2·1. Involutive Heegaard Floer homology and local equivalence group

In this subsection, we review the involutive Heegaard Floer theory and the notion of local equivalence. We will only consider the minus flavor of the Heegaard Floer chain complex $CF^-(Y)$ [ Reference Ozsváth and Szabó41 ], which is a graded $\mathbb{F}[U]$ -module, where U is a formal variable of grading $-2$ .

Ozsváth and Szabó [ Reference Ozsváth and Szabó41 ] defined the Heegaard Floer homology $HF^-(Y)$ for a closed 3-manifold Y, and proved that it is a diffeomorphism invariant. Hendricks and Manolescu [ Reference Hendricks and Manolescu18 ] defined a homotopy involution $\iota$ on a Heegaard Floer chain complex by composing the map which arises from exchanging $\alpha$ -curves and $\beta$ -curves in the Heegaard diagram with the canonical chain homotopy equivalence based on the naturality [ Reference Juhász, Thurston and Zemke22 ]. Then they introduced the involutive Heegaard Floer homology $HFI^-(Y)$ as the homology of the mapping cone of $1+\iota$ on $CF^-(Y)$ . When Y is a homology sphere, $HFI^-(Y)$ gives rise to two homology cobordism invariants $\underline{d\mkern-2mu}\mkern2mu (Y)$ and $\bar{d}(Y)$ .

Instead of computing $HFI^- (Y)$ directly, Hendricks, Manolescu and Zemke [ Reference Hendricks, Manolescu and Zemke19 ] defined an equivalence relation of the involutive Heegaard Floer chain complexes of homology spheres. The equivalence class is a homology cobordism invariant. Moreover, the classes form an abelian group $\mathfrak{I}$ , called the local equivalence group, together with the homomorphism $h\;:\;\Theta_\mathbb{Z}^3\rightarrow \mathfrak{I}$ as mentioned in Section 1.

Let $Y_1$ and $Y_2$ be two homology spheres. Recall from [ Reference Hendricks and Manolescu18 ] that a cobordism W from $Y_1$ to $Y_2$ with a self-conjugate spin $^c$ -structure $\mathfrak{t}$ induces the $\iota$ -equivariant $\mathbb{F}[U]$ -chain map $F_{W, \mathfrak{t}}\;:\; CF^-(Y_1)\rightarrow CF^-(Y_2)$ . Moreover, we have:

The local equivalence group is modeled on the homology cobordism group in the sense of Theorem 2·1.

Definition 2·2. Let C be a free, $\mathbb{Z}$ -graded and finitely generated chain complex over $\mathbb{F}[U]$ with a grading-preserving $\mathbb{F}[U]$ -equivariant homotopy involution $\iota$ . The pair $(C, \iota)$ is called an $\iota$ -complex if the localisation of its homology is isomorphic to $\mathbb{F}[U, U^{-1}]$ , i.e.,

\[U^{-1}H_*(C) \cong \mathbb{F}[U, U^{-1}].\]

Note that the involutive Heegaard Floer chain complex $(CF^-(Y), \iota)$ for a homology sphere Y is an $\iota$ -complex.

It is proved in [ Reference Hendricks, Manolescu and Zemke19 ] that the set of $\iota$ -complexes modulo local equivalence with the operation tensor product forms an abelian group $\mathfrak{I}$ , called the local equivalence group. The identity element of $\mathfrak{I}$ is given by the trivial complex consisting of a single $\mathbb{F}[U]$ -tower starting at grading zero, together with the trivial $\iota$ . Inverse elements in $\mathfrak{I}$ are obtained by dualising $\iota$ -complexes. Moreover, Hendricks, Manolescu and Zemke [ Reference Hendricks, Manolescu and Zemke19 ] also proved that there is an $\iota$ -equivariant chain homotopy equivalence:

\[(CF^-(Y_1\#Y_2),\iota) \simeq ((CF^-(Y_1)\otimes CF^-(Y_2))[\!-\!2], \iota_1\otimes \iota_2),\]

where $[\!-\!2]$ denotes the grading shift by $-2$ . By Theorem 2·1, the local equivalence class of Y is a homology cobordism invariant, and from the connected sum formula above we have the homomorphism defined as:

\[h\;:\; \Theta_\mathbb{Z}^3 \to \mathfrak{I}, \ \ h(Y) = (CF^-(Y)[\!-\!2], \iota),\]

where the d-invariant [ Reference Ozsváth and Szabó39 ], the $\underline{d\mkern-2mu}\mkern2mu $ - and $\bar{d}$ -invariants [ Reference Hendricks and Manolescu18 ], and the $\phi_n$ -invariants for all $n \geq 1$ [ Reference Dai, Hom, Stoffregen and Truong5 ] factor through $\mathfrak{I}$ .

2·2. Monotone graded subroots for Seifert fibered spheres

In this subsection, we explain the roles of graded roots and their monotone graded subroots to describe local equivalence classes of Seifert fibered spheres. We follow the references [ Reference Dai and Manolescu6, Reference Dai and Stoffregen8, Reference Karakurt and Şavk24, Reference Némethi35 ].

In [ Reference Némethi35 ], Némethi introduced two combinatorial objects to compute Heegaard Floer homology of Seifert fibered spheres effectively. The first object is called a graded root $(R,\nu)$ equipped with a grading function $\nu$ on the vertex set of R, where R is an upwards-opening tree with an infinite downwards stem defined from the negative definite plumbing graph for a Seifert fibered sphere. The second one is a graded $\mathbb{F}[U]$ -module, called the lattice homology $\mathbb{H} ^- (R)$ . It is combinatorially defined from $(R,\nu)$ , whose grading is induced by $\nu$ with the formal variable U of grading $-2$ . The grading on $\mathbb{H}^-(R)$ is denoted by the same symbol $\nu$ by abuse of notation. See [ Reference Dai and Manolescu6, Reference Némethi35 ] for precise definitions.

As observed in [ Reference Dai4, Reference Dai and Manolescu6 ], for a Seifert fibered sphere, there is an involution J on the vertex set of R that reflects a graded root along the central vertical axis. We call the triple $(R,\nu,J)$ a symmetric graded root, and an example is depicted in Figure 1. Moreover, there is an $\mathbb{F}[U]$ –equivariant action $J_*$ induced by J on $\mathbb{H} ^- (R)$ . See [ Reference Dai4 , section 2·1] for further discussion.

Figure 1.

The graded root R with the involution J for the Brieskorn sphere $\Sigma(3,4,13)$ . The leaves and angles of R are labelled by $v_1, \ldots v_6$ and $\alpha_1, \ldots \alpha_5$ , respectively.

By combining the results of Ozsváth and Szabó [ Reference Ozsváth and Szabó40 ], Némethi [ Reference Némethi35 ] and Dai and Manolescu [ Reference Dai and Manolescu6 ], we have the following graded $\mathbb{F}[U]$ -module isomorphism:

\[F\;:\; HF^-(Y) \xrightarrow{\cong} \mathbb{H}^-(R)[\sigma] \quad \text{and} \quad F \circ \iota_* = J_* \circ F,\]

where $\sigma$ denotes a grading shift Footnote 1 and $\iota_*$ is the induced involution by $\iota$ on $HF^-(Y)$ .

Following [ Reference Dai and Manolescu6 ], we next construct a finitely generated free graded $\mathbb{F}[U]$ -chain complex whose homology is the same as $\mathbb{H}^-(R)$ . See [ Reference Dai and Manolescu6 , lemma 4·3] for a proof of this fact.

Let $v_1, v_2, \ldots, v_n$ (resp. $\alpha_1, \alpha_2, \ldots, \alpha_{n-1}$ ) be the leaves (resp. the upward-opening angles) of R, enumerated from left to right in lexicographic order. The grading of vertex $v_i$ and the grading of the vertex supporting angle $\alpha_i$ are denoted by $\nu(v_i)$ and $\nu(\alpha_i)$ , respectively; see Figure 1.

Then the generators of $C_*(R)$ are given as follows. For each leaf $v_i$ , we place a single generator (also denoted by $v_i$ by abuse of notation) in grading $\nu(v_i)$ , so that we introduce an entire tower of generators $\mathbb{F}[U]v_i$ . For each angle $\alpha_i$ , we similarly place a single generator (denoted by also $\alpha_i$ by abuse of notation) in grading $\nu(\alpha_i) + 1$ . We next define our differential to be identically zero on $v_i$ , and set

\[\partial \alpha_i = U^{(\nu(v_i)-\nu(\alpha_i))/2}v_i + U^{(\nu(v_{i+1})-\nu(\alpha_i))/2}v_{i+1}\]

for $\alpha_i$ . Finally, we extend it to the entire complex linearly and $\mathbb{F}[U]$ -equivariantly.

There is an involution J on $C_*(R)$ induced by J on R, and it is given by sending $v_i$ to $v_{n-i+1}$ , $\alpha_i$ to $\alpha_{n-i}$ , and extending linearly and $\mathbb{F}[U]$ -equivariantly. See Figure 2. One can check that the pair $(C_*(R), J)$ forms an $\iota$ -complex. Moreover, J also induces the involution $J_*$ on $\mathbb{H}^-(R)$ . See [ Reference Dai and Manolescu6 , section 4] and [ Reference Dai and Stoffregen8 , section 2·3] for more details.

Figure 2.

The standard complex $C_*(R)$ with the involution J that captures the lattice homology $\mathbb{H}^- (\Sigma(3,4,13))$ . Here, the solid (resp. the dashed) lines represent the action of U (resp. $\partial$ ).

In [ Reference Dai and Manolescu6 , section 6], Dai and Manolescu introduced the notion of a monotone graded subroot of a symmetric graded root R, to simply determine the local equivalence class of a Seifert fibered sphere.

Let $(R, \nu, J)$ be a symmetric graded root, and let $\mathcal{G}_R$ be the image of the grading function $\nu\;:\; R \to \mathbb{Z}$ . For a positive integer n, let $h_1, \ldots, h_n$ (resp. $r_1, \ldots, r_n$ ) be a sequence of decreasing (resp. increasing) even integers in $\mathcal{G}_R$ such that $h_n \geq r_n$ . Then a monotone graded subroot

\[M=M(h_1,r_1; \ldots; h_n,r_n)\]

of R equipped with the same involution J is constructed in the following fashion:

1. (1) form the stem of M by drawing a single infinite tower with the uppermost vertex in grading $r_n$ ;

2. (2) if $h_n \gt r_n$ , use leaves $v_i$ and $J v_i$ of R to introduce leaves $v_i$ and $J v_i$ of M in grading $h_i$ for each $1 \leq i \lt n$ ;

3. (3) connect $v_i$ and $J v_i$ to the stem by using two paths meeting the stem in grading $r_i$ for each $1 \leq i \lt n$ ;

4. (4) if $h_n = r_n$ , then set $v_n = J v_n$ at grading $r_n$ in the second step.

By construction, the standard complex $(C_*(M), J)$ of a monotone graded subroot M is a subcomplex of $(C_*(R), J)$ , and it also forms an $\iota$ -complex itself, see [ Reference Dai and Manolescu6 , section 6]. However, this subcomplex is sufficient to describe the local equivalence class of a Seifert fibered sphere completely, as proven by Dai and Manolescu.

We now describe the parameterisation of a monotone graded subroot for a Seifert fibered sphere by using involutive correction terms $\underline{d}$ and $\overline{d}$ . See Figure 3 for an example.

Figure 3.

For the Brieskorn sphere $\Sigma(3,4,13)$ , the monotone graded subroot $M = M(0,-2)$ with the involution J is drawn in black, compare with Figure 1.

Theorem 2·5 ([ Reference Dai and Manolescu6 , section 8]; [ Reference Dai and Stoffregen8 , theorem 4·4]). Let Y be a Seifert fibered sphere with a graded root R. Then there exists a monotone graded subroot $M = M(h_1,r_1; \ldots; h_n,r_n)$ of R such that

\[h_1 = \overline{d}(Y) \quad \text{and} \quad r_n = \underline{d}(Y).\]

By following the convention in [ Reference Dai and Stoffregen8 ], from now on, we will refer to the standard complex $(C_* (M), J)$ as just the monotone graded subroot M for simplicity.

Now we present a theorem, provided by Dai and Stoffregen [ Reference Dai and Stoffregen8 ], to describe how a monotone graded subroot decomposes into simpler ones up to local equivalence.

Theorem 2·6 ([ Reference Dai and Stoffregen8 , theorem 4·2]). For any monotone graded subroot $M=M(h_1,r_1; \ldots;\; h_n,r_n)$ , we have the local equivalence:

\[M = \left ( \sum_{i=1}^{n} M(h_i,r_i) \right ) - \left ( \sum_{i=1}^{n-1} M(h_{i+1},r_i) \right ) .\]

Relying on the above results, Dai and Stoffregen further proved the following theorem. Recall that $B(0) = \Sigma(2,3,5)$ , and $B(n) = \Sigma(2n+1, 4n+1, 4n+3)$ for $n \geq 1$ .

Theorem 2·7 ([ Reference Dai and Stoffregen8 , theorem 1·1]). The image $\mathfrak{I}_{SF}$ of the subgroup $\Theta_{SF}^3$ generated by the Seifert fibered spheres under the homomorphism h is infinitely generated, i.e.,

\[\mathfrak{I}_{SF} = \langle h(B(n)) \rangle \cong \mathbb{Z}^\infty.\]

For any Seifert fibered sphere Y, there exists a unique homology sphere $Z_Y$ expressed as the linear combination of the basis elements $\{B(n)\}$ such that $h(Y)=h(Z_Y)$ , i.e., the homology sphere $Y \# -Z_Y$ lies in $\operatorname{ker}(h)$ . We call such $Z_Y$ the associated homology sphere to Y with respect to $\{B(n)\}$ . We omit the reference to the basis when it is clear from the context. In the final section of this article, beyond merely establishing the existence of $Z_Y$ and the linear independence of Y and $Z_Y$ in $\Theta_\mathbb{Q}^3$ , we will explicitly construct $Z_Y$ for a specific family Y by applying Theorem 2·6.

2·3. Local equivalence in Seiberg–Witten Floer theory

Based on the work of Manolescu [ Reference Manolescu32 ], Stoffregen [ Reference Stoffregen50 ] previously defined an analogous notion of the local equivalence class for homology spheres in Seiberg–Witten Floer theory, and computed these classes for Seifert fibered spheres. In this subsection, we present the equivalence between two notions of the local equivalence class in Seiberg–Witten Floer theory and the one in involutive Heegaard Floer theory for each Seifert fibered sphere.

As described in [ Reference Stoffregen50 ], the local equivalence class of the $\operatorname{Pin}(2)$ -equivariant Seiberg–Witten Floer stable homotopy type SWF(Y) and of its associated chain complex of its spectrum also form abelian groups $\mathfrak{LE}$ and $\mathfrak{CLE}$ , respectively, together with the homomorphisms:

\[\Theta_\mathbb{Z}^3 \rightarrow \mathfrak{LE}\rightarrow\mathfrak{CLE},\]

where:

1. (1) the $\kappa$ -invariant [ Reference Manolescu32 ] and $\kappa o_i$ -invariants [ Reference Lin31 ] for $i = 0, \ldots, 7$ factor through $\mathfrak{LE}$ ;

2. (2) the $\delta$ -invariant [ Reference Frøyshov15 ], the $\alpha$ -, $\beta$ -, and $\gamma$ -invariants [ Reference Manolescu33 ], and the $\underline{\delta}$ - and $\overline{\delta}$ -invariants [ Reference Stoffregen49 ] factor through $\mathfrak{CLE}$ .

See [ Reference Manolescu34 , section 3] for further details.

Recall that SWF(Y) recovers the $\operatorname{Pin}(2)$ -equivariant Seiberg–Witten Floer homology of Y.Footnote 2 Motivated from the facts that $\mathbb{H}^-(R)$ is isomorphic to $HF^-(Y)$ [ Reference Némethi35 ], and hence to the Seiberg–Witten Floer homology [ Reference Kutluhan, Lee and Taubes27, Reference Lidman and Manolescu29 ], Dai, Sasahira and Stoffregen constructed the $\operatorname{Pin}(2)$ -equivariant spectrum $\mathcal{H}(R)$ of the graded root R of Y, and proved the following:

Theorem 2·8 ([ Reference Dai, Sasahira and Stoffregen7 , theorems 1·2 and 7·3]). Let Y be a Seifert fibered sphere with a graded root R. Then there is a $\operatorname{Pin}(2)$ -equivariant homotopy equivalence:

\[\mathcal{H}(R)\simeq SWF(Y).\]

In particular, for two Seifert fibered spheres $Y_1$ and $Y_2$ , the following are equivalent:

1. (i) $h(Y_1)=h(Y_2)$ in $\mathfrak{I}$ ;

2. (ii) $[SWF(Y_1)] = [SWF(Y_2)]$ in $\mathfrak{LE}$ .

In particular, if the associated homology sphere $Z_Y$ to a Seifert fibered sphere Y is itself a Seifert fibered sphere, then the homology sphere $Y \# -Z_Y$ in $\ker (h)$ also lies in the kernel of $\Theta^3_\mathbb{Z} \to \mathfrak{LE}$ .

2·4. Filtered instanton Floer homology and $r_s$ -invariants

In this subsection, we briefly recall instanton Floer homology and its filtered version by Nozaki, Sato, and Taniguchi [ Reference Nozaki, Sato and Taniguchi38 ]. We follow the conventions and notations in [ Reference Nozaki, Sato and Taniguchi38 ].

For a given homology sphere Y, the Chern–Simons functional $cs_Y$ is defined on the space A(Y) of SU(2)-connections on the product principal SU(2)-bundle $P_Y = Y \times SU(2)$ .Footnote ³ Since $cs_Y$ is invariant under the action of the group $\operatorname{Map}_0(Y, SU(2))$ consisting of degree 0 maps in the gauge group, $cs_Y\;:\;A(Y)\rightarrow \mathbb{R}$ factors through the configuration space $\widetilde{B}(Y) = A(Y) / \operatorname{Map}_0(Y, SU(2))$ . We consider the subset $\widetilde{B}^*(Y)$ which consists of the orbits of irreducible connections, namely the stabiliser subgroup is $\{\pm I\}$ where I denotes the constant map to the identity in SU(2).

We fix a Riemannian metric on Y and an appropriate perturbation $\pi$ of $cs_Y$ . Then we construct an infinite-dimensional analogue of Morse chain complex of $\widetilde{B}^*(Y)$ using the perturbed Chern–Simons functional $cs_{Y,\pi}$ . The set $\widetilde{R}^*_\pi(Y)$ of the irreducible critical points of $cs_{Y,\pi}$ in $\widetilde{B}^*(Y)$ , consists of flat connections, namely, those with curvature $F_A=0$ . The integer-valued Floer index ind, which is an analogue of the ordinary Morse index, is also well-defined on $\widetilde{R}^*(Y)$ . Then the instanton Floer chain complex is defined as:

\[CI_i(Y)=\mathbb{Z}\{a \in \widetilde{R}_\pi^*(Y) \ | \ ind(a) = i\},\]

together with the boundary map $\partial\;:\; CI_i(Y) \rightarrow CI_{i-1}(Y)$ given by:

\[\partial (a) = \!\!\!\!\! \sum\limits_{{\substack{b \in \widetilde{R}^*_\pi(Y) \\ \text{ind}(b) = i-1}}} \!\!\!\!\!\!\#(M^Y(a, b)/\mathbb{R})b.\]

Here, $M^Y(a, b)$ denotes the 1-dimensional moduli space of instantons over the cylinder $\mathbb{R} \times Y$ where its restriction on $Y\times t$ is asymptotically a when t goes to $-\infty$ and b when t goes to $+\infty$ . Note that $\mathbb{R}$ acts on $M^Y(a,b)$ by the translation.

The boundary map satisfies $\partial^2 = 0$ , so we obtain the instanton Floer homology $I_*(Y)$ , which does not depend on the choice of Riemannian metric and perturbation, and is a diffeomorphism invariant of Y. By taking the dual complex $CI^i(Y)$ , one can also obtain the instanton Floer cohomology $I^*(Y)$ .

Note that the chain complex only involves irreducible critical points. Let $\theta \in A(Y)$ be the product SU(2)-connection on $P_Y$ so that $\theta$ is a reducible critical point. Donaldson [ Reference Donaldson9 ] introduced an obstruction cocycle $\theta_Y\;:\;CI_1(Y)\rightarrow \mathbb{Z}$ in the first instanton Floer cochain complex $CI^1(Y)$ to count the trajectories from irreducible solutions to $\theta$ as follows:

\[\theta_Y([a]) = \#(M^Y(a, \theta)/\mathbb{R}),\]

where $M^Y(a,\theta)$ is similarly defined to $M^Y(a,b)$ . See [ Reference Nozaki, Sato and Taniguchi38 ] for a precise definition. The cocycle $\theta_Y$ forms a well-defined cohomology class $[\theta_Y]\in I^1(Y)$ , which vanishes whenever Y bounds a negative definite 4-manifold.

On the other hand, Fintushel and Stern [ Reference Fintushel and Stern14 ] introduced the filtered instanton Floer homology $I_*^{[r,r+1]} (Y)$ for any fixed real number r whose filtrations are given by the Chern–Simons functional. We recall the recent work of Nozaki, Sato, and Taniguchi [ Reference Nozaki, Sato and Taniguchi38 ] which extends the work of Fintushel and Stern to the filtered instanton Floer homology $I_*^{[s,r]} (Y)$ for a more general interval [s, r]. We write the set of real numbers:

\[\Lambda_Y = cs_Y(\widetilde{R}_\pi(Y)), \quad \Lambda_Y^* = cs_Y(\widetilde{R}^*_\pi(Y)) \quad \text{and} \quad \mathbb{R}_Y = \mathbb{R} - \Lambda_Y,\]

where $\widetilde{R}_\pi(Y)$ is the set of the orbits of all critical points of $cs_{Y,\pi}$ , whether irreducible or reducible.

Definition 2·9. Let $s\in [\!-\!\infty, 0]$ , $r\in \mathbb{R}_Y$ and $\lambda_Y = \min\{|a-b| |a \neq b, a, b\in \Lambda_Y\}$ . The filtered instanton Floer chain complex is defined as:

\[CI_i^{[s, r]}(Y) = \begin{cases} \mathbb{Z}\{a \in \widetilde{R}_{\pi}^*(Y) \ | \ ind(a) = i, cs_{Y, \pi}(a)\in (s, r)\} & \text{if $s\in \mathbb{R}_Y$,}\\ \mathbb{Z}\{a \in \widetilde{R}_{\pi}^*(Y) \ | \ ind(a) = i, cs_{Y, \pi}(a)\in (s-\lambda_Y/2, r)\}& \text{if $s\in \Lambda_Y$,} \end{cases}\]

together with the boundary map given by the restriction of $\partial$ defined on the ordinary $CI_i(Y)$ .

Since $\partial^2=0$ , we have the filtered instanton Floer homology $I_*^{[s,r]}(Y)$ , which is a diffeomorphism invariant of Y. Technically, the chain complex can be defined for arbitrary intervals, however, we avoid the case when $r\in \Lambda_Y$ to make the canonically defined map $i_{[s, r]}^{[s',r']}\;:\;CI_i^{[s,r]}(Y)\rightarrow CI_i^{[s',r']}(Y)$ behave well. More precisely.

Lemma 2·10 ([ Reference Nozaki, Sato and Taniguchi38 , lemma 2·9]). Suppose $s\le s' \le 0 \le r \le r'$ where $r, r'\in \mathbb{R}_Y$ . If both [r, r’] and [s, s’] do not contain any value of $\Lambda_Y^*$ , then the map $i_{[s, r]}^{[s',r']}\;:\;CI_i^{[s,r]}(Y)\rightarrow CI_i^{[s',r']}(Y)$ defined by

\[ i_{[s,r]}^{[s',r']}(a)= \begin{cases} a &\text{if $a\in CI_i^{[s',r']}(Y)$,}\\ 0 &\text{otherwise,} \end{cases}\]

is a chain homotopy equivalence.

By taking duals of $CI_i^{[s,r]}(Y)$ and of the chain maps, we can also consider the filtered instanton Floer cochain complex $CI^i_{[s,r]}(Y)$ and its cohomology $I^i_{[s,r]}(Y)$ , and the dual maps on them.

Let $Y_1$ and $Y_2$ be two homology spheres. Now we consider an oriented cobordism W from $Y_1$ to $Y_2$ . As in other Floer theories, W induces a chain map between the instanton Floer chain complexes. In particular, when $b_1(W)=b_2^+(W)=0$ , the (co)chain maps preserve the Floer indices:

\begin{align*} CW_i^{[s, r]}&\;:\;CI_i^{[s,r]}(Y_1)\rightarrow CI_i^{[s,r]}(Y_2);\\ CW^i_{[s, r]}&\;:\;CI^i_{[s,r]}(Y_2)\rightarrow CI^i_{[s,r]}(Y_1).\end{align*}

Let $\theta_Y^{[s,r]}$ be the restriction of $\theta_Y$ to $CI_1^{[s,r]}(Y)$ . In [ Reference Nozaki, Sato and Taniguchi38 ], Nozaki, Sato and Taniguchi observed that the map $\theta_Y^{[s,r]}$ defines a cocycle in $CI^1_{[s,r]}(Y)$ . The cohomology class $[\theta_Y^{[s,r]}]\in I^1_{[s,r]}(Y)$ also behaves well under the induced map $IW^1_{[s,r]}$ by $CW^1_{[s,r]}$ on the first cohomology. More precisely,

Theorem 2·11 ([ Reference Nozaki, Sato and Taniguchi38 , lemma 2·13]). If there is a negative definite cobordism W from $Y_1$ to $Y_2$ with $H^1(W;\;\mathbb{R})=0$ , then for $r\in \mathbb{R}_{Y_1}\cap \mathbb{R}_{Y_2}$ ,

\[IW^1_{[s,r]}\big(\big[\theta_{Y_2}^{[s,r]}\big]\big) = |H_1(W;\;\mathbb{Z})|\big[\theta_{Y_1}^{[s,r]}\big].\]

In particular, if Y bounds a negative definite 4-manifold, then the obstruction class $[\theta_Y^{[s, r]}]\in I^1_{[s,r]} (Y)$ vanishes.

Let $i_{[s',r']}^{[s,r]}$ be the dual map of $i^{[s',r']}_{[s,r]}$ . We use the same notation for the induced maps on (co)homology by abusing notation. One can observe from Lemma 2·10 that:

Lemma 2·12 ([ Reference Nozaki, Sato and Taniguchi38 , lemma 2·15]). Suppose $s\le s' \le 0 \le r \le r'$ where $r, r'\in \mathbb{R}_Y$ . Then,

\[i_{[s',r']}^{[s,r]}\big[\theta_Y^{[s',r']}\big] = \big[\theta_Y^{[s,r]}\big].\]

Suppose $[\theta_Y^{[s, r]}]\neq 0$ . By definition of $i_{[s,r]}^{[s',r']}$ and Lemma 2·12, $[\theta_Y^{[s, r']}]$ vanishes for some $r'\lt r$ and then $[\theta_Y^{[s, r'']}]=0$ for any $r''\le r'$ . This “birth-death” property motivates to define the $r_s$ -invariant to measure when the obstruction class $[\theta_Y^{[s,r]}]$ begins to vanish. The coefficient ring is taken as $\mathbb{Q}$ , since one may want to divide $IW^1_{[s,r]}([\theta_{Y_2}^{[s,r]}])$ by the non-zero integer $|H_1(W;\;\mathbb{Z})|$ to obtain $[\theta_{Y_1}^{[s,r]}]$ as in Theorem 2·11.

Definition 2·13 ([ Reference Nozaki, Sato and Taniguchi38 , definition 3·2]). Let $s\in [\!-\!\infty, 0]$ . The $r_s$ -invariant of Y is defined as:

\[r_s (Y) = \sup \big\{ r \in (0,\infty] \ | \ \big[\theta_Y^{[s, r]}\otimes id_\mathbb{Q}\big] = 0 \in I^1_{[s,r]} (Y;\;\mathbb{Q}) \big\}.\]

By definition and the vanishing property of the obstruction class, if Y bounds a negative definite 4-manifold, then $r_s(Y) = \infty$ for all $s\in[\!-\!\infty, 0]$ . In particular, for any $s\in[\!-\!\infty, 0]$ and any Seifert fibered sphere $\Sigma(a_1,\ldots, a_n)$ , we have

\[r_s(\Sigma(a_1,\ldots,a_n))=\infty.\]

Nozaki, Sato and Taniguchi [ Reference Nozaki, Sato and Taniguchi38 , theorem 5·15] proved that the value $r_s(Y)$ is invariant under rational homology cobordism. They also obtained the following connected sum formula [ Reference Nozaki, Sato and Taniguchi38 , theorem 1·1(4)]

\[r_{s_1+s_2}(Y_1\#Y_2)\ge \min\{r_{s_1}(Y_1)+s_2, \ r_{s_2}(Y_2)+s_1\},\]

so it is useful to take $s=0$ to obtain linear independence in $\Theta_\mathbb{Q}^3$ .

Theorem 2·14 will play a crucial role in the proof of Theorem 1·2, coupled with the useful formula for $r_0(Y)$ for a Seifert fibered sphere Y such that $R(Y) \gt 0$ , where R is the integer defined for a Seifert fibered sphere by Fintushel and Stern [ Reference Fintushel and Stern13 ]. We close this section by presenting such a formula.

Theorem 2·15 ([ Reference Nozaki, Sato and Taniguchi38 , corollary 1·4]). Let Y be the Seifert fibered sphere $\Sigma(a_1,\ldots, a_n)$ . If $R(Y) \gt0$ , then for any $s\in [\!-\!\infty, 0]$ ,

\[r_s(\!-\!Y) = \frac{1}{4a_1\cdots a_n} \cdot\]