2.1 $G$-CW complexes

In this section we recall the definition of spaces of type SWF from [Reference ManolescuMan16], and introduce local equivalence. Spaces of type SWF are the output of the construction of the Seiberg–Witten Floer stable homotopy type of [Reference ManolescuMan03] and [Reference ManolescuMan16]; see § 4.

First, we recall some basics of equivariant algebraic topology from [Reference tom DiecktDie87]. The reader is encouraged to consult both [Reference ManolescuMan16] and [Reference tom DiecktDie87] for a fuller discussion. For now, $G$ will denote a compact Lie group. We define a $G$-equivariant $k$-cell as a copy of $G/H\times D^{k}$, where $H$ is a closed subgroup of $G$. A (finite) equivariant $G$-CW decomposition of a relative $G$-space $(X,A)$, where the action of $G$ takes $A$ to itself, is a filtration $(X_{n}|n\in \mathbb{Z}_{{\geqslant}0})$ such that:

• ∙ $A\subset X_{0}$ and $X=X_{n}$ for $n$ sufficiently large;

• ∙ the space $X_{n}$ is obtained from $X_{n-1}$ by attaching $G$-equivariant $n$-cells.

When $A$ is a point, we call $(X,A)$ a pointed $G$-CW complex. We always assume that $G$ acts on the left.

Let $EG$ be the total space of the universal bundle of $G$. For two pointed $G$-spaces $X_{1}$ and $X_{2}$, write

$$\begin{eqnarray}X_{1}\wedge _{G}X_{2}=(X_{1}\wedge X_{2})/(gx_{1}\times x_{2}\sim x_{1}\times g^{-1}x_{2}).\end{eqnarray}$$

The Borel homology of a pointed $G$-space $X$ is given by

$$\begin{eqnarray}\tilde{H}_{\ast }^{G}(X)=\tilde{H}_{\ast }(EG_{+}\wedge _{G}X),\end{eqnarray}$$

where $EG_{+}$ is $EG$ with a disjoint basepoint. Similarly, we define Borel cohomology:

$$\begin{eqnarray}\tilde{H}_{G}^{\ast }(X)=\tilde{H}^{\ast }(EG_{+}\wedge _{G}X).\end{eqnarray}$$

Additionally, we have a map given by projecting to the first factor:

$$\begin{eqnarray}f:EG_{+}\wedge _{G}X\rightarrow BG_{+}.\end{eqnarray}$$

From $f$ we have a map $p_{G}=f^{\ast }:H^{\ast }(BG)\rightarrow \tilde{H}_{G}^{\ast }(X)$. Then $H^{\ast }(BG)$ acts on $\tilde{H}_{\ast }^{G}(X)$, by composing $p_{G}$ with the cap product action of $\tilde{H}_{G}^{\ast }(X)$ on $\tilde{H}_{\ast }^{G}(X)$. We may also define the unpointed version of the above constructions in an apparent way.

As an example, consider the case $G=S^{1}$. Here $BS^{1}=\mathbb{C}P^{\infty }$, so $H^{\ast }(BS^{1})=\mathbb{F}[U]$, with $\text{deg}\,U=2$. Then $\mathbb{F}[U]$ acts on $H_{\ast }^{S^{1}}(X)$, for $X$ any $S^{1}$-space.

From now on we let $G=\text{Pin}(2)$. The group $G=\text{Pin}(2)$ is the set $S^{1}\cup jS^{1}\subset \mathbb{H}$, where $S^{1}$ is the unit circle in the $\langle 1,i\rangle$ plane. The group action of $G$ is induced from the group action of the unit quaternions. Manolescu shows in [Reference ManolescuMan16] that $H^{\ast }(BG)=\mathbb{F}[q,v]/(q^{3})$, where $\text{deg}\,q=1$ and $\text{deg}\,v=4$, so $\tilde{H}_{\ast }^{G}(X)$ is naturally an $\mathbb{F}[q,v]/(q^{3})$-module for $X$ a pointed $G$-space. Moreover $S^{\infty }=S(\mathbb{H}^{\infty })$ has a free action by the quaternions, making $S^{\infty }$ a free $G$-space. Since $S^{\infty }$ is contractible, we identify $EG=S^{\infty }$. We may view $EG=S^{\infty }$ also as $ES^{1}$ (as an $S^{1}$-space) by forgetting the action of $j$.

We will also need to relate $G$-Borel homology and $S^{1}$-Borel homology. Consider

$$\begin{eqnarray}f:\mathbb{C}P^{\infty }=BS^{1}\rightarrow BG,\end{eqnarray}$$

the map given by quotienting by the action of $j\in G$ on $BS^{1}=ES^{1}/S^{1}$. Then we have the following fact (for a proof, see [Reference ManolescuMan16, Example 2.11]).

Fact 2.1. The natural map

$$\begin{eqnarray}f^{\ast }=\text{res}_{S^{1}}^{G}:H^{\ast }(BG)\rightarrow H^{\ast }(BS^{1})\end{eqnarray}$$

is an isomorphism in degrees divisible by $4$, and zero otherwise. In particular, $v\rightarrow U^{2}$.

Moreover, for $X$ a $G$-space, we have a natural map

$$\begin{eqnarray}g:EG_{+}\wedge _{S^{1}}X\rightarrow EG_{+}\wedge _{G}X.\end{eqnarray}$$

The map $g$ induces a map

$$\begin{eqnarray}g_{\ast }=\text{cor}_{G}^{S^{1}}:\tilde{H}_{\ast }^{S^{1}}(X)\rightarrow \tilde{H}_{\ast }^{G}(X),\end{eqnarray}$$

called the corestriction map. As a Corollary of Fact 2.1, we have a relationship between the action of $U$ and $v$ (see [Reference tom DiecktDie87, § III.1]).

Fact 2.2. Let $X$ be a $G$-space. Then, for every $x\in H_{\ast }^{S^{1}}(X)$,

$$\begin{eqnarray}v(\text{cor}_{G}^{S^{1}}(x))=\text{cor}_{G}^{S^{1}}(U^{2}x).\end{eqnarray}$$

We shall use that Borel homology with $\mathbb{F}$ coefficients behaves well with respect to suspension. If $V$ is a finite-dimensional (real) representation of $G$, let $V^{+}$ be the one-point compactification, where $G$ acts trivially on $V^{+}-V$. Then $\unicode[STIX]{x1D6F4}^{V}X=V^{+}\wedge X$ will be called the suspension of $X$ by the representation $V$.

We mention the following representations of $G$:

• ∙ let $\tilde{\mathbb{R}}^{s}$ be the vector space $\mathbb{R}^{s}$ on which $j$ acts by $-1$, and $e^{i\unicode[STIX]{x1D703}}$ acts by the identity, for all $\unicode[STIX]{x1D703}$;

• ∙ we let $\tilde{\mathbb{C}}$ be the representation of $G$ on $\mathbb{C}$ where $j$ acts by $-1$, and $e^{i\unicode[STIX]{x1D703}}$ acts by the identity for all $\unicode[STIX]{x1D703}$;

• ∙ the quaternions $\mathbb{H}$, on which $G$ acts by multiplication on the left.

Remark 2.4. We list some examples of spaces of type SWF. The simplest space of type SWF is $S^{0}$. More interesting examples may be produced as follows. Let $X$ be a free, finite $G$-CW complex. Define

$$\begin{eqnarray}\tilde{\unicode[STIX]{x1D6F4}}X=X\times [0,1]/(0,x)\sim (0,x^{\prime })\quad \text{and}\quad (1,x)\sim (1,x^{\prime })\quad \text{for}\;\text{all}\;x,x^{\prime }\in X.\end{eqnarray}$$

We call $\tilde{\unicode[STIX]{x1D6F4}}X$ the unreduced suspension of $X$. Here $G$ acts on $\tilde{\unicode[STIX]{x1D6F4}}X$ by multiplication on the first factor. We fix one of the cone points as the base point. Then the $S^{1}$-fixed-point set is precisely $S^{0}\subset \tilde{\unicode[STIX]{x1D6F4}}X$, and $\tilde{\unicode[STIX]{x1D6F4}}X$ is a space of type SWF. As a particular example, let $X=G$, where $G$ acts on $X$ by multiplication on the left, as usual. Then $\tilde{\unicode[STIX]{x1D6F4}}X$ is, topologically, the suspension of two disjoint circles.

We also find it convenient to recall the definition of reduced Borel homology, for spaces $X$ of type SWF:

(8)$$\begin{eqnarray}\tilde{H}_{\ast ,\text{red}}^{S^{1}}(X)=\tilde{H}_{\ast }^{S^{1}}(X)/\text{Im}\,U^{N},\end{eqnarray}$$

for $N\gg 0$. Indeed, for all $N$ sufficiently large $\text{Im}\,U^{N}=\text{Im}\,U^{N+1}$, so $\tilde{H}_{\ast ,\text{red}}^{S^{1}}(X)$ is well defined.

Associated to a space $X$ of type SWF at level $s$, we take the Borel cohomology $\tilde{H}_{G}^{\ast }(X)$, from which we define $a(X),b(X),$ and $c(X)$ as in [Reference ManolescuMan16]:

(9)$$\begin{eqnarray}\displaystyle \begin{array}{@{}c@{}}a(X)=\text{min}\{r\equiv s\;\text{mod}\;4\mid \exists \,x\in \tilde{H}_{G}^{r}(X),v^{l}x\neq 0\;\text{for}\;\text{all}\;l\geqslant 0\},\\[2.0pt] b(X)=\text{min}\{r\equiv s+1\;\text{mod}\;4\mid \exists \,x\in \tilde{H}_{G}^{r}(X),v^{l}x\neq 0\;\text{for}\;\text{all}\;l\geqslant 0\}-1,\\[2.0pt] c(X)=\text{min}\{r\equiv s+2\;\text{mod}\;4\mid \exists \,x\in \tilde{H}_{G}^{r}(X),v^{l}x\neq 0\;\text{for}\;\text{all}\;l\geqslant 0\}-2.\end{array} & & \displaystyle\end{eqnarray}$$

The well-definedness of $a,b$, and $c$ follows from the equivariant localization theorem (see [Reference tom DiecktDie87] § III). We also list an equivalent definition of $a,b$, and $c$ from [Reference ManolescuMan16], using homology:

(10)$$\begin{eqnarray}\displaystyle \begin{array}{@{}c@{}}a(X)=\text{min}\;\{r\equiv s\;\text{mod}\;4\mid \exists \,x\in \tilde{H}_{r}^{G}(X),x\in \text{Im}\,v^{l}\;\text{for}\;\text{all}\;l\geqslant 0\},\\[2.0pt] b(X)=\text{min}\;\{r\equiv s+1\;\text{mod}\;4\mid \exists \,x\in \tilde{H}_{r}^{G}(X),x\in \text{Im}\,v^{l}\;\text{for}\;\text{all}\;l\geqslant 0\}-1,\\[2.0pt] c(X)=\text{min}\;\{r\equiv s+2\;\text{mod}\;4\mid \exists \,x\in \tilde{H}_{r}^{G}(X),x\in \text{Im}\,v^{l}\;\text{for}\;\text{all}\;l\geqslant 0\}-2.\end{array} & & \displaystyle\end{eqnarray}$$

Definition 2.6 (see [Reference ManolescuMan14]).

Let $X$ and $X^{\prime }$ be spaces of type SWF, $m,m^{\prime }\in \mathbb{Z}$, and $n,n^{\prime }\in \mathbb{Q}$. We say that the triples $(X,m,n)$ and $(X^{\prime },m^{\prime },n^{\prime })$ are stably equivalent if $n-n^{\prime }\in \mathbb{Z}$ and there exists a $G$-equivariant homotopy equivalence, for some $r\gg 0$ and some nonnegative $M\in \mathbb{Z}$ and $N\in \mathbb{Q}$:

(11)$$\begin{eqnarray}\unicode[STIX]{x1D6F4}^{r\mathbb{R}}\unicode[STIX]{x1D6F4}^{(M-m)\tilde{\mathbb{R}}}\unicode[STIX]{x1D6F4}^{(N-n)\mathbb{H}}X\rightarrow \unicode[STIX]{x1D6F4}^{r\mathbb{R}}\unicode[STIX]{x1D6F4}^{(M^{\prime }-m^{\prime })\tilde{\mathbb{R}}}\unicode[STIX]{x1D6F4}^{(N-n^{\prime })\mathbb{H}}X^{\prime }.\end{eqnarray}$$

Let $\mathfrak{E}$ be the set of equivalence classes of triples $(X,m,n)$ for $X$ a space of type $\text{SWF}$, $m\in \mathbb{Z}$, $n\in \mathbb{Q}$, under the equivalence relation of stable $G$-equivalence.Footnote ² The set $\mathfrak{E}$ may be considered as a subcategory of the $G$-equivariant Spanier–Whitehead category [Reference ManolescuMan16], by viewing $(X,m,n)$ as the formal desuspension of $X$ by $m$ copies of $\tilde{\mathbb{R}}^{+}$ and $n$ copies of $\mathbb{H}^{+}$. For $(X,m,n),(X^{\prime },m^{\prime },n^{\prime })\in \mathfrak{E}$, a map $(X,m,n)\rightarrow (X^{\prime },m^{\prime },n^{\prime })$ is simply a map as in (11) that need not be a homotopy equivalence. We define Borel homology for $(X,m,n)\in \mathfrak{E}$ by

(12)$$\begin{eqnarray}\tilde{H}_{\ast }^{G}((X,m,n))=\tilde{H}_{\ast }^{G}(X)[m+4n].\end{eqnarray}$$

In (12), we needed to use the suspension-invariance $\tilde{H}_{\ast }^{G}(\unicode[STIX]{x1D6F4}^{\tilde{\mathbb{R}}}X)=\tilde{H}_{\ast }^{G}(X)[-1]$, which only holds with $\mathbb{Z}/2$-coefficients. For $\mathbb{Z}$-coefficients and other cohomology theories there need not be suspension-invariance of the same form and $\mathfrak{E}$ must be replaced with a category involving suspension only by certain representations (cf. [Reference ManolescuMan14, Remark 4.4]).

Definition 2.7. We call $X_{1},X_{2}\in \mathfrak{E}$locally equivalent if there exist $G$-equivariant (stable) maps

$$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D719}:X_{1}\rightarrow X_{2}, & \displaystyle \nonumber\\ \displaystyle & \displaystyle \unicode[STIX]{x1D713}:X_{2}\rightarrow X_{1}, & \displaystyle \nonumber\end{eqnarray}$$

which are $G$-homotopy equivalences on the $S^{1}$-fixed-point set. For such $X_{1},X_{2}$, we write $X_{1}\equiv _{l}X_{2}$, and let $\mathfrak{LE}$ denote the set of local equivalence classes.

Local equivalence is easily seen to be an equivalence relation. The set $\mathfrak{LE}$ comes with an abelian group structure, with multiplication given by smash product. One may check that inverses are given by Spanier–Whitehead duality.

2.2 $G$-CW decompositions of $G$-spaces

Throughout this section $X$ will denote a space of type SWF. Here we will give example $G$-CW decompositions and construct a $G$-CW structure on smash products of $G$-spaces.

For $W$ a CW complex, we write $C_{\ast }^{\text{CW}}(W)$ for the corresponding cellular (CW) chain complex. We fix a convenient CW decomposition of $G$. The $0$-cells are the points $1,j,j^{2},j^{3}$ in $G$, and the $1$-cells are $s,js,j^{2}s,j^{3}s$, where $s=\{e^{i\unicode[STIX]{x1D703}}\in S^{1}\mid \unicode[STIX]{x1D703}\in (0,\unicode[STIX]{x1D70B})\}$. We identify each of the cells of this CW decomposition with its image in $C_{\ast }^{\text{CW}}(G)$, the corresponding $CW$ chain complex of $G$. Then $\unicode[STIX]{x2202}(s)=1+j^{2}$. To ease notation, we will refer to $C_{\ast }^{\text{CW}}(G)$ by ${\mathcal{G}}$.

We will use that this CW decomposition also induces a CW decomposition of $S^{1}$, for which $C_{\ast }^{\text{CW}}(S^{1})$ is the subcomplex of ${\mathcal{G}}$ generated by $1,j^{2},s,j^{2}s$.

A $G$-CW decomposition of $X$ also induces a CW decomposition of $X$, using the decomposition of $G$ into cells as above, which we will call a $G$-compatible CW decomposition of $X$.

Example 2.9. We find a CW decomposition for $\mathbb{H}^{+}$ as a $G$-space. We write elements of $\mathbb{H}$ as pairs of complex numbers $(z,w)=(r_{1}e^{i\unicode[STIX]{x1D703}_{1}},r_{2}e^{i\unicode[STIX]{x1D703}_{2}})$ in polar coordinates. The action of $j$ is then given by $j(z,w)=(-\bar{w},\bar{z})$. Fix the point at infinity as the base point. We let $(0,0)$ be the ($G$-invariant) $0$-cell labeled $r_{0}$. We let $y_{1}$ be the $G$-$1$-cell given by the orbit of $\{(r_{1},0)\mid r_{1}>0\}$:

$$\begin{eqnarray}\{(r_{1}e^{i\unicode[STIX]{x1D703}},r_{2}e^{i\unicode[STIX]{x1D703}})\mid \,r_{1}r_{2}=0\}.\end{eqnarray}$$

We take $y_{2}$ the $G$-$2$-cell given by the orbit of $\{(r_{1},r_{2})\mid r_{1}r_{2}\neq 0\}$:

$$\begin{eqnarray}\{(r_{1}e^{i\unicode[STIX]{x1D703}_{1}},r_{2}e^{i\unicode[STIX]{x1D703}_{2}})\mid \,\unicode[STIX]{x1D703}_{1}=\unicode[STIX]{x1D703}_{2}\hspace{0.2em}{\rm mod}\hspace{0.2em}\unicode[STIX]{x1D70B},r_{1}r_{2}\neq 0\}.\end{eqnarray}$$

Finally, $y_{3}$ consists of the orbit of $\{(r_{1}e^{i\unicode[STIX]{x1D703}_{1}},r_{2})\mid \unicode[STIX]{x1D703}_{1}\in (0,\unicode[STIX]{x1D70B}),\;r_{1}r_{2}\neq 0\}$:

$$\begin{eqnarray}\{(r_{1}e^{i\unicode[STIX]{x1D703}_{1}},r_{2}e^{i\unicode[STIX]{x1D703}_{2}})\mid \,\unicode[STIX]{x1D703}_{1}\neq \unicode[STIX]{x1D703}_{2}\;\text{mod}\;\unicode[STIX]{x1D70B},r_{1}r_{2}\neq 0\}.\end{eqnarray}$$

We now give $X_{1}\wedge X_{2}$ a $G$-CW structure for $X_{1}$ and $X_{2}$ spaces of type SWF. To do so, we proceed cell by cell on both factors, so we need only find a $G$-CW structure on $G\times G$, $\mathbb{Z}/2\times G$, and $\mathbb{Z}/2\times \mathbb{Z}/2$, each with the diagonal $G$-action. The space $\mathbb{Z}/2\times G$ has a $G$-CW decomposition as $G\amalg G$, as may be seen directly, and $\mathbb{Z}/2\times \mathbb{Z}/2$ may be written as a disjoint union of $G$-$0$-cells $\mathbb{Z}/2\amalg \mathbb{Z}/2$.

Example 2.10. The $G$-CW structure on $G\times G$ is more complicated. Note that the product CW decomposition on $G\times G$ is not equivariant. We choose a homotopy $\unicode[STIX]{x1D719}_{t}:G\times G\rightarrow G\times G$ as in Figure 1, with $t\in [0,1]$, $\unicode[STIX]{x1D719}_{0}=\text{Id},$ and $\unicode[STIX]{x1D719}_{1}(G\times G)$ shown. The arrows depict the action of $S^{1}$. On the left, the diagonal lines show the $G$-action before homotopy. For example, the homotopy $\unicode[STIX]{x1D719}$ takes the line $\ell =\{(e^{i\unicode[STIX]{x1D703}}\times e^{i\unicode[STIX]{x1D703}})\mid \unicode[STIX]{x1D703}\in (0,\unicode[STIX]{x1D70B})\}$, the first half of the diagonal in $S^{1}\times S^{1}$, to the sum of CW cells:

$$\begin{eqnarray}s\otimes 1+j^{2}\otimes s.\end{eqnarray}$$

The arrows on the right show the $G$-action on $G\times G$ given by

(13)$$\begin{eqnarray}g(g_{1}\times g_{2})=\unicode[STIX]{x1D719}_{1}(g\unicode[STIX]{x1D719}_{1}^{-1}(g_{1}\times g_{2})).\end{eqnarray}$$

The action (13) is clearly cellular with respect to the product CW structure of $G\times G$. Then $G\times G$ admits a $G$-CW-decomposition so that the induced CW decomposition is the product CW decomposition of $G\times G$.

Now, let $X_{1}$ and $X_{2}$ be spaces of type SWF. We then give $X_{1}\wedge X_{2}$ a $G$-CW decomposition proceeding cell-by-cell. That is, for $G$-cells $e_{1}\subseteq X_{1},e_{2}\subseteq X_{2}$ we give $e_{1}\wedge e_{2}$ the appropriate $G$-CW decomposition as constructed above. This is possible because the cells $e_{i}$ are necessarily of the form $D^{k}$, $\mathbb{Z}/2\times D^{k}$, or $G\times D^{k}$. In particular, the construction of a $G$-CW structure on $X_{1}\wedge X_{2}$ gives us a $G$-CW structure for suspensions. For $V$ a finite-dimensional $G$-representation which is a direct sum of copies of $\mathbb{R},\tilde{\mathbb{R}},$ and $\mathbb{H}$, we have $\unicode[STIX]{x1D6F4}^{V}X=V^{+}\wedge X$, and so we give $\unicode[STIX]{x1D6F4}^{V}X$ the smash product $G$-CW decomposition.

Figure 1.

Homotopy of the action of $G$ on $G\times G$.

Finally, we construct a CW structure for the $G$-smash product $X_{1}\wedge _{G}X_{2}=(X_{1}\wedge X_{2})/G$ (where $G$ acts diagonally). More generally, we describe a CW structure for the quotient $W/G$ for $W$ a $G$-CW complex. Indeed, let $W=\bigcup e_{i}$ a $G$-CW complex, where $e_{i}=G/H_{i}\times D^{k(i)}$ are equivariant $G$-cells for some function $k$, and $H_{i}\subseteq G$ are subgroups. Then $W/G$ admits a CW decomposition given by $W/G=\bigcup e_{i}/G=\bigcup D^{k(i)}$.

2.3 Modules from $G$-CW decompositions

Throughout this section $X$ will denote a space of type SWF. Here we will show that the CW chain complex of $X$ inherits a module structure from the action of $G$, and we will define chain local equivalence.

From the group structure of $G$, $C_{\ast }^{\text{CW}}(G)={\mathcal{G}}$ acquires an algebra structure. Namely, the multiplication map $G\times G\rightarrow G$ gives a map $C_{\ast }^{\text{CW}}(G)\otimes _{\mathbb{F}}C_{\ast }^{\text{CW}}(G)\rightarrow C_{\ast }^{\text{CW}}(G)$. Here, we have used the product $G$-CW decomposition of $G\times G$, from Example 2.10, for which the multiplication map is cellular. A small calculation yields

$$\begin{eqnarray}C_{\ast }^{\text{CW}}(G)\cong \mathbb{F}\langle s,j\rangle /(sj=j^{3}s,s^{2}=0,j^{4}=1).\end{eqnarray}$$

For any $G$-compatible decomposition of $X$, the relative CW chain complex $C_{\ast }^{\text{CW}}(X,\text{pt})$, which we will call the reduced $G$-CW chain complex of $X$, inherits the structure of a ${\mathcal{G}}$-chain complex, as the (cellular) map $G\times X\rightarrow X$ gives a map ${\mathcal{G}}\times C_{\ast }^{\text{CW}}(X,\text{pt})\rightarrow C_{\ast }^{\text{CW}}(X,\text{pt})$. That is, $C_{\ast }^{\text{CW}}(X,\text{pt})$ is a module over ${\mathcal{G}}$, such that, for $z\in C_{\ast }^{\text{CW}}(X,\text{pt})$, and $a\in {\mathcal{G}}$, $\unicode[STIX]{x2202}(az)=a\unicode[STIX]{x2202}(z)+\unicode[STIX]{x2202}(a)z$. The assertion that $(gh)x=g(hx)$ for $g,h\in {\mathcal{G}}$ and $x\in C_{\ast }^{\text{CW}}(X,\text{pt})$ follows from the fact that the map $G\times X\rightarrow X$ is a group action. More concretely, having fixed a $G$-compatible CW decomposition of $X$, $C_{\ast }^{\text{CW}}(X)$ is a ${\mathcal{G}}$-module with basis (over ${\mathcal{G}}$) given by the set of $G$-cells.

We find the module structure for the Examples 2.8–2.10 of § 2.2.

Example 2.12. We also find the ${\mathcal{G}}$-chain complex structure of $C_{\ast }^{\text{CW}}(\mathbb{H}^{+},\text{pt})$ from Example 2.9. We choose CW cells $r_{0}^{\prime },y_{i}^{\prime }$, as in Example 2.9, whose orbits are $r_{0},y_{i}$. One may check that the differentials are given by

(14)$$\begin{eqnarray}\unicode[STIX]{x2202}(r_{0}^{\prime })=0,\quad \unicode[STIX]{x2202}y_{1}^{\prime }=r_{0}^{\prime },\quad \unicode[STIX]{x2202}y_{2}^{\prime }=(1+j)y_{1}^{\prime },\quad \text{and}\quad \unicode[STIX]{x2202}y_{3}^{\prime }=sy_{1}^{\prime }+(1+j)y_{2}^{\prime }.\end{eqnarray}$$

The ${\mathcal{G}}$-action on the fixed-point set, $r_{0}^{\prime }$, is necessarily trivial. However, elsewhere the $G$-action on $(\mathbb{H}^{+},\text{pt})$ is free, and so the submodule (not a subcomplex, however) of $C_{\ast }^{\text{CW}}(\mathbb{H}^{+},\text{pt})$ generated by $y_{1}^{\prime },y_{2}^{\prime },y_{3}^{\prime }$ is ${\mathcal{G}}$-free, specifying the ${\mathcal{G}}$-module structure of $C_{\ast }^{\text{CW}}(\mathbb{H}^{+},\text{pt})$.

Example 2.13. The CW chain complex of the usual product CW structure on $G\times G$ becomes a ${\mathcal{G}}$-module via

$$\begin{eqnarray}C_{\ast }^{\text{CW}}(G\times G)=C_{\ast }^{\text{CW}}(G)\otimes _{\mathbb{ F}}C_{\ast }^{\text{CW}}(G),\end{eqnarray}$$

where the action of ${\mathcal{G}}$ is given by

(15)$$\begin{eqnarray}\displaystyle \begin{array}{@{}c@{}}s(a\otimes b)=sa\otimes b+j^{2}a\otimes sb,\\ j(a\otimes b)=ja\otimes jb.\end{array} & & \displaystyle\end{eqnarray}$$

The differentials are induced by those of the usual product CW structure on $G\times G$.

For $X_{1}\wedge X_{2}$ with the $G$-CW decomposition described in § 2.2, we have

(16)$$\begin{eqnarray}C_{\ast }^{\text{CW}}(X_{1}\wedge X_{2},\text{pt})=C_{\ast }^{\text{CW}}(X_{1},\text{pt})\otimes _{\mathbb{F}}C_{\ast }^{\text{CW}}(X_{2},\text{pt}),\end{eqnarray}$$

as ${\mathcal{G}}$-chain complexes, using (15).

Furthermore the CW chain complex for the $G$-smash product $X_{1}\wedge _{G}X_{2}$ is given by

(17)$$\begin{eqnarray}C_{\ast }^{\text{CW}}(X_{1}\wedge _{G}X_{2},\text{pt})\simeq C_{\ast }^{\text{CW}}(X_{1}\wedge X_{2},\text{pt})/{\mathcal{G}}.\end{eqnarray}$$

Here, the right-hand side is defined by quotienting by the action of $(1+j)$ and $s$. One may check directly that the result is a chain complex.

We will write elements of the latter as $x_{1}\otimes _{{\mathcal{G}}}x_{2}$. Note that Borel homology $\tilde{H}_{\ast }^{G}(X)$ is calculated using a $G$-smash product, and so may be computed from the following chain complex:

(18)$$\begin{eqnarray}\tilde{H}_{\ast }^{G}(X)=H((C_{\ast }^{\text{CW}}(EG)\otimes _{\mathbb{ F}}C_{\ast }^{\text{CW}}(X,\text{pt}))/{\mathcal{G}},\unicode[STIX]{x2202}).\end{eqnarray}$$

In (18), we choose some fixed $G$-CW decomposition of $EG$ to define $C_{\ast }^{\text{CW}}(EG)$. Following (18), we make a definition.

Definition 2.14. Let $Z$ a ${\mathcal{G}}$-chain complex. We define the $G$-Borel homology of $Z$ by

(19)$$\begin{eqnarray}H_{\ast }^{G}(Z)=H(C_{\ast }^{\text{CW}}(EG)\otimes _{{\mathcal{G}}}Z,\unicode[STIX]{x2202}),\end{eqnarray}$$

and similarly for $S^{1}$-Borel homology:

(20)$$\begin{eqnarray}H_{\ast }^{S^{1}}(Z)=H(C_{\ast }^{\text{CW}}(EG)\otimes _{C_{\ast }^{\text{CW}}(S^{1})}Z,\unicode[STIX]{x2202}),\end{eqnarray}$$

where $C_{\ast }^{\text{CW}}(S^{1})$ is viewed as a subcomplex of ${\mathcal{G}}$.

Note then that the ${\mathcal{G}}$-module $C_{\ast }^{\text{CW}}(X,\text{pt})$ determines $\tilde{H}_{\ast }^{G}(X)$ for $X$ a space of type SWF.

Definition 2.15. A ${\mathcal{G}}$-chain complex $Z$of type SWF at level $s$ is a ${\mathcal{G}}$-chain complex $Z$ generated by

(21)$$\begin{eqnarray}\{c_{0},c_{1},c_{2},\ldots ,c_{s}\}\,\cup \,\mathop{\bigcup }_{i\in I}\{x_{i}\},\end{eqnarray}$$

subject to the following conditions (perhaps with an overall grading shift). The element $c_{i}$ is of degree $i$, and $I$ is some finite index set. The only relations are $j^{2}c_{i}=c_{i},sc_{i}=0,jc_{0}=c_{0}$. The differentials are given by $\unicode[STIX]{x2202}c_{0}=0$, $\unicode[STIX]{x2202}c_{1}=c_{0}$, and $\unicode[STIX]{x2202}c_{i}=(1+j)c_{i-1}$ for $2\leqslant i\leqslant s$. The submodule generated by $\{{x_{i}\}}_{i\in I}$ is free under the action of ${\mathcal{G}}$. We call the submodule generated by $\{{c_{i}\}}_{i}$ the fixed-point set (or fixed-point subcomplex) of $Z$.

Note that the fixed-point subcomplex is part of the data of a complex of type SWF.

Chain complexes of type SWF are to be thought of as reduced $G$-CW chain complexes of spaces of type SWF. Indeed, all spaces $X$ of type SWF have reduced $G$-CW chain complex homotopy equivalent to a complex of type SWF. To see this, we first choose a homotopy equivalence $X^{S^{1}}\simeq (\tilde{\mathbb{R}}^{s})^{+}$ and decompose the latter according to the CW decomposition of Example 2.8. We note that $X^{S^{1}}$ is a $G$-CW subcomplex of $X$, and all cells of $(X,X^{S^{1}})$ are free $G$-cells, since $X$ is a space of type SWF. Then $X$ is homotopy-equivalent to a space obtained by attaching $X/X^{S^{1}}$ to $(\tilde{\mathbb{R}}^{s})^{+}$. By a cellular approximation we obtain a chain map,

(22)$$\begin{eqnarray}C_{\ast }^{\text{CW}}(X/X^{S^{1}},\text{pt})\rightarrow C_{\ast }^{\text{CW}}(\unicode[STIX]{x1D6F4}^{\mathbb{R}}(\tilde{\mathbb{R}}^{s})^{+},\text{pt}),\end{eqnarray}$$

whose mapping cone is homotopy equivalent to $C_{\ast }^{\text{CW}}(X,\text{pt})[-1]$. However, any two equivariant CW complexes for $(\tilde{\mathbb{R}}^{s})^{+}$ are chain homotopy equivalent. To see this, first note that for any such CW decomposition, the standard CW chain complex $\langle c_{0},\ldots ,c_{s}\rangle$ admits a quasi-isomorphism to $C_{\ast }^{\text{CW}}((\tilde{\mathbb{R}}^{s})^{+},\text{pt})$. This induces a quasi-isomorphism $\langle c_{1},\ldots ,c_{s}\rangle \rightarrow C_{\ast }^{\text{CW}}((\tilde{\mathbb{R}}^{s})^{+},S^{0})$. That is, we have a quasi-isomorphism between free $\mathbb{Z}_{2}$-complexes. By [Reference WeibelWei94, Theorem 10.4.8], these complexes are in fact homotopy equivalent, from which it also follows that $C_{\ast }^{\text{CW}}((\tilde{\mathbb{R}}^{s})^{+},\text{pt})\simeq \langle c_{0},\ldots ,c_{s}\rangle$. Similarly, we obtain that any two equivariant CW complexes for $\unicode[STIX]{x1D6F4}^{\mathbb{R}}(\tilde{\mathbb{R}}^{s})^{+}$ are chain homotopy equivalent. Then we may replace (22) with the same sequence, using the standard CW complex structure on $(\tilde{\mathbb{R}}^{s})^{+}$, and we see that $C_{\ast }^{\text{CW}}(X,\text{pt})$ is homotopy-equivalent to a chain complex of type SWF.

Let $\tilde{\oplus }$ denote a direct sum of ${\mathcal{G}}$-modules that need not be a direct sum of chain complexes. Note that a choice of decomposition $R\,\tilde{\oplus }\,F$ of a chain complex of type SWF, where $R=\langle c_{0},\ldots ,c_{s}\rangle$ and $F=\text{span}_{i}\{x_{i}\}$, is not part of the data in the definition. However, for a space $X$ of type SWF, $C_{\ast }^{\text{CW}}(X,\text{pt})$ comes with a preferred fixed-subset (from fixed cells), and much of what follows could be rephrased where the decomposition is part of the data of a complex of type SWF (this will turn out to be somewhat immaterial; cf. Lemma 2.24).

To introduce chain local equivalence, we will consider the CW chain complexes coming from suspensions. For a module $M$ and a submodule $S\subseteq M$, we let $\langle S\rangle \subseteq M$ denote the subset generated by $S$.

Note that, by Example 2.11 and the $G$-CW decomposition constructed in § 2.2 for suspensions, for $X$ a space of type SWF:

(23)$$\begin{eqnarray}C_{\ast }^{\text{CW}}(\unicode[STIX]{x1D6F4}^{\tilde{\mathbb{R}}}X,\text{pt})=\langle c_{0},c_{1}\rangle \,\otimes _{\mathbb{F}}\,C_{\ast }^{\text{CW}}(X,\text{pt}),\end{eqnarray}$$

with relations $\unicode[STIX]{x2202}c_{1}=c_{0}$, $j^{2}c_{1}=c_{1},jc_{0}=c_{0},sc_{0}=sc_{1}=0$. The differential on the right is given by $\unicode[STIX]{x2202}(a\otimes b)=\unicode[STIX]{x2202}(a)\otimes b+a\otimes \unicode[STIX]{x2202}(b)$. Similarly, using Example 2.12:

$$\begin{eqnarray}C_{\ast }^{\text{CW}}(\unicode[STIX]{x1D6F4}^{\mathbb{H}}X,\text{pt})=\langle r_{0},y_{1},y_{2},y_{3}\rangle \,\otimes _{\mathbb{F}}\,C_{\ast }^{\text{CW}}(X,\text{pt}),\end{eqnarray}$$

with the product differential on the right, and differentials for the $y_{i}$ given as in Example 2.12.

For $V=\mathbb{H}$, $\tilde{\mathbb{R}}$, or $\mathbb{R}$, and $Z$ a ${\mathcal{G}}$-chain complex, we set

(24)$$\begin{eqnarray}\unicode[STIX]{x1D6F4}^{V}Z=C_{\ast }^{\text{CW}}(V^{+},\text{pt})\,\otimes _{\mathbb{ F}}\,Z,\end{eqnarray}$$

with ${\mathcal{G}}$-action given by

(25)$$\begin{eqnarray}\displaystyle \begin{array}{@{}c@{}}s(a\otimes b)=(sa\otimes b)+(j^{2}a\otimes sb),\\[2.0pt] j(a\otimes b)=ja\otimes jb.\end{array} & & \displaystyle\end{eqnarray}$$

The chain complexes $C_{\ast }^{\text{CW}}(\mathbb{H}^{+},\text{pt})$ and $C_{\ast }^{\text{CW}}(\tilde{\mathbb{R}}^{+},\text{pt})$ were given in Examples 2.11 and 2.12, respectively. Also, $C_{\ast }^{\text{CW}}(\mathbb{R}^{+},\text{pt})=\langle c_{1}\rangle$, where $jc_{1}=c_{1},sc_{1}=0$, and $\text{deg}\,c_{1}=1$. Hence, for example,

(26)$$\begin{eqnarray}\unicode[STIX]{x1D6F4}^{\mathbb{R}}Z=Z[-1].\end{eqnarray}$$

For $V=\mathbb{H}^{i}\oplus \tilde{\mathbb{R}}^{j}\oplus \mathbb{R}^{k}$ for some constants $i,j,k$, we define $\unicode[STIX]{x1D6F4}^{V}Z$ by

(27)$$\begin{eqnarray}\unicode[STIX]{x1D6F4}^{V}Z=(\unicode[STIX]{x1D6F4}^{\mathbb{H}})^{i}(\unicode[STIX]{x1D6F4}^{\tilde{\mathbb{R}}})^{j}(\unicode[STIX]{x1D6F4}^{\mathbb{R}})^{k}Z,\end{eqnarray}$$

where $(\unicode[STIX]{x1D6F4}^{\mathbb{H}})^{i}$ denotes applying $\unicode[STIX]{x1D6F4}^{\mathbb{H}}$$i$ times, and so for $\tilde{\mathbb{R}}$ and $\mathbb{R}$. Then

(28)$$\begin{eqnarray}\unicode[STIX]{x1D6F4}^{V}\unicode[STIX]{x1D6F4}^{W}Z\simeq \unicode[STIX]{x1D6F4}^{W}\unicode[STIX]{x1D6F4}^{V}Z,\end{eqnarray}$$

for two $G$-representations $V,W$, each a direct sum of copies of $\mathbb{H},\mathbb{R},\tilde{\mathbb{R}}$.

Definition 2.17. Let $Z_{i}$ be chain complexes of type SWF, $m_{i}\in \mathbb{Z},n_{i}\in \mathbb{Q}$, for $i=1,2$. We call $(Z_{1},m_{1},n_{1})$ and $(Z_{2},m_{2},n_{2})$chain stably equivalent if $n_{1}-n_{2}\in \mathbb{Z}$ and there exist $M\in \mathbb{Z},N\in \mathbb{Q}$ and maps

(29)$$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D6F4}^{(N-n_{1})\mathbb{H}}\unicode[STIX]{x1D6F4}^{(M-m_{1})\tilde{\mathbb{R}}}Z_{1}\rightarrow \unicode[STIX]{x1D6F4}^{(N-n_{2})\mathbb{H}}\unicode[STIX]{x1D6F4}^{(M-m_{2})\tilde{\mathbb{R}}}Z_{2}, & \displaystyle\end{eqnarray}$$

(30)$$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D6F4}^{(N-n_{1})\mathbb{H}}\unicode[STIX]{x1D6F4}^{(M-m_{1})\tilde{\mathbb{R}}}Z_{1}\leftarrow \unicode[STIX]{x1D6F4}^{(N-n_{2})\mathbb{H}}\unicode[STIX]{x1D6F4}^{(M-m_{2})\tilde{\mathbb{R}}}Z_{2}, & \displaystyle\end{eqnarray}$$

which are chain homotopy equivalences.

Chain stable equivalence is an equivalence relation, and we denote the set of chain stable equivalence classes by $\mathfrak{CE}$.

Proof. Say that $[(X_{1},m_{1},n_{1})]=[(X_{2},m_{2},n_{2})]\in \mathfrak{C}$ with $G$-CW decompositions $C_{i}$ of $X_{i}$. We will show that

(31)$$\begin{eqnarray}[(C_{\ast }^{\text{CW}}(X_{1},\text{pt}),m_{1},n_{1})]=[(C_{\ast }^{\text{CW}}(X_{2},\text{pt}),m_{2},n_{2})]\in \mathfrak{CE},\end{eqnarray}$$

where $C_{\ast }^{\text{CW}}(X_{i},\text{pt})$ is the ${\mathcal{G}}$-chain complex associated to the $G$-CW decomposition $C_{i}$ of $X_{i}$. (In the case $X_{1}\simeq X_{2}$, and $m_{1}=m_{2},n_{1}=n_{2}$, we are showing that the corresponding element in $\mathfrak{CE}$ does not depend on the choice of $G$-CW decomposition.) By hypothesis, there are homotopy equivalences $f$ and $g$:

$$\begin{eqnarray}\displaystyle & \displaystyle f:\unicode[STIX]{x1D6F4}^{(N-n_{1})\mathbb{H}}\unicode[STIX]{x1D6F4}^{(M-m_{1})\tilde{\mathbb{R}}}X_{1}\rightarrow \unicode[STIX]{x1D6F4}^{(N-n_{2})\mathbb{H}}\unicode[STIX]{x1D6F4}^{(M-m_{2})\tilde{\mathbb{R}}}X_{2}, & \displaystyle \nonumber\\ \displaystyle & \displaystyle g:\unicode[STIX]{x1D6F4}^{(N-n_{2})\mathbb{H}}\unicode[STIX]{x1D6F4}^{(M-m_{2})\tilde{\mathbb{R}}}X_{2}\rightarrow \unicode[STIX]{x1D6F4}^{(N-n_{1})\mathbb{H}}\unicode[STIX]{x1D6F4}^{(M-m_{1})\tilde{\mathbb{R}}}X_{1}. & \displaystyle \nonumber\end{eqnarray}$$

By the equivariant cellular approximation theorem (see [Reference WanerWan80]), we may homotope $f$ and $g$ to cellular maps (where the cell structures of suspensions are given as in (24)):

$$\begin{eqnarray}\displaystyle & \displaystyle f^{\text{CW}}:\unicode[STIX]{x1D6F4}^{(N-n_{1})\mathbb{H}}\unicode[STIX]{x1D6F4}^{(M-m_{1})\tilde{\mathbb{R}}}C_{1}\rightarrow \unicode[STIX]{x1D6F4}^{(N-n_{2})\mathbb{H}}\unicode[STIX]{x1D6F4}^{(M-m_{2})\tilde{\mathbb{R}}}C_{2}, & \displaystyle \nonumber\\ \displaystyle & \displaystyle g^{\text{CW}}:\unicode[STIX]{x1D6F4}^{(N-n_{2})\mathbb{H}}\unicode[STIX]{x1D6F4}^{(M-m_{2})\tilde{\mathbb{R}}}C_{2}\rightarrow \unicode[STIX]{x1D6F4}^{(N-n_{1})\mathbb{H}}\unicode[STIX]{x1D6F4}^{(M-m_{1})\tilde{\mathbb{R}}}C_{1}. & \displaystyle \nonumber\end{eqnarray}$$

Since $f$ and $g$ are homotopy equivalences, so are $f^{\text{CW}}$ and $g^{\text{CW}}$. The cellular maps $f^{\text{CW}}$ and $g^{\text{CW}}$ induce maps $f_{\ast }$ and $g_{\ast }$:

$$\begin{eqnarray}\displaystyle & \displaystyle f_{\ast }:\unicode[STIX]{x1D6F4}^{(N-n_{1})\mathbb{H}}\unicode[STIX]{x1D6F4}^{(M-m_{1})\tilde{\mathbb{R}}}C_{\ast }^{\text{CW}}(X_{1},\text{pt})\rightarrow \unicode[STIX]{x1D6F4}^{(N-n_{2})\mathbb{H}}\unicode[STIX]{x1D6F4}^{(M-m_{2})\tilde{\mathbb{R}}}C_{\ast }^{\text{CW}}(X_{2},\text{pt}), & \displaystyle \nonumber\\ \displaystyle & \displaystyle g_{\ast }:\unicode[STIX]{x1D6F4}^{(N-n_{2})\mathbb{H}}\unicode[STIX]{x1D6F4}^{(M-m_{2})\tilde{\mathbb{R}}}C_{\ast }^{\text{CW}}(X_{2},\text{pt})\rightarrow \unicode[STIX]{x1D6F4}^{(N-n_{1})\mathbb{H}}\unicode[STIX]{x1D6F4}^{(M-m_{1})\tilde{\mathbb{R}}}C_{\ast }^{\text{CW}}(X_{1},\text{pt}). & \displaystyle \nonumber\end{eqnarray}$$

These are chain homotopy equivalences, by construction, and so we obtain (31), as needed. ◻

Definition 2.20. Let $Z_{i}$ be chain complexes of type SWF, $m_{i}\in \mathbb{Z},n_{i}\in \mathbb{Q}$, for $i=1,2$. We call $(Z_{1},m_{1},n_{1})$ and $(Z_{2},m_{2},n_{2})$chain locally equivalent, written $(Z_{1},m_{1},n_{1})\equiv _{cl}(Z_{2},m_{2},n_{2})$, if there exist $M\in \mathbb{Z},N\in \mathbb{Q}$ and maps

(32)$$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D6F4}^{(N-n_{1})\mathbb{H}}\unicode[STIX]{x1D6F4}^{(M-m_{1})\tilde{\mathbb{R}}}Z_{1}\rightarrow \unicode[STIX]{x1D6F4}^{(N-n_{2})\mathbb{H}}\unicode[STIX]{x1D6F4}^{(M-m_{2})\tilde{\mathbb{R}}}Z_{2}, & \displaystyle\end{eqnarray}$$

(33)$$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D6F4}^{(N-n_{1})\mathbb{H}}\unicode[STIX]{x1D6F4}^{(M-m_{1})\tilde{\mathbb{R}}}Z_{1}\leftarrow \unicode[STIX]{x1D6F4}^{(N-n_{2})\mathbb{H}}\unicode[STIX]{x1D6F4}^{(M-m_{2})\tilde{\mathbb{R}}}Z_{2}, & \displaystyle\end{eqnarray}$$

which are chain homotopy equivalences on fixed-point sets.

We call a map as in (32) or (33) a chain local equivalence. By the condition ‘is a homotopy equivalence on fixed-point sets’ we mean that the maps restrict to homotopy equivalences of $\langle c_{0},\ldots ,c_{s}\rangle$ (in particular, we require that the chain maps in (32)–(33) send fixed-point sets to fixed-point sets).

Elements $Z_{1},Z_{2}\in \mathfrak{CE}$ are chain locally equivalent if and only if there are chain local equivalences $Z_{1}\rightarrow Z_{2}$ and $Z_{2}\rightarrow Z_{1}$. There are pairs of chain complexes with a chain local equivalence in one direction but not the other; these are not chain locally equivalent complexes. Chain local equivalence is an equivalence relation. Denote the set of chain local equivalence classes by $\mathfrak{CLE}$. The set $\mathfrak{CLE}$ is naturally an abelian group, with multiplication given by the tensor product (over $\mathbb{F}$, with ${\mathcal{G}}$-action as above). To discuss inverses in $\mathfrak{CLE}$, we take a brief discursion.

Note that it is not obvious that chain homotopy equivalent complexes of type SWF are chain locally equivalent, as there may be self-homotopy-equivalences of a complex that do not take fixed-point sets to fixed-point sets. Note, however, that any two ${\mathcal{G}}$-chain complexes coming from a space of type SWF with different $G$-CW decompositions are necessarily homotopy-equivalent by a chain map preserving the subcomplex of the fixed point set (coinciding with the notion of the ‘fixed-point set’ of a chain complex of type SWF), by the equivariant cellular approximation theorem. For more general ${\mathcal{G}}$-chain complexes, we have Lemma 2.26 below, which states that the chain homotopy type determines the chain local equivalence class.

Inverses are given by dual chain complexes. Note that $\text{Hom}(Z;\mathbb{F})$ will not usually be a chain complex of type SWF; however, some suspension of it will be chain homotopy equivalent to a chain complex of type SWF; see Lemma 2.27. In the case of chain complexes of type SWF arising as CW-chain complexes, inverses are more readily understood. Here, we recall from [Reference ManolescuMan16] that the Spanier–Whitehead dual of a space of type SWF is again a space of type SWF, from which the existence of group inverses (for chain complexes coming from spaces) follows.

2.4 Calculating the chain local equivalence class

In this section we will obtain a description of $\mathfrak{CLE}$ more amenable to calculations than the definition. Throughout this section $Z$ will denote a chain complex of type SWF. The main result is Lemma 2.24, which allows us to determine if $(Z_{1},m_{1},n_{1})$ and $(Z_{2},m_{2},n_{2})$ are chain locally equivalent without checking all possible $M,N$.

For $Z$ a chain complex of type SWF, we will let $Z$ also denote the element $(Z,0,0)\in \mathfrak{CE}$.

For $Z_{1},Z_{2}\in \mathfrak{CLE}$, we call a local equivalence $Z_{1}\rightarrow Z_{2}$ (without needing to suspend either complex) an unstable local map.

Proof. We may assume $s>0$, as otherwise the statement is vacuous. To construct $Z_{i}^{\prime }$, observe that the image $\text{Im}\,\unicode[STIX]{x2202}|_{F_{i}}\cap R_{i}$, must be either $\{(1+j)c_{s}\}$ or empty, and in the latter case the lemma is trivial, so we assume that the former holds. Then, construct a new complex $Z_{i}^{\prime }=\langle f\rangle \,\tilde{\oplus }\,F_{i}[s]$, where the boundary map $F_{i}[s]\rightarrow \langle f\rangle$ is obtained as the boundary map $F_{i}\rightarrow \langle (1+j)c_{s}\rangle$. It is clear that an unstable local map $Z_{1}\rightarrow Z_{2}$ induces an unstable local map $Z_{1}^{\prime }\rightarrow Z_{2}^{\prime }$, and we are left with showing $Z_{i}\simeq \unicode[STIX]{x1D6F4}^{\tilde{\mathbb{R}}^{s}}Z_{i}^{\prime }$.

To see this, we view $\unicode[STIX]{x1D6F4}^{\tilde{\mathbb{R}}^{s}}Z_{i}^{\prime }$ as the mapping cone of

(34)$$\begin{eqnarray}\unicode[STIX]{x1D6F4}^{\tilde{\mathbb{R}}^{s}}F_{i}[s]\rightarrow \unicode[STIX]{x1D6F4}^{\tilde{\mathbb{R}}^{s}}\langle f\rangle [-1].\end{eqnarray}$$

We claim $\unicode[STIX]{x1D6F4}^{\tilde{\mathbb{R}}^{s}}F_{i}[s]\simeq F_{i}$. To see this, construct a map $F_{i}\rightarrow \unicode[STIX]{x1D6F4}^{\tilde{\mathbb{R}}^{s}}F_{i}[s]$ by $x\rightarrow (1+j)c_{s}\otimes x[s]$; it is straightforward to check that this map is a quasi-isomorphism. By [Reference WeibelWei94, Theorem 10.4.8], and using that $\unicode[STIX]{x1D6F4}^{\tilde{\mathbb{R}^{s}}}F_{i}$ is free, this map admits a homotopy inverse, establishing the claim. Then, the mapping cone (34) is homotopy equivalent to the mapping cone of some $F_{i}\rightarrow \unicode[STIX]{x1D6F4}^{\tilde{\mathbb{R}}^{s}}\langle f\rangle [-1]\cong R_{i}[-1]$. In particular, the map $F_{i}\rightarrow \unicode[STIX]{x1D6F4}^{\tilde{\mathbb{R}}^{s}}F_{i}[s]$ constructed above commutes with the boundary maps in the construction of $Z_{i}^{\prime }$ and in (34). Then we have a map $\unicode[STIX]{x1D719}:Z_{i}\rightarrow \unicode[STIX]{x1D6F4}^{\tilde{\mathbb{R}}^{s}}Z_{i}^{\prime }$, which is a quasi-isomorphism; that is, the mapping cone $\text{Cone}(\unicode[STIX]{x1D719})$ is trivial in the derived category. Since $\text{Cone}(\unicode[STIX]{x1D719})$ is homotopy-equivalent to a free complex, it is also trivial in the homotopy category, and so $\unicode[STIX]{x1D719}$ is a homotopy equivalence, as needed.◻

Lemma 2.24. Let $Z_{1}$ and $Z_{2}$ be locally equivalent suspensionlike chain complexes of type SWF. Then there exist unstable local maps

(35)$$\begin{eqnarray}Z_{1}\leftrightarrow Z_{2},\end{eqnarray}$$

without needing to suspend $Z_{i}$.

Proof. By Lemma 2.23, we may assume that $Z_{1}$ and $Z_{2}$ are level $0$ and suspensionlike. It then suffices to show that if there are unstable local maps $\unicode[STIX]{x1D6F4}^{\mathbb{H}}Z_{1}\leftrightarrow \unicode[STIX]{x1D6F4}^{\mathbb{H}}Z_{2}$, then there are unstable local maps $Z_{1}\leftrightarrow Z_{2}$.

Let $Z_{i}=R_{i}\,\tilde{\oplus }\,F_{i}$. That is, $Z_{i}$ is the mapping cone of a map $F_{i}\rightarrow R_{i}[-1]$. Now, $\unicode[STIX]{x1D6F4}^{\mathbb{H}}Z_{i}=\unicode[STIX]{x1D6F4}^{\mathbb{H}}R_{i}\,\tilde{\oplus }\,\unicode[STIX]{x1D6F4}^{\mathbb{H}}F_{i}$. We claim $\unicode[STIX]{x1D6F4}^{\mathbb{H}}F_{i}\simeq F_{i}[-4]$. First, we have a quasi-isomorphism $F_{i}[-4]\rightarrow \unicode[STIX]{x1D6F4}^{\mathbb{H}}F_{i}$ by $x[-4]\rightarrow s(1+j)^{3}y_{3}\otimes x$. Further, any free finite ${\mathcal{G}}$-chain complex $F$ with $H(F)=0$ is nullhomotopic (the verification of this is similar to the construction of projective resolutions in abelian categories). It follows that free finite quasi-isomorphic ${\mathcal{G}}$-chain complexes are in fact homotopy equivalent, giving the claim. In particular, $\unicode[STIX]{x1D6F4}^{\mathbb{H}}Z_{i}$ is homotopy equivalent to the mapping cone of a map $F_{i}[-4]\rightarrow \unicode[STIX]{x1D6F4}^{\mathbb{H}}R_{i}[-1]$. However, since $Z_{i}$ is suspensionlike, we have that $F_{i}[-4]$ is concentrated in degree at least $5$, while $s(1+j)^{3}y_{3}[-1]$ is the only cycle in degree $5$. It is then readily checked that the map $F_{i}[-4]\rightarrow \langle s(1+j)^{3}y_{3}[-1]\rangle$ defining (a complex homotopy equivalent to) $\unicode[STIX]{x1D6F4}^{\mathbb{H}}Z_{i}$ is identical to the map $F_{i}\rightarrow R_{i}$ defining $Z_{i}$.

Now, say we have local equivalences $\unicode[STIX]{x1D719}:\unicode[STIX]{x1D6F4}^{\mathbb{H}}R_{1}\,\tilde{\oplus }\,F_{1}[-4]\rightarrow \unicode[STIX]{x1D6F4}^{\mathbb{H}}R_{2}\,\tilde{\oplus }\,F_{2}[-4]$. For grading reasons, $\unicode[STIX]{x1D719}$ must send $\unicode[STIX]{x1D6F4}^{\mathbb{H}}R_{1}\rightarrow \unicode[STIX]{x1D6F4}^{\mathbb{H}}R_{2}$. In particular, there is then a map of exact sequences as follows.

Any local map $\unicode[STIX]{x1D719}$ sends $s(1+j)^{3}y_{3}\otimes c_{0}\in \unicode[STIX]{x1D6F4}^{\mathbb{H}}R_{1}\rightarrow s(1+j)^{3}y_{3}\otimes c_{0}\in \unicode[STIX]{x1D6F4}^{\mathbb{H}}R_{2}$, as is a straightforward check. Then we inherit a local map $\unicode[STIX]{x1D713}:Z_{1}\rightarrow Z_{2}$ by defining it separately on the fixed point set and on $F_{1}$. That is, set $\unicode[STIX]{x1D713}(c_{0})=c_{0}$, and for $x\in F_{1}$, set $\unicode[STIX]{x1D713}(x)=\unicode[STIX]{x1D719}(x[-4])$. The map $\unicode[STIX]{x1D713}$ commutes with the boundary $F_{i}\rightarrow R_{i}[-1]$ using $\unicode[STIX]{x1D719}(s(1+j)^{3}y_{3}\otimes c_{0})=s(1+j)^{3}y_{3}\otimes c_{0}$ and the construction of the boundary map $F_{i}[-4]\rightarrow \langle s(1+j)^{3}y_{3}\rangle [-1]$. Then, $\unicode[STIX]{x1D713}$ is a local map, and we similarly construct a local map going the other direction, completing the proof.◻

Lemma 2.24 states that if $\unicode[STIX]{x1D6F4}^{(N_{0}-n_{i})\mathbb{H}}\unicode[STIX]{x1D6F4}^{(M_{0}-m_{i})\tilde{\mathbb{R}}}Z_{i}$ are suspensionlike, then if $(Z_{1},m_{1},n_{1})$ and $(Z_{2},m_{2},n_{2})$ are (stably) locally equivalent, there is a local equivalence realized by genuine (unstable) chain maps after suspending the complexes $Z_{i}$ by $(N_{0}-n_{i})\mathbb{H}\oplus (M_{0}-m_{i})\tilde{\mathbb{R}}$.

Finally, to take advantage of Lemma 2.24, we need to know that many complexes admit suspensionlike representatives.

Similarly, we can show the following result.

We have a result similar to Lemma 2.25 relating to duality.

Finally, we remark that we could simplify the discussion of duality by defining a variant of $\mathfrak{CLE}$ using only chain complexes that are homotopy-equivalent to the CW chain complex associated to actual $G$-spaces of type SWF.

2.5 Inessential subcomplexes and connected quotient complexes

In this section, we show how Lemma 2.24 allows for a convenient characterization of chain locally equivalent complexes. We then define connected $S^{1}$-homology of spaces of type SWF, which we will use later to define $\mathit{SWFH}_{\text{conn}}$ as in Corollary 1.5.

If $M$ is inessential, then $Z/M\equiv _{cl}Z$. We order inessential subcomplexes by inclusion, $N\leqslant M$ if $N\subseteq M$. We show that there is a unique ‘minimal’ model $Z/N$ locally equivalent to $Z$.

Proof. Indeed, if there exist maps $\unicode[STIX]{x1D6FC}:Z/M\rightarrow Z,$ and $\unicode[STIX]{x1D6FD}:Z/N\rightarrow Z$ as above, consider the composition

$$\begin{eqnarray}\unicode[STIX]{x1D719}:Z/N\rightarrow Z\rightarrow Z/M.\end{eqnarray}$$

In particular, we have a map $\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D719}:Z/N\rightarrow Z$, which is injective by Lemma 2.29. It then follows that $\unicode[STIX]{x1D719}$ is injective. We also have

$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D713}:Z/M\rightarrow Z\rightarrow Z/N. & & \displaystyle \nonumber\end{eqnarray}$$

As before, $\unicode[STIX]{x1D713}$ is injective. Then, since we have injective chain maps between $Z/N$ and $Z/M$, finite-dimensional $\mathbb{F}$-complexes, the two chain complexes must have the same dimension. An injective map between complexes of the same dimension is bijective, and, finally, a bijective ${\mathcal{G}}$-chain complex map is a ${\mathcal{G}}$-chain complex isomorphism.◻

Lemma 2.31. Let $Z$ be a chain complex of type SWF and $M$ a maximal inessential subcomplex of $Z$. We have a (noncanonical) decomposition of $Z$:

(36)$$\begin{eqnarray}Z=(Z/M)\oplus M,\end{eqnarray}$$

where the isomorphism class of $Z/M$ is an invariant of $Z$, independent of the choice of maximal inessential subcomplex $M\subseteq Z$.

Proof. By Lemma 2.31, we may write $Z=Z_{\text{conn}}\oplus Z_{\text{iness}},W=W_{\text{conn}}\oplus W_{\text{iness}}$, with $Z_{\text{iness}},W_{\text{iness}}$ maximal inessential subcomplexes. Say we have local equivalences (we need not consider suspensions, by Lemma 2.24)

$$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D719}:Z_{\text{conn}}\oplus Z_{\text{iness}}\rightarrow W_{\text{conn}}\oplus W_{\text{iness}}, & \displaystyle \nonumber\\ \displaystyle & \displaystyle \unicode[STIX]{x1D713}:W_{\text{conn}}\oplus W_{\text{iness}}\rightarrow Z_{\text{conn}}\oplus Z_{\text{iness}}. & \displaystyle \nonumber\end{eqnarray}$$

We restrict $\unicode[STIX]{x1D719}$ and $\unicode[STIX]{x1D713}$ to $Z_{\text{conn}}$ and $W_{\text{conn}}$, since it is clear that $Z_{\text{conn}}\oplus Z_{\text{iness}}$ is chain locally equivalent to $Z_{\text{conn}}$, and likewise for $W_{\text{conn}}$. Further, we project the image of $\unicode[STIX]{x1D719}$ and $\unicode[STIX]{x1D713}$ to $W_{\text{conn}}$ and $Z_{\text{conn}}$, respectively. Call the resulting maps $\unicode[STIX]{x1D719}_{0}$ and $\unicode[STIX]{x1D713}_{0}$. If $\unicode[STIX]{x1D719}_{0}$ had a nontrivial kernel, then we would obtain by composition a local equivalence:

$$\begin{eqnarray}\unicode[STIX]{x1D713}_{0}\unicode[STIX]{x1D719}_{0}:Z_{\text{conn}}/\text{ker}\,\unicode[STIX]{x1D719}_{0}\rightarrow Z_{\text{conn}}.\end{eqnarray}$$

Composing with the inclusion $\unicode[STIX]{x1D704}:Z_{\text{conn}}\rightarrow Z$ gives a chain local map $\unicode[STIX]{x1D704}\unicode[STIX]{x1D713}_{0}\unicode[STIX]{x1D719}_{0}:Z_{\text{conn}}/\text{ker}\;\unicode[STIX]{x1D719}_{0}\rightarrow Z$, so by Lemma 2.29, $\unicode[STIX]{x1D704}\unicode[STIX]{x1D713}_{0}\unicode[STIX]{x1D719}_{0}$ is injective. Thus, $\unicode[STIX]{x1D719}_{0}$ is injective. Similarly $\unicode[STIX]{x1D713}_{0}$ is injective, so we obtain an isomorphism of chain complexes $Z_{\text{conn}}\cong W_{\text{conn}}$. Conversely, a homotopy equivalence $Z_{\text{conn}}\rightarrow W_{\text{conn}}$ yields a local equivalence $Z\rightarrow W$ by the composition

$$\begin{eqnarray}Z\xrightarrow[{}]{\unicode[STIX]{x1D70B}}Z_{\text{conn}}\rightarrow W_{\text{conn}}\rightarrow W,\end{eqnarray}$$

where $\unicode[STIX]{x1D70B}:Z\rightarrow Z_{\text{conn}}$ is projection to the first summand.◻

The next corollary allows us to view the chain local equivalence type of a space of type SWF in $\mathfrak{CE}$ instead of $\mathfrak{CLE}$.

Proof. Fix $[(Z,m,n)]=[(Z^{\prime },m^{\prime },n^{\prime })]\in \mathfrak{CLE}$ with $Z$ and $Z^{\prime }$ suspensionlike; we will show that $[(Z_{\text{conn}},m,n)]=[(Z_{\text{conn}}^{\prime },m^{\prime },n^{\prime })]$ in $\mathfrak{CE}$. First, we observe that, for $V=\mathbb{H},\tilde{\mathbb{R}}$:

(37)$$\begin{eqnarray}\unicode[STIX]{x1D6F4}^{V}Z_{\text{conn}}\simeq (\unicode[STIX]{x1D6F4}^{V}Z)_{\text{conn}}.\end{eqnarray}$$

We have, for $M,N$ sufficiently large,

$$\begin{eqnarray}\unicode[STIX]{x1D6F4}^{(M-m)\tilde{\mathbb{R}}}\unicode[STIX]{x1D6F4}^{(N-n)\mathbb{H}}Z\leftrightarrows \unicode[STIX]{x1D6F4}^{(M-m^{\prime })\tilde{\mathbb{R}}}\unicode[STIX]{x1D6F4}^{(N-n^{\prime })\mathbb{H}}Z^{\prime }.\end{eqnarray}$$

Here the maps in both directions are local equivalences. Choosing $M\geqslant \text{max}\{m,m^{\prime }\}\,$ and $N\geqslant \text{max}\,\{n,n^{\prime }\}$ guarantees that both

$$\begin{eqnarray}\unicode[STIX]{x1D6F4}^{(M-m)\tilde{\mathbb{R}}}\unicode[STIX]{x1D6F4}^{(N-n)\mathbb{H}}Z\quad \text{and}\quad \unicode[STIX]{x1D6F4}^{(M-m^{\prime })\tilde{\mathbb{R}}}\unicode[STIX]{x1D6F4}^{(N-n^{\prime })\mathbb{H}}Z^{\prime }\end{eqnarray}$$

are suspensionlike. Then, by Theorem 2.33, we have a homotopy equivalence:

$$\begin{eqnarray}(\unicode[STIX]{x1D6F4}^{(M-m)\tilde{\mathbb{R}}}\unicode[STIX]{x1D6F4}^{(N-n)\mathbb{H}}Z)_{\text{conn}}\rightarrow (\unicode[STIX]{x1D6F4}^{(M-m^{\prime })\tilde{\mathbb{R}}}\unicode[STIX]{x1D6F4}^{(N-n^{\prime })\mathbb{H}}Z^{\prime })_{\text{conn}}.\end{eqnarray}$$

However, by (37), we obtain a homotopy equivalence:

$$\begin{eqnarray}\unicode[STIX]{x1D6F4}^{(M-m)\tilde{\mathbb{R}}}\unicode[STIX]{x1D6F4}^{(N-n)\mathbb{H}}(Z_{\text{conn}})\rightarrow \unicode[STIX]{x1D6F4}^{(M-m^{\prime })\tilde{\mathbb{R}}}\unicode[STIX]{x1D6F4}^{(N-n^{\prime })\mathbb{H}}(Z_{\text{conn}}^{\prime }).\end{eqnarray}$$

Then $[(Z_{\text{conn}},m,n)]=[(Z_{\text{conn}}^{\prime },m^{\prime },n^{\prime })]\in \mathfrak{CE}$, as needed. Finally, we show $B$ is injective. If $(Z_{\text{conn}},m,n)$ is stably equivalent to $(Z_{\text{conn}}^{\prime },m^{\prime },n^{\prime })$, then $(Z,m,n)$ and $(Z^{\prime },m^{\prime },n^{\prime })$ are locally equivalent, by Theorem 2.33 and (37).◻

By Corollary 2.34, instead of considering the relation given by chain local equivalence, we need only consider chain homotopy equivalences.