if the identity

$\Delta[f_{24}]\Delta[f_{12}]=\Delta[f_{34}]\Delta[f_{13}]$ (207)

holds, where $\Delta$ denotes the functor homotopy category $X\to
F_{\mathbb{R}\oplus V^{1}_{h}}\Sigma^{\mathbb{R}\oplus V^{1}_{h}}X$ on the
stable homotopy category. This latter identity holds in turn of we an find
representing maps for each of the $\Delta f_{ab}$ in the category of
orthogonal spectra and a homotopy between the composed maps. Now, we can apply
the functor $F_{\mathbb{R}\oplus V^{1}_{h}}$ to the maps in (206) and apply
the isomorphism

$F_{\mathbb{R}\oplus V^{1}_{h}}\Sigma^{\mathbb{R}\oplus
V^{1}_{a,\hat{s}}}\\{C_{a}\\}_{\mathfrak{H}_{a}}\simeq F_{\mathbb{R}\oplus
V^{1}_{a,\hat{s}}}\Sigma^{\mathbb{R}\oplus
V^{1}_{a,\hat{s}}}|\mathcal{C}(H_{a},J_{a})|_{\mathfrak{H}_{a}}$

to interpret these maps potentially being representing maps for the
$\Delta[f_{ab}]$; given this claim, the homotopy (204) establishes (207) and
thus (205). To see this latter claim about representing maps, we simply note
that applying $F_{V^{1}_{ab,s}}\Sigma^{V^{1}_{ab,s}}$ to the map
$\bar{f}_{ab}$ of (200) and applying isomorphisms in (206) to the maps inside
the desuspension functors, and the isomorphisms $V^{1}_{ab,s}\oplus
V^{1}_{ab}=V^{1}_{h}$ gives exactly $F_{\mathbb{R}\oplus V^{1}_{h}}$ applied
to the maps in (206). This proves the claim about representing maps.

It remains to justify (204). But this follows from Lemma 58. Indeed, given
maps $X\xrightarrow{f}Y\xrightarrow{g}Z$, an extension of $g$ to a map
$Cone(f)\to Z$ specifies a homotopy of $g\circ f\sim*$. This lemma states that
we are precisely in this position with
$f=\Sigma^{V^{1}_{12,s}}f_{12}\underline{\vee}\Sigma^{V^{1}_{13,s}}f_{13}$ and
$g=\Sigma^{V^{1}_{24,s}}f_{24}\wedge\Sigma^{V^{1}_{34,s}}f_{34}$. So we have a
homotopy
$\Sigma^{V^{1}_{24},s}f_{24}\Sigma^{V^{1}_{12},s}f_{12}\underline{\vee}(-\Sigma^{V^{1}_{34},s}f_{34})\Sigma^{V^{1}_{13},s}f_{13}\sim*$,
as desired. ∎

###  10.2 Comparison with Floer Homology

######  Lemma 59.

Let $\mathcal{C}$ be a virtually smooth semi-freely framed proper flow
category together with an embedding $\mathbf{E}$ and its extension
$\bar{\mathbf{E}}$. If $\mathcal{C}_{2}\subset\mathcal{C}$ is a downwards-
closed subposet of $\mathcal{C}$ that is sent to itself by the $G$-action,
then, letting $\mathcal{C}_{2}$ also denote the full virtually smooth flow
subcategory on the objects of $\mathcal{C}_{2}$ equipped with the restrictions
of $\mathbf{E}$ and $\bar{\mathbf{E}}$, we have a q-cofibration

$\Sigma^{V_{2}}\\{\mathcal{C}_{2}\\}\subset\\{\mathcal{C}\\}$ (208)

where $V_{2}$ is the perpendicular to the image of the shift space associated
to $\mathcal{C}_{2}$ in the shift space associated to $\mathcal{C}$.
Similarly, we have a level $q$-cofibration [73, Theorem 2.4]

$F_{V_{2}}\Sigma^{V_{2}}|\mathcal{C}_{2}|\subset|\mathcal{C}|.$ (209)

Moreover, we have that there is an (equivariant) homeomorphism

$\\{\mathcal{C}\\}/\Sigma^{V_{2}}\\{\mathcal{C}_{2}\\}\simeq\\{\mathcal{C}_{1}\\}$
(210)

where $\mathcal{C}_{1}$ is the full flow subcategory of $\mathcal{C}$ on the
objects $Ob(\mathcal{C}_{2})\setminus Ob(\mathcal{C}_{2})$, and is also
equipped with the restricted embedded framings, embeddings, and extensions of
embeddings.

Thus, in particular, given a grading $\alpha:\mathcal{C}\to\mathbb{Z}$ on the
poset $\mathcal{C}$, i.e. an integer-value function on the objects of
$\mathcal{C}$ such that $\alpha(x)\geq\alpha(y)$ if $x>y$ and
$\alpha(gx)=\alpha(x)$ for all $x\in\mathcal{C}$ and all $g\in G$, then there
is a filtration by sub-spectra, i.e. there is are diagrams of
$G-CW$-subspectra

$\\{\mathcal{C}\\}_{\leq k-1}\subset\\{\mathcal{C}\\}_{\leq
k}\subset\\{\mathcal{C}\\}_{\leq k+1}\subset\ldots,$ (211)

and a diagram of orthogonal $G$-spectra

$\cdots\to|\mathcal{C}|_{\leq k-1}\to|\mathcal{C}|_{\leq
k}\to|\mathcal{C}|_{\leq k+1}\to\cdots$ (212)

where each map is a level $q$-cofibration, and

$\\{\mathcal{C}\\}=\cup_{k}\\{\mathcal{C}\\}_{\leq
k},|\mathcal{C}|=\operatorname{colim}_{k}|\mathcal{C}|_{\leq k}.$

###### Proof.

The inclusion (208) follows from the construction, since, writing
$V^{1}_{\mathcal{C}}$ for the shift space associated to $\mathcal{C}$, we have
that $V^{1}_{\mathcal{C}_{2}}(x)^{\perp}\oplus
V_{2}=V^{1}_{\mathcal{C}}(x)^{\perp}$; thus the attaching maps (36) for the
the cells of $\\{\mathcal{C}\\}$ coming from objects of $\mathcal{C}_{2}$ are
therefore just suspensions of the corresponding attaching maps in
$\\{\mathcal{C}_{2}\\}$. The inclusion (209) then follows immediately from the
fact that $V_{2}\oplus V^{1}_{\mathcal{C}_{2}}=V^{1}_{\mathcal{C}}$. The claim
(210) follows from the fact that when defining $\\{\mathcal{C}_{1}\\}$ one
collapses all parts of the boundaries of the cells associated to objects of
$\mathcal{C}_{1}$ that would have mapped to the cells associated to objects of
$\mathcal{C}_{2}$ to basepoint instead (see Condition F of Definition 32 for
extensions of embeddings). The remaining claims follow immediately from the
definitions. ∎

######  Lemma 60.

In the setting of Lemma 59, there is a map defined up to contractible choice

$f:\\{\mathcal{C}_{1}\\}\to\Sigma^{\mathbb{R}\oplus
V_{2}}\\{\mathcal{C}_{2}\\}$

as well as

$\bar{f}=F_{V}f:|\mathcal{C}_{1}|\to F_{V_{2}}\Sigma^{\mathbb{R}\oplus
V_{2}}|\mathcal{C}_{2}|.$

###### Proof.

This follows from (209), (210) by composing the last map in the Puppe sequence

$\Sigma^{V_{2}}\\{\mathcal{C}_{2}\\}\to\\{\mathcal{C}\\}\to
Cone(\Sigma^{V_{2}}\\{\mathcal{C}_{2}\\}\to\\{\mathcal{C}\\})\to\Sigma^{\mathbb{R}V_{2}}\\{\mathcal{C}_{2}\\}$

with an equivariant homotopy inverse of the equivalence (210). ∎

######  Lemma 61.

Let $\mathcal{C}_{2}$ be a downwards-closed flow-subcategory of a virtually
smooth flow category $\mathcal{C}$ with an $E^{s}_{2}$ framing and an
extension of an embedding; equip $\mathcal{C}_{2}$ with the restrictions of
all of this data. Let $\mathcal{C}_{1}$ be the full subcategory of
$\mathcal{C}$ on $Ob(\mathcal{C})\setminus Ob(\mathcal{C}_{2})$, and equip it
with the restrictions of all the data associated to $\mathcal{C}$. Suppose
that we are given gradings $\alpha_{1}$ and $\alpha_{2}$ of the posets
$\mathcal{C}_{1}$ and $\mathcal{C}_{2}$, and $\mathcal{C}$ _respects_ these
gradings in the sense that if $x_{i}\in\mathcal{C}_{i}$ for $i=1,2$ and
$x_{i}>x_{2}$ in $\mathcal{C}$ then $\alpha_{1}(x_{1})\geq\alpha_{2}(x_{2})$.
Then the map

$f:\\{\mathcal{C}_{1}\\}\to\Sigma^{\mathbb{R}\oplus
V_{2}}\\{\mathcal{C}_{2}\\}$

defined by Lemma 60 where $V_{2}$ as in that lemma, respects the filtrations
on the domain and codomain induced by suspending the filtrations of Equation
211, Lemma 59. Moreover, the space of choices of such $f$ which respect these
filtrations is contractible.

###### Proof.

Define $\alpha:\mathcal{C}\to\mathbb{Z}$ by $\alpha(x)=\alpha_{1}(x)$ if
$x\in\mathcal{C}_{2}$, and otherwise $\alpha(x)=\alpha_{2}(x)$ if
$x\in\mathcal{C}_{2}$. Then we have a filtration of $\\{\mathcal{C}\\}$ (211)
associated to $\alpha$, and the restriction of this filtration to
$\Sigma^{V_{2}}\\{\mathcal{C}_{2}\\}$ is the suspension of the filtration by
$\alpha_{2}$ of $\\{\mathcal{C}_{2}\\}$. This defines a filtration by
$\mathbb{Z}$ of
$Z=Cone(\Sigma^{V_{2}}\\{\mathcal{C}_{2}\\}\to\\{\mathcal{C}\\})$ and the
projection (210) sends this filtration onto the filtration of
$\\{\mathcal{C}_{1}\\}$ associated to $\alpha_{1}$. Thus the inverse map
associated to (210) can be resepects these filtrations as well, and it is an
inverse to an explicit homeomorphism so there are no choices to make. One then
sees that the Puppe map $Z\to\Sigma^{\mathbb{R}\oplus
V_{2}}\\{\mathcal{C}_{2}\\}$ respects filtrations as well (with the latter
filtered by the suspension of the filtration associated to $\alpha_{2}$). The
space of choices of a Puppe map is contractible, thus proving the lemma, ∎

Recall that the homology groups of a cofibrant spectrum $X$ as
$H_{*}(X,\mathbb{Z})=\pi_{*}(X\wedge H\mathbb{Z})$, where $H\mathbb{Z}$ is the
Eilenberg-MacLane spectrum. Moreover, if $X$ is a colimit over $k$ of
q-cofibrant spectra $X_{\leq k}$ along q-cofibrations, then there is a
spectral sequence

$E^{1}_{s,t}=H_{s+t}(Cone(X_{\leq s-1}\to X_{\leq s}),\mathbb{Z})\Rightarrow
H_{*}(X,\mathbb{Z}),$ (213)

which converges if the colimit is finite.

An elementary computation shows that, using the notation of Lemma 59 if
$X=|\mathcal{C}|$, $X_{\leq k}=|\mathcal{C}|_{\leq k}$ as in (212), and
$\alpha$ is _stict_ (i.e. if $x>y$ then $\alpha(x)>\alpha(y)$), then there is
a level equivalence

$Cone(X_{\leq s-1}\to X_{\leq
s})\simeq\bigvee_{x\in\mathcal{C}:\alpha(x)=s}\SS^{\mu}(x)$ (214)

where $\mu(x)$ is the grading on $\mathcal{C}$. Indeed, we have that $X_{\leq
s}=F_{V^{1}_{max}}\\{\mathcal{C}\\}_{\leq s}$, and cone on the inclusion

$\\{\mathcal{C}\\}_{\leq s-1}\to\\{\mathcal{C}\\}_{\leq s}$

is

$\bigvee_{x\in\mathcal{C}:\alpha(x)=s}\SS^{\mathbb{R}^{\bar{A}(x)+1}\oplus
V^{0}(x)\oplus V^{1}(x)^{\perp}};$

by the grading condition (27) we have that

$\mu(x)=\dim V^{0}(x)-\dim V^{1}(x)+(\bar{A}(x)+1)=\bar{A}(x)+1+\dim
V^{0}(x)+\dim V^{1}(x)^{\perp}-\dim V^{1}_{max}$

which proves the equivalence (214).

Now, given an orientation of $V^{0}(x)$, $V^{1}(x)$, and $V^{1}_{max}$, we get
an isomorphism of $\tilde{H}_{*}(\SS^{\mathbb{R}^{\bar{A}(x)+1}\oplus
V^{0}(x)\oplus V^{1}(x)^{\perp}})$ (where $\tilde{H}$ denotes reduced
homology) with a copy of $\mathbb{Z}$ in the relevant degree.

Thus, after choosing _orientation data_ $\mathfrak{o}$, which consists of an
orientation of $V^{1}_{max}$ as well as orientations of $V^{0}(x)$ and
$V^{1}(x)$ for each $x$, the $E^{1}$ pages of the associated spectral sequence
takes the form

$E^{1}_{s,t}=\oplus_{x\in\mathcal{C}:\alpha(x)=s,\mu(x)=s+t}\mathbb{Z}.$ (215)

If we are in the situation of Lemma 61, the map $f$ induces a map
$\bar{f}=F_{V^{1}_{max}}f:|\mathcal{C}_{1}|\to
F_{V_{2}}\Sigma^{\mathbb{R}\oplus V_{2}}|\mathcal{C}_{2}|$, which in turn
induces a map of spectral sequences

$\bar{f}^{1}_{s,t}:(E^{1}_{\mathcal{C}_{1}})_{s,t}\to(E^{1}_{\mathcal{C}_{2}})_{s,t+1}.$
(216)

(We will find that this shift by $1$ in the grading cancels out when applying
this computation to flow categories associated to continuation maps.)

A choice of orientation data in turn gives orientations of $V^{i}(x,y)$ for
$x>y$ (by requiring that the map $V^{i}(x,y)\oplus V^{i}(y)\to V^{i}(x)$ is
orientation preserving). In particular, if $\mu(x)=\mu(y)+1$, then (27)
implies that the map $e_{xy}$ associated to the embedding $\mathbf{E}$ of
$\mathcal{C}$ is a map between equidimensional manifolds. The definition of
the attaching map $e_{x}$ then shows:

######  Lemma 62.

Suppose that for all $s$ and all $x\in\mathcal{C}_{1}$ and
$y\in\mathcal{C}_{2}$ with $\alpha_{1}(x)=\alpha_{2}(y)=s$, the map $e_{xy}$
associated to the embedding of $\mathcal{C}$ has isolated zeros in its
interior. Then the map $\bar{f}^{1}_{s,t}$ sends the fundamental class of
$\SS^{\mu(x)}$ to a sum of fundamental classes of $\SS^{\mu(y)+1}$ for
$y\in\mathcal{C}_{2}$ such that $\mu(y)=\mu(x)-1$ and
$\alpha_{1}(x)=\alpha_{1}(y)$, with the coefficient of the fundamental class
of $\SS^{\mu(y)+1}$ being exactly the signed number of zeros of $e_{x,y}$ on
the interior, with the signs defined via the standard orientation of
$\mathbb{R}^{\bar{A}(x,y)}$ and the orientations of $V^{i}(x,y)$ assigned
above. $\blacksquare$

Now, let us suppose that we are in a geometric setting, i.e.
$\mathcal{C}=\mathcal{C}(H,J)$ for some convex nondegenerate Floer data
$(H,J)$, or $\mathcal{C}=\mathcal{C}(H_{s},J_{s})$ for convex nondegenerate
continuation data $(H_{s},J_{s})$. Then, the orientation data $\mathfrak{o}$
actually trivializes the _orientation lines_ associated to the objects of
$\mathcal{C}$ ([4, 89, 42]); in other words, it defines orientation data in
the sense of Section 6.1), which lets us associate signs
$s_{u},s_{u^{\prime}}\in\pm$ to regular Floer trajectories
$u\in\mathcal{C}(H,J)(x,y)$ or regular continuation map trajectories
$u^{\prime}\in\mathcal{C}(H_{s},J_{s})(x^{\prime},y^{\prime})$ via usual
coherent orientations procedure [42] outline in Section 6.1. Indeed, the
orientation lines $o(x)$ of that section are canonically isomorphic to the
determinant lines of the operators $D_{x}$ (157); since
$D_{x}^{\mathfrak{P}^{\prime}}$ (166) is a stabilization of $D_{x}$ by
$V^{1}(x)$, one has that

$o(x)\simeq\det D_{x}\simeq\det D_{x}^{\mathfrak{P}}\otimes\det V^{1}(x)=\det
V^{0}(x)\otimes\det V^{1}(x).$

We are now able to explain how to compare usual computations in Floer homology
with the maps on homology of Floer homotopy defined above:

######  Theorem 15.

Suppose that the admissible Floer data $(H,J)$ are regular. In the setting of
Lemma 59, let us choose $\alpha$ to be the grading $\mu$ by the Conley-Zehnder
index $\tilde{CZ}$ (see Section 6.1). Then we can choose homotopy definition
data $\mathfrak{H}$ and orientation data $\mathfrak{o}$ for $\mathcal{C}(H,J)$
such that the differential on the first page of the first pace of the spectral
sequence (215) agrees with the Floer differential.

Similarly, suppose that we have a continuation map category
$\mathcal{C}(H_{s},J_{s})$ such that for some strict filtrations $\alpha_{i}$
of $\mathcal{C}(H_{i},J_{i})$ (with $(H_{-},J_{-})=(H_{1},J_{1})$ and
$(H_{+},J_{+})=(H_{2},J_{2})$) with $\mathcal{C}(H_{s},J_{s})$ respecting this
strict filtration, we have that for every $x\in\mathcal{C}(H_{1},J_{1})$ and
$y\in\mathcal{C}(H_{2},J_{2})$ with $\alpha_{1}(x)=\alpha_{2}(y)$ and
$\mu^{CZ}(x)=\mu^{CZ}(y)$, every unbroken continuation map trajectory
$u^{\prime}\in\mathcal{C}(H_{s},J_{s})(x,y)$ is regular. Then the coefficient
of the fundamental class of $\SS^{\mu(y)+1}$ in $\bar{f}^{1}([\SS^{\mu(x)}])$
is exactly $\sum_{u^{\prime}}s_{u^{\prime}}$ where the sum runs over these
trajectories (up to a global sign).

###### Proof.

We will prove the statement about the Floer differential; the proof of the
second statement is similar. Write
$\mathcal{C}^{\prime}(x,y)=(T(x,y),V(x,y),\sigma(x,y))$. There is a map
$\phi:\mathcal{C}(H,J)(x,y)\to T(x,y)$ which is a homeomorphism onto the zero-
set of $\sigma(x,y)$, and thus a map $\mathcal{C}(x,y)\to{\overline{M}}(x,y)$,
$u\mapsto a(\phi(u))$. In particular, regular index $1$ Floer trajectories $u$
are mapped by $\phi$ to isolated zeros of $\sigma(x,y)$, and to the top open
stratum of ${\overline{M}}(x,y)$. We will show the claim in the theorem under
the condition that the perturbation map $\lambda_{V^{1}(x,y)}$ (85) is
identically zero in an open neighborhood of $a(\phi(u))$, for every (regular)
index $1$ Floer trajectory $u$; this can be achieved in the proofs of
Propositions 7 (and 19 in the continuation map case). The choice of
orientation data means that there is an integer assigned to the zero of
$e_{x,y}$ corresponding to $u$; this is the same integer as one assigned to
the zero $\phi(u)$ of $\sigma(x,y)$, when we give $V(x,y)$ and $TT(x,y)$ the
orientations coming from the framings of these spaces and the orientation
data. We will first argue that this integer $s^{\prime}_{u}$ is $\pm 1$; we
will then argue that in fact $s^{\prime}_{u}$ agrees with $s_{u}$, the sign
assigned to $u$ by Floer homology.

Let us argue the first point. By Equation 155 and the condition on the
triviality of $\lambda_{V^{1}(x,y)}$ near $u$ we have an isomorphism

$\begin{gathered}TT(x,y)_{\phi(u)}=T(x,y)_{a(\phi(u))}\oplus
V^{1}(x,y)_{u}\oplus\ker D_{u}=\mathbb{R}^{\mathfrak{A}^{\prime}(x,y)}\oplus
V^{1}(x,y)\oplus\mathbb{R}\end{gathered}$ (217)

where $D_{u}$ is the linearization of the _unperturbed_ Floer equation. Now
one can produce a convenient chart around $\phi(u)=(\tilde{u},a(\phi(u)),0)$
with domain $V^{1}(x,y)\oplus\mathbb{R}^{\bar{A}(x,y)+1}$ by: translating the
map $\tilde{u}$ in the $s$ direction, moving the $\bar{A}(x,y)+1$ marked
points on its domain (up to global simultaneous translation), or deforming
$0\in V^{1}(x,y)$ to a nonzero element. In this chart, using the definition of
$\sigma(x,y)$ together with the description

$(V(x,y)^{1}_{T})_{u}=V^{1}(x,y)\oplus V^{1}_{can}(x,y)$ (218)

the $V^{1}(x,y)$ component of $\sigma(x,y)$ is simply the composition of the
projection to $V^{1}(x,y)$ with the identity map $V^{1}(x,y)\to V^{1}(x,y)$,
and the second component of $\sigma(x,y)$ is the projection to the
$\mathbb{R}^{\bar{A}(x,y)+1}$ component of the chart together with a map
$\mathbb{R}^{\bar{A}(x,y)+1}\to\mathbb{R}^{\bar{A}(x,y)+1}$ determined by
$\delta A$. This latter map is always manifestly orientation-preserving; thus,
we have that $s^{\prime}_{u}=\pm 1$, with the sign given by the difference in
orientations of $V^{1}(x,y)$ coming from the framing of the obstruction
bundles, and of $V^{1}(x,y)$ induced from the inclusion
$V^{1}(x,y)\oplus\mathbb{R}\simeq\ker D_{u}^{\mathfrak{P}^{\prime}}$.

We will now compare this sign with $s_{u}$. To do that we must consider how
the framings we produce orient $T(x,y)_{u}$ and $V(x,y)_{u}$. The latter are
produced from the orientation data and the construction of Section 8.6, for
which $V^{1}_{can}(x,y)$ and the component $\mathbb{R}^{\bar{A}(x,y)}$ (this
is all but the last two terms of (217)) are framed in a canonical way while
the terms $\ker D_{u}^{\mathfrak{P}}=V^{1}(x,y)_{u}\oplus\ker D_{u}$ and
$V^{1}(x,y)_{u}$ get induced orientations (from the framing) via the procedure
described in that section, starting from the orientations of of the $\ker
D_{x}^{\mathfrak{P}}$ and $V^{1}(x)$ – i.e., the orientation data. On the
other hand, in Floer homology, one has that the orientation line

$o(x)\simeq\det\ker D_{x}^{\mathfrak{P}}\otimes\det V^{1}(x)$

receives an orientation from the orientation data, and the sign $s_{xy}$ is
the comparison sign from comparing the orientation of $o(x)$ produced from
orientation data from the orientation of $o(x)$ produced from the canonical
orientation of the determinant line $\ker D_{u}$ of $u$ and the orientation of
$o(y)$ via the linear gluing map $\ker D_{u}\otimes o(y)\to o(x)$. But we have
that the framings $a_{xy}$ of Proposition LABEL:prop:framings-exist are chosen
precisely such they induce the orientations on $\ker
D_{u}^{\mathfrak{P}^{\prime}}$ induced via the stabilized gluing maps
$D^{\mathfrak{P}^{\prime}}_{V^{1}(x),V^{1}(y)}$ of (166); this shows that the
Floer comparison sign and the sign $s^{\prime}_{u}$ agree. The proof for
continuation maps maps is similar. ∎

######  Proposition 37.

Let $(H,J)$ be admissible regular Floer data. Then there exists homotopy
definition data $\mathfrak{H}$ such that if we define the Floer homotopy type
$|\mathcal{C}(H,J)|$ using $\mathfrak{H}$ we have an isomorphism

$\pi_{*}(|\mathcal{C}(H,J)|\wedge H\mathbb{Z})\simeq HF_{*}(H,J).$

In fact, recall that the Eilenberg-MacLane spectrum $H\mathbb{Z}$ has a single
nonzero homotopy group, namely $\pi_{0}(H\mathbb{Z})=\mathbb{Z}$; thus given
the chain complex $CF_{*}(H,\mathbb{Z})$ and choosing orientation data
$\mathfrak{o}$ so that we get isomorphisms
$CF_{k}(H,\mathbb{Z})=\mathbb{Z}^{a_{k}}$, there is an associated
$H\mathbb{Z}$-module spectrum which we denote $|CF_{*}(H,J)|$ which is built
by attaching wedges of suspensions of $H\mathbb{Z}$ via the maps given by the
coefficients of the differential, and one has a weak equivalence

$|\mathcal{C}(H,J)|\wedge H\mathbb{Z}\simeq|CF_{*}(H,J)|.$

###### Proof.

Choosing $\alpha$ to be the Conley-Zehnder index, this follows by induction
from (214) and (15). ∎

###  10.3 Choosing data compatibly with the cyclotomic action

We will show how to imitate the colimit (190) in the setting of Floer homotopy
types in the setting where none of the Floer data or continuation data are
required to be regular. First, fix a _complete $S^{1}$-universe_
$\mathfrak{U}$; by restriction of representations this universe is also a
complete $C_{k}$-universe for all $k$. Assume that the $S^{1}$-equivariant
polarization class $\rho\in KO^{1}_{S^{1}}(LM)$ vanishes, and inductively
choose a sequence of cyclotomically compatible trivializations of the
polarization map for all $(H^{\\#p^{k}},J^{\\#p^{k}})$ (see Lemma 39) once and
for all.

By repeatedly applying Proposition 33 , we produce a sequence of homotopy
definition data $\mathfrak{H}_{k}^{cont}$ for
$\mathcal{C}(H_{s}^{\\#p^{k}},J_{s}^{\\#p^{k}})$ which have the following
property. Write

$\mathfrak{H}^{+}_{k}=\mathfrak{H}_{k}^{cont}|_{\mathcal{C}(H^{\\#p^{k}},J^{\\#p^{k}})},$
$\mathfrak{H}^{-}_{k}=\mathfrak{H}_{k}^{cont}|_{\mathcal{C}(H^{\\#p^{k+1}},J^{\\#p^{k+1}})},$
$\mathfrak{H}_{k}=\mathfrak{H}^{-}_{k-1}\text{ for }k\geq
1,\mathfrak{H}_{0}=\mathfrak{H}^{+}_{0}.$

Proposition 33 allows us to require that for $k\geq 1$, $\mathfrak{H}_{k}^{+}$
is a $C_{k}$-equivariant stabilization of $\mathfrak{H}_{k-1}^{-}$ (where
$\mathfrak{H}_{k-1}$ was chosen such that $\mathfrak{H}_{k-1}^{-}$ has a
$C_{k}$-equivariant structure). Write

$\tilde{V}(\mathfrak{H}^{cont}_{k})=\tilde{V}(\mathfrak{H}_{k}^{-})=\tilde{V}(\mathfrak{H}_{k-1}^{-})\oplus\tilde{V}(\mathfrak{H}_{k-1}^{-},s)$
(219)

for the corresponding $C_{k}$-equivariant decomposition of shift spaces.

Using (197) and (198), the map associated to a Puppe sequence defines, for
every $k\geq 0$, a $C_{p^{k}}$-equivariant map

$g_{1}^{k}:\Sigma^{\mathbb{R}}\\{\mathcal{C}(H^{\\#p^{k+1}},J^{\\#p^{k+1}})\\}_{\mathfrak{H}^{-}_{k}}\to\Sigma\\{\mathcal{C}(H^{\\#p^{k}},J^{\\#p^{k}})\\}_{\mathfrak{H}^{+}_{k}}\simeq\Sigma^{\mathbb{R}\oplus\tilde{V}(\mathfrak{H}_{k-1}^{-},s)}\\{\mathcal{C}(H^{\\#p^{k}},J^{\\#p^{k}})\\}_{\mathfrak{H}_{k-1}}.$
(220)

where the second equivalence only makes sense for $k\geq 1$.

Using (179), the decompositions (219) define associated maps

$g_{2}^{k}:F_{\mathbb{R}\oplus\tilde{V}(\mathfrak{H}^{-}_{k})}\Sigma^{\mathbb{R}\oplus\tilde{V}(\mathfrak{H}_{k-1}^{-},s)}\\{\mathcal{C}(H^{\\#p^{k}},J^{\\#p^{k}})\\}_{\mathfrak{H}^{-}_{k-1}}.\to
F_{\mathbb{R}\oplus\tilde{V}(\mathfrak{H}_{k}-1)}\Sigma^{\mathbb{R}}\\{\mathcal{C}(H^{\\#p^{k}},J^{\\#p^{k}})\\}_{\mathfrak{H}^{-}_{k-1}}.$
(221)

Define

$|\mathcal{C}^{\prime}(H,J)|^{\prime}_{\mathfrak{H}}=F_{\mathbb{R}\oplus\tilde{V}^{1}_{\mathfrak{H}}}\Sigma^{\mathbb{R}}\\{\mathcal{C}^{\prime}(H,J)\\}$

The composition of (220) and (221) is a map

$g^{k}=g_{2}^{k}\circ(F_{\mathbb{R}\oplus\tilde{V}(\mathfrak{H}^{-}_{k})}g_{1}^{k}):|\mathcal{C}(H^{\\#p^{k}},J^{\\#p^{k}})|^{\prime}_{\mathfrak{H}_{k}}\to|\mathcal{C}(H^{\\#p^{k-1}},J^{\\#p^{k-1}})|^{\prime}_{\mathfrak{H}_{k}};$
(222)

this map makes sense even for $k=1$ by simply avoiding (221).

Dualizing, we get $C_{k-1}$-equivariant maps

$f_{k}:F(|\mathcal{C}(H^{\\#p^{k-1}},J^{\\#p^{k-1}})|^{\prime}_{\mathfrak{H}_{k}},\SS)\to
F(|\mathcal{C}(H^{\\#p^{k}},J^{\\#p^{k}})|^{\prime}_{\mathfrak{H}_{k}},\SS).$
(223)

for all $k\geq 1$.

Now, we have one more geometric proposition:

######  Proposition 38.

The homotopy continuation data $\mathfrak{H}_{k}$ can be chosen such that

$\\{\mathcal{C}(H_{s}^{\\#p^{k}},J_{s}^{\\#p^{k}})\\}^{C_{p}}\simeq\Sigma^{\tilde{V}(\mathfrak{H}^{-}_{k-1},s)^{C_{p}}}\\{\mathcal{C}(H_{s}^{\\#p^{k-1}},J_{s}^{\\#p^{k-1}})\\}.$

Using this proposition, we can choose the maps $g_{1}^{k}$ (which come from
Puppe sequences) such that

$(g_{1}^{k})^{C_{p}}=\Sigma^{\tilde{V}(\mathfrak{H}^{-}_{k-1},s)^{C_{p}}}g_{1}^{k-1}.$

Writing

$|H^{k}|:=|\mathcal{C}(H^{\\#p^{k-1}},J^{\\#p^{k-1}})|^{\prime}_{\mathfrak{H}_{k}},$

and

$\SS|H^{k}|:=F(|\mathcal{C}(H^{\\#p^{k-1}},J^{\\#p^{k-1}})|^{\prime}_{\mathfrak{H}_{k}},\SS)$

for short, we have a diagram of maps

$\SS|H^{\\#p^{0}}|\xrightarrow{t_{0}}\SS|H^{\\#p^{1}}|\xrightarrow{t_{1}}\SS|H^{\\#p^{2}}|\xrightarrow{t_{2}}\cdots.$
(224)

Here the map $t_{k}$ is $C_{p^{k}}$-equivariant.

###### Proof of Proposition 38.

We simply make sure that the integralization data for
$\mathcal{C}(H_{s},J_{s})$ is chosen such that it satisfies the conditions of
Lemma 33; this is always possible by choosing $c(s)$ to decrease a sufficient
amount. This statement is proven by induction on $k$ with the base case at
$k=0$ being Proposition 33. Given the case for $k-1$, the case for $k$ follows
from follows from Proposition 21, Lemma 36, Lemma 47, Proposition 5 (after
first using Lemma 20 and Lemma 17 to induce an embedding and the extension of
the embedding for
$\mathcal{C}^{\prime}(H_{s}^{\\#p^{k}},J_{s}^{\\#p^{k}})^{C_{p}}$ from that
chosen for $\mathcal{C}^{\prime}(H_{s}^{\\#p^{k-1}},J_{s}^{\\#p^{k-1}})$), and
subsequently using, Theorem 8, Lemma 18, and Proposition 6. One simply notes
that now the perturbation data satisfy the hypotheses of Lemma 33 for
$\mathcal{C}(H_{s}^{p^{k}},J_{s}^{\\#p^{k}})$, and one proceeds onwards with
the induction. ∎

###  10.4 Invariance of equivariant Floer homotopy

######  Proposition 39.

Suppose that $H_{\pm}$ are parts of admissible Floer data $(H_{\pm},J_{\pm})$,
with $H_{\pm}$ having the same slope at infinity if $M$ has contact boundary.
Then, choosing an admissible continuation datum $(H_{s},J_{s})$ between
$(H_{\pm},J_{\pm})$ (assuming the relevant condition on polarization classes)
choosing homotopy definition data $\mathfrak{H}_{s}$ as in Proposition 33, the
map $f$ defined by Proposition 34 on Floer homotopy types is an equivalence of
(nonequivariant) spectra.

###### Proof.

Choose a continuation homotopy datum $(H_{h},J_{h})$ such that
$(H_{1},J_{1})=(H_{3},J_{3})=(H_{4},J_{4})=(H_{-},J_{-}),(H_{2},J_{2})=(H_{+},J_{+})$,
and $(H_{13},J_{13})$ and $(H_{34},J_{34})$ are $s$-independent. By
Proposition 35 we can find a homotopy definition datum $\mathfrak{H}_{h}$ for
$\mathcal{C}(H_{h},J_{h})$ such that

  * •

The associated homotopy definition datum $\mathfrak{H}_{h,12}$ for
$\mathcal{C}(H_{12},J_{12})$ is a stabilizations of $\mathfrak{H}_{s}$,

  * •

For every $x\in Fix(H_{-})$, we get three distinct objects
$(x_{1},x_{3},x_{4})\in\mathcal{C}(H_{h},J_{h})$ with
$x_{i}\in\mathcal{C}(H_{i},J_{i})$ corresponding to the Hamiltonian trajectory
$x$ of $H_{i}=H_{-}$. Because the actions of $x_{i}$ are all the same, both
$\mathcal{C}(H_{13},J_{13})(x_{1},x_{3})$ and
$\mathcal{C}(H_{34},J_{34})(x_{3},x_{4})$ are each a single point
corresponding to the (regular) constant continuation map trajectory at $x$. We
can thus arrange for the perturbation function $\lambda_{V^{1}(x_{i},x_{j})}$
for the perturbation data $\mathfrak{P}_{h}$ underlying $\mathfrak{H}_{h}$
associated to $(x_{1},x_{3})$ and to $(x_{3},x_{4})$ to be zero; this is
possible because in the proof of Proposition 35 (or more saliently in the
proof of Proposition 22) there are no constraints arising from boundary strata
to $\mathcal{C}(x_{3},x_{4})$ or $\mathcal{C}(x_{1},x_{3})$ in the inductive
procedure for choosing perturbation data.

  * •

The homotopy definition data $\mathfrak{H}_{i}$ associated to
$\mathcal{C}(H_{i},J_{i})$ for $i=1,3,4$ are all stabilization of the same
homotopy definition datum $\mathfrak{H}_{-}$.

Choose a strict grading $\alpha_{-}$ on $\mathcal{C}(H_{-},J_{-})$, i.e. a map
$\alpha_{-}:\mathcal{C}(H_{-},J_{-})\to\mathbb{Z}$ such that if $x>y$ then
$\alpha_{-}(x)>\alpha_{-}(y)$. By Lemma 59 and Equation 214 this defines
filtrations of $|\mathcal{C}(H_{i},J_{i})_{\mathfrak{H},x}|$. An inductive
argument shows that the posets
$\mathcal{C}(H_{13},J_{13})\simeq\mathcal{C}(H_{34},J_{34})\simeq\mathcal{C}(H_{s},J_{s})$
respect these filtrations in the sense of Lemma 61. Write

$f_{ij}:F_{\mathbb{R}\oplus V^{1}_{i,s_{ij}}}\Sigma^{\mathbb{R}\oplus
V^{1}_{i,s_{ij}}}|\mathcal{C}(H_{i},J_{i})_{\mathfrak{H}_{i}}|\to
F_{\mathbb{R}\oplus V^{1}_{j,s_{ij}}}\Sigma^{\mathbb{R}\oplus
V^{1}_{j,s_{ij}}}|\mathcal{C}(H_{j},J_{j})_{\mathfrak{H}_{j}}|$

for a choice of such maps associated to $\mathfrak{H}_{ij}$ by Lemma 34; by
Lemma 61 these can be chosen to preserve the filtrations associated to the
gradings. In particular, we see that for both $f_{13}$ and $f_{34}$, the maps
(216) induced in the spectral sequences (213) are isomorphisms, since these
satisfy the condition Theorem 15 and the only trajectories contributing to the
maps on spectral sequences are the constant continuation map trajectories.
Thus the maps induced by
$-F_{V^{1}_{34,s}}\Sigma^{V^{1}_{34,s}}f_{34}F_{V^{1}_{12,s}}\Sigma^{V^{1}_{12,s}}f_{13}$
(see Proposition 36) are isomorphisms on integral homology; by Proposition 36,
that implies that the map induced by
$F_{V^{1}_{24,s}}\Sigma^{V^{1}_{24,s}}f_{24}F_{V^{1}_{12,s}}\Sigma^{V^{1}_{12,s}}{f_{12}}$
is also an isomorphism on integral homology. So $f_{12}$ induces an
isomorphism on integral homology, and thus a weak equivalence of spectra.

A similar argument but now with a choice of continuation homotopy datum
$H_{h}$ and homotopy definition datum $\mathfrak{H}_{h}$ such that
$(H_{1},J_{1})=(H_{3},J_{3})=(H_{4},J_{4})=(H_{+},J_{+})$, and
$(H_{2},J_{2})=(H_{-},J_{-})$, with $\mathfrak{H}_{34}$ a stabilization of
$\mathfrak{H}_{s}$, shows that the map $f$ is a surjection on integral
homology. So $f$ is an isomorphism on integral homology; but a map of finite
spectra which is an isomorphism on integral homology is a weak equivalence. ∎

######  Proposition 40.

Let $p$ be a prime. Suppose that we are in the setting of Proposition 39, but
now all data chosen are $C_{p^{k}}$-equivariant (and the corresponding
assumption on polarization classses holds). Then the map of
$C_{p^{k}}$-spectra $f$ defined by Proposition 34 on Floer homotopy types is
an equivalence of $C_{p^{k}}$-spectra.

###### Proof.

Write $(H_{0,\pm},J_{0,\pm},H_{s,0},J_{s,0})$ for the data which upon
$p^{k}$-iteration give $(H_{\pm},J_{\pm},H_{s},J_{s})$, respectively. We have
that $\mathcal{C}^{\prime}(H_{s},J_{s})^{C_{p^{r}}}$ for $0<r\leq k$ is
exactly $\mathcal{C}^{\prime}(H_{s,0}^{\\#p^{k-r}},J_{s,0}^{\\#p^{k-r}})$,
where the latter is obtained by taking the $C_{p^{r}}$ invariant parts of the
integralization data perturbation data in the natural fashion, with framing as
in Lemma 3; the latter category is equipped with embeddings and extensions of
embeddings by taking $C_{p^{r}}$-invariants. Similar results hold for
$(H_{s},J_{s},H_{s,0},J_{s,0})$ replaced by
$(H_{\pm},J_{\pm},H_{0,\pm},J_{0,\pm})$, respectively, and we see that
$V_{\pm,s}^{C_{p^{r}}}$ is exactly the $C_{p^{k-r}}$ representation
$V_{\pm,s}$ arising in Proposition 34 when applied to
$\mathcal{C}^{\prime}(H_{s,0}^{\\#p^{k-r}},J_{s,0}^{\\#p^{k-r}})$. Thus see in
this setting that $f^{\Phi C_{p^{r}}}$ is a map that would be assigned by
Proposition 34 to
$\mathcal{C}^{\prime}(H_{s,0}^{\\#p^{k-r}},J_{s,0}^{\\#p^{k-r}})$.

Thus the claim follows, from the above, together with Proposition 39 applied
to each of the maps $f^{\Phi C_{p^{r}}}$ (which is sensible by the previous
observation), together with the fact that a map of $C_{p^{k}}$-spectra that is
a weak equivalence on ordinary spectra of all geometric fixed point spectra is
a weak equivalence of $C_{p^{k}}$-spectra [88]. ∎

######  Remark 64.

We can easily upgrade this proposition to its $C_{k}$-equivariant analog by
choosing equivariant perturbation data in an inductive fashion indexed by the
lattice of subgroups in $C_{k}$, but we have not done so here.

###### Proof of Theorem 1.

This follows immediately from Proposition 40 and Proposition 37. ∎

###### Proof of Theorem 2.

This is implicit in the proof of Proposition 40. ∎

###  10.5 Constructing symplectic cohomology and the cyclotomic structure
maps.

We now wish to make sense of the homotopy colimit of the diagram (191) where
all spectra and maps are defined with the constructions of Section 10.1,
choosing the homotopy definition data $\mathfrak{H}_{k}$ such that the
statements in Proposition 38 hold. This would be the spectral analog of the
definition of symplectic cohomology. To to this we should take a “fibrant-
cofibrant” replacement of this diagram, and then subsequently form the mapping
telescope on it. However, this desire is complicated by the fact that we are
repeatedly changing groups throughout the diagram, so the diagram does not sit
in any one model category; and moreover that none of the function spectra we
have written are homotopical because $\SS$ is generally not fibrant. We will
make the desired “cofibrant-fibrant replacement” precise below.

Let $c_{k}$, $f_{k}$ denote the cofibrant and fibrant replacement functors in
the category of orthogonal $C_{p^{k}}$-spectra, and for $k>\ell$, let
$R^{k}_{\ell}$ be the restriction functors from orthogonal $C_{p^{k}}$-spectra
to orthogonal $C_{p^{\ell}}$-spectra. Explicitly, we will replace the (224)
with a digram for which the $k$-th term is $c_{k}f_{k}F(|H^{k}|,f_{k}\SS)$,
and then define the symplectic cohomology to be be a mapping telescope
assocaited to this diagram.

The basic challenge for performing this replacement

$c_{k-1}R^{k}_{k-1}\neq R^{k}_{k-1}c_{k},f_{k-1}R^{k}_{k-1}\neq
R^{k}_{k-1}f_{k}.$

However, let us recall Lemma 52: let $X$ be an $H$-spectrum, $Y$ be a
$G$-spectrum, and $t:X\to Y$ be an $H$-map of spectra. For precision we will
denote this by

$t:X\to R^{G}_{H}Y.$

Then we can form the following diagrams of spectra:

${c_{H}X}$${c_{H}R^{G}_{H}Y}$${cHR^{G}_{H}c_{G}Y}$${R^{G}_{H}c_{G}Y,}$${f_{H}X}$${f_{H}R^{G}_{H}Y}$${f_{H}R^{G}_{H}f_{G}Y}$${R^{G}_{H}f_{G}Y}$$\scriptstyle{c_{H}t}$$\scriptstyle{f_{H}t}$
(225)

Here $(c_{H},f_{H},c_{G},f_{G})$ denote the cofibrant and fibrant replacement
functors for orthogonal $H$ and $G$-spectra. Moreover, there is a diagram

${c_{H}f_{H}X}$${c_{H}f_{H}R^{G}_{H}Y}$${c_{H}f_{H}R^{G}_{H}f_{G}Y}$${c_{H}f_{H}R^{G}_{H}c_{G}f_{G}Y}$${f_{H}R^{G}_{H}c_{G}f_{G}Y}$${R^{G}_{H}c_{G}f_{G}Y}$${c_{H}X}$${c_{H}R^{G}_{H}Y}$${c_{H}R^{G}_{H}Y}$${c_{H}R^{G}_{H}c_{G}Y}$${R^{G}_{H}c_{G}Y}$${R^{G}_{H}c_{G}Y.}$
(226)

${\Phi^{C_{p}}c_{k}F(|H^{k}|,f_{k}\SS)}$${\Phi^{C_{p}}c_{k}F(|H^{k}|,f_{k}f_{k+1}\SS)}$${\Phi^{C_{p}}c_{k}F(|H^{k}|,f_{k+1}\SS)}$${\Phi^{C_{p}}c_{k}F(|H^{k+1}|,f_{k+1}\SS)}$${\Phi^{C_{p}}c_{k}c_{k+1}F(|H^{k+1}|,f_{k+1}\SS)}$${\Phi^{C_{p}}c_{k+1}F(|H^{k+1}|,f_{k+1}\SS)}$${\cdots}$${F(\Phi^{C_{p}}|H^{k}|,\Phi^{C_{p}}f_{k}\SS)}$${F(\Phi^{C_{p}}|H^{k}|,\Phi^{C_{p}}f_{k}f_{k+1}\SS)}$${F(\Phi^{C_{p}}|H^{k}|,\Phi^{C_{p}}f_{k+1}\SS)}$${F(\Phi^{C_{p}}|H^{k+1}|,\Phi^{C_{p}}f_{k+1}\SS)}$${F(\Phi^{C_{p}}|H^{k+1}|,\Phi^{C_{p}}f_{k+1}\SS)}$${F(\Phi^{C_{p}}|H^{k+1}|,\Phi^{C_{p}}f_{k+1}\SS)}$${\cdots}$${F(\Phi^{C_{p}}|H^{k}|,f_{k-1}\Phi^{C_{p}}f_{k}\SS)}$${F(\Phi^{C_{p}}|H^{k}|,f_{k-1}f_{k}\Phi^{C_{p}}f_{k}f_{k+1}\SS)}$${F(\Phi^{C_{p}}|H^{k}|,f_{k}\Phi^{C_{p}}f_{k+1}\SS)}$${F(\Phi^{C_{p}}|H^{k+1}|,f_{k}\Phi^{C_{p}}f_{k+1}\SS)}$${F(\Phi^{C_{p}}|H^{k+1}|,f_{k}\Phi^{C_{p}}f_{k+1}\SS)}$${F(\Phi^{C_{p}}|H^{k+1}|,f_{k}\Phi^{C_{p}}f_{k+1}\SS)}$${\cdots}$${c_{k-1}F(|H^{k-1}|,f_{k-1}\SS)}$${c_{k-1}F(|H^{k-1}|,f_{k-1}f_{k}\SS)}$${c_{k-1}F(|H^{k-1}|,f_{k}\SS)}$${c_{k-1}F(|H^{k}|,f_{k}\SS)}$${c_{k-1}c_{k}F(|H^{k}|,f_{k}\SS)}$${c_{k}F(|H^{k}|,f_{k}\SS)}$${\cdots}$
Figure 11: A large commutative diagram. The diagram continues on the right as
it starts on the left.

Below, we will discuss ‘diagrams’ several times; however, each time, we are
considering objects and arrows in a various homotopy categories satisfying a
relation that can be depicted in a diagrammatic fashion, rather than an actual
functor from some simple indexing category to a given category. We will
explain in each case what the relations are.

######  Theorem 16.

The diagram (224) canonically defines associated objects and morphisms,
depicted

$F(|H^{0}|,\SS)\xrightarrow{[t_{0}]}F(|H^{1}|,\SS)\xrightarrow{[t_{1}]}\cdots$
(227)

where $F(|H^{k}|,\SS)$ is thought of an object in $Ho(C_{p^{k}}-OSp)$, and
$[t_{k}]\in Ho(C_{p^{k}}-OSp)$ is a map $F(|H^{k}|,\SS)\to
R^{k+1}_{k}F(|H^{k}|,\SS)$. In fact all objects in the above diagram have
canonical representatives $c_{k}f_{k}F(|H^{k}|,f_{k}\SS)\in C_{p^{k}}-OSp$.

Similarly, there are objects in morphisms depicted as

${\Phi^{C_{p}}F(|H^{1}|,\SS)}$${\Phi^{C_{p}}F(|H^{2}|,\SS)}$${\cdots}$${F(|H^{0}|,\SS)}$${F(|H^{1}|,\SS)}$${\cdots}$$\scriptstyle{\Phi^{C_{p}}{[}t_{1}{]}}$$\scriptstyle{{[}\phi_{1}{]}}$$\scriptstyle{\Phi^{C_{p}}{[}t_{2}{]}}$$\scriptstyle{{[}\phi_{2}{]}}$$\scriptstyle{{[}t_{0}{]}}$$\scriptstyle{{[}t_{1}{]}}$
(228)

where the horizontal maps are induced from the maps (227) functorially,
vertical maps $[\phi_{k+1}]$ are all isomorphisms in $Ho(C_{p^{k}}-OSp)$, and

$[\phi_{k+1}]\Phi^{C_{p}}[t_{k+1}]=[t_{k}][\phi_{k+1}]\text{ in
}Ho(C_{p^{k}}-OSp).$ (229)

Choosing representatives $t^{\prime}_{k}$ of $[t_{k}]$ and defining
$SH(M,\SS)_{\geq r}$ to be the mapping telescope and choosing representatives
of $\phi_{k}$ of $[\phi_{k}]$ and homotopies representing (229), one has
stable equivalences

$(\tilde{\phi}^{r}:\Phi^{C_{p}}SH(M,\SS)_{\geq r}\to SH(M,\SS)_{\geq r-1})\in
C_{p^{r-1}}-OSp.$ (230)

Moreover, there are objects and morphisms, depicted pictorially as

${\Phi^{C_{p}}SH(M,\SS)_{\geq 1}}$${\Phi^{C_{p}}SH(M,\SS)_{\geq
2}}$${\Phi^{C_{p}}SH(M,\SS)_{\geq 3}}$${\cdots}$${SH(M,\SS)_{\geq
0}}$${SH(M,\SS)_{\geq 1}}$${SH(M,\SS)_{\geq 2}}$${\cdots}$ (231)

where the vertical arrows are the $\tilde{\phi}^{k}$, the horizontal arrows
are the inclusion maps $i_{k+1,k}$ on mapping telescopes (see (184)). This
diagram commutes strictly in the sense that
$\tilde{\phi}^{k}i_{k+1,k}=i_{k,k-1}\tilde{\phi}^{k+1}$ in $C_{p^{k-1}}-OSp$.

###### Proof.

The second diagram in (225) defines maps

$f_{k}\SS\to f_{k}f_{k+1}\SS\to f_{k+1}\SS.$ (232)

Above, and for the rest of this proof (including Figure 11) we will adopt the
convention that

  * •

We will drop all symbols denoting group restriction. Thus $f_{k}f_{k+1}\SS$ is
a $C_{p^{k}}$-spectrum, while $f_{k+1}\SS$ is a $C_{p^{k+1}}$-spectrum.

  * •

All maps are “as equivariant as possible”, in the sense that if we have a map
$f:X^{\prime}\to Y^{\prime}$ where $X^{\prime}$ is produced by some cofibrant
and fibrant replacements from $X^{\prime}$ and similarly $Y^{\prime}$ from
$Y$, then $f$ is equivariant with respect to the maximal subgroup of $S^{1}$
that it could be given the kinds of spectra that $X^{\prime}$ and $Y^{\prime}$
are.

All ambiguities regarding insertion of $R^{G}_{H}$ into formulae can be
resolved by Equations (180), (181), and (182).

The diagram (232) lets us replace each map in (224) with a zig-zag of maps:

${F(|H^{k}|,f_{k}\SS)}$${F(|H^{k}|,f_{k}f_{k+1}\SS)F(|H^{k}|,f_{k+1}\SS)}$${F(|H^{k+1}|,f_{k+1}\SS).}$
(233)

Using the top diagram of (225) then lets us produce the bottom row of Figure
11. Applying $\Phi^{C_{p}}$ to the bottom row and shifting the indices over
produces the top row of Figure 11.

Now there are canonical maps

$\Phi^{H}cF(X,Y)\to F(\Phi^{H}X,\Phi^{H}Y)$

adjoint to

$\Phi^{H}cF(X,Y)\wedge\Phi^{H}\to\Phi^{H}(cF(X,Y)\wedge
Y)\to\Phi^{H}(F(X,Y)\wedge Y)\to\Phi^{H}(Y).$

These maps are stable equivalences whenever $X$ is $F_{V}Y$ for some _finite_
$G$-CW complex $Y$ and $Y$ is fibrant. These maps produce maps to the second
row of Figure 11, and the maps from second row to the third row are produced
by maps to fibrant replacements.

The maps from the fourth row to the third row are built from the strictly
commutative diagram

${F(|H^{\\#p^{0}}|,\SS)}$${F(|H^{\\#p^{1}}|,\SS)}$${F(|H^{\\#p^{2}}|,\SS)}$${\cdots}$${F(\Phi^{C_{p}}|H^{\\#p^{1}}|,\SS)}$${F(\Phi^{C_{p}}|H^{\\#p^{2}}|,\SS)}$${F(\Phi^{C_{p}}|H^{\\#p^{3}}|,\SS)}$${\cdots}$
(234)

by performing the same operation to the bottom row of (234) as is used to
produce the bottom row of Figure 11 from the top row of (234), and then
applying maps from cofibrant and to fibrant replacements.

The fact that there is a diagram like (234) follows from Proposition 38 and
some careful index checking. Indeed, Proposition 38 implies that there is a
natural transformation of diagrams from the diagram with maps
$(F_{\mathbb{R}\oplus\tilde{V}(\mathfrak{H}^{-}_{k})}g_{1}^{k})^{\Phi C_{p}}$
to the diagram with maps
$(F_{\mathbb{R}\oplus\tilde{V}(\mathfrak{H}^{-}_{k-1})\oplus\tilde{V}(\mathfrak{H}^{-}_{k-1},s)^{C_{p}}}\Sigma^{\tilde{V}(\mathfrak{H}^{-}_{k-1},s)^{C_{p}}}g_{1}^{k-1})$,
and the maps (179) give a natural transformation from the latter diagram to
the diagram with maps
$(F_{\mathbb{R}\oplus\tilde{V}(\mathfrak{H}^{-}_{k-1})}g_{1}^{k-1})$.
Similarly, Proposition 38 gives a natural transformation from the diagram with
maps $(g_{2}^{k})^{\Phi^{k}_{2}}$ to the diagram with maps

$(F_{\tilde{V}(\mathfrak{H}_{k-1},s)^{C_{p}}}\Sigma^{\tilde{V}(\mathfrak{H}_{k-1},s)^{C_{p}}})g^{k-1}_{2}$

(the diagram where we insert the extra representation into the free spectrum
and the suspension) and then again (179) lets us remove the copies of
$F_{\tilde{V}(\mathfrak{H}_{k-1},s)^{C_{p}}}\Sigma^{\tilde{V}(\mathfrak{H}_{k-1},s)^{C_{p}}}$.
This pair of natural transformations fits together to define a natural
transformation from the diagram with maps $(g^{k})^{\Phi C_{p}}$ to the
diagram with maps $g^{k-1}$. One then gets the diagram by applying
$F(\cdot,\SS)$.

It is tedious but straightforward to check that the diagram Figure 11 commutes
strictly, with all vertical arrows weak equivalences. Using (226), after
extending the right most column of Figure 11 with three more identical
columns, one gets a bigger diagram where the bottom row maps into a row like
the current bottom row of Figure 11 but extended like the top row of (226)
such that all objects are cofibrant fibrant, and similarly the top row of the
new diagram admits a map to $\Phi^{H}$ applied to the bottom row. The whole
diagram still strictly commutes, and the objects on the top corners are
cofibrant and the objects on the bottom corners are fibrant. Looking at this
diagram in the homotopy category of $G$-spectra produces the diagrams (227),
(228).

Choosing representatives $t^{\prime}_{k}$ as in the statement of the lemma
defines representatives $\Phi^{C_{p}}t^{\prime}_{k}$ of
$\phi^{C_{p}}[t^{\prime}_{k}]$. Choosing representatives of $[\phi_{k}]$ and
the homotopies as in the lemma, and using the fact that mapping telescopes
commute with geometric fixed points, defines the maps $\tilde{\phi}^{k}$.
These are weak equivalences by Lemma 56.

The fact that these fit into (231) follows immediately from basic facts about
the mapping telescope reviewed in Section 9.3.

∎

###  10.6 Comparison with Nikolaus-Scholze.

Let $N$ denote the nerve of a $1$-category and let $N^{hc}$ denote the
homotopy-coherent nerve of a simplicially-enriched category. [72, Tag 00KM].
Given a category $\mathcal{C}$, there is a simplicially enriched category
associated to $\mathcal{C}$ with underlying category $\mathcal{C}$ and the
morphisms from $x$ to $y$ the simplicial set obtained by taking the constant
simplicial object on $\mathcal{C}(x,y)$; we will also denote this category by
$\mathcal{C}$. There is a map of simplicial sets [72, Tag 00KZ].

$N(\mathcal{C})\simeq N^{hc}(\mathcal{C}).$ (235)

Given a model category $\mathcal{C}$ with objects contained in the
Grothendieck universe $\overline{\mathcal{U}}$, we will denote the associated
hammock localization by $L^{H}(\mathcal{C})$ [39, 55]. Thus, writing
$L^{H}(\mathcal{C})^{f}$ for the Begner fibrant replacament [22], there is a
simplicial set $N^{hc}(L^{H}(\mathcal{C})^{f})$ which is an $\infty$-category,
and we call the _$\infty$ -category associated to $\mathcal{C}$._.

There is thus a functor of $\infty$-categories from $N(\mathcal{C})$ to the
$\infty$-category associated to $\mathcal{C}$. In fact, the latter is a
localization of the $\infty$-category $N(\mathcal{C})$ at the weak
equivalences [39, 54]. In particular, our notion of of the $\infty$-category
$C_{p^{n}}Sp$ underlying the homotopy theory of orthogonal $C_{p^{n}}$-spectra
agrees with that of [77].

######  Definition 91.

(Nikolaus-Scholze [77]) The $\infty$-category of genuine
$C_{p^{\infty}}$-spectra is the $\infty$-category

$C_{p^{\infty}}Sp:=\varprojlim C_{p^{k}}Sp$

where the inverse limit is taken along the group restriction functors
$F^{C_{p^{k+1}}}_{C_{p^{k}}}$ (or rather the functors obtained from applying
$N^{hc}(L^{H}(\cdot)^{f})$ to these functors).

The functor $\Phi^{C_{p}}$ induces functors

$\Phi^{C_{p}}:C_{p^{k}}Sp\to C_{p^{k}}/{C_{p}}Sp\simeq C_{p^{k-1}}Sp.$

for all $p$, and commutes with $F^{C_{p^{k+1}}}_{C_{p^{k}}}$. It thus induces
a functor

$C_{p^{\infty}}Sp\to(C_{p^{\infty}}/C_{p})Sp\simeq C_{p^{\infty}}Sp$

The $\infty$-category of genuine $p$-cyclotomic spectra is the equalizer of
the diagram

${C_{p^{\infty}}Sp}$${C_{p^{\infty}}Sp.}$$\scriptstyle{Id}$$\scriptstyle{\Phi^{C_{p}}}$
(236)

######  Theorem 17.

The data of Theorem 16 define an associated genuine $p$-cyclotomic spectrum in
the sense of Nikolaus-Scholze.

###### Proof.

This is essentially an exercise in standard $\infty$-categorical
combinatorics. We first recall two notions and some useful theorems.

First, recall that a category can be regarded as a simplicial category with
the simplicial sets of morphisms being discrete; we will use this
identification to speak about functors from categories to simplicial
categories. Equivalently, a functor from a category to a simplicial category
is a functor from the category to the underlying category of the simplicial
category.

The simplicial category $QCat$ has objects all small $\infty$-categories and
morphisms $QCat(X,Y)$ the core of the $\infty$-category of functors
$Fun(X,Y)$. The $\infty$-category of $\infty$-categories $\mathcal{QC}$ is the
homotopy-coherent nerve of $QCat$.

Suppose we have a category $\mathcal{C}$ and a functor
$\mathcal{F}:\mathcal{C}\to QCat$. To this data, one can associate the
weighted nerve $N_{\bullet}^{\mathcal{F}(\mathcal{C})}$ [72, Tag 025X], which
admits a map

$U:N_{\bullet}^{\mathcal{F}}(\mathcal{C})\to\mathcal{N}(\mathcal{C})$

which is a cocartesian fibration Moreover, objects of the $\infty$-category
$N_{\bullet}^{\mathcal{F}(\mathcal{C})}$ are tuples $(C,x)$, where $C\in
Ob(\mathcal{C})$ and $x$ is an object of $\mathcal{F}(C)$, and $1$-morphisms
are tuples $(f,e):(C,x)\to(D,y)$ where $f:C\to D$ is a morphism in
$\mathcal{C}$ and $e:\mathcal{F}(f)(x)\to y$ is a 1-morphism in
$\mathcal{F}(D)$. Finally, $(f,e)$ is $U$-cocartesian if and only if $e$ is an
isomorphism in the $\infty$-category $\mathcal{F}(D)$. [72, Tag 046Y].

The weighted nerve is an $\infty$-category, and there is a map of
$\infty$-categories

$N_{\bullet}^{\mathcal{F}}(\mathcal{C})\xrightarrow{\theta}\int_{\mathcal{N}(\mathcal{F})}\mathcal{N}^{hc}(F).$

which is an equivalence of $\infty$-categories and gives a bijection on
$U$-cocartesian morphisms [72, Tag 027J].

Now, Theorem 16 produces the data

$SH(M,\SS)_{0}\xleftarrow{i_{01}}SH(M,\SS)_{1}\xleftarrow{i_{12}}SH(M,\SS)_{2}\xleftarrow{i_{23}}\cdots$
(237)

where $SH(M,\SS)_{r}\in C_{p^{r}}-OSp$ and

$i_{ab}:F^{C_{p^{b}}}_{C_{p^{a}}}SH(M,\SS)_{a}\to SH(M,\SS)_{b}$

for $b>a$ are the corresponding inclusions of mapping telescopes, and are all
stable equivalences of orthogonal spectra.

Write $\mathbb{Z}_{\geq 0}$ for the category corresponding to the poset of
nonnegative integers ordered by magnitude There is a functor

$\mathcal{F}:\mathbb{Z}_{\geq 0}^{op}\to 1-Cat$

mapping $n$ to $C_{p^{n}}-OSp$ and mapping $n\to m$ to
$F^{C_{p^{n}}}_{C_{p^{m}}}$. This defines an associated functor of
$\infty$-categories

$N(F):\mathbb{Z}_{\geq 0}^{op}\to QCat$

which sends each $1$-morphism $n\to m$ to $N(F^{C_{p^{n}}}_{C_{p^{m}}})$.
There is another functor

$N^{hc}(L^{H}(\mathcal{F})^{h}):Z^{op}\to QCat$

which sends each $1$-morphism $n\to m$ to
$N^{hc}(L^{H}(F^{C_{p^{n}}}_{C_{p^{m}}})^{f})$ where we are are using the fact
that $F^{C_{p^{n}}}_{C_{p^{m}}}$ sends $C_{p^{n}}-$stable equivalences to
$C_{p^{m}}$-stable equivalences to define $L^{H}(F^{C_{p^{n}}}_{C_{p^{m}}})$.
Note that

$N^{hc}(L^{H}(\mathcal{F}^{h}))(n)=C_{p^{n}}Sp,$

the $\infty$-category underlying the homotopy theory of genuine
$C_{p^{n}}$-spectra. There is a natural transformation of functors

$N(\mathcal{F})\to N^{hc}(L^{H}(\mathcal{F}^{h}))$ (238)

induced by the maps

$N(\mathcal{C})\to N^{hc}(L^{H}(\mathcal{C})^{f}).$

This defines a functor

$N_{\bullet}^{N(\mathcal{F})}\to N_{\bullet}^{N^{hc}(L^{H}(\mathcal{F}^{h})}.$
(239)

The data (237) produces an element of
$Fun(N(Z^{op}),N_{\bullet}^{N(\mathcal{F})})$ which sends $n\to m$ to
$i_{nm}$. There is a corresponding element of

$Fun(N(Z^{op}),N_{\bullet}^{N^{hc}(L^{H}(\mathcal{F}^{h})})$

sending $n\to m$ to $i_{nm}$, related to the previous element by the functor
above between weighted nerves.

Each of these morphisms is $U$-cocartesian for

$U:N_{\bullet}^{N^{hc}(L^{H}(\mathcal{F}^{h})})\to N(Z^{op})$

since it is an isomorphism in the $\infty$-category $C_{p^{n}}-Sp$. Applying
$\theta$, we have produced an element of

$\underleftarrow{\text{holim}}(\mathcal{F}):=Fun^{U-coCart}_{\mathcal{N}(Z^{op})}(N(Z^{op}),N^{hc}(N^{hc}(L^{H}(\mathcal{F})^{h}).$
(240)

Now, $\underleftarrow{\text{holim}}(\mathcal{F})$ is a limit of the diagram
$N(N^{hc}(L^{H}(\mathcal{F}^{h})))$ [72, Tag 03AB]; we have produced an
element of $C_{p^{\infty}}Sp$.

To argue that the data of the cyclotomic structure maps enhances this to an
genuine $p$-cyclotomic spectrum, we argue similarly. We first detail the
construction of the functor

$\Phi^{C_{p}}:C_{p^{\infty}}Sp\to C_{p^{\infty}}Sp.$

Let $\mathcal{F}^{\prime}:Z_{\geq 0}^{op}\to 1-Cat$ be the unique functor
sending $0$ to $\Delta^{0}$, and otherwise $\mathcal{F}(n\to
m)=N^{hc}(L^{H}(\mathcal{F}^{\prime})^{h})(n+1\to m+1)$ for all
$n,m\in\mathbb{Z}$. There is a natural transformation

$N^{hc}(L^{H}(\mathcal{F})^{h})\to\mathcal{F}^{\prime}$

given by applyin $N^{hc}(L^{H}(\Phi^{C_{p}})^{f}):C_{p^{n}}Sp\to
C_{p^{n-1}}Sp$ for all $n\geq 1$, and applying the unique map
$C_{p}Sp\to\Delta^{0}$ for $n=0$. This induces a functor
$N^{N^{hc}(L^{H}(\mathcal{F})^{h})}_{\bullet}\to
N^{\mathcal{F}^{\prime}}_{\bullet}$ preserving $U$-cocartesian morphisms, and
by postcomposition this defines a map

$\underleftarrow{\text{holim}}(\mathcal{F})\to\underleftarrow{\text{holim}}(\mathcal{F^{\prime}})\simeq\underleftarrow{\text{holim}}(\mathcal{F})$

(see [72, Tag 03AC]); here the last isomorphism uses the fact that, writing
$Z_{\geq 1}\subset Z_{\geq 0}$ for the full subcategory on the strictly
positive integer, cocartesian sections of the cocartesian fibration
$N^{\mathcal{F}^{\prime}}_{\bullet}$ over $N(Z_{\geq 1}^{op})$ extend uniquely
to $N(Z_{\geq 0}^{op})$, and this induces an isomorphism of simplicial sets.

We now have the strictly commutative diagram of $\infty$-categories (236), and
we use the same method to produce an element of its limit. Explicitly, we can
produce a genuine $p$-cyclotomic spectrum by picking an element $X\in
C_{p^{\infty}}Sp$ and an isomorphism $\Phi^{C}_{p}X\to X$ in
$C_{p^{\infty}}Sp$. Specifically, the fact that there are equivalences
$\Phi^{C_{p}}(SH(M,\SS)_{\geq r})\to\Phi^{C_{p}}(SH(M,\SS)_{\geq r-1})$ such
that $\Phi^{C_{p}}i_{ab}=i_{a-1,b-1}$ as in Theorem 16 satisfying the
commutative diagram (231) defines a map in the category of sections
$Fun_{/N(Z_{\geq 0}^{op})}(N(Z_{\geq 0}^{op}),N^{N(\mathcal{F})})$, which
becomes an isomorphism in $C_{p^{\infty}}-Sp$ after applying the map (239)
[72, Tag 035R].

∎

Using this data, one can define the $p$-typical topological cyclic homology
groups $TR(SH(M,\SS),p),TC(SH(M,\SS),p)$ as in [24] (see [77, Definition
II.4.4]). There is a functor from genuine $p$-cyclotomic spectra to
$p$-cyclotomic spectra in the sense of Nikolaus-Scholze which is an
equivalence on the subcategory of bounded below objects; however, in general
symplectic cohomology is bounded neither above nor below.

###### Proof of Theorem 3.

It suffices to compute the homotopy groups of $H\mathbb{Z}\wedge
SH^{\bullet}(M,\SS)$ as an ordinary non-equivariant spectrum. But it is
constructed via the homotopy colimit (227); smashing commutes with homotopy
colimits, and on homology, by the functorial isomorphisms $H\mathbb{Z}\wedge
F(X,\SS)\simeq F(X,H\mathbb{Z})$ for finitely-built spectra $X$, Proposition
39 identifies $\pi_{*}H\mathbb{Z}\wedge F(|H^{i}|,\SS)$ with
$HF^{-*}(\widetilde{H^{\\#p^{k}}};\mathbb{Z})$, where
$\widetilde{H^{\\#p^{k}}}$ is a non-equivariant perturbation of $H^{p^{k}}$
which is now nondegenerate, and the new homotopy definition data for
$F(|widetilde{H^{\\#p^{k}}}|,\SS)$ are chosen such that that proposition
holds. To identify the action of $H\mathbb{Z}\wedge[t_{i}]$ on homotopy groups
with the corresponding action of continuation maps on
$HF^{-*}(H^{\\#p^{k}};\mathbb{Z})$, one constructs commutative diagrams via
Proposition 35 and Proposition (36) where the new horizontal maps are induced
by _regular, non-equivariant_ admissible continuation data, which satisfy the
regularity hypotheses of Theorem 15 letting us make the comparison with the
usual Floer-theoretic continuation map. ∎

######  Remark 65.

While this document does not contain a complete proof that the cyclotomic
spectrum $SH^{\bullet}(M,\SS)$ is independent of the choices made, verifying
this should simply be a slightly tedious exercise in the technology of this
paper. One first argues that $SH^{\bullet}(M,\SS)$ is unchanged up to
equivalence if one compatibly stabilizes all homotopy definition data (a
fairly simple type of stabilization, for which one allows only stabilization
of perturbation data by parameterizations with trivial perturbation functions,
is sufficient). One then uses Proposition 35, and an analog of Proposition 38
for homotopy definition data to build a ‘map of diagrams’ of the form (224).
Performing the functorial replacements of Proposition 16 then defines maps
which give maps of diagrams (227) satisfying compatibility conditions with
respect to the geometric fixed point functors, as in (229). These maps are
pieced together to define a map of the mapping telescopes defining equivariant
maps of the spectra $SH(M,\SS)_{\geq 0}$, and this should define exactly a map
in the corresponding infinity category of genuine $p$-cyclotomic spectra.
Checking that it is an equivalence is then done on homotopy groups, noting
that the cyclotomic structure and comparison with symplectic homology (Theorem
3) immediately implies that it is an equivalence on homology after taking
geometric fixed points for any $C_{p^{k}}$, and is thus an a map of genuine
$p$-cyclotomic spectra which is an equivalence of underlying
$C_{p^{\infty}}$-spectra, i.e. an equivalence of genuine $p$-cyclotomic
spectra.

##  11 Appendix

###  11.1 Classifying spaces for $K$ theory

#### (Equivariant) Grassmannians.

We first recall the construction of the space $BO$ and $BU$, as well as their
equivariant analogs. Given a map $A:V\to W$ between real inner product spaces,
let $A^{T}:W\to V$ denote its transpose. If instead $(V,W)$ are Hermitian
inner product spaces and $A$ is complex linear, let $A^{\dagger}:W\to V$
denote its Hermitian transpose. For a real inner product space $V$ and a
Hermitian inner product space $W$, let

$EO(k,V)=\\{A\in Hom(\mathbb{R}^{k},V)|A^{T}A=id\\},\;EU(k,W)=\\{A\in
Mat_{\mathbb{C}}(\mathbb{C}^{n},W)|A^{\dagger}A=id\\}.$ (241)

For $k\leq\dim V$ and $k\leq\dim_{\mathbb{C}}W$, there are fibrations

$EO(k,V)\to EO(k+1,V\oplus\mathbb{R})\to S^{n},EU(k,W)\to
EU(k+1,W\oplus\mathbb{C})\to S^{2n+1}$ (242)

show inductively that for any $C$ there is an $N=N(C,k)$ such that for $\dim
V>N$ or $\dim_{\mathbb{C}}W>N$, we have $\pi_{i}(EO(k,V))=\pi_{i}(EU(k,W)=0$
for $0\leq i\leq C$. There is a free $O(k)$ action on $EO(k,n)$ by
precomposition and a similar free $U(k)$ action on $EU(k,n)$. Thus

$EO(k)=\operatorname{colim}_{n}EO(k,\mathbb{R}^{n}),EU(k)=\operatorname{colim}_{n}EU(k,\mathbb{C}^{n})$
(243)

are contractible spaces (which can be given a CW stucture, since all above
maps are embeddings of manifolds) with free actions of $O(k)$ and $U(k)$.
Therefore

$BO(k)=EO(k)/O(k)=Gr(k,\mathbb{R}^{\infty}),BU(k)=EU(k)/U(k)=Gr_{\mathbb{C}}(k,\mathbb{C}^{\infty})$

are classifying spaces for $O(k)$ and $U(k)$, respectively. They manifestly
parameterize $k$-planes in $\mathbb{R}^{\infty}$ and $\mathbb{C}^{\infty}$,
respectively. In particular, there are $\mathbb{R}^{k}$ and $\mathbb{C}^{k}$
bundles $\eta_{k}^{\mathbb{R}}\to BO(k),\eta_{k}^{\mathbb{C}}\to BU(k)$ called
universal bundles.

The inclusions

$\mathbb{R}^{\infty}\simeq
0\oplus\mathbb{R}^{\infty}\subset\mathbb{R}^{\infty},\mathbb{C}^{\infty}\simeq
0\oplus\mathbb{C}^{\infty}\subset\mathbb{R}^{\infty}$ (244)

induce closed inclusions

$BO(k)\subset BO(k+1),BU(k)\subset BU(k+1)$ (245)

covered by inclusions

$EO(k)\subset EO(k+1),EU(k)\subset EU(k+1)$ (246)

where we send $A\in EO(k)$ to the map sending first $k$ coordinate vectors
$e_{1},\ldots,e_{k}$ to the images of $A(e_{1}),\ldots A(e_{k})$ under (244),
and send $e_{k+1}$ to the first coordinate vector in $\mathbb{R}^{\infty}$ or
$\mathbb{C}^{\infty}$ respectively. These inclusions equivariant with respect
to the inclusions of groups

$O(n)\subset O(n+1),U(n)\subset U(n+1)$ (247)

given on matrices by $A\mapsto A\oplus 1$.

by taking quotients by $O(k)\subset O(k+1)$, $U(k)\subset U(k+1)$ thought of
as matrices with the top left $k\times k$ entries nonzero. In particular, the
colimits

$EO=\operatorname{colim}_{k}EO(k),EU=\operatorname{colim}_{k}EU(k)$ (248)

are contractible spaces (which can be given a CW structure, using the
filtrations arising from (243)) with free actions of the topological groups

$O=\operatorname{colim}_{n}O(n),U=\operatorname{colim}_{n}U(n)$ (249)

and thus $BO=EO/O=\operatorname{colim}_{k}BO(k)$ and
$BU=EU/U=\operatorname{colim}_{k}BU(k)$ are classifying spaces (also CW
complexes) for $O$ and $U$. Manifestly, any map from $f:K\to BO$ from a
compact $CW$ complex $k$ factors through some $BO(k)$, and thus the pullback
$f^{*}\eta^{\mathbb{R}}_{k}$ to $K$ will be a vector bundle; the indeterminacy
of the choice of $k$ implies that the virtual vector bundle
$[f^{*}\eta^{\mathbb{R}}_{k}]-[\underline{\mathbb{R}}^{k}]$ is well defined.
The fact that vector bundles admit complements then implies that
$\mathbb{Z}\times BO$ and $\mathbb{Z}\times BU$ are classifying spaces for
$KO^{0}$ and $KU^{0}$, respectively.

An equivariant analog of the entire construction is straightforward. We will
only do the orthogonal case as the unitary case is completely analogous. Fix a
compact Lie group $G$, and let $\mathcal{U}$ be a complete $G$-universe. The
previous discussion are $(EO(k),EU(k))$ are free contractible $G$-CW complexes
with free actions of $O(k)$ and $U(k)$ commuting with the $G$-actions.

Fix an isomorphism

$\mathcal{U}\simeq V_{1}^{\mathbb{Z}}\oplus V_{2}^{\mathbb{Z}}\oplus\ldots$
(250)

where the $\\{V_{i}\\}_{i=1}^{\infty}$ are a choice of irreps of $G$ such that
each irreducible representation occurs exactly once. Write

$\mathcal{U}_{a_{1},\ldots,}\simeq V_{1}^{\mathbb{Z}_{\geq-a_{1}}}\oplus
V_{2}^{\mathbb{Z}_{\geq-a_{2}}}\oplus\ldots\subset\mathcal{U}.$

for $(a_{i})_{i=1}^{\infty}$ any sequence of nonnegative integers with only
finitely many nonzero terms.

Let $\mathcal{Z}$ be the poset of such sequences with
$(a_{i})_{i=1}^{\infty}\leq(b_{i})_{i=1}^{\infty}$ if $a_{i}\leq b_{i}$ for
all $i$. Then the inclusions
$\mathcal{U}_{(a_{i})}\subset\mathcal{U}_{(b_{i})}$ for $(a_{i})\leq(b_{i})$
define a functor

$F:\mathcal{Z}\to
G-Top,F((a_{i}))=E(\mathbb{R}^{a_{1}}\oplus\mathbb{R}^{a_{2}}\oplus\ldots,\mathcal{U}_{(a_{i})})$

with all morphisms closed inclusions (which can be refined to inclusions of
$G$-CW complexes). The colimit of this functor over this filtered diagram is a
space that is homeomoprhic to $EO$. This colimit is what we mean by $EO\in
G-Top$, and it is straightforward to show that it is a classifying space for
$KO^{0}_{G}$.

######  Lemma 63.

Let $\mathcal{H}$ be the completion of a complete $G$-universe. There is a
$G$-homotopy equivalence

$\mathbb{Z}\times BO\to\operatorname{Fred}H$

which induces a natural isomorphism between the functors (121) and (123).

###### Proof.

The argument for $G=1$ is straightforward: the proof of Theorem 9 allows one
to build a map $c(FredH)\to\mathbb{Z}\times BO$ which is a weak equivalence.
The result then follows by Theorem 12. For general $G$-the equivariant analog
of the proof of Theorem 9 and Theorem 8.2. ∎

## References

  * [1] Alberto Abbondandolo and Matthias Schwarz  “On the Floer homology of cotangent bundles”  In _Communications on Pure and Applied Mathematics: A Journal Issued by the Courant Institute of Mathematical Sciences_ 59.2 Wiley Online Library, 2006, pp. 254–316
  * [2] Mohammed Abouzaid  “A cotangent fibre generates the Fukaya category”  In _Advances in Mathematics_ 228.2 Elsevier, 2011, pp. 894–939
  * [3] Mohammed Abouzaid  “A geometric criterion for generating the Fukaya category”  In _Publications Mathématiques de l’IHÉS_ 112, 2010, pp. 191–240
  * [4] Mohammed Abouzaid  “Symplectic cohomology and Viterbo’s theorem”  In _arXiv preprint arXiv:1312.3354_ , 2013
  * [5] Mohammed Abouzaid and Andrew J Blumberg  “Arnold conjecture and Morava K-theory”  In _arXiv preprint arXiv:2103.01507_ , 2021
  * [6] Mohammed Abouzaid and Andrew J Blumberg  “Foundation of Floer homotopy theory I: Flow categories”  In _arXiv preprint arXiv:2404.03193_ , 2024
  * [7] Mohammed Abouzaid, Yoel Groman and Umut Varolgunes  “Framed $E^{2}$ structures in Floer theory”  In _arXiv preprint arXiv:2210.11027_ , 2022
  * [8] Mohammed Abouzaid and Thomas Kragh  “On the immersion classes of nearby Lagrangians”  In _Journal of Topology_ 9.1 Oxford University Press, 2016, pp. 232–244
  * [9] Mohammed Abouzaid, Mark McLean and Ivan Smith  “Complex cobordism, Hamiltonian loops and global Kuranishi charts”  In _arXiv preprint arXiv:2110.14320_ , 2021
  * [10] John Frank Adams  “Prerequisites (on equivariant stable homotopy) for Carlssons’s lecture”  In _Algebraic Topology Aarhus 1982: Proceedings of a conference held in Aarhus, Denmark, August 1–7, 1982_ , 2006, pp. 483–532  Springer
  * [11] John Frank Adams  “Stable homotopy and generalised homology”  University of Chicago press, 1974
  * [12] Pierre Albin and Richard Melrose  “Resolution of smooth group actions”  In _Contemp. Math_ 535, 2011, pp. 1–26
  * [13] Daniel Alvarez-Gavela, Yakov Eliashberg and David Nadler  “Positive arborealization of polarized Weinstein manifolds”  In _arXiv preprint arXiv:2011.08962_ , 2020
  * [14] Donald W Anderson and Luke Hodgkin  “The K-theory of Eilenberg-MacLane complexes”  In _Topology_ 7.3 Pergamon, 1968, pp. 317–329
  * [15] Natella Antonyan, Sergey A Antonyan and Alejandra Soria-Pérez  “Homotopy characterization of G-ANR’s”  In _Glasnik matematički_ 42.1 Hrvatsko matematičko društvo i PMF-Matematički odjel, Sveučilišta u Zagrebu, 2007, pp. 69–82
  * [16] SA Antonyan  “Equivariant generalization of Dugundji’s theorem”  In _Mathematical notes of the Academy of Sciences of the USSR_ 38 Springer, 1985, pp. 844–848
  * [17] Sergey A Antonyan and Erik Elfving  “The equivariant homotopy type of G-ANR’s for proper actions of locally compact groups”  In _Algebraic topology—old and new_ 85, 2009, pp. 155–178
  * [18] Michael Atiyah  “K-theory”  Philadelphia, PA: Westview Press, 1994
  * [19] Michael F Atiyah and Isadore M Singer  “Index theory for skew-adjoint Fredholm operators”  In _Publications Mathématiques de l’IHÉS_ 37, 1969, pp. 5–26
  * [20] Shaoyun Bai and Guangbo Xu  “Arnold conjecture over integers”  In _arXiv preprint arXiv:2209.08599_ , 2022
  * [21] Stefan Bauer and Mikio Furuta  “A stable cohomotopy refinement of Seiberg-Witten invariants: I”  In _Inventiones mathematicae_ 155 Springer, 2004, pp. 1–19
  * [22] Julia Bergner  “A model category structure on the category of simplicial categories”  In _Transactions of the American Mathematical Society_ 359.5, 2007, pp. 2043–2058
  * [23] Bhargav Bhatt, Matthew Morrow and Peter Scholze  “Topological Hochschild homology and integral p-adic Hodge theory”  In _Publications mathématiques de l’IHÉS_ 129.1 Springer, 2019, pp. 199–310
  * [24] Marcel Bökstedt, Wu Chung Hsiang and Ib Madsen  “The cyclotomic trace and algebraic K-theory of spaces”  In _Inventiones mathematicae_ 111 Springer, 1993, pp. 465–539
  * [25] Raoul Bott  “An application of the Morse theory to the topology of Lie-groups”  In _Bulletin de la Société Mathématique de France_ 84, 1956, pp. 251–281
  * [26] Kai Cieliebak, Klaus Mohnke, Kai Cieliebak and Klaus Mohnke  “Symplectic hypersurfaces and transversality in Gromov-Witten theory”  In _Journal of Symplectic Geometry_ 5.3 International Press of Boston, 2007, pp. 281–356
  * [27] Ralph L Cohen  “The Floer homotopy type of the cotangent bundle”  In _arXiv preprint math/0702852_ , 2007
  * [28] Ralph L Cohen and John DS Jones  “A homotopy theoretic realization of string topology”  In _Mathematische Annalen_ 324 Springer, 2002, pp. 773–798
  * [29] Ralph L Cohen, John DS Jones and Graeme B Segal  “Floer’s infinite dimensional Morse theory and homotopy theory”  In _The Floer memorial volume_ Springer, 1995, pp. 297–325
  * [30] Ralph L Cohen and Inbar Klang  “Twisted Calabi–Yau ring spectra, string topology, and gauge symmetry”  In _Tunisian Journal of Mathematics_ 2.1 Mathematical Sciences Publishers, 2019, pp. 147–196
  * [31] Ralph L Cohen and Alexander A Voronov  “Notes on string topology”  In _arXiv preprint math/0503625_ , 2005
  * [32] Charles Conley and Eduard Zehnder  “Morse-type index theory for flows and periodic solutions for Hamiltonian Equations”  In _Communications on pure and applied mathematics_ 37.2 Wiley Online Library, 1984, pp. 207–253
  * [33] Robert Connelly  “A New Proof of Brown’s Collaring Theorem”  In _Proceedings of the American Mathematical Society_ 27.1 JSTOR, 1971, pp. 180  DOI: 10.2307/2037284
  * [34] Laurent Côté and Yusuf Barış Kartal  “Equivariant Floer homotopy via Morse-Bott theory”  In _arXiv preprint arXiv:2309.15089_ , 2023
  * [35] Laurent Côté and Yusuf Barış Kartal  “Recovering generalized homology from Floer homology: the complex oriented case”  In _arXiv preprint arXiv:2404.02776_ , 2024
  * [36] Dieter Degrijse et al.  “Proper equivariant stable homotopy theory”  American mathematical society, 2023
  * [37] Aleksander Doan, Eleny-Nicoleta Ionel and Thomas Walpuski  “The Gopakumar-Vafa finiteness conjecture”  In _arXiv preprint arXiv:2103.08221_ , 2021
  * [38] Aleksander Doan and Thomas Walpuski  “Castelnuovo’s bound and rigidity in almost complex geometry”  In _Advances in Mathematics_ 379 Elsevier, 2021, pp. 107550
  * [39] William G Dwyer and Daniel M Kan  “Calculating simplicial localizations”  In _Journal of Pure and Applied Algebra_ 18.1 North-Holland, 1980, pp. 17–35
  * [40] Yakov Eliashberg and Nikolai M Mishachev  “Introduction to the $h$-principle”  American Mathematical Soc., 2002
  * [41] Yves Félix and Jean-Claude Thomas  “Rational BV-algebra in string topology”  In _Bulletin de la Société mathématique de France_ 136.2, 2008, pp. 311–327
  * [42] Andreas Floer and Helmut Hofer  “Coherent orientations for periodic orbit problems in symplectic geometry”  In _Mathematische Zeitschrift_ 212.1 Springer, 1993, pp. 13–38
  * [43] Kenji Fukaya and Yong-Geun Oh  “Zero-loop open strings in the cotangent bundle and Morse homotopy”  In _Asian Journal of Mathematics_ 1.1 International Press of Boston, 1997, pp. 96–180
  * [44] Sheel Ganatra  “Cyclic homology, $S^{1}$-equivariant floer cohomology, and calabi-yau structures”  In _arXiv preprint arXiv:1912.13510_ , 2019
  * [45] Sheel Ganatra, John Pardon and Vivek Shende  “Covariantly functorial wrapped Floer theory on Liouville sectors”  In _Publications mathématiques de l’IHÉS_ 131.1 Springer, 2020, pp. 73–200
  * [46] Sheel Ganatra et al.  “Integrality of mirror maps and arithmetic homological mirror symmetry for Greene–Plesser mirrors”, 2023  eprint: arXiv:2312.01949
  * [47] David B Gauld  “Topological properties of manifolds”  In _The American Mathematical Monthly_ 81.6 Taylor & Francis, 1974, pp. 633–636
  * [48] Viktor L Ginzburg  “The Conley conjecture”  In _Annals of mathematics_ JSTOR, 2010, pp. 1127–1180
  * [49] Viktor L Ginzburg and Başak Z Gürel  “Hamiltonian pseudo-rotations of projective spaces”  In _Inventiones mathematicae_ 214.3 Springer, 2018, pp. 1081–1130
  * [50] Detlef Gromoll and Wolfgang Meyer  “Periodic geodesics on compact Riemannian manifolds”  In _Journal of Differential Geometry_ 3.3-4 Lehigh University, 1969, pp. 493–510