3.1 Representations

Recall that we defined the group G to be the cyclic group of order 4 generated by $j \in \mathrm{Pin}(2)$ , i.e.

\[G = \{1, j, -1, -j\}. \]

Define a subgroup H of G by

\[H = \{1,-1\} \subset G. \]

Let $\mathbb{R}$ denote the trivial one-dimensional real representation of G. Let $\tilde{\mathbb{R}}$ be the one-dimensional real representation space of G defined by the surjection $G \to \mathbb{Z}_2=\{1,-1\}$ and the scalar multiplication of $\mathbb{Z}_2$ on $\mathbb{R}$ .

Let $\tilde{\mathbb{C}}$ be a one-dimensional complex representation of G defined via the surjection $G \to \mathbb{Z}_2$ and the scalar multiplication of $\mathbb{Z}_2$ on $\mathbb{C}$ . Note that for an even natural number s, say 2t, there is an isomorphism of real representations $\tilde{\mathbb{R}}^{2s} \cong \tilde{\mathbb{C}}^t$ .

We also introduce G-representations

\begin{align*} \mathbb{C}_+,\ \mathbb{C}_-, \end{align*}

where $\mathbb{C}_+, \mathbb{C}_-$ are complex one-dimensional representations defined by assigning to $j \in G$ multiplication of $i, -i$ , respectively. Note that we have the relation $\mathbb{C}_- = \tilde{\mathbb{C}} \cdot \mathbb{C}_+$ in the complex representation ring R(G).

We can think of these spaces

(3.1) \begin{align} \mathbb{R},\ \tilde{\mathbb{R}},\ \mathbb{C}_+,\ \mathbb{C}_-\end{align}

as H-representations through the inclusion $H \subset G$ , and they correspond to

\[ \mathbb{R},\ \mathbb{R},\ \tilde{\mathbb{C}},\ \tilde{\mathbb{C}} \]

as H-representations, respectively.

It is straightforward to check that the representation ring R(G) is given by

\[R(G) = \mathbb{Z}[w,z]/(w^2-2w, w-2z+z^2). \]

Here, the generators w,z are given as the K-theoretic Euler classes of $\tilde{\mathbb{C}}, \mathbb{C}_+$ , namely,

\[w=1-\tilde{\mathbb{C}},\quad z=1-\mathbb{C}_+. \]

The augmentation map $R(G) \to \mathbb{Z}$ is given by $w,z \mapsto 0$ , so the augmentation ideal is given by $(w,z) \subset R(G)$ .

Readers may compare this expression with the standard expression of R(G), given by $R(G)=\mathbb{Z}[t]/(t^4-1)$ , an isomorphism

\[\mathbb{Z}[w,z]/(w^2-2w, w-2z+z^2) \to \mathbb{Z}[t]/(t^4-1) \]

is given by $w \mapsto 1-t^2$ and $z \mapsto 1-t$ .

It is also straightforward to check that

(3.2)

In a finite-dimensional approximation of the configuration space on which the (finite-dimensional approximation of) G-equivariant Seiberg–Witten flow acts, only three of the representations (3.1) may appear: $\tilde{\mathbb{R}},\ \mathbb{C}_+,\ \mathbb{C}_-$ . Here $\mathbb{C}_+, \mathbb{C}_-$ appear in the following way. Originally a finite-dimensional approximation of the configuration space is of the form $\tilde{\mathbb{R}}^N \oplus \mathbb{H}^{N'}$ as $\mathrm{Pin}(2)$ -representation for large N, N’. Restricting attention to the I-invariant part, the remaining symmetry on the configuration space is given by G, and as G-representation space, $\mathbb{H}$ splits into $\mathbb{C}_+ \oplus \mathbb{C}_-$ . More concretely, let us write the set of quaternions by

(3.3) \begin{align}\mathbb{H}= \mathbb{C} \oplus j\mathbb{C}= \mathbb{R} \oplus i\mathbb{R} \oplus j\mathbb{R} \oplus (-k)\mathbb{R}\cong (\mathbb{R} \oplus j\mathbb{R}) \oplus i(\mathbb{R}\oplus(-j)\mathbb{R}).\end{align}

Let us equip $\mathbb{R} \oplus j\mathbb{R}$ with the complex structure by j, and $\mathbb{R} \oplus (-j)\mathbb{R}$ with the complex structure by $-j$ . Then the $(\mathbb{R} \oplus j\mathbb{R})$ -component of (3.3) corresponds to $\mathbb{C}_+$ , and the $(\mathbb{R} \oplus (-j)\mathbb{R})$ -component corresponds to $\mathbb{C}_-$ .

3.2 Spaces of type G-SWF and a numerical invariant

In this subsection, we first define a variant of spaces of type SWF introduced by Manolescu [Reference ManolescuMan16] in our setup, and then define a numerical invariant which is the main ingredient of the $\kappa$ -invariant of involutions.

In general, for a finite-dimensional vector space V, let $V^+$ denote the one-point compactification of V.

Let X be a space of type G-SWF at even level $s=2t$ . Let $\iota : X^H \to X$ denote the inclusion. The image of $\iota^\ast : \tilde{K}_G(X) \to \tilde{K}_G(X^H)$ is expressed in the form $\mathfrak{J}(X) \cdot b_{t\tilde{\mathbb{C}}}$ , where $\mathfrak{J}(X)$ is an ideal of R(G) and $b_{t\tilde{\mathbb{C}}}$ is the Bott element [Reference ManolescuMan14, § 2.1].

Repeating the proof of [Reference ManolescuMan14, Lemma 1], replacing $\mathrm{Pin}(2)$ and $S^1$ with G and H respectively, the G-equivariant localization theorem deduces the following lemma.

Definition 3.3. For a space of type G-SWF, define

\[k(X) := \min\{k \geqslant 0 \,|\, \exists\, x \in \mathfrak{J}(X), wx = 2^k w\}. \]

By Lemma 3.2, there exists $k \geqslant 0$ such that $wx = 2^k w$ for some $x \in \mathfrak{J}(X)$ , and thus k(X) is a well-defined natural number.

The rest of this subsection is devoted to establishing basic properties of the quantity k(X).

Lemma 3.4. Let X and X’ be spaces of type G-SWF at the same even level. Suppose that there exists a pointed G-equivariant map $f : X \to X'$ whose H-fixed-point set map is a G-homotopy equivalence. Then we have

\[k(X) \leqslant k(X'). \]

Proof. We have a commutative diagram

where $(f^H)^\ast$ is an isomorphism that is given by a multiplication of a unit. This implies $\mathfrak{J}(X') \subset \mathfrak{J}(X)$ , and hence $k(X) \leqslant k(X')$ .

We need to know the behavior of the ideal $\mathfrak{J}(X)$ under suspension to define the K-theoretic Frøyshov invariant $\kappa(Y,\iota)$ .

Lemma 3.5. Let X be a space of type G-SWF at an even level. Then we have

(3.4) \begin{align} \mathfrak{J}(\Sigma^{\tilde{\mathbb{C}}}X) = \mathfrak{J}(X),\quad \mathfrak{J}(\Sigma^{\mathbb{C}_+}X) = z\cdot \mathfrak{J}(X),\quad \mathfrak{J}(\Sigma^{\mathbb{C}_-}X) = (w+z-wz)\cdot \mathfrak{J}(X), \end{align}

and

(3.5) \begin{align} k(\Sigma^{\tilde{\mathbb{C}}}X) = k(X),\quad k(\Sigma^{\mathbb{C}_+ \oplus \mathbb{C}_-}X) = k(X)+1. \end{align}

Next, to prove the statement about $\mathbb{C}_+$ , consider the diagram induced from the inclusion $i : X \to \Sigma^{\mathbb{C}_+}X$ and the inclusions from the H-invariant part.

(3.6)

Since $(\Sigma^{\mathbb{C}_+}X)^H = X^H$ , the bottom map $(i^H)^\ast$ is the identity. By the Bott isomorphism, the top map $i^\ast$ is given by multiplication by the Euler class z associated to the representation $\mathbb{C}_+$ . This implies that

\[\mathfrak{J}(\Sigma^{\mathbb{C}_+} X) = z \cdot \mathfrak{J}(X) \]

in R(G).

Similarly, for the statement on $\mathbb{C}_-$ we have the following diagram.

The top map $i^\ast$ is given by multiplication by the Euler class $1-\mathbb{C}_-$ , and by the relation $\mathbb{C}_-=\tilde{\mathbb{C}} \cdot \mathbb{C}_-$ . This Euler class is given by

\[1-\mathbb{C}_-= 1-(1-w)(1-z)= w+z-wz. \]

Next we prove (3.5). The statement about $\tilde{\mathbb{C}}$ is clear by (3.4). To show the statement about $\mathbb{C}_+ \oplus \mathbb{C}_-$ , first note that we have

\[\mathfrak{J}(\Sigma^{\mathbb{C}_+ \oplus \mathbb{C}_-}X) = z(w+z-wz) \cdot \mathfrak{J}(X) \]

by (3.4). Using the relations $w^2-2w=0, w-2z+z^2=0$ , we deduce that

(3.7) \begin{align} w(z(w+z-wz)) =zw^2+wz^2-w^2z^2 =2zw-wz^2 =2zw-2zw+w^2 = 2w.\end{align}

For $x \in \mathfrak{J}(X)$ and $k \geqslant 0$ , it follows from (3.7) that $wx = 2^{k}w$ holds if and only if $w(z(w+z-wz))x = 2^{k+1}w$ holds. Thus, we have $k(\Sigma^{\mathbb{C}_+ \oplus \mathbb{C}_-}X) = k(X)+1$ .

Example 3.8. By definition, it is easy to see that $k(S^0)=0$ . Combining this with Lemma 3.5, we have that

(3.8) \begin{align}k((\tilde{\mathbb{C}}^s \oplus (\mathbb{C}_+\oplus \mathbb{C}_-)^l)^+)=l\end{align}

for $s, l \geqslant 0$ .

Lemma 3.9. Let X and X’ be spaces of type G-SWF at levels 2t and 2t’, respectively, such that $t \lt t'$ . Suppose that there exists a pointed G-equivariant map $f : X \to X'$ whose G-fixed-point set map is a homotopy equivalence. Then we have

\[k(X) + t \leqslant k(X') + t'. \]

(3.9)

Since $\tilde{K}_G((X')^H), \tilde{K}_G(X^H), \tilde{K}_G((X')^G), \tilde{K}_G(X^G)$ are free R(G)-modules of rank 1, we may regard the four maps among them as multiplication by elements of R(G). Note that $(f^G)^\ast$ is just the identity map, since f is supposed so that $f^G$ is a homotopy equivalence. The vertical maps $\tilde{K}_G((X')^H) \to \tilde{K}_G((X')^G)$ and $\tilde{K}_G(X^H) \to \tilde{K}_G(X^G)$ are given by multiplication with the K-theoretic Euler classes $w^{t'}, w^{t}$ , respectively. Thus we have that $(f^H)^\ast$ is given as multiplication by an element $y \in R(G)$ satisfying that

(3.10) \begin{align} w^{t} \cdot y = w^{t'}.\end{align}

We claim that $(f^H)^\ast$ in the diagram (3.9) is given by multiplication by $2^{t'-t-1}w$ ; in other words, we show that $y=2^{t'-t-1}w$ . First, since $t \lt t'$ , the map

\[(f^H)^\ast : \tilde{K}_H((X')^H) \to \tilde{K}_H(X^H) \]

is the zero map. Hence the image of y under the restriction $R(G) \to R(H)$ is zero. From this and (3.2) we deduce that $y = cw$ for some $c \in \mathbb{Z}$ . Then it follows from (3.10) and the relation $w^2=2w$ that

\[2^tcw = cw^{t+1} = w^{t'}= 2^{t'-1}w, \]

and thus we have $c=2^{t'-t-1}$ and $y=2^{t'-t-1}w$ , as claimed.

Take $x \in \mathfrak{J}(X')$ such that $wx=2^{k'}w$ , where $k'=k(X')$ . By the previous paragraph, we have

\[ (f^H)^\ast(x) = 2^{t'-t-1}wx=2^{k'+t'-t-1}w. \]

It follows from this that

\[w\cdot(f^H)^\ast(x) = 2^{k'+t'-t}w. \]

However, by the commutativity of the diagram (3.9), $(f^H)^\ast(x)$ belongs to $\mathfrak{J}(X)$ . Thus we obtain $k'+t'-t \geqslant k(X)$ .

Next, following [Reference ManolescuMan14], we consider equivariant Spanier–Whitehead duality in our setup. See also [Reference ManolescuMan16, § 2.2].

Here, we define the notion of stable commutability only using the representations $\tilde{\mathbb{R}}, \mathbb{C}_+, \mathbb{C}_-$ .

The following lemma is modeled on [Reference ManolescuMan14, Lemma 7].

Lemma 3.11. Let X and X’ be spaces of type G-SWF at levels 2t and 2t’, respectively. Suppose that X and X’ are equivariantly V-dual for some G-representation $V \cong \tilde{\mathbb{C}}^s \oplus (\mathbb{C}_+\oplus \mathbb{C}_-)^{l}$ with $s, l\geqslant 0$ . Then we have

\[k(X) + k(X') \geqslant l. \]

Proof. Let $\epsilon$ and $\eta$ be the maps associated to the V-duality between X and X’. The restrictions of $\epsilon$ and $\eta$ induce a $V^{H}$ -duality between $X^H$ and $(X')^{H}$ , and this implies that $t+t'=s$ . Think of $\epsilon^H$ and $\eta^H$ as $G/H \cong \mathbb{Z}_2$ -equivariant self-maps of $(\tilde{\mathbb{C}}^s)^+$ . The restriction of $\epsilon^H$ and $\eta^H$ on $G/H$ -fixed-point sets gives rise to a self-bijection of $S^0$ , which is a self-duality of $S^0$ . Thus we see that $\epsilon^H$ and $\eta^H$ are unreduced suspensions of self-maps of the unit sphere $S(\tilde{\mathbb{C}}^s)$ , up to $G/H$ -equivalence. Since self-maps of $S(\tilde{\mathbb{C}}^s)$ are determined by their degree, so are $\epsilon^H$ and $\eta^H$ . The duality diagrams imply that the product of the degree of $\epsilon^H$ and that of $\eta^H$ is $\pm1 \in \mathbb{Z}$ , and hence each degree is also 1 or $-1$ . This implies that $\epsilon^H$ and $\eta^H$ are $G/H$ -homotopy equivalence, which is equivalent to that they are G-homotopy equivalence. By Lemma 3.4 applied to $\epsilon$ and $\eta$ and (3.8), we have that

(3.11) \begin{align}k(X \wedge X') = k(V^+) = l.\end{align}

However, for $x \in \mathfrak{J}(X)$ and $x' \in \mathfrak{J}(X')$ with $wx=2^{k(X)}w$ and $wx'=2^{k(X')}w$ , we have that $xx' \in \mathfrak{J}(X \wedge X')$ , where xx’ is the image of $x \otimes x'$ under a natural map $\tilde{K}_G(X) \otimes \tilde{K}_G(X') \to \tilde{K}_G(X \wedge X')$ . We also have $w(xx') = 2^{k(X)}wx'=2^{k(X)+k(X')}w$ , and thus obtain $k(X \wedge X') \leqslant k(X)+k(X')$ . This combined with (3.11) implies the desired conclusion.

3.3 Doubling construction and spectrum classes

Given a spin rational homology 3-sphere $(Y,\mathfrak{t})$ with a smooth odd involution $\iota$ , in § 3.5 we shall define the Seiberg–Witten Floer K-theory and the K-theoretic Frøyshov invariant. This is based on a space-valued 3-manifold invariant constructed in § 3.4, which is an analog of Manolescu’s stable homotopy type. In this subsection, we prepare a set in which the space-valued invariant takes value.

Fixing an $\iota$ -invariant metric g on Y, we obtain a G-equivariant Conley index $I^\mu_\lambda (Y, \iota, g)$ as explained. One option of the definition of the Seiberg–Witten Floer K-theory is just the (certain degree shift of) K-cohomology of $I^\mu_\lambda (Y, \iota, g)$ , but it is not convenient to define the K-theoretic Frøyshov invariant. This is because if we consider the suspension by one of the representation spaces $\mathbb{C}_+$ and $\mathbb{C}_-$ , then it is difficult to describe the difference between k(X) and $k(\Sigma^{\mathbb{C}_{\pm}})$ for the topological space X. However, as we explain in Lemma 3.5 and Remark 3.7, we have $k(\Sigma^{\mathbb{C}_{\pm} \oplus \mathbb{C}_{\pm}}X)=k(\Sigma^{\mathbb{C}_{\pm} \oplus \mathbb{C}_{\mp}}X)=k(X)+1$ .

This observation leads us to consider the ‘complexification’ or ‘double’ of $I^\mu_\lambda (Y, \iota, g)$ and of the relative Bauer–Furuta invariant because if we consider the ‘double’, the suspension is always given by even-dimensional representation space.

To do this, let us define the double of a general space of type G-SWF. Define a group automorphism $\alpha : G \to G$ by $\alpha(j)=-j$ .

Definition 3.12. Let X be a space of type G-SWF at level t. Denote by $X^\dagger$ the space of type G-SWF at level t defined as the same topological space with X, but the G-action on X is given by composing the original G-action on X with $\alpha$ . Then $X \wedge X^\dagger$ is also a space of type G-SWF at level 2t. Define the space of type G-SWF D(X) by

\[D(X) = X \wedge X^\dagger, \]

which we call the double of X.

Similarly, for a real or complex representation V of G, define a representation $V^\dagger$ by the same vector space with V, but with G-action obtained by composing the original G-action on V with $\alpha$ . Define a representation D(V) of G by

\[D(V) = V \oplus V^\dagger. \]

Example 3.13. Since the automorphism $\alpha : G \to G$ does not affect the real representation $\tilde{\mathbb{R}}$ , we have

\[D(\tilde{\mathbb{R}}^t) = \tilde{\mathbb{R}}^{2t} \cong \tilde{\mathbb{C}}^t \]

for $t \geqslant 0$ .

However, $\alpha$ swaps $\mathbb{C}_+$ for $\mathbb{C}_-$ :

\begin{align*}(\mathbb{C}_+)^\dagger = \mathbb{C}_-,\quad(\mathbb{C}_-)^\dagger = \mathbb{C}_+.\end{align*}

Thus we have isomorphisms of G-representations

\begin{align*}&D(\mathbb{C}_+) \cong D(\mathbb{C}_-) \cong \mathbb{C}_+ \oplus \mathbb{C}_- \cong (\mathbb{C}_+)_\mathbb{C} \cong (\mathbb{C}_-)_\mathbb{C},\end{align*}

where $(\mathbb{C}_\pm)_\mathbb{C}$ denotes the complexification of $\mathbb{C}_\pm$ . More generally, we have

\[D(\mathbb{C}_+^m \oplus \mathbb{C}_-^n) \cong (\mathbb{C}_+\oplus \mathbb{C}_-)^{m+n} \]

for $m,n \geqslant 0$ .

Following [Reference ManolescuMan14, § 4], consider a triple (D,m,n), where D is a space of type G-SWF at an even level, and $m \in \mathbb{Z}$ and $n \in \mathbb{Q}$ . We have in mind the case that D is given as $D=D(X)$ for some X, not necessarily at an even level.

Definition 3.14. For such triples (D,m,n), (D’,m’,n’), we say that they are G-stably equivalent to each other if $n-n' \in \mathbb{Z}$ and there exist $M,N \geqslant 0$ and a pointed G-homotopy equivalence

\[\Sigma^{(M-m)\tilde{\mathbb{C}}} \Sigma^{(N-n)(\mathbb{C}_+ \oplus \mathbb{C}_-)}D\to \Sigma^{(M-m')\tilde{\mathbb{C}}} \Sigma^{(N-n')(\mathbb{C}_+ \oplus \mathbb{C}_-)}D'. \]

Define $\mathfrak{C}_{\iota}$ to be the set of G-stable equivalence classes of triples (D,m,n). An element of $\mathfrak{C}_{\iota}$ is called a spectrum class.

Informally, we may think of the triple (D,m,n) as the formal desuspension of X by $m \tilde{\mathbb{C}}$ and by $n(\mathbb{C}_+ \oplus \mathbb{C}_-)$ , so symbolically one may write

\[(D,m,n) = \Sigma^{-m \tilde{\mathbb{C}}}\Sigma^{-n(\mathbb{C}_+ \oplus \mathbb{C}_-)}D, \]

while we need to keep in mind that n may not be an integer.

As well as the non-equivariant case, we can define the notion of local equivalence, which was introduced by Stoffregen [Reference StoffregenSto20], in our G-equivariant setting.

Definition 3.15. Let (D,m,n), (D’,m’,n’) be triples as above. A G-stable map $(D,m,n) \to (D',m',n')$ is a pointed G-map

\[\Sigma^{(M-m)\tilde{\mathbb{C}}} \Sigma^{(N-n)(\mathbb{C}_+ \oplus \mathbb{C}_-)}D\to \Sigma^{(M-m')\tilde{\mathbb{C}}} \Sigma^{(N-n')(\mathbb{C}_+ \oplus \mathbb{C}_-)}D' \]

for some $M,N \geqslant 0$ . A G-stable map $(D,m,n) \to (D',m',n')$ is called a G-local map if it induces a G-homotopy equivalence on the H-fixed-point sets. We say that (D,m,n) and (D’,m’,n’) are G-locally equivalent if there exist G-local maps $(D,m,n) \to (D',m',n')$ and $(D',m',n') \to (D,m,n)$ .

The G-local equivalence is evidently an equivalence relation, and we call an equivalence class for this relation a G-local equivalence class. The set of G-local equivalence classes is denoted by $\mathcal{LE}_G$ . We write an element of $\mathcal{LE}_G$ as $[(D,m,n)]_{\mathrm{loc}}$ . Evidently the G-stable equivalence implies the G-local equivalence, and we have a natural surjection $\mathfrak{C}_{\iota} \to \mathcal{LE}_G$ .

Once we have a G-stable map $f: (D, m,n) \to (D', m',n')$ written as

\[f: \Sigma^{(M-m)\tilde{\mathbb{C}}} \Sigma^{(N-n)(\mathbb{C}_+ \oplus \mathbb{C}_-)}D\to \Sigma^{(M-m')\tilde{\mathbb{C}}} \Sigma^{(N-n')(\mathbb{C}_+ \oplus \mathbb{C}_-)}D', \]

by suspensions, we can enlarge M and N as arbitrary large positive integers. So we can define the formal (de)suspension $\Sigma^{ l (\mathbb{C}_+ \oplus \mathbb{C}_-) } f$ of $f: (D, m,n-l) \to (D', m',n'-l)$ by

\[\Sigma^{ l (\mathbb{C}_+ \oplus \mathbb{C}_-) } f : \Sigma^{(M-m)\tilde{\mathbb{C}}} \Sigma^{(N-n+l)(\mathbb{C}_+ \oplus \mathbb{C}_-)}D\to \Sigma^{(M-m')\tilde{\mathbb{C}}} \Sigma^{(N-n'+ l )(\mathbb{C}_+ \oplus \mathbb{C}_-)}D'. \]

For a triple (D,m,n) as defined above, we define

\[\tilde{K}_G^\ast(D,m,n) := \tilde{K}^{\ast+m+2n}(D) \]

and

\[k(D,m,n) := k(D) - n \in \mathbb{Q}. \]

The second statement of the following lemma is the main motivation that we work with $\mathfrak{C}_{\iota}$ , allowing only suspension by $\tilde{\mathbb{C}}$ and $\mathbb{C}_+ \oplus \mathbb{C}_-$ .

The statement for (2) in Lemma 3.17 can be improved as follows.

Proof. Given two triples (D, m, n), (D’, m’, n’), suppose that they are G-local equivariant. Let

\[\Sigma^{(M-m)\tilde{\mathbb{C}}} \Sigma^{(N-n)(\mathbb{C}_+ \oplus \mathbb{C}_-)}D\to \Sigma^{(M-m')\tilde{\mathbb{C}}} \Sigma^{(N-n')(\mathbb{C}_+ \oplus \mathbb{C}_-)}D' \]

be a G-local map from (D,m,n) to (D’,m’,n’), where $M,N \geqslant 0$ . Applying Lemma 3.4 to this, we obtain

\[k(\Sigma^{(M-m)\tilde{\mathbb{C}}} \Sigma^{(N-n)(\mathbb{C}_+ \oplus \mathbb{C}_-)}D)\leqslant k(\Sigma^{(M-m')\tilde{\mathbb{C}}} \Sigma^{(N-n')(\mathbb{C}_+ \oplus \mathbb{C}_-)}D'). \]

By Lemma 3.5, this is equivalent to that $k(D,m,n) \leqslant k(D',m',n')$ . Similarly, we obtain $k(D',m',n') \leqslant k(D,m,n)$ from a G-local map $(D', m', n') \to (D', m', n')$ , and thus have $k(D,m,n) = k(D',m',n')$ .

3.4 Doubled Seiberg–Witten Floer stable homotopy type for involutions

Now we are ready to construct an invariant of 3-manifolds with involution. Let $(Y,\mathfrak{t})$ be a spin rational homology 3-sphere and $\iota$ be a smooth orientation-preserving involution on Y. Suppose that $\iota$ also preserves the given spin structure $\mathfrak{t}$ on Y and is of odd type. Fix an $\iota$ -invariant metric g on Y, and choose a lift $\tilde{\iota}$ of $\iota$ of order 4 on the spin structure. Then we obtain a G-equivariant Conley index $I^\mu_\lambda (Y, \iota, \tilde{\iota}, g)$ once we fix $\lambda \ll 0 \ll \mu$ . Recall that the finite-dimensional approximation of the configuration space is decomposed into

(3.12) \begin{align}V_\lambda^\mu =V(\tilde{\mathbb{R}})^\mu_\lambda \oplus V(\mathbb{C}_+)^\mu_\lambda\oplus V(\mathbb{C}_-)^\mu_\lambda.\end{align}

Here, each direct summand is isomorphic to the direct sum of some copies of $\tilde{\mathbb{R}}, \mathbb{C}_+, and \mathbb{C}_-$ , respectively. The H-invariant part of $I^\mu_\lambda (Y, \iota, g)$ is given by $V(\tilde{\mathbb{R}})^\mu_\lambda$ .

First, we see that the dependence on the choice of lift $\tilde{\iota}$ . Recall that $\iota$ has exactly two lifts, and once we pick a lift $\tilde{\iota}$ , the other lift is $-\tilde{\iota}$ .

Lemma 3.19. The double $D(I^\mu_\lambda (Y,\mathfrak{t}, \iota, \tilde{\iota}, g))$ is independent of the choice of $\tilde{\iota}$ . Namely, there is a canonical G-equivariant homeomorphism

(3.13) \begin{align}D(I^\mu_\lambda (Y,\mathfrak{t}, \iota, \tilde{\iota}, g))\cong D(I^\mu_\lambda (Y,\mathfrak{t}, \iota, -\tilde{\iota}, g)).\end{align}

Proof. To record the dependence, denote the involution I introduced in (2.1) by $I^+$ if it is defined using $\tilde{\iota}$ , and by $I^-$ if it is defined using $-\tilde{\iota}$ . Then it is straightforward to check that

\[I^+(a,\phi i) = I^-(a,\phi)i \]

for all $(a,\phi) \in \Omega^1(Y) \oplus \Gamma(\mathbb{S})$ .

Recall also how the representations $\mathbb{C}_+, \mathbb{C}_-$ appear in the configuration space for the Seiberg–Witten Floer theory. Decompose $\mathbb{H}$ so that

\begin{align*}\mathbb{H}= \mathbb{C} \oplus j\mathbb{C}= \mathbb{R} \oplus i\mathbb{R} \oplus j\mathbb{R} \oplus (-k)\mathbb{R}\cong (\mathbb{R} \oplus j\mathbb{R}) \oplus i(\mathbb{R}\oplus(-j)\mathbb{R}).\end{align*}

Then $(\mathbb{R} \oplus j\mathbb{R}, j)$ and $(i(\mathbb{R}\oplus(-j)\mathbb{R}), -j)$ are isomorphic to $\mathbb{C}_+$ and $\mathbb{C}_-$ , respectively. Note that the right multiplication of i gives complex linear isomorphisms

\[(\mathbb{R} \oplus j\mathbb{R}, j) \to (i(\mathbb{R}\oplus(-j)\mathbb{R}), -j),\quad (i(\mathbb{R}\oplus(-j)\mathbb{R}), -j) \to (\mathbb{R} \oplus j\mathbb{R}, j). \]

Denote a finite-dimensional approximation of the $I^\pm$ -invariant part of the configuration space by

\[V(\tilde{\mathbb{R}})^\mu_\lambda \oplus V_\pm(\mathbb{C}_+)^\mu_\lambda\oplus V_\pm(\mathbb{C}_-)^\mu_\lambda. \]

(Note that the choice of lift of $\iota$ does not affect the H-invariant part $V(\tilde{\mathbb{R}})^\mu_\lambda$ .) By the above two paragraphs, we see that the right multiplication of i gives complex linear isomorphisms

\[V_+(\mathbb{C}_+)^\mu_\lambda \to V_-(\mathbb{C}_-)^\mu_\lambda,\quad V_+(\mathbb{C}_-)^\mu_\lambda \to V_-(\mathbb{C}_+)^\mu_\lambda. \]

Therefore, the right multiplication of i induces a G-equivariant homeomorphism

\[I^\mu_\lambda (Y,\mathfrak{t}, \iota, \tilde{\iota}, g)\to I^\mu_\lambda (Y, \mathfrak{t},\iota, -\tilde{\iota}, g)^\dagger, \]

and this gives rise to the desired G-equivariant homeomorphism (3.13).

In view of Lemma 3.19, henceforth we drop from our notation the choice of lift of $\iota$ .

We also need a kind of correction term to absorb the dependence of the invariant metric. Recall a correction term introduced in [Reference ManolescuMan03]:

\[n(Y, \mathfrak{t},g) := \tfrac{1}{2}(\eta_{\mathrm{dir}}(Y, \mathfrak{t},g)-\dim_{\mathbb{C}}\mathop{\mathrm{Ker}}\nolimits D_{(Y, \mathfrak{t},g)}-\tfrac{1}{4}\eta_{\mathrm{sign}}(Y,g)) \in \mathbb{Q}. \]

Here, $\eta_{\mathrm{dir}}(Y, \mathfrak{t},g)$ and $\eta_{\mathrm{sign}}(Y,g)$ are the eta invariants of the Dirac operator and the signature operator, respectively, and $D_{(Y, \mathfrak{t},g)}$ is the Dirac operator on $(Y, \mathfrak{t},g)$ . Alternatively, we can write $n(Y, \mathfrak{t},g)$ as

(3.14) \begin{align}n(Y, \mathfrak{t},g) = \mathop{\mathrm{ind}}\nolimits_\mathbb{C} D_W + \frac{\sigma (W)}{8},\end{align}

where W is a compact spin Riemann 4-manifold bounded by (Y, g), and $\mathop{\mathrm{ind}}\nolimits_\mathbb{C} D_W$ is the complex spin Dirac index on W with spectral boundary condition. From the description (3.14), we have that $n(Y, \mathfrak{t},g) \in \frac{1}{8}\mathbb{Z}$ . Regarding the orientation reversal property, we have

(3.15) \begin{align}n(Y, \mathfrak{t},g) + n(-Y, \mathfrak{t},g) = \dim_{\mathbb{C}}\mathop{\mathrm{Ker}}\nolimits D_{(Y, \mathfrak{t},g)},\end{align}

where $D_{(Y, \mathfrak{t},g)}$ is the three-dimensional Dirac operator on $(Y, \mathfrak{t},g)$ (see [Reference ManolescuMan16, p. 167]).

In our G-equivariant setting, a direct analog of Manolescu’s Floer stable homotopy type is a triple

(3.16) \begin{align}(\Sigma^{-V(\tilde{\mathbb{R}})^0_\lambda}\Sigma^{-V(\mathbb{C}_+)^0_\lambda}\Sigma^{-V(\mathbb{C}_-)^0_\lambda}I^\mu_\lambda (Y, \mathfrak{t}, \iota, g),0,n(Y, \mathfrak{t},g)/4),\end{align}

where the division by 4 for the last factor shall be explained in Remark 3.21. However, the triple (3.16) does not lie in $\mathfrak{C}_{\iota}$ . So, instead, we consider the ‘double’ of this triple:

(3.17) \begin{align}(\Sigma^{-D(V(\tilde{\mathbb{R}})^0_\lambda)}\Sigma^{-D(V(\mathbb{C}_+)^0_\lambda)}\Sigma^{-D(V(\mathbb{C}_-)^0_\lambda)}D(I^\mu_\lambda (Y, \mathfrak{t}, \iota, g)),0,n(Y, \mathfrak{t},g)/2).\end{align}

Rewriting desuspensions to make (3.17) precise, in view of Example 3.13, we arrive at the following definition.

Definition 3.20. Given $Y, \mathfrak{t}, \iota, g$ as above, define an element $\textit{DSWF}_G(Y, \mathfrak{t},\iota) \in \mathfrak{C}_{\iota}$ by

\begin{align*}\textit{DSWF}_G(Y, \mathfrak{t},\iota):=[(D(I^\mu_\lambda (Y, \mathfrak{t}, \iota, g)),\dim_{\mathbb{R}}V(\tilde{\mathbb{R}})^0_\lambda,\dim_{\mathbb{C}}V(\mathbb{C}_+)^0_\lambda+\dim_{\mathbb{C}}V(\mathbb{C}_-)^0_\lambda+n(Y, \mathfrak{t},g)/2)].\end{align*}

We call $\textit{DSWF}_G(Y, \mathfrak{t},\iota)$ the doubled G-equivariant Seiberg–Witten Floer stable homotopy type or doubled Seiberg–Witten Floer G-spectrum class.

Recall that, symbolically, $\textit{DSWF}_G(Y, \mathfrak{t},\iota)$ can be thought of as

\[\Sigma^{-(\dim_{\mathbb{R}}V(\tilde{\mathbb{R}})^0_\lambda)\tilde{\mathbb{C}}}\Sigma^{-(\dim_{\mathbb{C}}V(\mathbb{C}_+)^0_\lambda+\dim_{\mathbb{C}}V(\mathbb{C}_-)^0_\lambda+n(Y, \mathfrak{t},g)/2)(\mathbb{C}_+ \oplus \mathbb{C}_-)}D(I^\mu_\lambda (Y, \mathfrak{t}, \iota, g)). \]

Remark 3.21. The number $n(Y, \mathfrak{t},g)/4$ in (3.16) was chosen by the following observation. First, since I and i anti-commute, the dimension of the I-invariant part of the kernel/cokernel of the Dirac operator is half of the dimension of the original kernel/cokernel. Second, passing to the double, the dimension turns into the double of the original one. Last, for suspensions corresponding to the third factor of triples, we use the direct sum $\mathbb{C}_+ \oplus \mathbb{C}_-$ rather than a one-dimensional complex vector space. So, in summation,

\[\tfrac{1}{2}\cdot2\cdot \tfrac{1}{2}n(Y, \mathfrak{t},g) = \tfrac{1}{2}n(Y, \mathfrak{t},g) \]

should be put at the third factor of the triple (3.17). Adopting this number is important to prove the invariance of the doubled G-Floer homotopy type shown in the following proposition.

Next, we shall show the invariance of $\textit{DSWF}_G(Y, \mathfrak{t},\iota)$ . Before that, we note a technicality about trivializations of representations. Since $\tilde{\mathbb{C}}$ is a complex representation of G and $GL(N,\mathbb{C})$ is connected for all N, for any triple (D,m,n) and for any complex N-dimensional representation V of G which is isomorphic to the direct sum of copies of $\tilde{\mathbb{C}}$ , we have a G-stable equivalence

\[(V^+ \wedge D, N + m, n) \simeq (D, m, n), \]

and this G-stable equivalence is canonical up to homotopy. A similar remark applies also to the desuspension by $\mathbb{C}_+\oplus \mathbb{C}_-$ since $\mathbb{C}_+\oplus \mathbb{C}_-$ is also a complex representation.

Proof. We basically follow the original argument by Manolescu [Reference ManolescuMan14, proof of Theorem 1]. First, we fix g and show the independence on $\lambda, \mu$ . Recall the behavior of the Conley index under suspension. That is, for $\lambda<\lambda'\ll0\ll \mu'<\mu$ ,

\begin{align*} I^{\mu}_{\lambda} = I^{\mu'}_{\lambda},\quad I^{\mu}_{\lambda} \cong I^{\mu}_{\lambda'} \wedge (V^{\lambda'}_{\lambda})^+ = I^{\mu}_{\lambda'} \wedge (V(\tilde{\mathbb{R}})^{\lambda'}_{\lambda} \oplus V(\mathbb{C}_+)^{\lambda'}_{\lambda}\oplus V(\mathbb{C}_-)^{\lambda'}_{\lambda})^+.\end{align*}

This combined with Example 3.13 implies that

\begin{align*} D(I^{\mu}_{\lambda}) &= D(I^{\mu'}_{\lambda}),\\ D(I^{\mu}_{\lambda}) &\cong D(I^{\mu}_{\lambda'}) \wedge (D(V^{\lambda'}_{\lambda}))^+\\ &\cong D(I^{\mu}_{\lambda'}) \wedge (\widetilde{\mathbb{C}}^{\dim_{\mathbb{R}}V(\tilde{\mathbb{R}})^{\lambda'}_{\lambda}} \oplus (\mathbb{C}_+ \oplus \mathbb{C}_-)^{\dim_{\mathbb{C}}V(\mathbb{C}_+)^{\lambda'}_{\lambda}+ \dim_{\mathbb{C}}V(\mathbb{C}_-)^{\lambda'}_{\lambda}})^+,\end{align*}

and from this the independence on $\lambda, \mu$ follows.

Next, we show the independence on g. Fix $\lambda, \mu$ , and to record the choice of g used in the construction, we temporarily denote by $\textit{DSWF}_G(Y, \mathfrak{t},\iota,g)$ the doubled G-spectrum class constructed using $\lambda, \mu$ . Take two $\iota$ -invariant metrics $g_0$ and $g_1$ . Since the space of G-invariant metrics is contractible, we may take a path of $\iota$ -invariant metrics $\{g_t\}_{t \in [0,1]}$ between $g_0$ and $g_1$ . The assumption that $b_1(Y)=0$ tells us that the de Rham operator is not involved here, as in the original case by Manolescu [Reference ManolescuMan03], and we focus on the family of Dirac operators $D_{t}$ associated with this path of metrics.

Recall that the I-invariant part of each Dirac operator $D_t^I$ is G-equivariant, and we obtain the G-equivariant spectral flow

\[\operatorname{sf}_G(\{D_{t}^I\}) \in R(G), \]

which can be written as a linear combination only of $\mathbb{C}_+, \mathbb{C}_-$ . Under the natural map $R(G) \to \mathbb{Z}$ induced from taking the dimension of the representation, the image of the equivariant spectral flow $\operatorname{sf}_G(\{D_{t}^I\})$ is given by the usual (non-equivariant) spectral flow $\operatorname{sf}(\{D_{t}^I\}) \in \mathbb{Z}$ . The difference between the doubled G-spectrum classes $\textit{DSWF}_G(Y, \mathfrak{t},\iota,g_0)$ and $\textit{DSWF}_G(Y, \mathfrak{t},\iota,g_1)$ is given by the ‘doubled’ equivariant spectral flow

\[D(\operatorname{sf}_G(\{D_{t}^I\})) \in R(G), \]

which is given by

(3.18) \begin{align} (\mathbb{C}_+ \oplus \mathbb{C}_-)^{\operatorname{sf}(\{D_{t}^I\})}.\end{align}

Recall that we have the formula

(3.19) \begin{align}\operatorname{sf}(\{D_{t}\}) = n(Y, \mathfrak{t},g_1) - n(Y, \mathfrak{t},g_0),\end{align}

and thus have

(3.20) \begin{align}\operatorname{sf}(\{D_{t}^I\}) = n(Y, \mathfrak{t},g_1)/2 - n(Y, \mathfrak{t},g_0)/2.\end{align}

Note that $\operatorname{sf}(\{D_{t}\})$ is an even integer because of the existence of the action I. Thus we finally get

\[D(\operatorname{sf}_G(\{D_{t}^I\}))= (\mathbb{C}_+ \oplus \mathbb{C}_-)^{n(Y, \mathfrak{t},g_1)/2 - n(Y, \mathfrak{t},g_0)/2}\in R(G), \]

and this completes the proof.

3.5 Seiberg–Witten Floer K-theory for involutions

As in the last subsection, let $(Y, \mathfrak{t})$ be a spin rational homology 3-sphere and $\iota$ be a smooth orientation-preserving involution $\iota$ . Suppose that $\iota$ also preserves the given spin structure $\mathfrak{t}$ and is of odd type.

Definition 3.24. Define the doubled Seiberg–Witten Floer K-cohomology by

\[\textit{DSWFK}_G(Y, \mathfrak{t},\iota) := \tilde{K}_G(\textit{DSWF}_G(Y, \mathfrak{t},\iota)), \]

defined as the isomorphism class of an R(G)-graded module. We define also the K-theoretic Frøyshov invariant by

\[\kappa(Y, \mathfrak{t},\iota) := k(\textit{DSWF}_G(Y, \mathfrak{t},\iota)). \]

Note that

\[\kappa(Y, \mathfrak{t},\iota) \in \tfrac{1}{16}\mathbb{Z} \]

since we have that $n(Y, \mathfrak{t},g) \in \frac{1}{8}\mathbb{Z}$ .

Using additional non-topological data $g, \lambda, \mu$ , the invariant $\kappa(Y,\mathfrak{t},\iota)$ is concretely described as

(3.21) \begin{align}\kappa(Y,\mathfrak{t},\iota)= k(D(I^\mu_\lambda(Y,\mathfrak{t},\iota,g)))-\dim_\mathbb{C}(V(\mathbb{C}_+)^0_\lambda)-\dim_\mathbb{C}(V(\mathbb{C}_-)^0_\lambda)-n(Y,\mathfrak{t},g)/2.\end{align}

Proposition 3.26. Let $(Y,\mathfrak{t},\iota)$ be an oriented spin rational homology sphere with odd involution. Then we have

\[\kappa(Y,\mathfrak{t},\iota) + \kappa(-Y,\mathfrak{t},\iota)\geqslant 0. \]

Proof. Denote by $\bar{V}_{-\lambda}^\mu$ the finite-dimensional vector space for $-Y$ used in the definition of the I-invariant part Conley index. Set $V(\mathbb{C})^{\mu}_{\lambda} = V(\mathbb{C}_+)^\mu_{\lambda}\oplus V(\mathbb{C}_-)^\mu_{\lambda}$ in the expression (3.12). Note that we have

\[\dim_{\mathbb{C}} V(\mathbb{C})^0_{\lambda} + \dim_{\mathbb{C}} \bar{V}(\mathbb{C})^0_{-\mu} + \tfrac{1}{2}\dim_{\mathbb{C}}\mathop{\mathrm{Ker}}\nolimits D_{(Y,\mathfrak{t},g)} = \dim_{\mathbb{C}} V(\mathbb{C})^\mu_{\lambda}, \]

where $D_{(Y, \mathfrak{t},g)}$ is the three-dimensional Dirac operator on $(Y, \mathfrak{t},g)$ . (The coefficient $1/2$ emerges from that we took the I-invariant part.) Combining this with (3.15), we obtain

(3.22) \begin{align}\dim_{\mathbb{C}} V(\mathbb{C})^0_{\lambda} + \dim_{\mathbb{C}} \bar{V}(\mathbb{C})^0_{-\mu} + \tfrac{1}{2}(n(Y,\mathfrak{t},g)+n(-Y,\mathfrak{t},g)) = \dim_{\mathbb{C}} V(\mathbb{C})^\mu_{\lambda}.\end{align}

Now, in expression (3.12), take $\mu=-\lambda$ . As described in [Reference ManolescuMan16, § 3.6] and [Reference Khandhawit, Lin and SasahiraKLS23, § 4.4], the doubled I-invariant part Conley indices $D(I_{-\mu}^\mu(Y, \mathfrak{t}, \iota,g))$ and $D(I_{-\mu}^\mu(-Y, \mathfrak{t}, \iota,g))$ are equivariantly $D(V^\mu_{-\mu})$ -dual. It follows from this combined with Lemma 3.11, (3.21), and (3.22) that

\begin{align*}\kappa(Y,\mathfrak{t},\iota)\, + &\,\kappa(-Y,\mathfrak{t},\iota)\\=&\, k(D(I^\mu_{-\mu}(Y,\mathfrak{t},\iota,g)))+k(D(I^\mu_{-\mu}(-Y,\mathfrak{t},\iota,g)))\\&-(\dim_\mathbb{C}(V(\mathbb{C})^0_{-\mu})+\dim_\mathbb{C}(\bar{V}(\mathbb{C})^0_{-\mu}))-(n(Y,\mathfrak{t},g)+n(-Y,\mathfrak{t},g))/2\\= &\,k(D(I^\mu_{-\mu}(Y,\mathfrak{t},\iota,g)))+k(D(I^\mu_{-\mu}(-Y,\mathfrak{t},\iota,g)))-\dim_\mathbb{C}(V(\mathbb{C})^\mu_{-\mu}) \\\geqslant & \dim_\mathbb{C}(V(\mathbb{C})^\mu_{-\mu}) -\dim_\mathbb{C}(V(\mathbb{C})^\mu_{-\mu})=0.\end{align*}

This completes the proof.

3.6 Cobordisms

Let $(Y_0, \mathfrak{t}_0)$ and $(Y_1, \mathfrak{t}_1)$ be spin closed 3-manifolds with $b_1(Y_i)=0$ . We do not assume that $Y_0$ and $Y_1$ are connected. Suppose that we have an involution $\iota_i$ on each $Y_i$ that preserves the components and the orientation and spin structure, and suppose that $\iota_i$ is of odd type. Let $(W,\mathfrak{s})$ be a smooth spin four-dimensional oriented cobordism with $b_1(W)=0$ . We assume that there is a involution $\iota$ on W such that $\iota|_{Y_i}=\iota_i$ for $i=0,1$ , and suppose that $\iota$ preserves $\mathfrak{s}$ and is of odd type. We may take an $\iota$ -invariant Riemannian metric g on W so that g is a cylindrical metric near $\partial W$ . Then the metrics defined by $g_i=g|_{Y_i}$ on $Y_i$ are $\iota_i$ -invariant metrics. Then $\iota$ lifts to some $\tilde{\iota}$ , which is a $\mathbb{Z}_4$ -lift of $\iota$ to the spinor bundle of $\mathfrak{s}$ . As well as the case of dimension 3, following [Reference KatoKat22], we may define the involutions

\begin{align*}I : \Omega^\ast(W) \to \Omega^\ast(W),\quad I : \Gamma(\mathbb{S}^{\pm}) \to \Gamma(\mathbb{S}^{\pm})\end{align*}

by

\begin{align*}I(a)=-\iota^{\ast}a,\quad I(\phi) = \tilde{\iota}( \phi)\cdot j,\end{align*}

where $\mathbb{S}^{\pm}$ are positive and negative spinor bundles. Here we consider the Sobolev norms $L^2_{k}$ for the spaces $\Omega^\ast(W)$ and $\Gamma(\mathbb{S}^{\pm})$ obtained from $\iota$ -invariant metrics and $\iota$ -invariant connections for a fixed integer $ k \geqslant 3$ . The relative Bauer–Furuta invariant of W introduced by Manolescu [Reference ManolescuMan03] gives a map between the Seiberg–Witten Floer stable homotopy types of $Y_0$ and $Y_1$ . This is obtained from the Seiberg–Witten map on W, which is given as a finite-dimensional approximation of a map

(3.23) \begin{align} SW : \Omega_{CC}^1(W) \times \Gamma(\mathbb{S}^+)\to \Omega^+(W) \times \Gamma(\mathbb{S}^-) \times \hat{V}(-Y_0)^{\mu}_{-\infty} \times \hat{V}(Y_1)^{\mu}_{-\infty}\end{align}

for large $\mu$ . Here,

is the space of 1-forms satisfying the double Coulomb condition. This is introduced by Khandhawit in [Reference KhandhawitKha15]. Here, for a general rational spin homology sphere Y, $\hat{V}(Y, \mathfrak{t})^\mu_{-\infty}$ is a subspace of $\hat{V}(Y, \mathfrak{t}) = \operatorname{Ker}d^* \times \Gamma(\mathbb{S})$ , which is defined as the direct sum of eigenspaces whose eigenvalues are less than $\mu$ . The $\Omega^+(W) \times \Gamma(\mathbb{S}^-)$ -factor of the map SW is given as the Seiberg–Witten equations, and the $\hat{V}(-Y_0)^{\mu}_{-\infty} \times \hat{V}(Y_1)^{\mu}_{-\infty}$ -factor is given, roughly, as the restriction of four-dimensional configurations to three-dimensional ones. Taking the I-invariant part of (3.23), we obtain a G-equivariant map, and a finite-dimensional approximation of this gives us a G-equivariant map of the form

(3.24) \begin{align}f : \Sigma^{m_0\tilde{\mathbb{R}}}\Sigma^{n_0^+\mathbb{C}_+}\Sigma^{n_0^-\mathbb{C}_+}I_{-\mu}^{-\lambda} (Y_0) \to \Sigma^{m_1\tilde{\mathbb{R}}}\Sigma^{n_1^+\mathbb{C}_+}\Sigma^{n_1^-\mathbb{C}_+}I^\mu_\lambda (Y_1),\end{align}

where $I^\mu_\lambda (Y_i) = I^\mu_\lambda (Y_i, \mathfrak{t}_i, \iota_i, g_i)$ , and $m_i, n_i^\pm \geqslant 0$ and $-\lambda, \mu$ are sufficiently large. Taking the double of f, we obtain the ‘doubled cobordism map’ or the ‘doubled relative Bauer–Furuta invariant’

(3.25) \begin{align}D(f) :\Sigma^{m_0\tilde{\mathbb{C}}}\Sigma^{n_0(\mathbb{C}_+ \oplus \mathbb{C}_-)}D(I_{-\mu}^{-\lambda} (Y_0)) \to \Sigma^{m_1\tilde{\mathbb{C}}}\Sigma^{n_1(\mathbb{C}_+ \oplus \mathbb{C}_-)}D(I^\mu_\lambda (Y_1)),\end{align}

where $n_i=n_i^+ + n_i^-$ .

Denote by $V_i(\tilde{\mathbb{R}})_\lambda^\mu$ the vector space $V(\tilde{\mathbb{R}})_\lambda^\mu$ , a finite-dimensional approximation of the I-invariant part, for $Y_i$ , and let us use similar notation for other representations.

Remark 3.27. Note that, since the Dirac index $\mathop{\mathrm{ind}}\nolimits_{\mathbb{C}}D$ is even by the existence of the quaternionic structure, we have that

\[- \frac{\sigma(W)}{16}+ \frac{n(Y_1, \mathfrak{t}_1, g_1)}{2}-\frac{n(Y_0, \mathfrak{t}_0, g_0)}{2} \]

is an integer.

Lemma 3.28. We have

(3.26) \begin{align}m_0-m_1=&\dim_\mathbb{R} (V_1(\tilde{\mathbb{R}})^0_{\lambda})-\dim_\mathbb{R}(V_0(\tilde{\mathbb{R}})^0_{-\mu}) - b^+(W) + b^+_\iota(W),\end{align}

(3.27) \begin{align}n_0-n_1=&\dim_\mathbb{C}(V_1(\mathbb{C}_+)^0_{\lambda}) +\dim_\mathbb{C}(V_1(\mathbb{C}_-)^0_{\lambda})\\&-\dim_\mathbb{C}(V_0(\mathbb{C}_+)^0_{-\mu})-\dim_\mathbb{C}(V_0(\mathbb{C}_-)^0_{-\mu})\\&- \frac{\sigma(W)}{16}+ \frac{n(Y_1, \mathfrak{t}_1, g_1)}{2}-\frac{n(Y_0, \mathfrak{t}_0, g_0)}{2}.\end{align}

Proof. It is sufficient to show the equality when $\mu$ and $\lambda$ are not eigenvalues. The proof of this lemma is parallel to the proof of [Reference KhandhawitKha15, Proposition 2]. We set

\[U_i =\mathrm{Im}(d : \Omega^0(Y_i) \to \Omega^1(Y_i))^I. \]

Denote by $\Gamma(Y_i, \mathbb{S}_i)$ the space of sections of the spinor bundle on $Y_i$ . Let $r_i$ be the restriction from W to $Y_i$ , and

\[\Pi^- : \Omega^1(Y_i)^I \times \Gamma(Y_i, \mathbb{S}_i)^I \to V(Y_i)^{0}_{-\infty} \]

and

\[ \Pi_2 : \Omega^1(Y_i)^I \times \Gamma(Y_i, \mathbb{S}_i)^I \to U_i \]

be projections. Let $\textbf{t}a$ be the tangent component of the restriction of a 1-form a to the boundary.

When $\mu=0$ , the index of the linearization of the I-invariant part of the map (3.23) coincides with the index of the map

(3.28) \begin{align} \begin{split} F \colon& \Omega^1(W)^I \times \Gamma(\mathbb{S}^+)^I\\&\to\Omega^+(W)^I \times \Gamma(\mathbb{S}^-)^I \times \Omega^0(W)^I \times V(-Y_0)^{0}_{-\infty}\times V(Y_1)^{0}_{-\infty}\times U_0 \times U_1 \end{split}\end{align}

given by

\[F(a, \phi)=(d^+a,\; D\phi,\; d^{\ast}a,\; \Pi^- \circ r_0(a,\phi),\; \Pi^- \circ r_1(a, \phi),\; \Pi_2 \circ r_0(a, \phi), \;\Pi_2 \circ r_1(a, \phi)) \]

since the kernel of the map

(3.29) \begin{align}d^{\ast}\oplus \Pi_2 \circ r_0 \oplus \Pi_2 \circ r_1:\Omega^1(W)^I \times \Gamma(\mathbb{S}^+)^I\to \Omega^0(W)^I \times U_0 \times U_1\end{align}

coincides with $\Omega_{CC}^1(W)^I \times \Gamma(\mathbb{S}^+)^I$ and the cokernel of this map (3.29) is 0 since constant functions on W are not fixed by $-\iota^{\ast}$ . Note that $\int_{Y_i} \ast \textbf{t}a=0$ for all $a \in \Omega^1(W)^I$ .

Let $L_1$ be the self-operator acting on $ U_i \times\Omega^0(Y_i)^I$ by

Let $\Pi^-_1$ be the projection to the non-negative eigenspace of the operator $L_1$ . The kernel of the projection $\Pi_2 |_{\Omega^1(Y_i)^I}$ is $\text{Ker}(d^{\ast})^I_{Y_i}$ and the image of the $\Pi_1^-$ is

\[\{(b, d^{\ast}(dd^{\ast})^{-1/2}b) \mid b \in \Omega^1 (Y_i)^I \}. \]

The kernel of $(\Pi_2|_{\Omega^1(Y_i)^I})\circ r_i$ is complementary to the non-positive eigenspace of $L_1$ in the space $\Omega^1(Y_i)^I \times \Omega^0(Y_i)^I$ . Let $\hat{U}^0_{-\infty}(-Y_0)$ and $\hat{U}^0_{-\infty}(Y_1)$ denote the sums of the non-positive eigenspaces in

\[\Omega_1(Y_0)^I \times \Omega_0(Y_0)^I \times \Gamma(Y_0, \mathbb{S}_0)^I \]

and

\[\Omega_1(Y_1)^I \times \Omega_0(Y_1)^I \times \Gamma(Y_1, \mathbb{S}_1)^I \]

of $(L_1, D)^I$ , respectively. By [Reference Kronheimer and MrowkaKM07, Proposition 17.2.6], we have that the operator F is Fredholm and the index coincides with the index of the operator

(3.30) \begin{align} F' \colon \Omega^1(W)^I \times \Gamma(\mathbb{S}^+)^I\to\Omega^+(W)^I \times \Gamma(\mathbb{S}^-)^I \times \Omega^0(W)^I \times\hat{U}^0_{-\infty}(-Y_0) \times \hat{U}^0_{-\infty}(Y_1),\end{align}

where the components of F’ for $\hat{U}^0_{-\infty}(-Y_0)$ and $\hat{U}^0_{-\infty}(Y_1)$ are defined as the compositions of the restriction to $Y_i$ and the projections to $\hat{U}^0_{-\infty}(-Y_0)$ and $\hat{U}^0_{-\infty}(Y_1)$ . We also use the decomposition of

\[F' = F'_\mathbb{R} \oplus F'_\mathbb{C}, \]

where $F'_\mathbb{R}$ and $F'_\mathbb{C}$ are the real part and complex part of F’, respectively. We have that the index of $F'_\mathbb{C}$ is the half of the Atiyah–Patodi–Singer index of the $\mathrm{Spin}^c$ Dirac operator since I anti-commutes with i. We see that the index of $F'_\mathbb{R}$ is given by $- b^+(W) + b^+_\iota(W)$ as follows. Elements in the kernel and the cokernel of $F'_\mathbb{R}$ are able to be extended to $L^2$ -harmonic forms of

\[\hat{W}=((-\infty, 0] \times Y_0 ) \cup_{Y_0} W \cup_{Y_1} ([0, \infty) \times Y_1) \]

since $Y_i$ are rational homology spheres. We see that the space of $L^2$ -harmonic forms on $\hat{W}$ coincides with the de Rham cohomology of $\hat{W}$ from [Reference Atiyah, Patodi and SingerAPS75, Proposition 4.9] and $F'_\mathbb{R}$ is injective and its cokernel coincides with the ( $-\iota^{\ast}$ )-invariant part of the space of self-dual $L^2$ -harmonic forms. Thus we have that the index of $F'_\mathbb{R}$ is given by $b^+(W) - b^+_{\iota}(W)$ . When $\eta=0$ , the formula (3.27) follows from Atiyah–Patodi–Singer index theorem for Spin $^c$ Dirac operators.

In the case $\mu >0$ , we have that

(3.31) \begin{align}\mathop{\mathrm{ind}}\nolimits_\mathbb{C}{F'_\mathbb{C}}=\tfrac{1}{2}\mathrm{ind}_{\mathbb{C}}(D^+)-\dim_{\mathbb{C}}(V_0(\mathbb{C}_+ \oplus \mathbb{C}_-)^{0}_{-\mu})-\dim_{\mathbb{C}}(V_1(\mathbb{C}_+ \oplus \mathbb{C}_-)^{\mu}_0),\end{align}

and the index of $F'_\mathbb{R}$ is

(3.32) \begin{align}\mathop{\mathrm{ind}}\nolimits_\mathbb{R}{F'_\mathbb{R}}=- b^+(W) + b^+_\iota(W)-\dim(V_0(\tilde{\mathbb{R}})^{0}_{-\mu})-\dim(V_1(\tilde{\mathbb{R}})^{\mu}_0).\end{align}

Recall that the map (3.24) is given in the same way in [Reference ManolescuMan16, § 3.6] and that $V^\mu_\lambda(-Y_0)=V_{-\mu}^{-\lambda}(Y_0)$ . From the construction of the map (3.25), we easily see that

(3.33) \begin{align}\begin{split}&\mathop{\mathrm{ind}}\nolimits F'_\mathbb{R}+ \dim(V_0(\tilde{\mathbb{R}})_{-\mu}^{-\lambda})-\dim(V_1(\tilde{\mathbb{R}})_{0}^{\mu})-\dim(V_0(\tilde{\mathbb{R}})^{0}_{-\mu})\\&\quad =\mathop{\mathrm{ind}}\nolimits F'_\mathbb{R}+ \dim(V_0(\tilde{\mathbb{R}})_{0}^{-\lambda})-\dim(V_1(\tilde{\mathbb{R}})_{0}^{\mu})\\&\quad =m_0+\dim(V_0(\tilde{\mathbb{R}})_{-\mu}^{-\lambda})-m_1-\dim(V_1(\tilde{\mathbb{R}})^{\mu}_{\lambda})\end{split}\end{align}

and

(3.34) \begin{align}\begin{split}&\mathop{\mathrm{ind}}\nolimits_\mathbb{C} F'_\mathbb{C}+\dim_\mathbb{C}(V_0(\mathbb{C}_+)_{-\mu}^{-\lambda})+\dim_\mathbb{C}(V_0(\mathbb{C}_-)_{-\mu}^{-\lambda})\\&\qquad-\dim_\mathbb{C}(V_0(\mathbb{C}_+ \oplus \mathbb{C}_-)_{-\mu}^{0}) -\dim_\mathbb{C}(V_1(\mathbb{C}_+ \oplus \mathbb{C}_-)^{\mu}_{0})\\&\quad=\mathop{\mathrm{ind}}\nolimits_\mathbb{C} F'_\mathbb{C}+\dim_\mathbb{C}(V_0(\mathbb{C}_+ \oplus\mathbb{C}_-)_{0}^{-\lambda}) -\dim_\mathbb{C}(V_1(\mathbb{C}_+\oplus \mathbb{C}_-)^{\mu}_{0})\\&\quad=n_0+\dim_\mathbb{C}(V_0( \mathbb{C}_+\oplus \mathbb{C}_-)_{-\mu}^{-\lambda})-n_1-\dim_\mathbb{C}(V_1(\mathbb{C}_+ \oplus \mathbb{C}_-)^{\mu}_{\lambda}).\end{split}\end{align}

Thus we have the formula (3.26) from (3.32) and (3.33), and the formula (3.27) from (3.31) and (3.34).

3.7 Proof of Theorem 1.1

First, we state and prove the main result of this paper in the most general form.

Theorem 3.29. Let $(Y_0, \mathfrak{t}_0)$ , $(Y_1, \mathfrak{t}_1)$ be spin rational homology 3-spheres. Let $\iota_0, \iota_1$ be smooth involutions on $Y_0, Y_1$ . Suppose that $\iota_0, \iota_1$ preserve the given orientations and spin structures $\mathfrak{t}_0, \mathfrak{t}_1$ on $Y_0, Y_1$ , respectively, and suppose that $\iota_0, \iota_1$ are of odd type. Let $(W,\mathfrak{s})$ be a smooth compact oriented spin cobordism with $b_1(W)=0$ from $(Y_0, \mathfrak{t}_0)$ to $(Y_1, \mathfrak{t}_1)$ . Suppose that there exists a smooth involution $\iota$ on W such that $\iota$ preserves the given orientation and spin structure $\mathfrak{s}$ on W, and that the restriction of $\iota$ to the boundary is given by $\iota_0, \iota_1$ . Then we have

(3.35) \begin{align}-\frac{\sigma(W)}{16} + \kappa (Y_0, \mathfrak{t}_0, \iota_0 )\leqslant b^+(W)-b^+_{\iota}(W) + \kappa (Y_1, \mathfrak{t}_1, \iota_1 ).\end{align}

First, consider the case that $b^+(W) - b^+_\iota(W)>0$ . Then, by (3.26), the level of the domain of the doubled cobordism map (3.25) is smaller than that of the codomain of (3.25). Therefore, we can apply Lemma 3.9 to (3.25), and obtain that

\begin{align*} &k(\Sigma^{m_0\tilde{\mathbb{C}}}\Sigma^{n_0(\mathbb{C}_+ \oplus \mathbb{C}_-)}D(I^\nu_{-\nu} (Y_0))) + m_0 + \dim_\mathbb{R}(V_0(\tilde{\mathbb{R}})^0_{-\nu})\\ &\quad\leqslant k(\Sigma^{m_1\tilde{\mathbb{C}}}\Sigma^{n_1(\mathbb{C}_+ \oplus \mathbb{C}_-)}D(I^\nu_{-\nu} (Y_1))) + m_1 + \dim_\mathbb{R}(V_1(\tilde{\mathbb{R}})^0_{-\nu}).\end{align*}

By Lemma 3.5, this is equivalent to

\begin{align*} \begin{split} &k(D(I^\nu_{-\nu}(Y_0)))+n_0+m_0+\dim_\mathbb{R}(V_0(\tilde{\mathbb{R}})^0_{-\nu})\\ &\quad\leqslant k(D(I^\nu_{-\nu}(Y_1)))+n_1+m_1+\dim_\mathbb{R}(V_1(\tilde{\mathbb{R}})^0_{-\nu}). \end{split}\end{align*}

From this combined with (3.21), (3.26) and (3.27), we obtain the desired inequality (1.1).

Next, consider the case that $b^+(W) - b^+_\iota(W)=0$ . Then, by (3.26), the level of the domain of the doubled cobordism map (3.25) is the same as that of the codomain of (3.25). Then we can apply Lemma 3.4 to (3.25), instead of Lemma 3.9, and obtain the desired inequality (1.1) by the same argument for the case that $b^+(W) - b^+_\iota(W)>0$ above.

Next, we prove property (ii). Denote by $I_\iota$ the involution on the configuration space discussed in § 2 defined by $\iota$ and an $\iota$ -invariant metric g. Denote the configuration space by $\mathcal{C}(Y,g)=\mathop{\mathrm{Ker}}\nolimits(d^{*_g} \times \Gamma(\mathbb{S}))$ defined by the metric g. Similarly, denote by $I_{f^{-1} \circ \iota \circ f}$ the involution defined by $f^{-1} \circ \iota \circ f$ and the metric $f^{\ast}g$ . Then the pull-back under f induces a G-homeomorphism

\[\mathcal{C}(Y,g) \to \mathcal{C}(Y,f^{\ast}g), \]

which is equivariant under the action of $I_\iota$ and $I_{f^{-1} \circ \iota \circ f}$ . This induces a G-equivariant homeomorphism

\begin{align*}I^\mu_\lambda (Y, \mathfrak{t},\iota, g)\to I^\mu_\lambda (Y, \mathfrak{t}, f^{-1} \circ \iota \circ f, f^{\ast}g),\end{align*}

and an equivalence between representatives of $\textit{DSWF}_G(Y, \mathfrak{t},\iota)$ and of $\textit{DSWF}_G(Y, \mathfrak{t},f^{-1} \circ \iota \circ f)$ . Thus we have that

\[\textit{DSWF}_G(Y, \mathfrak{t},\iota)= \textit{DSWF}_G(Y, \mathfrak{t},f^{-1} \circ \iota \circ f) \]

in $\mathfrak{C}_{\iota}$ . This and Lemma 3.17 imply property (ii) of Theorem 1.1.

Property (iii) was proved in Proposition 3.26.

Last, we give the proof of property (iv). In general, for a spin manifold of $\dim \leqslant 4$ and an involution $\iota$ on this manifold preserving the spin structure with non-empty fixed-point set, the condition that $\iota$ is of odd type is equivalent to that the fixed-point set of $\iota$ is of codimension-2 [Reference Atiyah and BottAB68, Proposition 8.46]. Thus property (iv) immediately follows from Theorem 3.29.

Note that a statement similar to Theorem 3.29 for a spin 4-manifold W with one boundary component follows from Theorem 3.29 by removing a ball in W near a fixed point, as far as the involution has non-empty fixed-point set. But, for free involutions, it seems that the case of one boundary component is not deduced from the two-component case of Theorem 3.29. However, we can also carry out the proof of Theorem 3.29 for the one boundary component case without any essential change.

We record this as a statement as follows. We wish to thank David Baraglia for pointing this out. See also Nakamura’s work [Reference NakamuraNak13] in his $\mathrm{Pin}^-(2)$ -monopole setting for free involutions.

Theorem 3.30. Let $(Y, \mathfrak{t})$ be a spin rational homology 3-sphere. Let $\iota$ be a smooth involution on Y. Suppose that $\iota_Y$ preserves the given orientation and spin structure $\mathfrak{t}$ , respectively, and suppose that $\iota_Y$ is of odd type. Let $(W,\mathfrak{s})$ be a smooth compact oriented spin 4-manifold $b_1(W)=0$ bounded by $(Y, \mathfrak{t})$ . Suppose that there exists a smooth involution $\iota$ on W such that $\iota$ preserves the given orientation and spin structure $\mathfrak{s}$ on W, and that the restriction of $\iota$ to the boundary is given by $\iota_Y$ . Then we have

(3.36) \begin{align}-\frac{\sigma(W)}{16} \leqslant b^+(W)-b^+_{\iota}(W) + \kappa (Y, \mathfrak{t}, \iota_Y).\end{align}

Here, we summarize the proof. We shall obtain an $\mathbb{Z}_4$ -equivariant map

(3.37) \begin{align}D(f) :\Sigma^{m_0\tilde{\mathbb{C}}}\Sigma^{n_0(\mathbb{C}_+ \oplus \mathbb{C}_-)}S^0 \to \Sigma^{m_1\tilde{\mathbb{C}}}\Sigma^{n_1(\mathbb{C}_+ \oplus \mathbb{C}_-)}D(I^\mu_\lambda (Y))\end{align}

instead of (3.25) as a finite-dimensional approximation of the real part of the Seiberg–Witten equation as constructed in § 3.6, where the notation is the same as in § 3.6. By combining Lemma 3.9 applied to (3.37) with the fact that $k (D(S^0)) = 0$ , we get the desired conclusion.

3.8 Stronger 10/8-inequalities for SWF-spherical triples

For some class of spin rational homology spheres with odd involution, we can refine the 10/8-type inequality stated in Theorem 1.1. In order to do this, we need to recall our construction of the Seiberg–Witten Floer stable homotopy type. Let $(Y,\mathfrak{t})$ be a spin rational homology sphere with odd involution $\iota$ . Fix an $\iota$ -invariant metric g on Y and a lift $\tilde{\iota}$ of $\iota$ to the spin structure. Then we consider a finite-dimensional approximation of the I-invariant part of the configuration space on which we have the formal gradient flow for the Chern–Simons–Dirac functional, and obtain the Conley index $I^\mu_\lambda (Y, \mathfrak{t}, \iota, \tilde{\iota}, g)$ . Recall that $\mathbb{C}_+$ is the standard cyclic one-dimensional complex representation of $G=\mathbb{Z}_4$ .

The definition of SWF-spherical triples may seem to be modeled on Seiberg–Witten Floer stable homotopy type of 3-manifolds with positive scalar curvature metric. However, in our setup with involution, many Seifert homology spheres are examples of this kind. In Example 3.46 and Theorem 3.48 we shall see that some natural involutions on $S^3$ , lens spaces, and Seifert homology spheres give examples of SWF-spherical triples.

Lemma 3.32. If $(Y, \mathfrak{t}, \iota)$ is SWF-spherical so that, for an $\iota$ -invariant metric g, we have $I^\mu_\lambda (Y, \mathfrak{t}, \iota, \tilde{\iota}, g) \cong V^+$ for some G-representation space $V\cong \tilde{\mathbb{R}}^l \oplus \mathbb{C}_-^m \oplus \mathbb{C}_+^n$ . Then we have

\[\kappa (Y, \mathfrak{t}, \iota) = -n(Y, \mathfrak{t},g)/2. \]

Proof. We have

\[\dim_{\mathbb{C}}V(\mathbb{C}_+)^0_\lambda+\dim_{\mathbb{C}}V(\mathbb{C}_-)^0_\lambda=n+m=:n', \]

and thus obtain

\begin{align*}\textit{DSWF}_G(Y, \mathfrak{t},\iota)=&\,[(\tilde{\mathbb{R}}^{2l}\oplus(\mathbb{C}_+ \oplus \mathbb{C}_-)^{n'},2l,\dim_{\mathbb{C}}V(\mathbb{C}_+)^0_\lambda+\dim_{\mathbb{C}}V(\mathbb{C}_-)^0_\lambda+n(Y, \mathfrak{t},g)/2)]\\=&\,[(S^0,0,n(Y, \mathfrak{t},g)/2)].\end{align*}

The assertion of the lemma follows from this.

The main input in this subsection is the following Borsuk–Ulam-type theorem. For $N \in \mathbb{Z}$ , define $A(N) \in \{1,2,3\}$ by

(3.38) \begin{align}A(N) =\begin{cases}1, \quad N=0,2 \operatorname{mod} 8,\\2, \quad N=1,3,4,5,7 \operatorname{mod} 8,\\3, \quad N=6 \operatorname{mod} 8.\end{cases}\end{align}

Denote by $\tilde{\mathbb{R}}$ the non-trivial real one-dimensional representation of G, and by $\mathbb{C}$ the complex one-dimensional representation of G given by composing the natural inclusion $G \subset U(1)$ and the multiplication of U(1) on $\mathbb{C}$ .

Theorem 3.33 (Cf. Crabb [Reference CrabbCra89] and Stolz [Reference StolzSto89]). Suppose that $N \geqslant 2, N' \geqslant 1$ . Then there exists a pointed continuous G-map

\[f : (\mathbb{C}^N)^+ \to (\tilde{\mathbb{R}}^{N'})^+ \]

whose G-fixed-point set map is G-homotopy equivalence if and only if

\[N+A(N) \leqslant N'. \]

Remark 3.34. Note that, from the argument of the proof of Lemma 3.9, we already have the following Borsuk–Ulam-type theorem: if there exists a pointed continuous G-map

\[f : (\mathbb{C}^{n_0} \oplus \tilde{\mathbb{R}}^{m_0})^+ \to (\mathbb{C}^{n_1} \oplus \tilde{\mathbb{R}}^{m_1})^+ \]

that induces a homotopy equivalence on G-fixed-point sets, then we have

(3.39) \begin{align}n_0 - n_1 \leqslant m_1-m_0.\end{align}

Theorem 3.33 means that, if f is ‘desuspended’ so that f is a map from the complex representation space $\mathbb{C}^N$ to the real representation space $\tilde{\mathbb{R}}^{N'}$ , then we may get a stronger constraint.

Let us also recall a classical fact from equivariant stable homotopy theory. This is a version of the equivariant Freudenthal suspension theorem.

Theorem 3.35 (See, for example, [Reference Greenlees and MayGM95, Theorem 1.5]). Let G’ be a compact Lie group, $X_0, X_1$ be pointed G’-CW complexes, and V be a representation of G’. Denote by $\mathrm{Sub}(G')$ the set of subgroups of G’ and by $\mathrm{conn}(X)$ the connectivity of a given space X. Suppose that there exists a conjugation-invariant function

\[\mathcal{N} : \mathrm{Sub}(G') \to \{-1, 0, 1, 2, \ldots, +\infty\} \]

that satisfies the following properties.

1. (i) For all $H \in \mathrm{Sub}(G')$ with $V^H\neq \{0\}$ , we have $\mathcal{N}(H) \leqslant 2\mathrm{conn}(X_1^H)+1$ .

2. (ii) For all $H \in \mathrm{Sub}(G')$ and subgroup $K \subset H$ with $V^K \neq V^H$ , we have $\mathcal{N}(H) \leqslant \mathrm{conn}(X_1^K)$ .

3. (iii) For all $H \in \mathrm{Sub}(G')$ , we have $\dim X_0^H \leqslant \mathcal{N}(H)$ , where $\dim X_0^H$ denotes the dimension as a CW complex.

Then the suspension map

\[\Sigma^{V} : [X_0, X_1]_{G'} \to [\Sigma^{V}X_0, \Sigma^{V}X_1]_{G'} \]

is surjective.

The next proposition refines the weak Borsuk–Ulam theorem (3.39).

Proposition 3.37. Suppose that there exists a pointed continuous G-map

\[f : (\mathbb{C}^{n_0} \oplus \tilde{\mathbb{R}}^{m_0})^+ \to (\mathbb{C}^{n_1} \oplus \tilde{\mathbb{R}}^{m_1})^+ \]

that induces a homotopy equivalence on the G-fixed-point sets for natural numbers $n_0, n_1, m_0, m_1$ . Suppose that $n_0-n_1 \geqslant 2$ and $m_1-m_0 \geqslant 1$ . Then we have

(3.40) \begin{align}n_0-n_1+A(n_0-n_1) \leqslant m_1-m_0.\end{align}

\[\mathcal{N}(H)=\begin{cases}m_1-m_0-2, & H=\mathbb{Z}_2 \text{ or } G,\\2(m_1-m_0)-2, &H=\{1\}.\end{cases} \]

We shall apply Theorem 3.35 to this $\mathcal{N}$ and

\[G'=G,\quad X_0 = (\mathbb{C}^{n_0-n_1})^+,\quad X_1 = (\tilde{\mathbb{R}}^{m_1-m_0})^+, \quad V=\mathbb{C}^{n_1}\oplus \tilde{\mathbb{R}}^{m_0}. \]

Let us check that these satisfy the assumptions of Theorem 3.35. For $H=\mathbb{Z}_2$ or G, it is straightforward to check that assumptions (i), (ii) and (iii) in Theorem 3.35 are satisfied. For $H=\{1\}$ , again it is easy to check that assumptions (i) and (ii) are satisfied, and assumption (iii) is equivalent to that

\[n_0 - n_1 \leqslant m_1 -m_0 -1. \]

But this inequality follows from the weaker Borsuk–Ulam-type theorem mentioned in Remark 3.34. Thus we can apply Theorem 3.35 and deduce that there exists a pointed continuous G-map

\[f' : (\mathbb{C}^{n_0-n_1})^+ \to (\tilde{\mathbb{R}}^{m_1-m_0})^+ \]

that induces a homotopy equivalence on the G-fixed-point sets. By applying Theorem 3.33 to this map f’, we obtain the inequality (3.40).

Let us apply the refined Borsuk–Ulam-type theorem Proposition 3.37 to our setup.

Theorem 3.38. Let $(Y_0, \mathfrak{t}_0)$ , $(Y_1, \mathfrak{t}_1)$ be spin rational homology 3-spheres. Let $\iota_0, \iota_1$ be smooth involutions on $Y_0, Y_1$ . Suppose that $\iota_0, \iota_1$ preserve the given orientations and spin structures $\mathfrak{t}_0, \mathfrak{t}_1$ on $Y_0, Y_1$ respectively, and suppose that $\iota_0, \iota_1$ are of odd type. Suppose that $(Y_0, \mathfrak{t}_0, \iota_0)$ , $(Y_1, \mathfrak{t}_1,\iota_1)$ are SWF-spherical. Let $(W,\mathfrak{s})$ be a smooth compact oriented spin cobordism with $b_1(W)=0$ from $(Y_0, \mathfrak{t}_0)$ to $(Y_1, \mathfrak{t}_1)$ . Suppose that there exists a smooth involution $\iota$ on W such that $\iota$ preserves the given orientation and spin structure $\mathfrak{s}$ on W, and that the restriction of $\iota$ to the boundary is given by $\iota_0, \iota_1$ . Set

\[N(W, Y_0, \mathfrak{t}_0, \iota_0, Y_1, \mathfrak{t}_1, \iota_1):=-\frac{\sigma(W)}{16} + \kappa (Y_0, \mathfrak{t}_0, \iota_0 ) - \kappa (Y_1, \mathfrak{t}_1, \iota_1). \]

Then $N(W, Y_0, \mathfrak{t}_0, \iota_0, Y_1, \mathfrak{t}_1, \iota_1)$ is an integer, and if we have

\[N(W, Y_0, \mathfrak{t}_0, \iota_0, Y_1, \mathfrak{t}_1, \iota_1) \geqslant 2 \quad \text{and} \quad b^+(W) - b^+_\iota(W) \geqslant 1, \]

then the inequality

(3.41) \begin{align}\begin{split}-&\frac{\sigma(W)}{16} + \kappa (Y_0, \mathfrak{t}_0, \iota_0 ) + A(N(W, Y_0, \mathfrak{t}_0, \iota_0, Y_1, \mathfrak{t}_1, \iota_1))\\&\leqslant b^+(W)-b^+_{\iota}(W) + \kappa (Y_1, \mathfrak{t}_1, \iota_1 )\end{split}\end{align}

holds.

Proof. Since we suppose that $(Y_0, \mathfrak{t}_0, \iota_0)$ , $(Y_1, \mathfrak{t}_1,\iota_1)$ are SWF-spherical, the I-invariant-relative Bauer–Furuta invariant (3.24) is given by

\[f : (\mathbb{C}^{n_0} \oplus \tilde{\mathbb{R}}^{m_0})^+ \to (\mathbb{C}^{n_1} \oplus \tilde{\mathbb{R}}^{m_1})^+ \]

with

\begin{align*}m_1 - m_0 &= b^+(W) - b^+_\iota(W),\\n_0 - n_1 &= - \frac{\sigma(W)}{16}+ \frac{n(Y_1, \mathfrak{t}_1, g_1)}{2}-\frac{n(Y_0, \mathfrak{t}_0, g_0)}{2}= N(W, Y_0, \mathfrak{t}_0, \iota_0, Y_1, \mathfrak{t}_1, \iota_1).\end{align*}

By Remark 3.27 and this expression of $n_0-n_1$ , $N(W, Y_0, \mathfrak{t}_0, \iota_0, Y_1, \mathfrak{t}_1, \iota_1)$ is an integer. The assertion of Theorem 3.38 follows from Proposition 3.37 applied to this f.

3.9 Connected sum formula

First, we recall the operation of an equivariant connected sum. Let $(Y_0, \mathfrak{t}_0, y_0)$ and $ (Y_1,\mathfrak{t}_1, y_1)$ be spin rational homology 3-spheres with base points. Let $\iota_0$ and $\iota_1$ be involutions on $Y_0$ and $Y_1$ , respectively. Suppose that the fixed-point set of $\iota_i$ are codimension-2 and $\iota_i(y_i)=y_i$ , and that $\iota_i$ preserves the spin structure $\mathfrak{t}_i$ for $i=0, 1$ . We give an orientation $o(\iota_i)$ of the set of fixed points of the involution $\iota_i$ . For these data, the equivariant connected sum

\[(Y_0, \mathfrak{t}_0, y_0, \iota_0, o(\iota_0)) \# (Y_1, \mathfrak{t}_1, y_1, \iota_1, o(\iota_1)) \]

is defined.

If the fixed-point set of $\iota_i$ is connected for $i=0, 1$ , we have that the $\mathbb{Z}/2$ -equivariant isotopy class of the connected sum $(Y_0, \mathfrak{t}_0, y_0,\iota_0, o(\iota_0)) \# (Y_1, \mathfrak{t}_1, y_1, \iota_1, o(\iota_1))$ is independent of the choice of base points $y_0$ and $y_1$ , and we sometimes drop $y_0, y_1$ from our notation. Similarly, if the orientations $o(\iota_i)$ of the fixed-point sets are given, we drop $o(\iota_i)$ from our notation. By construction, for oriented knots K and K’ in $S^3$ , there is a $\mathbb{Z}_2$ -equivariant orientation-preserving diffeomorphism between $(\Sigma (K\# K'), \iota_{K\# K'} )$ and $(\Sigma (K), \iota_{K} )\# (\Sigma (K'), \iota_{K'} )$ , where $\Sigma (K)$ denotes the double branched covering space along K and $\iota_K$ is the covering involution on $\Sigma (K)$ .

The standard 3-handle cobordism between the connected sum and disjoint union of two manifolds can be considered equivariantly. Namely, we have an oriented homology cobordism $W_{01}$ from $(Y_0, \mathfrak{t}_0, \iota_0, o(\iota_0)) \# (Y_1, \mathfrak{t}_1, \iota_1, o(\iota_1))$ to $(Y_0, \mathfrak{t}_0, \iota_0, o(\iota_0)) \sqcup (Y_1, \mathfrak{t}_1, \iota_1, o(\iota_1))$ that is equipped with an involution $\iota$ whose restriction to the boundary is given by $\iota_0, \iota_1$ . We suppose that $W_{01}$ also admits a spin structure whose restriction to the boundary is given by $\mathfrak{t}_0, \mathfrak{t}_1$ , but, in all examples in this paper, we consider only the case that $Y_i$ are a $\mathbb{Z}_2$ -homology 3-spheres, and hence $W_{01}$ admits a unique spin structure whose restriction to the boundary is given by $\mathfrak{t}_0, \mathfrak{t}_1$ .

We prove a certain connected sum formula of $[\textit{DSWF}(Y, \mathfrak{t}, \iota )]_{\operatorname{loc}}$ . See [Reference StoffregenSto20, Fact 4.4] for the original connected sum formula of local equivalence classes of usual Seiberg–Witten Floer stable homotopy types.

Theorem 3.40. Let $(Y_0, \mathfrak{t}_0, \iota_0)$ and $ (Y_1,\mathfrak{t}_1, \iota_1)$ be spin $\mathbb{Z}_2$ -homology 3-spheres with odd involutions $\iota_0$ and $\iota_1$ . Suppose that the fixed-point set of $\iota_i$ is non-empty and connected for each $i=0,1$ . Then we have

\[[\textit{DSWF}_G (Y_0, \mathfrak{t}_0, \iota_0) \wedge \textit{DSWF}_G (Y_1, \mathfrak{t}_1, \iota_1) ]_{\operatorname{loc}} = [\textit{DSWF}_G(Y^\#, \mathfrak{t}^\#, \iota^\#)]_{\operatorname{loc}}, \]

where $(Y^\#, \mathfrak{t}^\#, \iota^\#) := (Y_0, \mathfrak{t}_0,\iota_0) \# (Y_1, \mathfrak{t}_1, \iota_1)$ .

Proof. Noting that we have $b^+(W_{01})=0$ , and hence $b^+_\iota(W_{01})=0$ , as well as [Reference StoffregenSto20, Fact 4.4], the doubled relative Bauer–Furuta invariant for the cobordism $W_{01}$ constructed in (3.25) gives rise to a local map

\[f : \textit{DSWF} (Y_0, \mathfrak{t}_0, \iota_0) \wedge \textit{DSWF} (Y_1, \mathfrak{t}_1, \iota_1) \to \textit{DSWF}(Y^\#, \mathfrak{t}^\#, \iota^\#), \]

and similarly the induced map for the cobordism $-W_{01}$ gives rise to a local map

\[f' : \textit{DSWF}(Y^\#, \mathfrak{t}^\#, \iota^\#) \to \textit{DSWF} (Y_0, \mathfrak{t}_0, \iota_0) \wedge \textit{DSWF} (Y_1, \mathfrak{t}_1, \iota_1). \]

This completes the proof.

Let us introduce a slight variant of the notion of SWF-spherical.

Definition 3.41. We say that a spin rational homology 3-sphere $(Y, \mathfrak{t}, \iota)$ with odd involution preserving the spin structure is locally DSWF-spherical if we have

\[[\textit{DSWF}_G(Y, \mathfrak{t}, \iota) ]_{\mathrm{loc}} = [(S^0,0, n)]_{\mathrm{loc}} \]

for some n. Note that, in this case, we have $\kappa (Y, \mathfrak{t}, \iota) = -n$ .

The difference between locally DSWF-spherical and SWF-spherical is that we consider the double and local equivalence class in the definition of locally DSWF-spherical. It is clear that if $(Y, \mathfrak{t}, \iota)$ is SWF-spherical, then this triple is locally DSWF-spherical. As a corollary of Theorem 3.40, we prove the following.

Proof. The first claim follows from Theorem 3.40. We prove the latter statement on orientation reversal. Doing a surgery on $[0,1] \times Y/\iota$ and the fixed-point set and taking the double branched cover, we have an equivariant $\mathbb{Z}_2$ -homology cobordism W from $(Y, \mathfrak{t}, \iota)\#(-Y, \mathfrak{t}, \iota)$ to $(S^3,\mathfrak{t}_0 ,\iota_0)$ , where $\mathfrak{t}_0$ is the unique spin structure on $S^3$ and $\iota_0$ is restriction of the complex conjugation on $\mathbb{C}^2 (\supset S^3)$ . It follows from the existence of the doubled relative Bauer–Furuta invariants of W and $-W$ that

(3.42) \begin{align}[\textit{DSWF} ( (Y, \mathfrak{t}, \iota)\#(-Y, \mathfrak{t}, \iota) )]_{\mathrm{loc}} = [(S^0, 0, 0)]_{\mathrm{loc}}.\end{align}

Let $(I(-Y), m',n' )$ be a representative of $\textit{DSWF}(-Y, \mathfrak{t}, \iota)$ . Since we supposed that $( Y, \mathfrak{t}, \iota)$ is locally DSWF-spherical, $\textit{DSWF} ( Y, \mathfrak{t}, \iota)$ is represented by $(S^0,0,n)$ for some n. It follows from conn sum of Theorem 3.40 and (3.42) that

\[[ ( I(-Y), m',n' + n )]_{\mathrm{loc}} =[\textit{DSWF} ( (Y, \mathfrak{t}, \iota)\wedge \textit{DSWF}(-Y, \mathfrak{t}, \iota) )]_{\mathrm{loc}} = [(S^0, 0, 0)]_{\mathrm{loc}}. \]

Thus we have

(3.43) \begin{align}[ \textit{DSWF}(-Y, \mathfrak{t}, \iota)]_{\mathrm{loc}} = [ ( I(-Y), m',n' )]_{\mathrm{loc}} = [(S^0, 0, -n)]_{\mathrm{loc}}\end{align}

from Remark 3.16.

Corollary 3.43. Let $(Y, \mathfrak{t})$ be a spin rational homology sphere equipped with an odd involution $\iota$ with non-empty fixed-point set. Suppose that $(Y, \mathfrak{t}, \iota)$ is locally DSWF-spherical. Then, for any spin rational homology 3-sphere $(Y', \mathfrak{t}')$ equipped with an odd involution $\iota'$ with non-empty fixed-point set, we have

\[\kappa ( Y\# Y', \mathfrak{t} \# \mathfrak{t}' , \iota\# \iota') = \kappa (Y, \mathfrak{t}, \iota) + \kappa (Y', \mathfrak{t}', \iota'). \]

Here, the connected sum of $(Y, \mathfrak{t}, \iota)$ and $(Y', \mathfrak{t}', \iota')$ can be taken along any fixed points. We also have that

\[\kappa(-Y,\mathfrak{t},\iota)=-\kappa(Y,\mathfrak{t},\iota). \]

Proof. By Theorem 3.40, we see

\[[\textit{DSWF} (Y, \mathfrak{t}, \iota) \wedge \textit{DSWF}(Y', \mathfrak{t}', \iota') ]_{\mathrm{loc}} = [\textit{DSWF}( Y\# Y', \mathfrak{t} \# \mathfrak{t}' , \iota\# \iota')]_{\mathrm{loc}}. \]

Thus, we have

\[\kappa (\textit{DSWF} (Y, \mathfrak{t}, \iota) \wedge \textit{DSWF} (Y', \mathfrak{t}', \iota')) = \kappa (\textit{DSWF}( Y\# Y', \mathfrak{t} \# \mathfrak{t}' , \iota\# \iota')). \]

Since we assumed that $[\textit{DSWF}(Y, \mathfrak{t}, \iota) ]_{\mathrm{loc}} = [(S^0,0, m )]_{\mathrm{loc}}$ for some m, and hence $\kappa(Y, \mathfrak{t}, \iota)=-m$ , we see that

\[[\textit{DSWF} (Y, \mathfrak{t}, \iota) \wedge \textit{DSWF} (Y', \mathfrak{t}', \iota')]_{\mathrm{loc}} = [(I(Y', \mathfrak{t}', \iota'),0,m'+ m )]_{\mathrm{loc}}, \]

where $I(Y', \mathfrak{t}', \iota')$ and m’ represent $\textit{DSWF} (Y', \mathfrak{t}', \iota')$ , i.e.

\[[(I(Y', \mathfrak{t}', \iota'), 0 , m')] = \textit{DSWF} (Y', \mathfrak{t}', \iota'). \]

Also, we have

\[\kappa (I(Y', \mathfrak{t}', \iota'),0 ,m'+ m )= \kappa (Y', \mathfrak{t}', \iota') - m \]

by the definition of $\kappa$ . This completes the proof.

The statement about the orientation reversal $-Y$ follows from (3.43) in the proof of Corollary 3.42).

We also note the stronger 10/8-inequality, Theorem 3.38, holds also for equivariant connected sums of SWF-spherical triples.

Theorem 3.44. For $N, N' \geqslant 0$ and $i \in \{1, \ldots, N\}, j \in \{1, \ldots, N'\}$ , let $(Y_i, \mathfrak{t}_i, \iota_i), (Y_j', \mathfrak{t}_j', \iota_j')$ be spin rational homology spheres with odd involution. Suppose that $(Y_i, \mathfrak{t}_i, \iota_i), (Y_j', \mathfrak{t}_j', \iota_j')$ are SWF-spherical and $\iota_i, \iota_j'$ have a non-empty fixed-point set. Form the equivariant connected sums

\[(Y, \mathfrak{t}, \iota)= \#_{i=1}^N (Y_i, \mathfrak{t}_i, \iota_i),\quad(Y', \mathfrak{t}', \iota')= \#_{j=1}^{N'} (Y_j', \mathfrak{t}_j', \iota_j') \]

along fixed points. Suppose that $(Y, \mathfrak{t}, \iota)$ , $(Y', \mathfrak{t}',\iota')$ are SWF-spherical. Let $(W,\mathfrak{s})$ be a smooth compact oriented spin cobordism with $b_1(W)=0$ from $(Y, \mathfrak{t})$ to $(Y', \mathfrak{t}')$ . Suppose that there exists a smooth involution $\iota$ on W such that $\iota$ preserves the given orientation and spin structure $\mathfrak{s}$ on W, and that the restriction of $\iota$ to the boundary is given by $\iota, \iota'$ . Set

\[N(W, Y, \mathfrak{t}, \iota, Y', \mathfrak{t}', \iota'):=-\frac{\sigma(W)}{16} + \kappa (Y, \mathfrak{t}, \iota) - \kappa (Y', \mathfrak{t}', \iota'). \]

Then $N(W, Y, \mathfrak{t}, \iota, Y', \mathfrak{t}', \iota')$ is an integer, and if we have

\[N(W, Y, \mathfrak{t}, \iota, Y', \mathfrak{t}', \iota') \geqslant 2 \quad \text{and} \quad b^+(W) - b^+_\iota(W) \geqslant 1, \]

then the inequality

\begin{align*}-\frac{\sigma(W)}{16} + \kappa (Y, \mathfrak{t}, \iota ) + A(N(W, Y, \mathfrak{t}, \iota, Y', \mathfrak{t}', \iota'))\leqslant b^+(W)-b^+_{\iota}(W) + \kappa (Y', \mathfrak{t}', \iota' )\end{align*}

holds.

Proof. As noted in Remark 3.39, Theorem 3.38 holds also for disjoint unions of spin rational homology spheres with odd involution that are SWF-spherical. Applying this to $\tilde{Y} = \bigsqcup_{i=1}^N (Y_i, \mathfrak{t}_i, \iota_i)$ and $\tilde{Y}' =\bigsqcup_{j=1}^{N'} (Y_j', \mathfrak{t}_j', \iota_j')$ , we have

\begin{align*}-\frac{\sigma(W)}{16} + \kappa (\tilde{Y}, \mathfrak{t}, \iota ) + A(N(W, \tilde{Y}, \mathfrak{t}, \iota, \tilde{Y}', \mathfrak{t}', \iota'))\leqslant b^+(W)-b^+_{\iota}(W) + \kappa (\tilde{Y}', \mathfrak{t}', \iota' ).\end{align*}

Since these disjoint unions $\tilde{Y}, \tilde{Y}'$ are equivariantly homology cobordant to Y and Y’, they induce local equivalences between $\textit{DSWF}$ . Since $\kappa$ is local equivalence invariant, this completes the proof.

3.10 Calculations

We carry out calculations of the doubled Seiberg–Witten Floer stable homotopy type $\textit{DSWF}_G(Y, \mathfrak{t},\iota)$ and the invariant $\kappa(Y, \mathfrak{t},\iota)$ for some $(Y,\mathfrak{t},\iota)$ . The most general statement on our calculation in this subsection is summarized in Theorem 3.52.

Proposition 3.45. Let $(Y,\mathfrak{t})$ be a spin rational homology 3-sphere equipped with an odd involution $\iota$ preserving $\mathfrak{t}$ . Assume that Y admits an $\iota$ -invariant positive scalar curvature metric g. Construct $\textit{DSWF}_G(Y, \mathfrak{t},\iota)$ using g. Then we have that

(3.44) \begin{align}\textit{DSWF}_G(Y, \mathfrak{t},\iota)=[(S^0,0,n(Y, \mathfrak{t},g)/2)]\end{align}

and

\[\kappa(Y, \mathfrak{t}, \iota) = -n(Y, \mathfrak{t},g)/2\quad{\text and}\quad\kappa(-Y, \mathfrak{t}, \iota) = -\kappa(Y, \mathfrak{t}, \iota). \]

Moreover, $(Y, \mathfrak{t}, \iota)$ is SWF-spherical.