for all }\;\;\gamma=\left(\begin{array}[]{cc}a&b\\\
-\overline{b}&\overline{a}\end{array}\right)\in\Gamma\subset\operatorname{SU}(2),$

where on the left hand side $\gamma$ is regarded as an isometry
$\mathscr{H}\rightarrow\mathscr{H}$. Using the above description and the
moment map equations (4.3)

$\displaystyle\mu_{\mathbb{C}}(\tau_{1},\tau_{2},\pi,\sigma)=$
$\displaystyle[\tau_{1},\tau_{2}]+\sigma\pi$
$\displaystyle\mu_{\mathbb{R}}(\tau_{1},\tau_{2},\pi,\sigma)=$
$\displaystyle[\tau_{1},\tau_{1}^{*}]+[\tau_{2},\tau_{2}^{*}]+\sigma\sigma^{*}-\pi^{*}\pi$

one easily verifies that $\mu_{\mathbb{R}}$ and $\mu_{\mathbb{C}}$ map
$N=M^{\Gamma}$ into $\mathfrak{u}(\mathscr{H})^{\Gamma}$ and
$\operatorname{End}_{\Gamma}(\mathscr{H})$, respectively. By the uniqueness of
the moment map it follows that $\mu_{N}$ is the restriction of $\mu$ to
$N=M^{\Gamma}$. This proves the first point.

Before we proceed to the second point recall that $U_{reg}\subset M$ was
defined to be the set of $m=(\tau_{1},\tau_{2},\pi,\sigma)$ for which

$\displaystyle A_{z}$
$\displaystyle=(\tau_{1}-z_{1},\tau_{2}-z_{2},\pi)^{t}\colon\mathscr{H}\oplus\mathscr{H}\oplus
E_{\infty}\rightarrow\mathscr{H}$ $\displaystyle B_{z}$
$\displaystyle=(-(\tau_{2}-z_{2}),\tau_{1}-z_{1},\sigma)\colon\mathscr{H}\oplus\mathscr{H}\oplus
E_{\infty}\rightarrow\mathscr{H}$

were injective and surjective, respectively, for all
$z=(z_{1},z_{2})\in\mathbb{C}^{2}$ (see (4.4)). Observe that the equation
$\mu_{\mathbb{R}}(m)=0$ may be rewritten as

$\tau_{1}\tau_{1}^{*}+\tau_{2}\tau_{2}^{*}+\sigma\sigma^{*}=\tau_{1}^{*}\tau_{1}+\tau_{2}^{*}\tau_{2}+\pi^{*}\pi,$

or equivalently $B_{0}B_{0}^{*}=A_{0}^{*}A_{0}$. From this it follows that

$\operatorname{Ker}(A_{0})=\operatorname{Ker}(A_{0}^{*}A_{0})=\operatorname{Ker}(B_{0}B_{0}^{*})=\operatorname{Ker}(B_{0}^{*}).$

Hence, $A_{0}$ is injective if and only if $B_{0}$ is surjective. In view of
the easily verified fact that for any $(z_{1},z_{2})\in\mathbb{C}^{2}$ we have
$\mu(\tau_{1}-z_{1},\tau_{2}-z_{2},\pi,\sigma)=\mu(\tau_{1},\tau_{2},\pi,\sigma)$,
the above argument applies to show that if $m\in\mu^{-1}(0)$, then for any
$z\in\mathbb{C}^{2}$, $A_{z}$ is injective if and only if $B_{z}$ is
surjective.

We may now verify the second claim as follows. As
$U_{reg}\cap\mu_{N}^{-1}(0)\subset\mu_{N}^{-1}(0)-\\{0\\}$, it suffices to
show that if $m=(\tau_{1},\tau_{2},\pi,\sigma)\in\mu_{N}^{-1}(0)-U_{reg}$ then
$m=0$. To this end assume that $m\in\mu_{N}^{-1}(0)-U_{reg}$. In view of the
above considerations this means that there exists
$z=(z_{1},z_{2})\in\mathbb{C}^{2}$ and $h\neq 0\in\mathscr{H}$ such that
$A_{z}(h)=0$. Then $h$ belongs to $\operatorname{Ker}\pi$ and is a common
eigenvector for $\tau_{1},\tau_{2}$ with eigenvalues $z_{1},z_{2}$
respectively. Using the equations in (4.11) we deduce that $\gamma\cdot h$ is
a common eigenvector for $\tau_{1},\tau_{2}$ with eigenvalues

$\left(\begin{array}[]{c}z_{1}^{\gamma}\\\
z_{2}^{\gamma}\end{array}\right)\coloneqq\left(\begin{array}[]{cc}a&b\\\
-\overline{b}&\overline{a}\end{array}\right)\left(\begin{array}[]{c}z_{1}\\\
z_{2}\end{array}\right)\;\;\mbox{ for all
}\;\;\gamma=\left(\begin{array}[]{cc}a&b\\\
-\overline{b}&\overline{a}\end{array}\right)\in\Gamma\subset\operatorname{SU}(2).$

Let $L\coloneqq\operatorname{Span}_{\mathbb{C}}\\{\gamma
h:\gamma\in\Gamma\\}$. This is a nontrivial $\Gamma$-invariant subspace of
$\mathscr{H}$ satisfying $\tau_{i}(L)\subset L$ for $i=1,2$ and $\pi(L)=0$.
Moreover, since $\gamma\cdot
h\in\operatorname{Ker}(A_{(z_{1}^{\gamma},z_{2}^{\gamma})})=\operatorname{Ker}(B_{(z_{1}^{\gamma},z_{2}^{\gamma})}^{*})$,
it follows that $\tau_{i}^{*}(\gamma h)=\overline{z_{i}^{\gamma}}h$ for
$i=1,2$ and $\sigma^{*}(\gamma h)=0$. Therefore, $\tau_{i}^{*}(L)\subset L$
for $i=1,2$ and $\sigma^{*}(L)=0$ as well. Hence, $\tau_{i}(L^{\perp})\subset
L^{\perp}$ for $i=1,2$ and $\sigma(E_{\infty})\subset L^{\perp}$.

To summarize, we have proved that there is an orthogonal splitting of
$\Gamma$-modules $\mathscr{H}=L\oplus L^{\perp}$ such that
$\tau_{i}=\tau_{i}^{\prime}\oplus\tau_{i}^{\prime\prime}$ preserves the
splitting for $i=1,2,$, $\pi(L)=0$ and $\operatorname{Im}(\sigma)\subset
L^{\perp}$. Pick $\lambda\neq 1\in U(1)$ and define $g\in
U(\mathscr{H})^{\Gamma}$ by setting $g=\lambda\operatorname{id}$ on $L$ and
$g=\operatorname{id}$ on $L^{\perp}$. Then
$(g\tau_{1}g^{-1},g\tau_{2}g^{-1},\pi
g^{-1},g\sigma)=(\tau_{1},\tau_{2},\pi,\sigma)$, so as $g\neq 1$ and
$U(\mathscr{H})^{\Gamma}$ acts freely on $\mu_{N}^{-1}(0)-\\{0\\}$ we conclude
that $m=0$ as desired. ∎

###### Proof of Proposition 4.1.

Given an adjacent pair $\alpha,\beta\in\mathcal{C}$ let $E\rightarrow S^{4}$
be the unique $\Gamma$-equivariant $\operatorname{SU}(2)$-bundle with
$[E_{N}]=\alpha$ and $[E_{S}]=\beta$ and minimal $c_{2}(E)[S^{4}]>0$. Then by
the above proposition the moduli space
$\widetilde{\mathcal{M}}_{\Gamma,\infty}(E)$ of $\Gamma$-invariant instantons
framed at $\infty=N$ may be identified with $\mathbb{R}\times
Y_{\Gamma^{\prime}}$ for some other finite subgroup
$\Gamma^{\prime}\subset\operatorname{SU}(2)$. Using the correspondence between
invariant instantons in $E\rightarrow S^{4}$ and instantons over
$\mathbb{R}\times\overline{Y}_{\Gamma}$, taking the framing and orientation
reversal into account, one obtains

$\mathbb{R}\times Y_{\Gamma^{\prime}}\cong
e_{-}^{-1}(*)\subset\widetilde{\mathcal{M}}_{z}(\alpha,\beta),$

where $e_{-}\colon\widetilde{\mathcal{M}}_{z}(\alpha,\beta)\rightarrow\alpha$
is the end-point map and $z$ is the homotopy class from $\alpha$ to $\beta$
corresponding to the relative Chern number $\hat{c}_{2}=c_{2}(E)/|\Gamma|$.
According to [Aus95] the natural translation action on the left hand side
corresponds to the translation action on the right hand side. Hence,

$Y_{\Gamma^{\prime}}\cong
e_{-}^{-1}(*)\subset\widetilde{\mathcal{M}}_{z}^{0}(\alpha,\beta).$

Since the end-point map is $\operatorname{SO}(3)$-equivariant it follows that
$\widetilde{\mathcal{M}}_{z}^{0}(\alpha,\beta)=e_{-}^{-1}(*)\cdot\operatorname{SO}(3)$
is compact. Therefore,
$\widetilde{\mathcal{M}}_{z}^{0}(\alpha,\beta)=\overline{\mathcal{M}}_{z}(\alpha,\beta)$,
and as
$\operatorname{gr}_{z}(\alpha,\beta)=\operatorname{dim}(e_{-}^{-1}(*))+1=4$,
we conclude that $z$ is indeed the unique homotopy class with
$-2\leq\operatorname{gr}_{z}(\alpha,\beta)\leq 5$.

To complete the proof we have to verify that
$\overline{\mathcal{M}}_{z}(\alpha,\beta)$ is empty when $\alpha$ and $\beta$
are not adjacent in $\mathcal{S}_{\Gamma}$. Let
$\alpha=\alpha_{0},\alpha_{1},\cdots,\alpha_{s}=\beta$ be the vertices of the
minimal edge path connecting $\alpha$ to $\beta$ in $\mathcal{S}_{\Gamma}$.
Let $z_{i}$ denote the unique homotopy class connecting $\alpha_{i}$ to
$\alpha_{i+1}$ with $\operatorname{gr}_{z_{i}}(\alpha_{i},\alpha_{i+1})=4$ for
each $0\leq i<s$. Let $\mathscr{H}_{i}$ be the minimal positive solution of
$(2-Q)\mathscr{H}_{i}=\alpha_{i}-\alpha_{i+1}$ for $0\leq i<s$. By Corollary
4.14 the minimal positive solution of $(2-Q)\mathscr{H}=\alpha-\beta$ is given
by $\mathscr{H}=\bigoplus_{i=0}^{s-1}\mathscr{H}_{i}$. Put
$k=\epsilon(\mathscr{H})$ and let $E^{l}\rightarrow S^{4}$ denote the
$\Gamma$-equivariant $\operatorname{SU}(2)$-bundle with invariants
$([E^{l}_{N}],[E^{l}_{S}],c_{2}(E^{l})[S^{4}])=(\alpha,\beta,k+|\Gamma|l)$ for
$l\in\mathbb{Z}$. The representation $\mathscr{H}_{l}$ used in the description
of the moduli space $\widetilde{\mathcal{M}}_{\Gamma,\infty}(E^{l})$ in the
equivariant ADHM correspondence is then given by
$\mathscr{H}_{l}=\mathscr{H}+lR$, where $R$ is the regular representation. The
dimension of the moduli space increases linearly with $l$ so, as
$\mathscr{H}_{0}=\mathscr{H}$ is the minimal positive solution of
$(2-Q)\mathscr{H}=\alpha-\beta$, we conclude that
$\widetilde{\mathcal{M}}_{\Gamma,\infty}(E^{l})$ is empty for $l<0$. This
space corresponds to the fiber
$e_{-}^{-1}(*)\subset\widetilde{\mathcal{M}}_{w}(\alpha,\beta)$ where
$w=z_{0}*z_{1}*\cdots*z_{s-1}$. By Theorem 2.1 part $(a)$ we have

$\operatorname{gr}_{w}(\alpha,\beta)=\sum_{i=0}^{s-1}\operatorname{gr}_{z_{i}}(\alpha_{i},\alpha_{i+1})=4s>4,$

since by assumption $s>1$. Consequently, the fiber
$e_{-}^{-1}(*)\subset\widetilde{\mathcal{M}}_{z}(\alpha,\beta)$ where $z$ is
the unique homotopy class with $\operatorname{gr}_{z}(\alpha,\beta)=0,4$,
corresponds to $\widetilde{\mathcal{M}}_{\Gamma,\infty}(E^{l})$ for some $l<0$
and must therefore be empty. This implies that the whole moduli space must be
empty and the proof is complete. ∎

###  4.3. The Structure of $DCI(\overline{Y}_{\Gamma})$

In this section we determine the complexes $DCI(\overline{Y}_{\Gamma})$
explicitly. Let $\Gamma\subset\operatorname{SU}(2)$ be a finite subgroup and
let $\mathcal{C}$ denote the set of gauge equivalence classes of flat
connections in the trivial $\operatorname{SU}(2)$-bundle over
$\overline{Y}_{\Gamma}$. In this section we make the convention that the
relative and absolute gradings

$\operatorname{gr}\colon\mathcal{C}\times\mathcal{C}\rightarrow\mathbb{Z}/8\;\;\mbox{
and }\;\;j\colon\mathcal{C}\rightarrow\mathbb{Z}/8,$

introduced after Theorem 2.1, are always defined with respect to the
orientation $\overline{Y}_{\Gamma}$ of $S^{3}/\Gamma$. Furthermore, the
vertices of the graph $\mathcal{S}_{\Gamma}$, introduced in Definition 3.9,
i.e., the set of isomorphism classes of $\operatorname{SU}(2)$-representations
of $\Gamma$, will be identified freely with the set $\mathcal{C}$.

######  Lemma 4.17.

For any pair $\alpha,\beta\in\mathcal{C}$ that are adjacent in
$\mathcal{S}_{\Gamma}$ it holds true that $\operatorname{gr}(\alpha,\beta)=4$.
For each $\alpha\in\mathcal{C}$ let $p(\alpha)$ denote the number of edges in
the minimal edge path connecting the trivial representation $\theta$ to
$\alpha$. Then $j(\alpha)\equiv 4p(\alpha)\ (\mathrm{mod}\ 8)$ for all
$\alpha\in\mathcal{C}$.

###### Proof.

Suppose that $\alpha$ and $\beta$ are adjacent in $\mathcal{S}_{\Gamma}$ and
let $z$ be the unique homotopy class with
$-2\leq\operatorname{gr}_{z}(\alpha,\beta)\leq 5$. Then according to Theorem
4.1 the fiber of
$e_{-}\colon\overline{\mathcal{M}}_{z}(\alpha,\beta)\rightarrow\alpha$ has
dimension $3$. Using the dimension formula (2.1) we deduce that

$\operatorname{gr}(\alpha,\beta)=\operatorname{dim}\overline{\mathcal{M}}_{z}(\alpha,\beta)-\operatorname{dim}(\alpha)+1=\operatorname{dim}e_{-}^{-1}(*)+1=4$

proving the first assertion. For the second let $\alpha\in\mathcal{C}$ be
arbitrary and let $\theta=\alpha_{0},\alpha_{1},\cdots,\alpha_{s}=\alpha$
denote the vertices in the minimal edge path connecting $\theta$ to $\alpha$
in $\mathcal{S}_{\Gamma}$. Then $p(\alpha)=s$ and by the additivity of the
grading

$j(\alpha)=\operatorname{gr}(\alpha,\theta)=\sum_{i=0}^{s-1}\operatorname{gr}(\alpha_{i+1},\alpha_{i})=4s\in
4\mathbb{Z}/8.$

Consequently, $j(\alpha)\equiv 4p(\alpha)\ (\mathrm{mod}\ 8)$ as required. ∎

To determine the differentials in $DCI(\overline{Y}_{\Gamma})$ we need the
following result, which is contained in the proof of [Aus95, Lemma 5.1].

######  Lemma 4.18.

Let $\alpha,\beta\in\mathcal{C}$ be adjacent in $\mathcal{S}_{\Gamma}$ with
$\beta$ irreducible, let $\mathscr{H}\in R(\Gamma)$ be the solution of
$(2-Q)\mathscr{H}=\alpha-\beta$ given by Proposition 4.12 and let
$\Gamma^{\prime}\subset\operatorname{SU}(2)$ be the corresponding finite
subgroup. Let $z$ be the unique homotopy class with
$\operatorname{gr}_{z}(\alpha,\beta)=4$. Then the degree of the map

$e_{-}^{-1}(*)=Y_{\Gamma^{\prime}}\rightarrow\beta=\operatorname{SO}(3)$

is up to a sign given by $2\dim_{\mathbb{C}}\mathscr{H}/|\Gamma^{\prime}|$.

######  Remark 4.19.

We are not aware of a procedure to pin down the orientation of
$Y_{\Gamma^{\prime}}$ in the above statement. For this reason we are only able
to determine the degree up to a sign. As we will see, this inaccuracy will not
affect our calculations in any relevant way.

For $i=0,1$ let $\mathcal{C}^{i}=\\{\alpha\in\mathcal{C}:j(\alpha)\equiv
4i\mod{8}\\}$ such that $\mathcal{C}=\mathcal{C}^{0}\cup\mathcal{C}^{i}$ is a
disjoint union. Let us fix generators $b_{\alpha}\in H_{0}(\alpha)$ for all
$\alpha\in\mathcal{C}$, and $t_{\alpha}\in H_{2}(\alpha)$ for all reducible
$\alpha$ and $t_{\alpha}:=b_{\alpha}u\in H_{3}(\alpha)$ for all irreducible
$\alpha$.

######  Definition 4.20.

Let $\mathcal{C}^{irr}\subset\mathcal{C}$ be the subset of irreducibles. For
each adjacent pair $\alpha,\beta\in\mathcal{C}$ with
$\beta\in\mathcal{C}^{irr}$ define $n_{\beta\alpha}$ to be the integer
$2\operatorname{dim}_{C}\mathscr{H}/|\Gamma^{\prime}|$ where $\mathscr{H}\in
R(\Gamma)$ is the minimal positive solution of the equation
$(2-Q)\mathscr{H}=\alpha-\beta$. If $\alpha$ and $\beta$ are not adjacent in
$\mathcal{S}_{\Gamma}$ we define $n_{\beta\alpha}=0$.

######  Theorem 4.21.

Let $\Gamma\subset\operatorname{SU}(2)$ be a finite subgroup. Then the
multicomplex $(DCI(\overline{Y}_{\Gamma})_{*,*},\\{\partial^{r}\\}_{r\geq 0})$
is given by

$DCI_{8s,*}=\bigoplus_{\alpha\in\mathcal{C}^{0}}H_{*}(\alpha)\mbox{ and
}DCI_{8s+4,*}=\bigoplus_{\beta\in\mathcal{C}^{1}}H_{*}(\beta)$

for all $s\in\mathbb{Z}$ and $DCI_{s,*}=0$ otherwise. The differentials are
given by $\partial^{r}=0$ for $r\neq 4$ and $\partial^{4}\colon
DCI_{4s,0}\rightarrow DCI_{4(s-1),3}$ is determined on generators by

$\partial^{4}(b_{\beta})=\sum_{\alpha\in\mathcal{C}^{irr}}n_{\alpha\beta}t_{\alpha}$

where the integers $n_{\alpha\beta}$ are defined above.

###### Proof.

The description of $DCI(Y,E)$ as a graded module is a simple consequence of
the definition and Lemma 4.17. By the final part of Theorem 4.1 the moduli
space $\overline{\mathcal{M}}_{z}(\alpha,\beta)$, where $z$ is the homotopy
class with $-2\leq\operatorname{gr}_{z}(\alpha,\beta)\leq 5$, is empty unless
$\alpha$ and $\beta$ are adjacent in $\mathcal{S}_{\Gamma}$. The statement
concerning the differentials therefore follows from Lemma 2.7 and Lemma 4.18.
∎

We will end this section by giving a few applications of Proposition 4.12 and
show how we can easily extract both the grading and the integers
$n_{\alpha\beta}$ provided we know where the one dimensional quaternionic
representations are placed in the graph $\overline{\Delta}_{\Gamma}$. All the
necessary information is contained in Appendix $A$.

######  Example 4.22.

($\Gamma=O^{*}$) There are four $1$-dimensional quaternionic representations;
two of real type $\theta,\eta$ and two of quaternionic type $\alpha$, $\beta$.
The graph $\overline{\Delta}_{O^{*}}$ is shown below.

$1$$2$$3$$4$$3$$2$$1$$2$$\theta$$\alpha$$\beta$$\eta$

The integers are the dimensions of the corresponding irreducible complex
representations and we have labeled the vertices corresponding to the
$1$-dimensional quaternionic representation. Here there are no representations
of complex type, in particular
$\overline{\Delta}_{\Gamma}=\overline{\Delta}_{\Gamma}/(\iota)$. One may
immediately conclude that the indexing is given by $j(\theta)=j(\beta)=0$ and
$j(\alpha)=j(\eta)=4$. To determine the integer $n_{\beta\eta}$ we need to
solve the equation $(2-Q)\mathscr{H}=\eta-\beta$ in $R(O^{*})$. According to
the recipe given in Proposition 4.12 we first delete the vertex corresponding
to $\beta$ and let $\Delta_{\mathscr{H}}$ be the component containing $\eta$;
that is, $\Delta_{\mathscr{H}}$ is just the single vertex corresponding to
$\eta$. We then recognize this graph as $\Delta_{C_{2}}$, from which we deduce
that $\Gamma^{\prime}=C_{2}$ and $\mathscr{H}=\frac{1}{2}\eta$. Therefore,
$n_{\beta\eta}=2\operatorname{dim}_{\mathbb{C}}\mathscr{H}/|\Gamma^{\prime}|=2/2=1$.
The situation with $\theta$ and $\alpha$ is completely symmetric so
$n_{\alpha\theta}=1$ as well.

The case of $n_{\beta\alpha}$ is more interesting. If we delete $\alpha$, the
component $\Delta_{\mathscr{H}}$ containing $\beta$ is given by

$1$$2$$2$$2$$1$$1$

We recognize this as the Dynkin graph $D_{6}$ which corresponds to the group
$D_{4}^{*}$ of order $16$. We find the integral weights of $\mathscr{H}$, as
given in the above graph, by knowledge of the dimensions of the irreducible
representations of $D_{4}^{*}$ (see Appendix $A.2$). From this we calculate

$\operatorname{dim}_{\mathbb{C}}\mathscr{H}=3\cdot 1+4\cdot 2+2\cdot 1+3\cdot
2+2\cdot 2+1\cdot 1=24$

and hence $n_{\beta\alpha}=2\cdot 24/16=3$. By symmetry $n_{\alpha\beta}=3$ as
well.

The calculation of $n_{\alpha\theta}$ and $n_{\beta\eta}$ in the above example
generalizes.

######  Lemma 4.23.

Let $\Gamma\subset\operatorname{SU}(2)$ be a finite subgroup and let $Q$ be
the canonical representation. Suppose $\rho$ is a $1$-dimensional complex
representation of real type and that $\rho\otimes Q$ is irreducible so that
$\eta=2\rho$ and $\beta=\rho\otimes Q$ correspond to a pair of distinct
vertices in $\mathcal{S}_{\Gamma}$. Then $n_{\beta\eta}=1$.

###### Proof.

We trivially have $(2-Q)\rho=2\rho-Q\otimes\rho=\eta-\beta$ so that
$\mathscr{H}=\rho$. This means that $\Delta_{\mathscr{H}}$ consists of a
single vertex and that the associated subgroup is $\Gamma^{\prime}=C_{2}$. It
follows that
$n_{\beta\eta}=2\operatorname{dim}_{\mathbb{C}}\mathscr{H}/|\Gamma^{\prime}|=2/2=1$.
∎

We also consider the most involved case $\Gamma=D_{n}^{*}$. The
$1$-dimensional quaternionic representations and the graph
$\overline{\Delta}_{D_{n}^{*}}$ are given in Appendix $A.2$. For $n$ even we
label the quaternionic representations by

$\theta,\eta_{1},\eta_{2},\eta_{3},\alpha_{1},\alpha_{2},\cdots,\alpha_{n/2}$

and for $n$ odd by

$\theta,\eta,\alpha_{1},\alpha_{2},\cdots,\alpha_{(n-1)/2},\lambda.$

Here $\theta,\eta$ and the $\eta_{i}$ are fully reducible, the $\alpha_{i}$
are irreducible and $\lambda$ is reducible.

######  Lemma 4.24.

In the above situation we have
$n_{\alpha_{i},\alpha_{i+1}}=n_{\alpha_{i+1},\alpha_{i}}=2$ for all $i$.

###### Proof.

Due to the symmetry in the graph $\overline{\Delta}_{D_{n}^{*}}$ it suffices
to show that $n_{\alpha_{i+1},\alpha_{i}}=2$ for all $i$. We therefore have to
solve $(2-Q)\mathscr{H}=\alpha_{i}-\alpha_{i+1}$ for $\mathscr{H}$. The
relevant portion of the graph $\overline{\Delta}_{D_{n}^{*}}$ is shown below

$1$$1$$2$$2$$2$$2$$2$$\alpha_{i}$$\alpha_{i+1}$

Note that $\alpha_{i}$ is the $2i-1$’th vertex labeled with $2$ from left to
right. Following the description in Proposition 4.12 we delete the vertex
corresponding to $\alpha_{i+1}$ and let $\Delta_{\mathscr{H}}$ be the
component containing $\alpha_{i}$. This graph has a total of $2(i+1)$ vertices
so we recognize $\Delta_{\mathscr{H}}=D_{2(i+1)}=\Delta_{D^{*}_{2i}}$. Thus
$\Gamma^{\prime}=D_{2i}^{*}$ of order $8i$. The weights of $\mathscr{H}$ are
therefore given by attaching the integer $2$ to each internal vertex (of
degree $\geq 2$) and $1$ to the rest. This yields

$n_{\alpha_{i+1}\alpha_{i}}=2\operatorname{dim}_{\mathbb{C}}\mathscr{H}/|D_{2i}^{*}|=2(1+1+4(2i-1)+2)/8i=2$

as required. ∎

We note that for cyclic groups there are no irreducible
$\operatorname{SU}(2)$-representations, from which it follows that
$DCI(\overline{Y}_{C_{m}})$ has trivial differential for each $m$. There are
now only a finite number of $n_{\alpha\beta}$ left to calculate. We state the
calculations of these below without proof and trust that the interested reader
will verify these for themselves.

For a finite subgroup $\Gamma\subset\operatorname{SU}(2)$ we attach labels to
the edges in the graph $\mathcal{S}_{\Gamma}$ to express all the information
needed to set up $DCI(\overline{Y}_{\Gamma})$. If $\alpha$ and $\beta$ are
adjacent and both irreducible we attach the symbol
$(n_{\alpha\beta}|n_{\beta\alpha})$ to the edge connecting $\alpha$ to $\beta$
such that $n_{\alpha\beta}$ is closest to $\alpha$. If $\alpha$ and $\beta$
are adjacent and only $\alpha$ is irreducible we attach $n_{\alpha\beta}$ to
the edge.

######  Proposition 4.25.

The graphs $\mathcal{S}_{\Gamma}$ with labels for the finite subgroups
$\Gamma\subset\operatorname{SU}(2)$ are given below.

$\theta$$\alpha$$\beta$$1$$(3|4)$$\theta$$\alpha$$\beta$$\eta$$1$$(3|3)$$1$$\theta$$\alpha$$\lambda$$1$$3$$\mathcal{S}_{I^{*}}$$\mathcal{S}_{O^{*}}$$\mathcal{S}_{T^{*}}$

$\theta$$\lambda_{1}$$\lambda_{2}$$\lambda_{m-1}$$\eta$$\theta$$\lambda_{1}$$\lambda_{2}$$\lambda_{m-1}$$\lambda_{m}$$\mathcal{S}_{C_{2m}}$$\mathcal{S}_{C_{2m+1}}$

$\mathcal{S}_{D^{*}_{2m}}$$\alpha_{1}$$\alpha_{2}$$\alpha_{m-1}$$\alpha_{m}$$\theta$$\eta_{1}$$\eta_{2}$$\eta_{3}$$(2|2)$$(2|2)$$1$$1$$1$$1$

$\mathcal{S}_{D^{*}_{2m+1}}$$\alpha_{1}$$\alpha_{2}$$\alpha_{m-1}$$\alpha_{m}$$\lambda$$\theta$$\eta$$(2|2)$$(2|2)$$2$$1$$1$

In all cases $\theta$ denotes the trivial representation. The letters $\alpha$
and $\beta$ are reserved for representation of quaternionic type, $\lambda$
for representations of complex type and $\eta$ for representations of real
type.

The notation for the $1$-dimensional quaternionic representations is
compatible with the notation in Appendix $A$.

######  Example 4.26.

The multicomplexes $(DCI(\overline{Y}_{\Gamma})_{*,*},\partial^{4})$ for
$\Gamma=T^{*},O^{*},I^{*}$ are shown below for $(s,t)$ with $0\leq s\leq 8$
and $0\leq t\leq 3$.

$1$$2$$3$$0$$R^{2}$$R$$0$$0$$0$$R$$0$$0$$R$$4$$R^{2}$$R$$0$$0$$8$$(1,3)$$(0)$$DCI(\overline{Y}_{T^{*}};R)$

$1$$2$$3$$0$$R^{2}$$0$$0$$R$$0$$R^{2}$$0$$0$$R$$4$$R^{2}$$0$$0$$R$$8$$(1,3)$$(1,3)$$DCI(\overline{Y}_{O^{*}};R)$

$1$$2$$3$$0$$R^{2}$$0$$0$$R$$0$$R$$0$$0$$R$$4$$R^{2}$$0$$0$$R$$8$$(1,3)$$(4)$$DCI(\overline{Y}_{I^{*}};R)$

##  5\. The Algebraic Construction of Equivariant Floer Homology

The equivariant instanton Floer homology associated with the trivial
$\operatorname{SU}(2)$-bundle over a closed oriented $3$-manifold $Y$ comes in
three flavors: $I^{+}(Y)$, $I^{-}(Y)$, $I^{\infty}(Y)$, the positive, negative
and Tate homology, respectively. These are all constructed algebraically from
the complex $\widetilde{CI}(Y)$, the associated action of
$C_{*}^{gm}(\operatorname{SO}(3);R)$ and the index filtration
$\\{F_{p}\widetilde{CI}(Y)\\}_{p\in\mathbb{Z}}$.

The construction proceeds as follows. First, following [Eis19, Appendix A], we
review the construction of four functors $C^{+}_{A}$, $C^{-}_{A}$,
$C^{(+,tw)}$ and $C^{\infty}_{A}$ from the category of right modules over a
differential graded $R$-algebra $A$ to the category of differential graded
$R$-modules. In fact, $C^{-}_{A}(R)$ will carry the structure of a
differential graded algebra and the three functors take values in the
subcategory of left $C^{-}_{A}(R)$-modules. Under the assumptions that the
ground ring $R$ is a principal ideal domain and the algebra $A$ is degreewise
free over $R$, the four functors will be exact and preserve quasi-
isomorphisms.

Next, we promote the functors to the category of filtered right $A$-modules.
If $M$ is a right $A$-module equipped with an increasing filtration

$\cdots\subset F_{p}M\subset F_{p+1}M\subset\cdots\subset M$

of $A$-submodules, the exactness of the functors ensure that
$C^{\bullet}_{A}(M)$ is naturally filtered by
$F_{p}C^{\bullet}_{A}(M):=C^{\bullet}_{A}(F_{p}M)$ for
each$\bullet\in\\{+,-,(+,tw),\infty\\}$. However, this filtration may not be
very well behaved. To remedy this issue we pass to the so-called full
completion, thereby obtaining four new functors $\hat{C}^{\bullet}_{A}$,
$\bullet\in\\{(+,-,(+,tw),\infty\\}$ from the category of filtered right
$A$-modules to the category of filtered left $C^{-}_{A}(R)$-modules; to be
specific,

$\hat{C}^{\bullet}_{A}(M)=\operatorname{lim}_{q}\operatorname{colim}_{p}C^{\bullet}_{A}(F_{p}M/F_{q}M).$

For $A=C^{gm}_{*}(\operatorname{SO}(3);R)$ the groups $I^{\bullet}(Y)$ for
$\bullet\in\\{+,-,\infty\\}$ are obtained by applying $\hat{C}^{\bullet}_{A}$
to $M=\widetilde{CI}(Y)$ equipped with the index filtration and then passing
to homology. These groups will then be modules over $H(C^{-}_{A}(R))$, which,
provided $\frac{1}{2}\in R$, is isomorphic to the polynomial algebra $R[U]$
with a single generator $U$ of degree $-4$. The mod $8$ periodicity of $M$
will be preserved so that $I^{\bullet}_{n}\cong I^{\bullet}_{n+8}$ for all
$n\in\mathbb{Z}$.

The above constructions are also functorial in a suitable sense depending on
$\bullet\in\\{+,-,\infty\\}$, in the differential graded algebra $A$.
Moreover, this functoriality will preserve quasi-isomorphisms in the situation
of interest; that is, we may equally well calculate $I^{\bullet}(Y)$ using
$A=\Lambda_{R}[u]$ and $M=DCI(Y)$. When we proceed to the calculations for
binary polyhedral spaces in the next section, we will indeed work in this
context.

###  5.1. Conventions

Throughout $R$ will be a fixed principal ideal domain. A module or algebra
will always mean an $R$-module or $R$-algebra, and the same applies in the
graded or differential graded setting. We will work with homological gradings
and differentials $\partial$ of degree $-1$. We follow the conventions on
graded and differential graded (DG) modules given in [Mac95, Chapter VI]. We
briefly recall the most central concepts. If $M$ and $N$ are DG modules then
$M\otimes N$ and $\operatorname{Hom}(M,N)$ are the DG modules given by

$(M\otimes N)_{n}=\bigoplus_{p+q=n}M_{n}\otimes N_{q}\;\mbox{ and
}\;\operatorname{Hom}(M,N)_{n}=\prod_{q-p=n}\operatorname{Hom}(M_{p},N_{q})$

with differentials $\partial_{M\otimes N}=\partial_{M}\otimes
1+1\otimes\partial_{N}$ and
$\partial_{\operatorname{Hom}(M,N)}=\operatorname{Hom}(1,\partial_{N})-\operatorname{Hom}(\partial_{M},1)$.
This means that $\partial_{M\otimes N}(a\otimes b)=\partial_{M}a\otimes
b+(-1)^{|a|}a\otimes\partial_{N}b$ and
$\partial_{\operatorname{Hom}(M,N)}(f)=\partial_{N}\circ
f-(-1)^{|f|}f\circ\partial_{M}$, where we write $|x|$ for the degree of an
element in a graded module. There is a natural interchange isomorphism
$\tau:M\otimes N\cong N\otimes M$ given by $m\otimes
n\mapsto(-1)^{|m||n|}n\otimes m$.

For each integer $p\in\mathbb{Z}$ let $R[p]$ denote the complex with a single
copy of $R$ concentrated in degree $p$. For a DG module $M$ the shifted
complex $M[p]$ is defined to be $R[p]\otimes M$. Therefore, $M[p]_{n+p}=M_{n}$
for all $n\in\mathbb{Z}$ with differential
$\partial_{M[p]}=(-1)^{p}\partial_{M}$. It is also possible to define
$M[p]=M\otimes R[p]$, but this will result in different sign conventions.

A differential graded algebra is a graded algebra $A$ equipped with a
differential satisfying the Leibniz rule
$\partial_{A}(ab)=\partial_{A}(a)b+(-1)^{|a|}a\partial_{A}b$ for all $a,b\in
A$. We will always assume that $A$ is unital, i.e., there is a unit $1\in
A_{0}$ for the multiplication. A left or right $A$-module is a graded left or
right $A$-module equipped with a differential satisfying a corresponding
Leibniz rule. This ensures that $H(M)$ is a graded $H(A)$-module. If $M$ is a
left or right $A$-module then so is $M[p]$. For $a\in A$ and $x\in
M[p]_{n}=M_{n-p}$ the induced product is given by $a\cdot x=(-1)^{|a|p}ax$ in
the left module context and by $x\cdot a=xa$ in the right module context. This
is a consequence of our left shifting (suspension) convention.

We also mention that there is a natural adjunction isomorphism of DG modules

$\operatorname{Hom}(M\otimes
N,P)\cong\operatorname{Hom}(M,\operatorname{Hom}(N,P))$

given by $(f\colon M\otimes N\rightarrow P)\mapsto(g\colon
M\rightarrow\operatorname{Hom}(N,P))$ where $f(m\otimes n)=g(m)(n)$. This
isomorphism admits various natural generalizations to the case where $M$, $N$
and $P$ carry additional module structures over various DG algebras, see
[Mac95, Chapter VI.8]. All the signs occurring in the upcoming theory can be
deduced from these conventions: the interchange morphism, the natural
adjunction and the left shift convention.

Finally, a homomorphism $f\colon M\rightarrow N$ of DG modules will mean a
degree $0$ chain map unless otherwise stated. We say that $f$ is a quasi-
isomorphism if the induced map in homology, $H(f)\colon H(M)\rightarrow H(N)$,
is an isomorphism.

###  5.2. The Bar Construction and the Functors $C^{\pm}_{A}$

Let $A$ be a unital DG algebra and assume that $A$ is equipped with an
augmentation $\epsilon\colon A\rightarrow R$, that is, $\epsilon$ is a map of
DG algebras where $R$ is regarded as a DG algebra concentrated in degree $0$.
The augmentation gives $R$ the structure of an $A$-bimodule. We define
$\bar{A}=A/R\cdot 1$ and note that there is a canonical identification of
$\bar{A}$ with the augmentation ideal $\operatorname{Ker}(\epsilon)$. Given a
right $A$-module $M$ and a left $A$-module $N$ define the bigraded module
$B_{*,*}(M,A,N)$ by

$B_{p,q}(M,A,N)=(M\otimes\bar{A}^{\otimes p}\otimes N)_{q}.$

We adopt the standard notation and write

$m[a_{1}|a_{2}|\cdots|a_{p}]n\coloneqq m\otimes a_{1}\otimes
a_{2}\otimes\cdots\otimes a_{p}\otimes n\in B_{p,q}(M,A,N).$

Here, $p$ is the simplicial degree,
$q=|m|+\left(\sum_{i=1}^{p}|a_{i}|\right)+|n|$ is the internal degree and
$p+q$ is the total degree. Let $\partial^{s}\colon B_{p,q}(M,A,N)\rightarrow
B_{p-1,q}(M,A,N)$ and $\partial^{i}\colon B_{p,q}(M,A,N)\rightarrow
B_{p,q-1}(M,A,N)$ act on $x=m[a_{1}|\cdots|a_{p}]n$ by

$\displaystyle\partial^{s}(x)=$
$\displaystyle\;ma_{1}[a_{2}|\cdots|a_{p}]n+\sum_{i=1}^{p-1}(-1)^{i}m[a_{1}|\cdots|a_{i}a_{i+1}|\cdots|a_{p}]n$
$\displaystyle\;+$ $\displaystyle(-1)^{p}m[a_{1}|\cdots|a_{p-1}]a_{p}n$
$\displaystyle\partial^{i}(x)=$
$\displaystyle\;(-1)^{p}(\partial_{M}m[a_{1}|\cdots|a_{p}]n+\sum_{i=1}^{p}(-1)^{\epsilon_{i-1}}m[a_{1}|\cdots|\partial
a_{i}|\cdots|a_{p}]n$ $\displaystyle+$
$\displaystyle\;(-1)^{\epsilon_{p}}m[a_{1}|\cdots|a_{p}]\partial_{N}n),$

where $\epsilon_{i}=|m|+|a_{1}|+\cdots+|a_{i}|$. Note that $\partial^{i}$ is
the usual tensor product differential in $M\otimes\bar{A}^{\otimes p}\otimes
N$ twisted by a sign depending on the simplicial degree. We call
$\partial^{s}$ the simplicial differential and $\partial^{i}$ the internal
differential. It holds true that $(\partial^{i})^{2}=0$,
$(\partial^{s})^{2}=0$ and
$\partial^{i}\partial^{s}+\partial^{s}\partial^{i}=0$. This ensures that the
triple $(B_{*,*}(M,A,N),\partial^{s},\partial^{i})$ is a double complex.

######  Definition 5.1.

[GM74, Appendix A] The bar construction $B(M,A,N)$ is the total complex
associated with the double complex
$(B_{*,*}(M,A,N),\partial^{s},\partial^{i})$ introduced above, that is,

$B_{k}(M,A,N)=\bigoplus_{p+q=n}B_{p,q}(M,A,N)\;\mbox{ and
}\;\partial=\partial^{s}+\partial^{i}.$

The bar construction is functorial in the triple $(M,A,N)$. For a triple
$(f,g,h)\colon(M,A,N)\rightarrow(M^{\prime},A^{\prime},N^{\prime})$ where $g$
is a map of augmented DG algebras and $f,h$ are chain maps linear along $g$,
there is an induced map $B(f,g,h)\colon B(M,A,N)\rightarrow
B(M^{\prime},A^{\prime},N^{\prime})$ of DG modules. There is also a natural
augmentation $B(M,A,N)\rightarrow M\otimes_{A}N$ given by
$m[a_{1}|\cdots|a_{p}]n\mapsto 0$ for $p>0$ and $m[]n\mapsto m\otimes n$.

######  Remark 5.2.

The DG module $B(M,A,A)$ inherits a right $A$-module structure from the last
factor. The augmentation $B(M,A,A)\rightarrow M$ is a map of right $A$-modules
and should be regarded as an analogue of a projective resolution of $M$.
Moreover, $B(M,A,N)\cong B(M,A,A)\otimes_{A}N$, so that $B(M,A,N)$ is a
derived tensor product $M\otimes^{L}_{A}N$ in a certain sense. These
statements are made precise in the language of model categories in [BMR14,
Section 10.2].

Following [Eis19] we introduce the notation

$BA\coloneqq B(R,A,R)\;\;\mbox{ and }\;\;EA\coloneqq B(R,A,A).$

Furthermore, we write $[a_{1}|a_{2}|\cdots|a_{p}]\coloneqq
1_{R}[a_{1}|a_{2}|\cdots|a_{p}]1_{R}\in BA$ and similar conventions apply for
$B(R,A,N)$ and $B(M,A,R)$.

######  Lemma 5.3.

For any right $A$-module $M$ and left $A$-module $N$ there is a natural map of
DG modules $\psi_{M,N}\colon B(M,A,N)\rightarrow B(M,A,R)\otimes B(R,A,N)$
given by

$m[a_{1}|\cdots|a_{p}]n\mapsto\sum_{i=0}^{p}(-1)^{(p-i)\epsilon_{i}}m[a_{1}|\cdots|a_{i}]\otimes[a_{i+1}|\cdots|a_{p}]n,$

where $\epsilon_{i}=|m|+|a_{1}|+\cdots+|a_{i}|$. The map $\psi_{R,R}\colon
BA\rightarrow BA\otimes BA$ along with the augmentation $BA\rightarrow R$ give
$BA$ the structure of a DG coalgebra. The map $\psi_{M,R}$ gives $B(M,A,R)$
the structure of a right DG $BA$-comodule and the map $\psi_{R,N}$ gives
$B(R,A,N)$ the structure of a left DG $BA$-comodule.

######  Definition 5.4.

[Eis19, Definition A.5] Let $A$ be an augmented DG algebra and let $M$ be a
right $A$-module. Define the positive and negative $A$-chains of $M$ to be

$C^{+}_{A}(M)\coloneqq B(M,A,R)\;\;\mbox{ and
}\;\;C^{-}_{A}(M)\coloneqq\operatorname{Hom}_{A}(EA,M).$

The homology of $C^{\pm}_{A}(M)$ is denoted by $H^{\pm}_{A}(M)$ and is called
the positive and negative $A$-homology of $M$, respectively.

The above constructions are functorial in the following sense. A homomorphism
$f\colon A\rightarrow B$ of augmented DG algebras induces maps

$f_{*}\colon C^{+}_{A}(f^{-1}M)\rightarrow C^{+}_{B}(M)\;\mbox{ and
}\;f^{*}\colon C^{-}_{B}(M)\rightarrow C^{-}_{A}(f^{-1}M)$

where $M$ is a $B$-module and $f^{-1}M$ is the $A$-module obtained from $M$ by
restriction along $f$. Furthermore, a homomorphism $g\colon M\rightarrow N$ of
$A$-modules induces maps

$g^{+}\colon C^{+}_{A}(M)\rightarrow C^{+}_{A}(N)\;\mbox{ and }\;g^{-}\colon
C^{-}_{A}(M)\rightarrow C^{-}_{A}(N).$

The following invariance result is proven in [Eis19, Theorem A.1,A.2]; it
relies on the standing assumption that $R$ is a PID.

######  Proposition 5.5.

Let $f\colon A^{\prime}\rightarrow A$ is a homomorphism of augmented DG
algebras and let $g\colon M\rightarrow M^{\prime}$ be a homomorphism of
$A$-modules.

  1. (i)

If $A$ and $A^{\prime}$ are degreewise free and $f$ is a quasi-isomorphism,
then $f_{*}$ and $f^{*}$ are quasi-isomorphisms.

  2. (ii)

If $A$ is degreewise free and $g$ is a quasi-isomorphism, then $g^{+}$ and
$g^{-}$ are quasi-isomorphisms.

The DG modules $C^{\pm}_{A}(M)$ carry additional structure. First, we have the
following sequence of isomorphisms of DG modules

$C_{A}^{-}(R)=\operatorname{Hom}_{A}(EA,\epsilon^{-1}(R))\cong\operatorname{Hom}_{R}(EA\otimes_{A}R,R)\cong\operatorname{Hom}(BA,R),$

which identifies $C_{A}^{-}(R)$ with the $R$-dual of $BA$. The DG coalgebra
structure of $BA$ dualizes to give $C_{A}^{-}(R)$ the structure of a DG
algebra. Next, for any right $A$-module $M$ the DG modules $C^{\pm}_{A}(M)$
obtain left $C_{A}^{-}(R)$-module structures in the following way. Write
$BA^{\vee}=\operatorname{Hom}_{R}(BA,R)=C_{A}^{-}(R)$, then the structure map
$C_{A}^{-}(R)\otimes C^{+}_{A}(M)\rightarrow C^{+}_{A}(M)$ is given by the
following composition

(5.1) ${BA^{\vee}\otimes C^{+}_{A}(M)}$${BA^{\vee}\otimes C^{+}_{A}(M)\otimes
BA}$${C^{+}_{A}(M)\otimes BA^{\vee}\otimes
BA}$${C^{+}_{A}(M)}$$\scriptstyle{1\otimes\psi_{M}}$$\scriptstyle{\tau\otimes
1}$$\scriptstyle{1\otimes\operatorname{ev}}$

where $\psi_{M}=\psi_{M,R}$ is the comodule structure map of
$C^{+}_{A}(M)=B(M,A,R)$ in Lemma 5.3, $\tau$ is the interchange isomorphism
$a\otimes b\mapsto(-1)^{|a||b|}b\otimes a$ and $\operatorname{ev}\colon
BA^{\vee}\otimes BA\rightarrow R$ is the evaluation map. In the other case the
structure map $C_{A}^{-}(R)\otimes C^{-}_{A}(M)\rightarrow C_{A}^{-}(M)$ is
given by the following composition

(5.2)
${BA^{\vee}\otimes\operatorname{Hom}_{A}(EA,M)}$${\operatorname{Hom}_{A}(BA\otimes
EA,M)}$${\operatorname{Hom}_{A}(EA,M)}$$\scriptstyle{\zeta}$$\scriptstyle{\psi_{EA}^{*}}$

where $\zeta(f\otimes g)(a\otimes b)=(-1)^{|a||g|}f(a)\cdot g(b)$ and
$\psi_{EA}^{*}=\operatorname{Hom}_{A}(\psi_{R,A},1)$ is the dual of the
comodule structure map $\psi_{R,A}\colon EA\rightarrow BA\otimes EA$.

The properties of the functors $C^{\pm}_{A}$ that will be important for us are
summarized in the following theorem. Here the morphisms in the category of
right or left modules over a DG algebra $A$ are taken to be the $A$-linear
chain maps of degree $0$.

######  Theorem 5.6.

Let $A$ be a degreewise $R$-free augmented DG algebra. Then the assignments

$M\mapsto C^{\pm}_{A}(M)\;\mbox{ and }\;(g\colon M\rightarrow
M^{\prime})\mapsto(g^{\pm}\colon C^{\pm}_{A}(M)\rightarrow
C^{\pm}_{A}(M^{\prime})$

define functors $C^{\pm}_{A}$ from the category of right $A$-modules to the
category of left $C_{A}^{-}(R)$-modules. Moreover, these functors preserve
quasi-isomorphisms and short exact sequences.

We will now treat the case $A=\Lambda_{R}[u]$ with $|u|=3$ in detail. These
calculations will be important for later purposes and will serve as an
excellent example of how the above constructions work. We should note that all
of these calculations easily generalize to the case where the degree of $u$ is
any odd number.

We will write $R[U]$ for the graded polynomial algebra on a single generator
$U$ of even non-zero degree. This is also a coalgebra with counit given by the
identity in degree $0$ and diagonal determined by

$U^{p}\mapsto\sum_{i=0}^{p}U^{i}\otimes U^{p-i}$

where $U^{0}=1$. This structure is compatible with the algebra structure and
turns $R[U]$ into a Hopf algebra, but this will not be important for us.

######  Lemma 5.7.

Let $A=\Lambda_{R}[u]$ with $|u|=3$. Then there is an isomorphism of DG
coalgebras $BA\cong R[V]$ where $|V|=4$, and an isomorphism of DG algebras
$C_{A}^{-}(R)=BA^{\vee}\cong R[U]$ where $|U|=-4$. In particular, these
complexes have trivial differentials so that $H^{+}_{A}(R)=R[V]$ and
$H^{-}_{A}(R)=R[U]$. The first isomorphism is given by

$V^{p}\mapsto(-1)^{p(p+1)/2}[u|\cdots|u]\;\;\mbox{ (p times) }$

for $p\geq 0$. The second is obtained by dualization of the first where
$U^{p}$ is determined by the evaluation $U^{p}(V^{p})=1$ for all $p\geq 0$.
Furthermore, the left $BA^{\vee}$-module structure of $BA$ is given by
$U^{s}\cdot V^{p}=V^{p-s}$, where we interpret $V^{i}=0$ for $i<0$.

###### Proof.

We have $A=\Lambda_{R}[u]$ so that $A_{0}=R$, $A_{3}=Ru$ and $A_{n}=0$
otherwise. The augmentation is necessarily given by the identity in degree
$0$. Hence, $\overline{A}=Ru$ concentrated in degree $3$. From the definition
of $B(R,A,R)$ we find that $BA_{p,3p}=R\cdot Z^{p}$ for $p\geq 0$, where
$Z^{p}=[u|\cdots|u]$ with $u$ repeated $p$ times, and $B_{p,q}=0$ otherwise.
It follows immediately that the differential vanishes identically. Using the
formula for the coalgebra structure map in Lemma 5.3 we find that

$\psi_{R,R}(Z^{p})=\sum_{i=0}^{p}(-1)^{(p-i)3i}Z^{i}\otimes Z^{p-i}.$

Define $V^{p}=(-1)^{p(p+1)/2}Z^{p}$ for $p\geq 0$. Then using the identity

$\frac{p(p+1)}{2}=(p-i)i+\frac{i(i+1)}{2}+\frac{(p-i)(p-i+1)}{2}$

we deduce that the above formula transforms into
$\psi_{R,R}(V^{p})=\sum_{i=0}^{p}V^{i}\otimes V^{p-i}$. Hence, $BA\cong R[V]$
as DG coalgebras with trivial differentials.

Define $U^{p}\in BA^{\vee}$ to be the dual of $V^{p}$, i.e., $U^{p}\colon
BA\rightarrow R$ has degree $-4p$ and acts by $U^{p}(V^{p})=1$ and vanishes
otherwise. The differential is still of course trivial. The product
$U^{s}\cdot U^{t}$ has degree $-4(s+t)$ and must therefore be an $R$-multiple
of $U^{s+t}$. The calculation

$(U^{s}\cdot U^{t})(V^{s+t})=(U^{s}\otimes
U^{t})\left(\sum_{i=0}^{s+t}V^{i}\otimes
V^{s+t-i}\right)=U^{s}(V^{s})U^{t}(V^{t})=1$

shows that $U^{s}\cdot U^{t}=U^{s+t}$. Hence, $C_{A}^{-}(R)\cong
BA^{\vee}\cong R[U]$ as DG algebras with trivial differentials. Finally, we
need to check that the left module structure $BA^{\vee}\otimes BA\rightarrow
BA$ is given by $U^{s}\cdot V^{p}=V^{p-s}$. Using the definition we find

$U^{s}\cdot V^{p}=\sum_{i=0}^{p}(-1)^{|U^{s}||V^{i}|}V^{i}\cdot
U^{s}(V^{p-i})=\left\\{\begin{array}[]{cl}V^{p-s}&\mbox{ if }p\geq s\\\
0&\mbox{ otherwise.}\end{array}\right.$

as required. ∎

We will also need explicit calculations of $H^{\pm}_{A}(M)$ when
$M=H_{*}(\alpha;R)$ for $\alpha=\operatorname{SO}(3)/\operatorname{SO}(2)$ or
$\alpha=\operatorname{SO}(3)$, under the assumption $\frac{1}{2}\in R$. In
other words, $M=R\oplus R[2]$ with the trivial $A$-module structure or
$M=A=\Lambda_{R}[u]$ (see (2.2)). For this purpose we give concrete models for
the complexes $C^{\pm}_{A}(M)$. Note that a right $\Lambda_{R}[u]$-module $M$
is simply a DG module equipped with a degree $3$ map $u:M\rightarrow M$
satisfying $u^{2}=0$ and $u\partial_{M}=\partial_{M}u$ (because $A$ acts from
the right).

######  Proposition 5.8.

Let $M$ be a right $A=\Lambda_{R}[u]$-module. Define bigraded modules
$D^{\pm}_{*,*}$ by

$D^{+}_{p,q}=\left\\{\begin{array}[]{cl}M_{q-3p}&p\geq 0\\\
0&p<0\end{array}\right.\;\mbox{ and
}\;D^{-}_{p,q}=\left\\{\begin{array}[]{cl}M_{q-3p}&p\leq 0\\\
0&p>0\end{array}\right..$

Define $U\colon D^{\pm}_{p,q}\rightarrow D^{\pm}_{p-1,q-3}$ to be the identity
$M_{q-3p}\rightarrow M_{q-3p}$ when $p\geq 1$ and $p\leq 0$, respectively, and
otherwise to be zero. Define

$\displaystyle\partial^{\prime}=(-1)^{p+q+1}u\colon D^{\pm}_{p,q}$
$\displaystyle=M_{q-3p}\rightarrow M_{q-3p+3}=D^{\pm}_{p-1,q}$
$\displaystyle\partial^{\prime\prime}=\partial_{M}\colon D^{\pm}_{p,q}$
$\displaystyle=M_{q-3p}\rightarrow M_{q-3p-1}=D^{\pm}_{p,q-1},$

in the range it makes sense. Then
$(D^{+}_{*,*},\partial^{\prime},\partial^{\prime\prime})$ and
$(D^{-}_{*,*},\partial^{\prime},\partial^{\prime\prime})$ are double complexes
and there are natural isomorphisms of DG $R[U]$-modules

$\operatorname{Tot}^{\oplus}(D^{+}_{*,*},\partial^{\prime},\partial^{\prime\prime})\cong
C^{+}_{A}(M)\;\;\mbox{ and
}\;\;\operatorname{Tot}^{\Pi}(D^{-}_{*,*},\partial^{\prime},\partial^{\prime\prime})\cong
C^{-}_{A}(M).$

###### Proof.

We have $\bar{A}=R\cdot u$ with $u$ in degree $3$. Therefore,
$B_{p,q}=(M\otimes\bar{A}^{\otimes p})_{q}\cong
M_{q-3p}\otimes\bar{A}^{\otimes p}_{3p}\cong M_{q-3p}$. The isomorphism is
given by $m\mapsto m[u|u|\cdots|u]$ with $u$ repeated $p$ times. For
simplicity of notation write this as $m\otimes Z^{p}$. Then, using the
formulas for the differentials, we find $\partial^{s}(m\otimes
Z^{p})=mu\otimes Z^{p-1}$ and $\partial^{i}(m\otimes
Z^{p})=(-1)^{p}\partial_{M}m\otimes Z^{p}$. The action of $U$ is given by
first applying the comodule structure map to $m\otimes Z^{p}$, and then
evaluating $U$ on the right factors in the resulting sum ($U$ has even degree
so the interchange introduces no sign). We have

$\psi_{M,R}(m\otimes Z^{p})=\sum_{i=0}^{p}(-1)^{(p-i)(|m|+3i)}(m\otimes
Z^{i})\otimes Z^{p-i}$

so that $U\cdot(m\otimes Z^{p})=(-1)^{|m|+p-1}(m\otimes Z^{p-1})\cdot
U(Z^{1})=(-1)^{|m|+p}m\otimes Z^{p-1}$ as $U$ was defined to be the dual of
$V^{1}=-Z^{1}$. It is now easy to check that when we introduce the sign change
$m\in D^{+}_{p,q}=M_{q-3p}\mapsto(-1)^{p(p+1)/2+p|m|}m\otimes Z^{p}$ the $U$
action and the differentials are given as in the proposition.

In the second case, $EA=B(R,A,A)$ has generators over $R$ of the form
$Z^{p}=[u|u|\cdots|u]$ and $Z^{p}\otimes u=[u|u|\cdots|u]u$ in degrees $4p$
and $4p+3$ respectively. It is convenient as earlier to introduce the sign
twist $V^{p}=(-1)^{p(p+1)/2}Z^{p}$, since then the differential is given by
$\partial(V^{p})=V^{p-1}\otimes u$ and $\partial(V^{p}\otimes u)=0$ and the
comodule structure map $\psi_{R,A}\colon EA\rightarrow BA\otimes EA$ is given
by

$\psi_{R,A}(V^{p})=\sum_{i=0}^{p}V^{i}\otimes V^{p-i}.$

An element $f\in\operatorname{Hom}_{A}(EA,M)_{n}$ is by $A$-linearity uniquely
determined by the elements $f(V^{p})\in M_{n+4p}$ for $p\geq 0$. The
differential is given by

$(\partial f)(V^{p})=\partial_{M}(f(V^{p}))-(-1)^{|f|}f(\partial
V^{p})=\partial_{M}(f(V^{p}))-(-1)^{|f|}f(V^{p-1})u$

and the action of $U$ is given by

$(U\cdot f)(V^{p})=(U\otimes f)(\psi_{R,A}(V^{p}))=(U\otimes f)(V\otimes
V^{p-1})=f(V^{p-1}).$

It is now easy to check that the complex
$C^{-}_{A}(M)=\operatorname{Hom}_{A}(EA,M)$ can be described as the product
totalization of the double complex given in the proposition. ∎

For later reference we record the following facts from the above proof. The
isomorphism
$\operatorname{Tot}^{\oplus}(D^{+},\partial^{\prime},\partial^{\prime\prime})\cong
C^{+}_{A}(M)$ is given by sending an element $m\in M_{q-3p}=D^{+}_{p,q}$ to

(5.3) $(-1)^{p(q-p)+p(p+1)/2}m\overbrace{[u|u|\cdots|u]}^{p}\in
B(M,A,R)_{p,q}$

and the isomorphism
$C^{-}_{A}(M)\cong\operatorname{Tot}^{\Pi}(D^{-},\partial^{\prime},\partial^{\prime\prime})$
is given by sending an element $f\in\operatorname{Hom}_{A}(EA,M)_{n}$ to

(5.4) $((-1)^{p(p+1)/2}f(\overbrace{[u|u|\cdots|u]}^{p}))_{p\geq
0}\in\prod_{p\geq 0}M_{n+4p}=\prod_{p\geq 0}D^{-}_{-p,n+p}.$

It should be noted that there are other choices of signs possible. We have
chosen the signs that ensure that the action of $U$ corresponds to the
identity map. A part of the double complex
$(D^{+}_{*,*},\partial^{\prime},\partial^{\prime\prime})$ corresponding to a
right $A$-module $M$ is shown below

${\vdots}$${\vdots}$${\vdots}$${0}$${M_{2}}$${M_{-1}}$${M_{-4}}$${\cdots}$${0}$${M_{1}}$${M_{-2}}$${M_{-5}}$${\cdots}$${0}$${M_{0}}$${M_{-3}}$${M_{-6}}$${\cdots}$${0}$${M_{-1}}$${M_{-4}}$${M_{-7}}$${\cdots.}$${\vdots}$${\vdots}$${\vdots}$$\scriptstyle{\partial_{M}}$$\scriptstyle{u}$$\scriptstyle{\partial_{M}}$$\scriptstyle{-u}$$\scriptstyle{\partial_{M}}$$\scriptstyle{\partial_{M}}$$\scriptstyle{\partial_{M}}$$\scriptstyle{-u}$$\scriptstyle{\partial_{M}}$$\scriptstyle{u}$$\scriptstyle{\partial_{M}}$$\scriptstyle{u}$$\scriptstyle{\partial_{M}}$$\scriptstyle{-u}$$\scriptstyle{\partial_{M}}$$\scriptstyle{-u}$$\scriptstyle{u}$

The dotted arrows represent the action of $U$. An element $x$ of
$C^{+}_{A}(M)_{n}$ can be represented by a sequence $(m_{n-4p})_{p\geq 0}$,
where almost all terms vanish. This element $x$ is a cycle if and only if
$\partial_{M}m_{n-4p}=(-1)^{n}m_{n-4(p-1)}u$ for each $p\geq 0$. Each column
is a shifted copy of the original complex $M$. The double complex
$(D^{-}_{*,*},\partial^{\prime},\partial^{\prime\prime})$ is obtained by
shifting the whole diagram from the right half plane over to the left half
plane. A more concise representation of this double complex is given in the
following diagram

${\cdots}$${M[-9[}$${M[-6]}$${M[-3]}$${M}$${0.}$$\scriptstyle{\pm
u}$$\scriptstyle{\mp u}$$\scriptstyle{\pm u}$

Note that in this case the internal differential in $M[-3n]$, $n\geq 0$, is
not twisted by a sign, and that the $\pm u$ notation means that the sign
depends on the internal degree.

With these concrete models in hand it is easy to calculate the following.

######  Corollary 5.9.

We have $H^{+}_{A}(A)\cong R$, $H^{-}_{A}(A)\cong R[3]$, with necessarily
trivial $R[U]$-module structure, and $H^{+}_{A}(R\oplus R[2])\cong R[W]$ with
$|W|=2$, $H^{-}_{A}(R\oplus R[2])=R[Z][2]$ with $|Z|=-2$. The $R[U]$-module
structures are determined by $U\cdot W^{p}=W^{p-2}$ and $U\cdot
Z^{p}=Z^{p+2}$.

######  Remark 5.10.

The notation $R[W]$ and $R[Z][2]$ in the above result is only used to
effectively express the underlying graded modules and the action of $R[U]$, it
should not be interpreted to mean that $H^{\pm}_{A}(R\oplus R[2])$ carry any
(co)algebra structure.

###  5.3. Tate Homology

We will now proceed to the construction of $C^{\infty}_{A}(M)$.

######  Definition 5.11.

For an augmented $DG$ algebra $A$ define the dualizing object by

$D_{A}\coloneqq C^{-}_{A}(A)=\operatorname{Hom}_{A}(EA,A).$

The dualizing object is a (left-left) $A-C^{-}_{A}(R)$-bimodule. The left
$C^{-}_{A}(R)$-module structure is the one already given in (5.2), while the
left $A$-module structure is obtained from the second factor in
$D_{A}=\operatorname{Hom}_{A}(EA,A)$, that is, $(a\cdot f)(x):=af(x)$ for all
$a\in A$, $f\in\operatorname{Hom}_{A}(EA,A)$ and $x\in EA$. This justifies the
following definition.

######  Definition 5.12.

For a right $A$-module $M$ define the twisted positive $A$-chains of $M$ to be
the $DG$ module

$C^{+,tw}_{A}(M)\coloneqq B(M,A,D_{A}).$

The homology of this complex is denoted by $H^{+,tw}_{A}(M)$ and is called the
positive twisted $A$-homology of $M$.

The complex $C^{+,tw}_{A}(M)$ carries a left $C^{-}_{A}(R)$-module structure
induced from the left $C^{-}_{A}(R)$-module structure of $D_{A}$.

######  Definition 5.13.

The norm map $N_{M}\colon C^{+,tw}_{A}(M)\rightarrow C^{-}_{A}(M)$ is for each
right $A$-module $M$ defined to be the composite

${B(M,A,D_{A})}$${M\otimes_{A}D_{A}}$${\operatorname{Hom}_{A}(EA,M),}$$\scriptstyle{\epsilon}$$\scriptstyle{\xi}$

where $\epsilon$ is the augmentation and $\xi$ is the adjoint of

${M\otimes_{A}(D_{A}\otimes
EA)}$${M\otimes_{A}A=M.}$$\scriptstyle{1\otimes\operatorname{ev}}$

Explicitly, for $m[a_{1}|\cdots|a_{p}]v\in B(M,A,D_{A})$ and $x\in EA$,

(5.5) $N_{M}(m[a_{1}|\cdots|a_{p}]v)(x)=\left\\{\begin{array}[]{cc}m\cdot
v(x)&\mbox{ for }p=0\\\ 0&\mbox{ for }p>0.\end{array}\right.$

######  Lemma 5.14.

The norm map $N_{M}\colon C^{+,tw}_{A}(M)\rightarrow C^{-}_{A}(M)$ is
$C^{-}_{A}(R)$-linear and natural in $M$.

###### Proof.

The naturality in $M$ is clear from the definition. To verify the
$C^{-}_{A}(R)$-linearity let $f\in C^{-}_{A}(R)$ and
$w=m[a_{1}|\cdots|a_{p}]u\in C^{+,tw}_{A}(M)=B(M,A,D_{A})$. Then $f\cdot
w=(-1)^{\eta}m[a_{1}|\cdots|a_{p}](f\cdot w)$ for
$\eta=p+|m|+|a_{1}|+\cdots+|a_{p}|$ and therefore $N_{M}(f\cdot w)=0=f\cdot
N_{M}(w)$ provided $p>0$. We may therefore assume that $p=0$ such that
$w=m[]u$. Let $v=N_{M}(w)$ and note that $|v|=|m|+|u|$. Let $x\in EA$ and
write $\psi_{R,A}(x)=\sum_{i}y_{i}\otimes z_{i}\in BA\otimes EA$ where
$\psi_{R,A}$ is the map of Lemma 5.3. Then by definition $(f\cdot
v)(x)=(f\otimes v)(\psi_{R,A}(x))$, which equals

(5.6)
$\sum_{i}(-1)^{|v||y_{i}|}f(y_{i})v(z_{i})=\sum_{i}(-1)^{|m||y_{i}|+|u||y_{i}|}m\cdot
f(y_{i})u(z_{i}).$

On the other hand, $N_{M}(f\cdot m[]u)=(-1)^{|f||m|}N_{M}(m[](f\cdot u))$, and
by (5.5) this coincides with

$(-1)^{|f||m|}\sum_{i}m\cdot(f\otimes u)(y_{i}\otimes
z_{i})=\sum_{i}(-1)^{|f||m|+|u||y_{i}|}m\cdot f(y_{i})u(z_{i}).$

The final expression is equal to the final expression in (5.6) because
$f\colon BA\rightarrow R$ so that $f(y_{i})$ can only be nonzero if
$|f|=-|y_{i}|$. Hence, $f\cdot N_{M}(w)=N_{M}(f\cdot w)$ and the proof is
complete. ∎

The Tate complex $C^{\infty}_{A}(M)$ will be defined to be the mapping cone
complex of the norm map. At a later stage we will also have to consider the
cone of a map of nonzero degree, so we fix our conventions in the following
definition.

######  Definition 5.15.

Let $B$ be a $DG$ algebra and suppose that $f\colon M\rightarrow N$ is a
degree $d$ chain map of left $B$-modules. This means that

  1. (a)

$f(M_{n})\subset N_{n+d}$ for all $n\in\mathbb{Z}$,

  2. (b)

$f(bm)=(-1)^{|b|d}bf(m)$ for all $b\in B$ and $m\in M$ and

  3. (c)

$f\circ\partial_{M}=(-1)^{d}\partial_{N}\circ f$.

The left $B$-module $\operatorname{Cone}(f)$ is defined by
$\operatorname{Cone}(f)_{k}=N_{k}\oplus M_{k-d-1}$ for all $k\in\mathbb{Z}$,
with differential

$\partial_{\operatorname{Cone}(f)}\cdot\left(\begin{array}[]{c}n\\\
m\end{array}\right)=\left(\begin{array}[]{cc}\partial_{N}&(-1)^{d}f\\\
0&(-1)^{d+1}\partial_{M}\end{array}\right)\left(\begin{array}[]{c}n\\\
m\end{array}\right)=\left(\begin{array}[]{c}\partial_{N}n+(-1)^{d}fm\\\
(-1)^{d+1}\partial_{M}m\end{array}\right)$

and $B$-module structure $b\cdot(n,m)=(bn,(-1)^{|b|(d+1)}bm)$.

######  Remark 5.16.

If $B=R$ is concentrated in degree $0$ and $f$ has degree $0$, then this
definition agrees with [Mac95, p. 46]. The above definition is what we get
when we regard $f$ as a degree $0$ map $f\colon M[d]\rightarrow N$ and take
into account the sign twist in the $B$-action upon shifting $M$.

The cone complex sits in a natural short exact sequence

${0}$${N}$${\operatorname{Cone}(f)}$${M[d+1]}$${0}$

of $B$-modules, whose connecting homomorphism $\delta\colon
H(M)_{n}\rightarrow H_{n+d}(N)$ recovers $H(f)$ up to a sign. In particular,
there is a natural exact triangle in homology.

######  Definition 5.17.

Let $M$ be a right $A$-module. Then the Tate complex of $M$ is defined to be
the $C^{-}_{A}(R)$-module

$C^{\infty}_{A}(M)\coloneqq\operatorname{Cone}(N_{M}\colon
C^{+,tw}_{A}(M)\rightarrow C^{-}_{A}(M)).$

The homology of this complex is denoted by $H^{\infty}_{A}(M)$ and is called
the Tate homology of $M$.

Given a map $g\colon M\rightarrow M^{\prime}$ of a right $A$-modules, there
are induced maps

$g^{+,tw}\colon C^{+,tw}_{A}(M)\rightarrow C^{+,tw}_{A}(M^{\prime})\;\;\mbox{
and }\;\;g^{\infty}\colon C^{\infty}_{A}(M)\rightarrow
C^{\infty}_{A}(M^{\prime})$

of $C^{-}_{A}(R)$-modules. The key properties of the functors $C^{+,tw}_{A}$
and $C^{\infty}_{A}$ are summarized in the following theorem.

######  Theorem 5.18.

Let $A$ be a degreewise $R$-free augmented DG algebra. Then the assignments

$(M\mapsto C^{\bullet}_{A}(M))\;\mbox{ and }\;(g\colon M\rightarrow
M^{\prime})\mapsto(g^{\bullet}\colon C^{\bullet}_{A}(M)\rightarrow
C^{\bullet}_{A}(M^{\prime}))$

for $\bullet\in\\{(+,tw),\infty\\}$, define functors $C^{+,tw}_{A}$,
$C^{\infty}_{A}$ from the category of right $A$-modules to the category of
left $C^{-}_{A}(R)$-modules. They are both exact and preserve quasi-
isomorphisms. Moreover, there is a natural exact triangle of
$H^{-}_{A}(R)$-modules

${H^{+,tw}_{A}(M)}$${H^{-}_{A}(M)}$${H^{\infty}_{A}(M).}$$\scriptstyle{H(N_{M})}$$\scriptstyle{[-1]}$

Suppose $f\colon A\rightarrow B$ is a quasi-isomorphism of augmented DG
algebras. For a given right $B$-module $M$ there is no induced map between
$C^{\bullet}_{A}(f^{-1}M)$ and $C^{\bullet}_{B}(M)$ for
$\bullet\in\\{(+,tw),\infty\\}$. Rather one has to introduce an intermediate
object to compare them.

######  Definition 5.19.

Let $f\colon A\rightarrow B$ be a homomorphism of augmented DG algebras.
Define the relative dualizing object to be the (left-left)
$C^{-}_{A}(R)-B$-bimodule

$D_{f}\coloneqq C^{-}_{A}(f^{-1}B)=\operatorname{Hom}_{A}(EA,f^{-1}B),$

where $f^{-1}B$ is regarded as a (right-left) $A-B$-bimodule.

Given a right $B$-module $M$ define the relative positive twisted complex to
be $C^{+,tw}_{f}(M)\coloneqq B(M,B,D_{f})$ equipped with the left
$C^{-}_{A}(R)$-module structure induced from $D_{f}$.

The relative positive twisted complex is a functor from the category of right
$B$-modules to the category of left $C^{-}_{A}(R)$-modules. Given a map
$g\colon M\rightarrow N$ of right $B$-modules the induced map is given by
$B(g,1,1)\colon B(M,B,D_{f})\rightarrow B(N,B,D_{f})$.

######  Lemma 5.20.

In the situation of the above definition, if $A$ and $B$ are degreewise free,
the relative functor $C^{+,tw}_{f}$ preserves quasi-isomorphisms and short
exact sequences.

###### Proof.

The assumptions that $A$ and $B$ are degreewise free along with the fact that
$R$ is PID ensure that the relative dualizing object $D_{f}$ is degreewise
flat. This is verified as in [Eis19, Lemma A.10]. It then follows from the
invariance theorem [Eis19, Theorem A.1] that $C^{+,tw}_{f}=B(-,B,D_{f})$ is an
exact functor preserving quasi-isomorphisms. ∎

######  Proposition 5.21.

Let $f\colon A\rightarrow B$ be a homomorphism of augmented DG algebras and
let $M$ be a right $B$-module. There is a relative norm map $N_{M}^{f}\colon
C^{+,tw}_{f}(M)\rightarrow C^{-}_{A}(f^{-1}M)$ fitting into the following
commutative diagram, natural in $M$,

${C^{+,tw}_{B}(M)}$${C^{+,tw}_{f}(M)}$${C^{+,tw}_{A}(f^{-1}M)}$${C^{-}_{B}(M)}$${C^{-}_{A}(f^{-1}M)}$${C^{-}_{A}(f^{-1}M).}$$\scriptstyle{\beta_{M}}$$\scriptstyle{N_{M}^{B}}$$\scriptstyle{N^{f}_{M}}$$\scriptstyle{N^{A}_{M}}$$\scriptstyle{\alpha_{M}}$$\scriptstyle{f^{*}}$$\scriptstyle{=}$

All of the maps in the diagram are $C^{-}_{A}(R)$-linear,
$C^{-}_{B}(R)$-linear or linear along $f^{*}:C^{-}_{B}(R)\rightarrow
C^{-}_{A}(R)$, as appropriate.

Moreover, if $f$ is a quasi-isomorphism and $A$ and $B$ are degreewise
$R$-free, then all the horizontal maps in the diagram are quasi-isomorphisms.

###### Proof.

Define $g\colon D_{B}\rightarrow D_{f}$ and $h\colon D_{A}\rightarrow D_{f}$
to be the maps

$g=f^{*}\colon C^{-}_{B}(B)\rightarrow C^{-}_{A}(f^{-1}B)\;\;\mbox{ and
}\;\;h=f^{-}\colon C^{-}_{A}(A)\rightarrow C^{-}_{A}(f^{-1}B).$

Note that $D_{A}$ is a $C^{-}_{A}(R)-A$-bimodule, $D_{f}$ is a
$C^{-}_{A}(R)-B$-bimodule and $D_{B}$ is a $C^{-}_{B}(R)-B$-bimodule.
Moreover, observe that $g$ is a homomorphism along
$(f^{*},1)\colon(C^{-}_{B}(R),B)\rightarrow(C^{-}_{A}(R),B)$ and that $h$ is a
homomorphism along $(1,f)\colon(C^{-}_{A}(R),A)\rightarrow(C^{-}_{A}(R),B)$.
We may therefore define the upper horizontal maps in the diagram by

$\displaystyle\beta_{M}\coloneqq B(1,1,g)$ $\displaystyle\colon
B(M,B,D_{B})\rightarrow B(M,B,D_{f})$ $\displaystyle\alpha_{M}\coloneqq
B(1,f,h)$ $\displaystyle\colon B(f^{-1}M,A,D_{A})\rightarrow B(M,B,D_{f}).$

It is clear by construction that these maps are natural in $M$. Furthermore,
by the above observations $\beta_{M}$ is linear along $f^{*}\colon
C^{-}_{B}(R)\rightarrow C^{-}_{A}(R)$ and $\alpha_{M}$ is
$C^{-}_{A}(R)$-linear.

We define the relative norm map $N^{f}_{M}\colon B(M,B,D_{f})\rightarrow
C^{-}_{A}(f^{-1}M)$ to be the composition of the augmentation
$B(M,B,D_{f})\rightarrow M\otimes_{B}D_{f}$ and the map $\kappa\colon
M\otimes_{B}D_{f}\rightarrow C^{-}_{A}(M)$ which is given as the adjoint of

${M\otimes_{B}(\operatorname{Hom}_{A}(EA,f^{-1}B)\otimes
EA)}$${M\otimes_{B}f^{-1}B\cong
f^{-1}M.}$$\scriptstyle{1\otimes\operatorname{ev}}$

Then $N^{B}_{M}$ is $C^{-}_{B}(R)$-linear while $N_{M}^{f}$ and $N_{M}^{A}$
are $C^{-}_{A}(R)$-linear. Consider the diagram

${B(M,B,D_{B})}$${B(M,B,D_{f})}$${B(f^{-1}M,A,D_{A})}$${M\otimes_{B}D_{B}}$${M\otimes_{B}D_{f}}$${f^{-1}M\otimes_{A}D_{A}}$${\operatorname{Hom}_{B}(EB,B)}$${\operatorname{Hom}_{A}(EA,f^{-1}M)}$${\operatorname{Hom}_{A}(EA,f^{-1}M),}$$\scriptstyle{\epsilon}$$\scriptstyle{B(1,1,g)}$$\scriptstyle{\epsilon}$$\scriptstyle{B(1,f,h)}$$\scriptstyle{\epsilon}$$\scriptstyle{1\otimes_{B}g}$$\scriptstyle{\xi}$$\scriptstyle{\kappa}$$\scriptstyle{1\otimes_{f}h}$$\scriptstyle{\xi}$$\scriptstyle{Ef^{*}}$$\scriptstyle{=}$

where $\epsilon$ denotes the augmentations and $\xi$ are the natural maps in
the definition of the norm map (see 5.13). In particular, the outer
compositions $\xi\circ\epsilon$ are the respective norm maps and by definition
$\kappa\circ\epsilon=N^{f}_{M}$. We wish to verify that the diagram commutes.
The two upper squares commute by the naturality of the augmentation in the bar
construction. To verify that the lower squares commute, consider the diagram
obtained by passing to the adjoints of the vertical maps $\xi$, $\kappa$ and
$\xi$ in the bottom half of the diagram. The resulting diagram is then seen to
commute using the fact that each of the maps $\xi$, $\kappa$ and $\xi$ was
defined as the adjoint of $1\otimes\operatorname{ev}$ for a suitable
evaluation. By omitting the middle row of the above diagram, we obtain the
commutative diagram in the statement of the proposition.

Finally, we need to verify that all the horizontal maps in the diagram are
quasi-isomorphisms provided that $f\colon A\rightarrow B$ is a quasi-
isomorphism of degreewise $R$-free algebras. We already know that this is true
for $f^{*}\colon C^{-}_{B}(M)\rightarrow C^{-}_{A}(f^{-1}M)$ by Proposition
5.5. For the upper horizontal maps, note first that $g=f^{*}$ and $h=f^{-}$
are quasi-isomorphisms by the same result. Then, since $D_{A}$, $D_{B}$ and
$D_{f}$ are degreewise flat, one concludes by [Eis19, Theorem A.1] that
$\beta_{M}=B(1,1,g)$ and $\alpha_{M}=B(1,f,g)$ are quasi-isomorphisms as well.
This completes the proof. ∎

From this result one quickly obtains the required invariance result.
Nevertheless, we will need the more refined statement of the above proposition
in the next section.

######  Corollary 5.22.

Let $f\colon A\rightarrow B$ be a quasi-isomorphism of degreewise $R$-free
augmented DG algebras. Then for every right $B$-module $M$ there are natural
isomorphisms of $H^{-}_{A}(R)\cong H^{-}_{B}(R)$ modules

$H^{+,tw}_{A}(f^{-1}M)\cong H^{+,tw}_{B}(M)\;\mbox{ and
}\;H^{\infty}_{A}(f^{-1}M)\cong H^{\infty}_{B}(M).$

Moreover, these isomorphisms are compatible with the exact triangles of
Theorem 5.18.

A DG algebra $A$ is said to satisfy Poincaré duality of degree
$d\in\mathbb{Z}$ provided there is a weak equivalence
$A\simeq\operatorname{Hom}(A,R[d])$ of $A-A$-bimodules, where the right and
left actions on the target are induced by the left action on $A$ and the
trivial left action on $R[d]$, respectively. Here, weak equivalence means that
there exist a finite number
$A=X_{0},X_{1},\cdots,X_{r}=\operatorname{Hom}(A,R[d])$ of $A$-bimodules and
for each $0\leq i<r$ there is a quasi-isomorphism $X_{i}\rightarrow X_{i+1}$
or $X_{i+1}\rightarrow X_{i}$ of $A$-bimodules. According to [Eis19, Theorem
A.19] there is under this hypothesis, an isomorphism $H^{+,tw}_{A}(M)\cong
H^{+}_{A}(M)[n]$ of $H^{-}_{A}(R)$-modules. We will give a different proof of
this result for $A=\Lambda_{R}[u]$ with $|u|=3$. This case is sufficient for
the application to equivariant instanton Floer homology when $\frac{1}{2}\in
R$.

We will now consider these constructions for $A=\Lambda_{R}[u]$ with $|u|=3$
as we did for $C^{\pm}_{A}$. The Tate complex $C^{\infty}_{A}(M)$, even in
this simple case, is big and difficult to handle explicitly. To remedy this we
will construct a much simpler and very explicit complex that also computes the
Tate homology $H^{\infty}_{A}(M)$. Indeed, we will show that if we take one of
the double complexes
$(D^{\pm}_{*,*},\partial^{\prime},\partial^{\prime\prime})$ of Proposition 5.8
and extend them in the natural way to the whole plane, then a suitable
totalization of this complex will do the trick.

######  Lemma 5.23.

Let $A=\Lambda_{R}[u]$ with $|u|=3$. Then there is an isomorphism of left
$A$-modules $A\otimes R[\alpha]\cong D_{A}$ where $|\alpha|=-4$. Under this
isomorphism the differential on $A\otimes R[\alpha]$ is given by
$\partial(1\otimes\alpha^{p})=-u\otimes\alpha^{p+1}$ and
$\partial(u\otimes\alpha^{p})=0$ for $p\geq 0$, and the action of
$R[U]=C^{-}_{A}(R)$ is given by
$U\cdot(a\otimes\alpha^{p})=a\otimes\alpha^{p+1}$ for $a\in A$ and $p\geq 0$.

Furthermore, the map $\rho\colon R\rightarrow A\otimes R[\alpha]$ of degree
$3$ given by $1\mapsto u\otimes\alpha^{0}$ is an $A$-linear quasi-isomorphism.

###### Proof.

Using the explicit model of Theorem 5.8 we see that

$(D_{A})_{n}=C^{-}_{A}(A)_{n}\cong\left\\{\begin{array}[]{cl}A_{0}&\mbox{ for
}n=4m,m\leq 0\\\ A_{3}&\mbox{for }n=4m+3,m\leq 0\\\ 0&\mbox{
otherwise}\end{array}\right.$

and a quick check shows that these identifications are compatible with the
left $A$-module structure. The required isomorphism is therefore obtained by
letting $1\otimes\alpha^{p}$ and $u\otimes\alpha^{p}$ correspond to the
generators $1\in A_{0}\cong C^{-}_{A}(A)_{-4p}$ and $u\in
A_{3}=C^{-}_{A}(A)_{-4p+3}$, respectively, for $p\geq 0$. The same result
shows that the differential and $R[U]$-action are determined by the formulas
$\partial(u\otimes\alpha^{p})=0$,
$\partial(1\otimes\alpha^{p})=-u\otimes\alpha^{p+1}$ and
$U\cdot(a\otimes\alpha^{p})=a\otimes\alpha^{p+1}$ for $a\in A$ and $p\geq 0$.

From this description we find that
$\partial\colon(D_{A})_{-4p}\rightarrow(D_{A})_{-4(p+1)-3}$ is an isomorphism
for each $p\geq 0$. Therefore, the map $\rho\colon R\rightarrow A\otimes
R[\alpha]$ of degree $3$ given by $1\mapsto u\otimes\alpha^{0}$ is an
$A$-linear quasi-isomorphism. ∎

In the following result we establish the isomorphism $H^{+,tw}_{A}(M)\cong
H^{+}_{A}(M)[3]$ of $R[U]$-modules for each right $A$-module $M$. It also
contains the key ingredients needed to establish the simplified model for
$C^{\infty}_{A}(M)$.

######  Lemma 5.24.

Let $A=\Lambda_{R}[u]$ with $|u|=3$ and let $M$ be a right $A$-module. Then
there is a quasi-isomorphism $f_{M}\colon C^{+}_{A}(M)\rightarrow
C^{+,tw}_{A}(M)$ of degree $3$ and a chain homotopy $s_{M}\colon
C^{+}_{A}(M)\rightarrow C^{+,tw}_{A}(M)$ of degree $0$ such that

$f_{M}\circ U-U\circ f_{M}=\partial\circ s_{M}-s_{M}\circ\partial.$

In particular, $H(f_{M})\colon H^{+}_{A}(M)\rightarrow H^{+,tw}_{A}(M)$ is a
degree $3$ isomorphism of $R[U]$-modules. Furthermore, $f_{M}$ and $s_{M}$ are
natural in the module $M$, and in terms of the explicit models
$C^{+}_{A}(M)\cong\operatorname{Tot}^{\oplus}(D^{+}_{*,*},\partial^{\prime},\partial^{\prime\prime})$
and
$C^{-}_{A}(M)\cong\operatorname{Tot}^{\Pi}(D^{-}_{*,*},\partial^{\prime},\partial^{\prime\prime})$
of Proposition 5.8 the compositions $N_{M}\circ f_{M}$ and $N_{M}\circ s_{M}$
are given in degree $n$ by

${\bigoplus_{p\geq 0}M_{n-4p}}$${M_{n}}$${M_{n+3}}$${\prod_{q\geq
0}M_{n+3+4q}}$${\bigoplus_{p\geq 0}M_{n-4p}}$${M_{n}}$${\prod_{q\geq
0}M_{n+4q},}$$\scriptstyle{\pi}$$\scriptstyle{(-1)^{n}u}$$\scriptstyle{\iota}$$\scriptstyle{\pi}$$\scriptstyle{\iota}$

respectively.

###### Proof.

Recall that $C^{+}_{A}(M)=B(M,A,R)$ and $C^{+,tw}_{A}(M)=B(M,A,D_{A})$. It is
convenient to introduce the notation

$\displaystyle m\otimes V^{p}\coloneqq$
$\displaystyle(-1)^{p(p+1)/2}m[u|u|\cdots|u]\in B(M,A,R)$ $\displaystyle
m\otimes V^{p}\otimes\psi\coloneqq$
$\displaystyle(-1)^{p(p+1)/2}m[u|u|\cdots|u]\psi\in B(M,A,D_{A})$

for $m\in M$ and $\psi\in D_{A}$, where $u$ is repeated $p$ times in both
formulas. In terms of this notation the differentials and actions of $U\in
R[U]$ of $B(M,A,R)$ and $B(M,A,D_{A})$ are given by the formulas

$\displaystyle U(m\otimes V^{p})=$ $\displaystyle\;(-1)^{|m|}m\otimes V^{p-1}$
$\displaystyle U(m\otimes V^{p}\otimes\psi)=$ $\displaystyle\;m\otimes
V^{p}\otimes(U\psi)$ $\displaystyle\partial(m\otimes V^{p})=$
$\displaystyle\;(-1)^{p}(\partial_{M}m+mu\otimes V^{p-1})$
$\displaystyle\partial(m\otimes V^{p}\otimes\psi)=$
$\displaystyle\;(-1)^{p}(\partial_{M}m\otimes V^{p}+mu\otimes
V^{p-1})\otimes\psi$ $\displaystyle\;+m\otimes(V^{p-1}\otimes
u\psi+(-1)^{|m|}V^{p}\otimes\partial_{D_{A}}\psi).$

By the previous lemma there is an isomorphism $D_{A}\cong A\otimes R[\alpha]$
with $|\alpha|=-4$. To simplify the notation write $a\alpha^{p}\coloneqq
a\otimes\alpha^{p}$ for $a\in A$. There is also an $A$-linear quasi-
isomorphism $\rho:R\rightarrow D_{A}$ of degree $3$ given by
$\rho(1)=u\alpha^{0}$. This map induces a quasi-isomorphism

$f_{M}\coloneqq B(1,1,\rho)\colon B(M,A,R)\rightarrow B(M,A,D_{A})$

of degree $3$, which in terms of the above notation is given by
$f_{M}(m\otimes V^{p})=(-1)^{|m|}m\otimes V^{p}\otimes u\alpha^{0}$ (the sign
is forced upon us by the fact that we regard $\rho$ as a map of degree $3$).
Using the above formulas we find that

(5.7) $(f_{M}\circ U-U\circ f_{M})(m\otimes V^{p})=m\otimes V^{p-1}\otimes
u\alpha^{0}-(-1)^{|m|}m\otimes V^{p}\otimes u\alpha^{1}.$

Define $s_{M}\colon B(M,A,R)\rightarrow B(M,A,D_{A})$ of degree $0$ by
$s_{M}(m\otimes V^{p})=m\otimes V^{p}\otimes\alpha^{0}$. A straightforward
calculation using the above formulas and the formula for the differential in
$D_{A}\cong A\otimes R[\alpha]$ given in the previous lemma shows that

$\displaystyle\partial s_{M}(m\otimes V^{p})=$
$\displaystyle\;(-1)^{p}(\partial_{M}m\otimes V^{p}\otimes\alpha^{0}+mu\otimes
V^{p-1}\otimes\alpha^{0})$ $\displaystyle\;+m\otimes V^{p-1}\otimes
u\alpha^{0}-(-1)^{|m|}m\otimes V^{p}\otimes u\alpha^{1}$ $\displaystyle
s_{M}\partial(m\otimes V^{p})=$ $\displaystyle(-1)^{p}(\partial_{M}m\otimes
V^{p}\otimes\alpha^{0}+mu\otimes V^{p-1}\otimes\alpha^{0}).$

Combining this with (5.7) we obtain $f_{M}\circ U-U\circ f_{M}=\partial
s_{M}-s_{M}\partial$ as required.

Finally, we need to consider the compositions of $f_{M}$ and $s_{M}$ with the
norm map $N_{M}\colon C^{+,tw}_{A}(M)\rightarrow C^{-}_{A}(M)$ in terms of the
models given in Proposition 5.8. By (5.3) $m\in M_{q-3p}=D^{+}_{p,q}$
corresponds to $(-1)^{|m|p}m\otimes V^{p}$, which is mapped by $f_{M}$ to
$(-1)^{|m|(p+1)}m\otimes V^{p}\otimes u\alpha^{0}$. The norm map kills off all
such elements with $p>0$ and sends $m\otimes V^{0}\otimes u\alpha^{0}\mapsto
mu\cdot\alpha^{0}$. This is the functional that sends $V^{0}$ to $mu\in
M_{q+3}$ and vanishes on $V^{q}$ for $q>0$. By (5.4) the identification
$C^{-}_{A}(M)_{n}\cong\prod_{q\geq 0}M_{n+4q}$ is given by
$\psi\mapsto(\psi(V^{q}))_{q\geq 0}$. We conclude that $N_{M}\circ f_{M}$ is
given by the composition in the statement of the lemma. The verification for
$N_{M}\circ s_{M}$ is analogous. ∎

We need two simple but slightly technical results before we can reach our
goal.

######  Lemma 5.25.

Let $X$ and $Y$ be DG $R[U]$-modules and suppose that $f\colon X\rightarrow Y$
is a degree $d$ map of $R[U]$-modules up to homotopy; that is, $f$ is a chain
map of degree $d$ and there is a chain homotopy $s\colon X\rightarrow Y$ such
that $fU-Uf=\partial_{Y}s+(-1)^{d}s\partial_{X}$. Let $\operatorname{Cone}(f)$
be given as in Definition 5.15, except that we redefine the $R[U]$-module
structure by setting

$U\cdot\left(\begin{array}[]{c}y\\\
x\end{array}\right)=\left(\begin{array}[]{cc}U&(-1)^{|s|}s\\\
0&U\end{array}\right)\left(\begin{array}[]{c}y\\\
x\end{array}\right)=\left(\begin{array}[]{c}Uy-(-1)^{d}sx\\\
Ux\end{array}\right)$

for all $(y,x)\in Y_{n}\oplus X_{n-d-1}=\operatorname{Cone}(f)_{n}$. Then
$\operatorname{Cone}(f)$ is a DG $R[U]$-module and the maps in the natural
short exact sequence

${0}$${Y}$${\operatorname{Cone}(f)}$${X[d+1]}$${0}$

are (strict) maps of $R[U]$-modules.

###### Proof.

It suffices to check that
$\partial_{\operatorname{Cone}(f)}U=U\partial_{\operatorname{Cone}(f)}$. This
follows easily from the relations $\partial_{X}U=U\partial_{X}$,
$\partial_{Y}U=U\partial_{Y}$ and $fU-Uf=\partial_{Y}s+(-1)^{d}s\partial_{X}$.
Next, the maps in the natural exact sequence are given degreewise by the
inclusion $Y_{n}\rightarrow Y_{n}\oplus X_{n-d-1}$ and the projection
$Y_{n}\oplus X_{n-d-1}\rightarrow X_{n-d-1}$ and are seen to commute with the
action of $U$. ∎

######  Lemma 5.26.

Let $f\colon X\rightarrow Y$ be a degree $3$ map of DG $R[U]$-modules up to
homotopy, and let $g\colon Y\rightarrow Z$ be a degree $0$ map of DG
$R[U]$-modules. Then $g\circ f:X\rightarrow Z$ is a degree $3$ map of
$R[U]$-modules up to homotopy and there is a commutative diagram

${0}$${Z}$${\operatorname{Cone}(g\circ
f)}$${X[4]}$${0}$${0}$${Z}$${\operatorname{Cone}(g)}$${Y[1]}$${0}$$\scriptstyle{=}$$\scriptstyle{\operatorname{Cone}(f,1)}$$\scriptstyle{f[1]}$

with short exact rows. If $\operatorname{Cone}(g\circ f)$ is given the DG
$R[U]$-module structure of the previous lemma, then the maps
$\operatorname{Cone}(f,1)$ and $f[1]$ are maps of DG $R[U]$-modules up to
homotopy and all the other maps in the diagram are strict maps of DG
$R[U]$-modules.

###### Proof.

By assumption there is a chain homotopy $s\colon X\rightarrow Y$ of degree $0$
such that $fU-Uf=\partial s-s\partial$. Then for $h\coloneqq g\circ f$ and
$t\coloneqq g\circ s$ we have $hU-Uh=\partial t-t\partial$ as well. This shows
that $h$ is a map of $R[U]$-modules up to homotopy. By functoriality of the
cone construction we obtain a commutative diagram of DG modules with short
exact rows

${0}$${Z}$${\operatorname{Cone}(g\circ
f)}$${X[4]}$${0}$${0}$${Z}$${\operatorname{Cone}(g)}$${Y[1]}$${0.}$$\scriptstyle{=}$$\scriptstyle{\operatorname{Cone}(f,1)}$$\scriptstyle{f[1]}$

Here, $\operatorname{Cone}(f,1)\colon\operatorname{Cone}(g\circ
f)_{n}=Z_{n}\oplus X_{n-4}\rightarrow Z_{n}\oplus
Y_{n-1}=\operatorname{Cone}(g)_{n}$ is given by $(z,x)\mapsto(z,-fx)$, and
$f[1]\colon X[4]_{n}=X_{n-4}\rightarrow Y_{n-1}=Y[1]_{n}$ is given by
$x\mapsto-fx$. Equip $\operatorname{Cone}(g\circ f)$ with the adjusted
$R[U]$-module structure given in the previous lemma. Then all the horizontal
maps in the diagram are strict maps of $R[U]$-modules. The right hand vertical
map is a map of $R[U]$-modules up to homotopy, so the only thing we have to
verify is that $\operatorname{Cone}(f,1)$ is a map of $R[U]$-modules up to
homotopy.

We claim that $v\colon\operatorname{Cone}(h)\rightarrow\operatorname{Cone}(g)$
given by $v(z,x)=(0,sx)$ is the required homotopy. Indeed, using matrix
notation for the various maps,

$\displaystyle C(f,1)U-UC(f,1)$ $\displaystyle=\left(\begin{array}[]{cc}1&0\\\
0&-f\end{array}\right)\left(\begin{array}[]{cc}U&t\\\
0&U\end{array}\right)-\left(\begin{array}[]{cc}U&0\\\
0&U\end{array}\right)\left(\begin{array}[]{cc}1&0\\\ 0&-f\end{array}\right)$
$\displaystyle=\left(\begin{array}[]{cc}0&t\\\ 0&Uf-fU\end{array}\right)$
$\displaystyle\partial_{\operatorname{Cone}(g)}v+v\partial_{\operatorname{Cone}(h)}$
$\displaystyle=\left(\begin{array}[]{cc}\partial_{Z}&g\\\
0&-\partial_{Y}\end{array}\right)\left(\begin{array}[]{cc}0&0\\\
0&s\end{array}\right)+\left(\begin{array}[]{cc}0&0\\\
0&s\end{array}\right)\left(\begin{array}[]{cc}\partial_{Z}&-h\\\
0&\partial_{X}\end{array}\right)$
$\displaystyle=\left(\begin{array}[]{cc}0&g\circ s\\\
0&s\partial_{X}-\partial_{Y}s\end{array}\right),$

where $C(f,1)$ is shorthand for $\operatorname{Cone}(f,1)$. Therefore, as
$t=g\circ s$ and $\partial_{Y}s-s\partial_{X}s=fU-Uf$, we deduce that
$\operatorname{Cone}(f,1)$ is $R[U]$-linear up to homotopy. ∎

Given a sequence of $R$-modules $\\{D_{s}\\}_{s\in\mathbb{Z}}$ we write
$\prod_{s\to-\infty}D_{s}$ for the submodule of $\prod_{s}D_{s}$ consisting of
the sequences $(x_{s})_{s\in\mathbb{Z}}$ for which there exists $s_{0}$ such
that $x_{s}=0$ for all $s\geq s_{0}$. Similarly, $\prod_{s\to\infty}D_{s}$ is
the submodule of $(x_{s})_{s\in\mathbb{Z}}$ with $x_{s}=0$ for all
sufficiently small $s$.

######  Definition 5.27.

Given a double complex
$E=(D_{*,*},\partial^{\prime},\partial^{\prime\prime})$, define
$\operatorname{Tot}^{\Pi,-\infty}(E)$ and
$\operatorname{Tot}^{\Pi,+\infty}(E)$ to be the complexes that in degree $n$
are given by

$\prod_{s\to-\infty}D_{s,n-s}\;\mbox{ and }\;\prod_{s\to\infty}D_{s,n-s}$

respectively, with differential
$\partial=\partial^{\prime}+\partial^{\prime\prime}$.

######  Theorem 5.28.

Let $A=\Lambda_{R}[u]$ with $|u|=3$ and let $M$ be a right DG $A$-module.
Define $D^{\infty}_{p,q}=M_{q-3p}$ for all $p,q\in\mathbb{Z}$. Set

$\displaystyle\partial^{\prime}=(-1)^{p+q+1}u\colon D^{\infty}_{p,q}$
$\displaystyle=M_{q-3p}\rightarrow M_{q-3p+3}=D^{\infty}_{p-1,q}$
$\displaystyle\partial^{\prime\prime}=\partial_{M}\colon D^{\infty}_{p,q}$
$\displaystyle=M_{q-3p}\rightarrow M_{q-3p-1}=D^{\infty}_{p,q-1}$

and let $U\colon D^{\infty}_{p,q}=M_{q-3p}\rightarrow
M_{q-3p}=D^{\infty}_{p-1,q-3}$ be the identity for all $p,q$. Then there is a
quasi-isomorphism, natural in $M$,

$\operatorname{Tot}^{\Pi,-\infty}(D_{*,*},\partial^{\prime},\partial^{\prime\prime})\rightarrow
C^{\infty}_{A}(M),$

which is a homomorphism of $R[U]$-modules up to homotopy.

###### Proof.

By Lemma 5.24 there is a quasi-isomorphism $f=f_{M}\colon
C^{+}_{A}(M)\rightarrow C^{+,tw}_{A}(M)$ and a homotopy $s=s_{M}\colon
C^{+}_{A}(M)\rightarrow C^{+,tw}_{A}(M)$ such that $fU-Uf=\partial
s-s\partial$. We apply Lemma 5.26 for $f\colon C^{+}_{A}(M)\rightarrow
C^{+,tw}_{A}(M)$ and the norm map $N_{M}\colon C^{+,tw}_{A}(M)\rightarrow
C^{-}_{A}(M)$ to obtain the commutative diagram with exact rows

(5.8) ${0}$${C^{-}_{A}(M)}$${\operatorname{Cone}(N_{M}\circ
f)}$${C^{+}_{A}(M)[4]}$${0}$${0}$${C^{-}_{A}(M)}$${C^{\infty}_{A}(M)}$${C^{+,tw}_{A}(M)[1]}$${0,}$$\scriptstyle{=}$$\scriptstyle{\operatorname{Cone}(f,1)}$$\scriptstyle{f[1]}$

where we recall that $C^{\infty}_{A}(M)=\operatorname{Cone}(N_{M})$. By the
same result $\operatorname{Cone}(f,1)$ is a map of $R[U]$-modules up to
homotopy. As both the outer vertical maps are quasi-isomorphisms, it follows
by passage to the long exact sequences in homology and the $5$-lemma that
$\operatorname{Cone}(f,1)$ is a quasi-isomorphism. Thus,
$\operatorname{Cone}(f,1)$ is the required map. This quasi-isomorphism is
natural in $M$, since both $f$ and $N_{M}$ are natural transformations in $M$
and the cone construction is functorial.

The remaining task is to establish an isomorphism
$\operatorname{Cone}(N_{M}\circ
f)\cong\operatorname{Tot}^{\Pi,-\infty}(D^{\infty}_{*,*},\partial^{\prime},\partial^{\prime\prime})$.
Using the identifications of Proposition 5.8 we obtain

$\operatorname{Cone}(N_{M}\circ f)_{n}=C^{-}_{A}(M)_{n}\oplus
C^{+}_{A}(M)_{n-4}\cong\left(\prod_{q\geq
0}M_{n+4q}\right)\oplus\left(\bigoplus_{p\geq 0}M_{n-4(p+1)}\right)$

for each $n\in\mathbb{Z}$. Moreover, by Lemma 5.24 $h\coloneqq N_{M}\circ f$
and $t\coloneqq N_{M}\circ s$ are given on $(x_{p})_{p\geq 0}\in
C^{+}_{A}(M)_{n}$ by $h((x_{p})_{p})=((-1)^{|x_{0}|}x_{0}u,0,\cdots)$ and
$t((x_{p})_{p\geq 0})=(x_{0},0,0,\cdots)$. Therefore, using the formulas

$\partial_{\operatorname{Cone}(h)}=\left(\begin{array}[]{cc}\partial_{C^{-}_{A}(M)}&-h\\\
0&\partial_{C^{+}_{A}(M)}\end{array}\right)\;\mbox{ and
}\;U=\left(\begin{array}[]{cc}U&t\\\ 0&U\end{array}\right),$

as well as the description of the differentials and the $U$-action in terms of
the models for $C^{\pm}_{A}(M)$, we deduce that
$\operatorname{Cone}(h)\cong\operatorname{Tot}^{\Pi,-\infty}(D^{\infty}_{*,*},\partial^{\prime},\partial^{\prime\prime})$.
∎

A part of the double complex
$(D^{\infty}_{*,*},\partial^{\prime},\partial^{\prime\prime})$ is shown below.

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