2.1 Hamiltonian circle actions, decorated graphs, and Delzant polygons

In the forthcoming sections, we will study loops in the fundamental group of $\operatorname {\mathrm{Symp}}_h({M}_{\mu ,c_1,c_2,c_3,c_4})$ . Since these loops will appear as Hamiltonian circle actions, we will make extensive use of Karshon’s classification of Hamiltonian circle actions and Delzant’s classification of toric actions on symplectic manifolds. For convenience, we give a quick overview on how these classifications work.

Karshon’s classification [Reference Karshon10] yields a bijection between certain decorated graphs and $4$ -tuples $(M^4,\omega ,\rho ,\Phi )$ consisting of a symplectic $4$ -manifold $(M^4,\omega )$ , and an effective Hamiltonian circle action $\rho $ with a given moment map $ \Phi : M \rightarrow \mathbb {R}$ . Given such a tuple $ (M^4, \omega , \rho , \Phi ) $ , the associated decorated graph is constructed as follows. Each component C of the fixed point set is either a single point or a symplectic surface, and fixed points on which the moment map is not extremal are isolated. For each such component C, there is a vertex $\langle C \rangle $ , labeled by the real number $\Phi (C)$ . A vertex that corresponds to a fixed symplectic surface is said to be “fat” and is given two more labels: the area label $\frac {1}{2 \pi } \int _{C} \omega $ , and the genus g of the surface. A $\mathbb {Z}_{k}$ -sphere is a gradient sphere in M on which $ S^{1} $ acts with isotropy $ \mathbb {Z}_{k} $ , $k\geq 2$ . For each $\mathbb {Z}_{k}$ -sphere containing two fixed points $ p $ and $ q $ , the graph has an edge connecting the vertices $ \langle p \rangle $ and $ \langle q \rangle $ labeled by the integer $ k $ .

Labeled graphs associated with effective Hamiltonian circle actions are characterized by the following properties. If we order the vertices according to their moment map labels, then:

• – there are exactly two extremal vertices;

• – fat vertices are extremal, and if the graph contains two fat vertices, then their genus label must coincide;

• – the area label of any fat vertex must be strictly positive;

• – a vertex is connected to no more than two edges, and no edge is connected to a fat vertex;

• – the moment map labels must be strictly monotone along each chain of edges;

• – if $e_{1},\ldots ,e_{\ell }$ is a chain of edges, and if $k_{1},\ldots ,k_{\ell }$ are the orders of their stabilizers, then $\gcd (k_{i}, k_{i+1}) = 1$ for $i = 1,\ldots \ell -1$ , and $(k_{i-1} + k_{i+1})/k_{i}$ is an integer for $i= 2,\ldots ,\ell -1$ .

We call such graphs admissible.

Furthermore, it can be shown that each Hamiltonian action on a four-dimensional manifold can be obtained from a circle action on a symplectic ruled surface by performing a sequence of $S^1$ -equivariant symplectic blowups. At the graph level, equivariant symplectic blowups correspond to the simple transformations pictured in Figures 1 and 2. Together with Lalonde–McDuff–Li–Liu’s uniqueness theorem [Reference Lalonde and McDuff14, Reference Li and Liu19] stating that any two cohomologous symplectic forms on blowups of ruled surfaces are diffeomorphic, this gives an effective algorithm to enumerate all effective circle actions on any given $4$ -manifold.

Figure 1:

Blowing up at a point inside an invariant surface at the minimum value of $ \Phi $ .

Figure 2:

Blowing up at an interior fixed point.

Since we are mainly interested in the continuous map $\rho :S^1\to \operatorname {\mathrm {Ham}}(M,\omega )$ , we do not need to keep track of the moment map associated with a Hamiltonian circle action. As any two moment maps only differ by a constant, we can either consider graphs only up to a uniform translation of their moment map labels, or normalize the moment map by setting $\min _{x\in M} \Phi (x) =0$ . Finally, note that the reparameterization of the circle $t\mapsto -t$ corresponds to changing the signs of the moment map labels.

There is an analogous classification of Hamiltonian toric actions that we now briefly describe in the special case of $4$ -manifolds. Given a $4$ -tuple $(M^{4},\omega ,\rho ,\Phi )$ consisting of a symplectic $4$ -manifold $(M,\omega )$ , an effective Hamiltonian toric action $\rho :T^2\to \operatorname {\mathrm {Ham}}(M,\omega )$ , and a moment map $\Phi :M\to \mathfrak {t}^*\simeq \mathbb {R}^2$ , the image $\Phi (M)$ is always a Delzant polygon, that is, a polygon satisfying the following three properties:

• – simplicity, i.e., there are two edges meeting each vertex;

• – rationality, i.e., the edges meeting at the vertex $ p $ are rational in the sense that each edge is of the form $ p + t u_{i} $ , $ t \in [0,\ell _i] $ , where $ \ell _i \in \mathbb {R}$ and $ u_{i} \in \mathbb {Z}^{2} $ ;

• – smoothness, i.e., for each vertex, the corresponding $ u_{1} $ , $ u_{2} $ can be chosen to be a $ \mathbb {Z} $ -basis of $ \mathbb {Z}^{2} $ .

Moreover, the preimage in the manifold of a vertex of the polygon $ \Phi (M) $ is a fixed point for the torus action, whereas the preimage of an edge is an invariant 2-sphere. The preimage of the interior of the polygon consists of free torus orbits. These facts are explained in [Reference Delzant8].

Delzant’s classification [Reference Delzant8] states that equivalence classes of $4$ -tuple $(M^{4},\omega ,\rho ,\Phi )$ up to equivariant symplectomorphisms that preserve the moment maps are classified by Delzant polygons in $\mathbb {R}^2$ . If we disregard the moment map and only consider an effective toric action as an injective homomorphism $\rho :\mathbb {T}^2\to \operatorname {\mathrm {Ham}}(M,\omega )$ , it is natural to declare two actions as equivalent if they only differ by a reparameterization of the torus or by a conjugation by an element of $\operatorname {\mathrm{Symp}}(M,\omega )$ . In this setting, the classification theorem yields a bijection

$$ \begin{gather*} \{\text{Conjugacy classes of toric actions on}~4\text{-manifolds up to reparameterizations}\}\\ \updownarrow\\ \{\text{Delzant polygons in~}\mathbb{R}^2 \text{~up to~} \operatorname{\mathrm{AGL}}(2;\mathbb{Z}) \text{~action}\}. \end{gather*} $$

If we restrict the action to the subcircle $\{ e\} \times S^1$ , we get a compact four-dimensional $S^1$ -space. The moment map for the $S^1$ -action is the composition of the $\mathbb {T}^{2}$ -moment map with the projection $\mathbb {R}^2 \to \mathbb {R}$ to the second coordinate. The fixed surfaces are the preimages, under the $\mathbb {T}^{2}$ -moment map, of the horizontal edges of the Delzant polygon. Such a surface has genus zero, and its normalized symplectic area is equal to the length of the corresponding horizontal edge. The isolated fixed points are the preimages of those vertices of the polygon that do not lie on horizontal edges. The $\mathbb {Z}_k$ -spheres, $k\geq 2$ , are the preimages of edges with slope $\pm k/b$ in a reduced form, where b is relatively prime to k. With this information, we can construct the graph for the $S^1$ -space out of the Delzant polygon. This is explained by Karshon in [Reference Karshon10, Section 2.2]. Note that, similarly, we can restrict the toric action to the subcircle $S^1 \times \{ e\}$ in order to obtain another compact four-dimensional $S^1$ -space. In this case, the moment map for the $S^1$ -action is the composition of the $\mathbb {T}^{2}$ -moment map with the projection $\mathbb {R}^2 \to \mathbb {R}$ to the first coordinate. This relation between polygons and decorated graphs will be particularly useful in the following subsections.

2.2 Quantum homology and Seidel morphism

Following [Reference McDuff and Salamon23], consider the (small) quantum homology ring ${QH}_*(M; \Pi ) = H_*(M, \mathbb {Q}) \otimes _{\mathbb {Q}} \Pi $ with coefficients in the ring $\Pi := \Pi ^{\mathrm {univ}}[q,q^{-1}]$ where the q is a polynomial variable of degree 2 and $ \Pi ^{\mathrm {univ}},$ called the universal Novikov ring, is a generalized Laurent series ring in a variable t of degree 0:

(2.1) $$ \begin{align} \Pi^{\mathrm{univ}}:= \left\{ \sum_{\kappa \in \mathbb{R}} r_\kappa t^\kappa \,\big|\, r_\kappa \in \mathbb{Q}, \ \#\{ \kappa> c\mid r_\kappa \neq 0\} < \infty, \forall c \in \mathbb{R} \right \}\!. \end{align} $$

The quantum homology ${QH}_*(M; \Pi )$ is $\mathbb {Z}$ -graded so that $\deg (a \otimes q^d t^\kappa )= \deg (a) +2d$ with $a \in H_*(M)$ . The quantum intersection product $a*b \in {QH}_{i+j -\dim M}(M; \Pi )$ , of classes $a \in H_i(M)$ and $b \in H_j(M)$ , has the form

$$ \begin{align*}a*b = \sum_{B \in H_2^S(M;\mathbb{Z})} (a*b)_B \otimes q^{-c_1(B)}t^{-\omega(B)},\end{align*} $$

where $H_2^S(M;\mathbb {Z})$ is the image of $\pi _2(M)$ under the Hurewicz map. The homology class $(a*b)_B \in H_{i+j-\dim M +2c_1(B)}(M)$ is defined by the requirement that

$$ \begin{align*}(a*b)_B \cdot_M c = \mathrm{GW}^M_{B,3}(a,b,c) \quad \mbox{ for all } c \in H_*(M).\end{align*} $$

In this formula, $ \mathrm {GW}^M_{B,3}(a,b,c) \in \mathbb {Q}$ denotes the Gromov–Witten invariant that counts the number of spheres in M in class B that meet cycles representing the classes $a,b,c \in H_*(M)$ . The product $*$ is extended to ${QH}_*(M)$ by linearity over $\Pi $ , and is associative (see [Reference McDuff and Salamon23, Proposition 11.1.9] for a proof of this fact). It also respects the $\mathbb {Z}$ -grading and gives ${QH}_*(M)$ the structure of a graded commutative ring, with unit $[M]$ .

The Seidel morphism is a homomorphism $\mathcal {S}$ from $\pi _1 (\operatorname {\mathrm {Ham}} (M, \omega ))$ to the degree $2n$ multiplicative units ${QH}_{2n}(M)^{\times }$ of the small quantum homology, first introduced by Seidel in [Reference Seidel27]. One way of thinking of it is to say that it “counts” pseudoholomorphic sections of the bundle $M_\Lambda \to S^2$ associated with the loop $\Lambda \subset \operatorname {\mathrm {Ham}}(M,\omega )$ via the clutching construction (as in [Reference McDuff and Tolman24, Section 2]): let $(M, \omega )$ be a closed symplectic manifold and let $\Lambda =\{ \Lambda _\theta \}$ be a loop in $\operatorname {\mathrm {Ham}}(M, \omega )$ based on identity. Denote by $M_\Lambda $ the total space of the fibration over $S^2$ with fiber M which consists of two trivial fibrations over 2-discs, glued along their boundary via $\Lambda $ . Namely, we consider $S^2$ as the union of the two 2-discs $D_0$ and $D_\infty $ such that $D_0$ is the closed unit disk centered at 0 in the Riemann sphere $S^2 = \mathbb {C} \cup \{ \infty \}$ and $D_\infty $ is another copy of this disk, embedded in $S^2 = \mathbb {C} \cup \{ \infty \}$ , via the orientation reversing map $r\, e^{i\theta } \mapsto r^{-1} \, e^{i\theta }$ . The total space is

This construction only depends on the homotopy class of $\Lambda $ . Moreover, the family (parameterized by $S^2$ ) of symplectic forms of the fibers can be extended to give a closed form, $\Omega $ , on $M_\Lambda $ (see [Reference Sternberg29]). By adding to $\Omega $ the pullback of a suitable area form on the base, we get a nondegenerate form. More precisely, $\omega _{\Lambda ,\kappa } = \Omega + \kappa \cdot \pi ^*(\omega _0)$ is symplectic, where $\omega _0$ is the standard symplectic form on $S^2$ (with area 1), $\pi $ is the projection to the base of the fibration, and $\kappa $ is a big enough constant to make $\omega _{\Lambda ,\kappa }$ nondegenerate. (Once chosen, $\kappa $ will be omitted from the notation.)

So we end up with the following Hamiltonian fibration:

In [Reference McDuff and Tolman24], McDuff and Tolman observed that, when $\Lambda $ is a circle action (with associated moment map $\Phi _\Lambda $ ), the clutching construction can be simplified since, then, $M_\Lambda $ can be seen as the quotient of $M \times S^3$ by the diagonal action of $S^1$ , $e^{2\pi i \theta }\cdot (x,(z_1,z_2))=(\Lambda _\theta (x),(e^{2\pi i \theta } z_1, e^{2\pi i \theta } z_2))$ . The symplectic form also has an alternative description in $M \times _{S^1} S^3$ . Let $\alpha \in \Omega ^1(S^3)$ be the standard contact form on $S^3$ such that $d\alpha = \chi ^*(\omega _0)$ where $\chi : S^3 \to S^2$ is the Hopf map and $\omega _0$ is the standard area form on $S^2$ with total area 1. For all $c\in \mathbb {R}$ , $\omega +cd\alpha -d(\Phi _\Lambda \alpha )$ is a closed 2-form on $M \times S^3$ which descends through the projection, $p: M \times S^3 \rightarrow M \times _{S^1} S^3$ , to a closed 2-form on $M_\Lambda $ :

(2.2) $$ \begin{align} \omega_c = p( \omega + c d\alpha -d(\Phi_{\Lambda} \alpha)), \end{align} $$

which extends $\Omega $ . Now, if $c>\max \,\Phi _\Lambda $ , $\omega _c$ is nondegenerate and coincides with $\omega _{\Lambda ,\kappa }$ for some big enough $\kappa $ .

A quantum class lying in the image of $\mathcal S$ is called a Seidel element. In [Reference McDuff and Tolman24], McDuff and Tolman were able to calculate the leading term of Seidel’s elements associated with Hamiltonian circle actions whose maximal fixed point component, $F_{\max }$ , is semi-free, that is, the action is semi-free on some neighborhood of $F_{\max }$ . Recall that a circle action is semi-free if the stabilizer of every point is trivial or the whole circle. Moreover, when the codimension of $F_{\max }$ is 2, their result immediately ensures that if there exists an invariant almost complex structure J on M so that $(M,J)$ is Fano, i.e., so that there are no J-pseudoholomorphic spheres in M with nonpositive first Chern number, all the lower-order terms vanish. In the presence of J-pseudoholomorphic spheres with vanishing first Chern number, there is a priori no reason why arbitrarily large multiple coverings of such objects should not contribute to the Seidel elements. In fact, as explained in [Reference Anjos and Leclercq4], when the almost complex manifold $(M,J)$ is only numerically effective (NEF), i.e., $c_1(B) \geq 0$ for every class $B \in H_2(M)$ with a J-holomorphic sphere representative, and not Fano, then there are indeed infinitely many contributions to the Seidel elements. More precisely, it is shown in [Reference Anjos and Leclercq4] that if M is a 4-toric manifold, then these quantum classes can still be expressed by explicit closed formulas. Moreover, these formulas only depend on the relative position of representatives of elements of $\pi _2(M)$ with vanishing first Chern number as edges of the moment polygon. In particular, they are directly readable from the polygon.

We now recall the precise results from [Reference Anjos and Leclercq4] that we will use in the forthcoming sections. Consider a four-dimensional closed symplectic manifold $(M,\omega )$ , endowed with a toric structure $(\rho ,\Phi )$ . Suppose that the associated Delzant polygon $P = \Phi (M)$ has $m \geq 4$ edges, and consider a Hamiltonian circle action $\Lambda $ on $(M, \omega )$ , with moment map $\Phi _\Lambda $ , such that $\Lambda $ is a subcircle of the toric action $\rho :\mathbb {T}^{2}\to \operatorname {\mathrm {Ham}}(M,\omega )$ .

We assume additionally that the fixed point component of $\Lambda $ on which $\Phi _\Lambda $ is maximal is a 2-sphere, $F_{\max } \subset M$ , whose momentum image is an edge D of P. We denote by $A \in H_2(M;\mathbb {Z})$ the homology class of $F_{\max }$ and by $\Phi _{\max } = \Phi _\Lambda (F_{\max })$ .

In this case, McDuff–Tolman’s result [Reference McDuff and Tolman24, Theorem 1.10] ensures that the Seidel element associated with $\Lambda $ is

(2.3) $$ \begin{align} \mathcal S(\Lambda) = A\otimes q t^{\Phi_{\max}} + \sum_{B \in H_2^S\!(M;\mathbb{Z})^{>0}} a_B \otimes q^{1-c_1(B)} t^{\Phi_{\max}-\omega(B)}, \end{align} $$

where $H_2^S(M;\mathbb {Z})^{>0}$ consists of the spherical classes of positive symplectic area, that is, $\omega (B)>0$ and $a_B\in H_*(M;\mathbb {Z})$ denotes the contribution of B. As mentioned above, when $(M,J)$ is Fano for some $S^1$ -invariant $\omega $ -compatible almost complex structure J, then all the lower-order terms vanish and we end up with $ \mathcal S(\Lambda )=A \otimes qt^{\Phi _{\max }}$ .

In the non-Fano case, one has to be careful about the number and relative position of edges, in the vicinity of D, corresponding to spheres in M with vanishing first Chern number. We denote the number of such edges by $\mathcal {\#}\{c_1=0\}$ . We denote the edges and the corresponding homology classes in M in a cyclic way, that is, D, which, we denote by $D_m$ below, has neighboring edges $D_{m-1}$ on one side and $D_{m+1}=D_1$ on the other, and they, respectively, induce classes $A_m$ , $A_{m-1}$ , and $A_{m+1}=A_1$ in $H_2(M;\mathbb {Z})$ .

Figure 3 shows the relevant parts of the different polygons we need to consider. Dotted lines represent edges with positive first Chern number, and we indicate near each edge with nontrivial contribution the homology class of the corresponding sphere in M. For example, in Case (3c), only three homology classes contribute: $A_{m-1}$ , $A_m$ , and $A_1$ ; $A_{m-1}$ and $A_1$ have vanishing first Chern number while $c_1(A_m) \neq 0$ .

Figure 3:

Cases appearing in Theorem 2.2.

Now, the following theorem gives the explicit expression of the Seidel element associated with $\Lambda $ when $\mathcal {\#}\{c_1=0\} \leq 2$ .

2.3 The fundamental group of $\operatorname {\mbox{Symp}} ( {M}_{\mu ,c_1,c_2,c_3,c_4})$

In this section, we recall the main results obtained by Li et al. [Reference Li, Li and Wu18] on the Torelli symplectic mapping class group and on the rank of the fundamental group of the group Symp $_{h}({M}_{\mu ,c_1,c_2,c_3,c_4}) $ of symplectomorphisms that act trivially on homology, for any given symplectic form.

First, note that diffeomorphic symplectic forms define symplectomorphism groups that are homeomorphic, and that symplectomorphism groups are invariant under rescalings of symplectic forms. Consequently, we can restrict ourselves to symplectic forms belonging to a fundamental domain for the action of $ \operatorname {\mathrm {Diff}} \times \mathbb {R_*} $ on the space $ \Omega _{+} $ of orientation-compatible symplectic forms defined on the n-fold blowup $ \mathbb {X}_{n} $ . The cohomology class of a reduced class $\omega $ (see Definition 1.1 in the Introduction) is $\nu L - \delta _1 V_1 - \cdots - \delta _n V_n$ . Let $\mathcal J_\omega $ be the space of compatible almost complex structures on $\mathbb {X}_n$ . For any $J \in \mathcal J_\omega $ on $\mathbb {X}_n$ , the first Chern class $c_1:= c_1 (TM) \in H^2(\mathbb {X}_n; \mathbb {Z})$ is the Poincaré dual to $K:= 3L - \sum _i V_i$ . Let $\mathcal K$ be the symplectic cone of $\mathbb {X}_n$ , that is,

$$ \begin{align*}\mathcal{K} = \{ A \in H^2(\mathbb{X}_n; \mathbb{Z}) \, | \, A = [\omega] \ \mbox{for some symplectic form} \ \omega \in \Omega _{+} \}.\end{align*} $$

Now, if C stands for the Poincaré dual of the symplectic cone of $ \mathbb {X}_{n} $ , then by uniqueness of symplectic blowups proved by McDuff in [Reference McDuff22], the diffeomorphism class of the form $ \omega $ only depends on its cohomology class. Therefore, it is enough to describe a fundamental domain of the action of $ \operatorname {\mathrm {Diff}} \times \mathbb {R_*} $ on C. Moreover, the canonical class K is unique up to orientation-preserving diffeomorphisms [Reference Li and Liu20], so it suffices to describe the action of the diffeomorphisms fixing K, $\operatorname {\mathrm {Diff}}_K$ , on

$$ \begin{align*}C_{K} = \{ A \in H_{2} ( \mathbb{X}_{n}; \mathbb{R} ) \ : \ A = PD [ \omega ] \ \mbox{for some} \ \omega \in \Omega_{K}\},\end{align*} $$

where $ \Omega _{K}$ is the set of orientation-compatible symplectic forms with K as the symplectic canonical class. By the results in [Reference Li and Liu20], the set of reduced classes is a fundamental domain of $C_{K} ( \mathbb {X}_{n} )$ under the action of $ \operatorname {\mathrm {Diff}}_{K}$ . A proof of this result is also given in [Reference Karshon and Kessler11, Theorem 1.4]. We now consider the following change of basis in $H_2(\mathbb {X}_n; \mathbb {Z})$ . Consider the symplectic manifold $(S^2 \times S^2, \mu \sigma \oplus \sigma )$ where the homology class of the base $ B \in H _{2} ( S ^{2} \times S ^{2} )$ represented by $ S ^{2} \times \{ pt \} $ has area $ \mu , $ and the homology class of the fiber $ F \in H _{2} ( S ^{2} \times S ^{2} )$ represented by $ \{ pt \} \times S ^{2} $ has area $ 1 $ . Recall that ${M}_{\mu ,c_1,\ldots ,c_{n-1}}=(S^2 \times S^2 \#\, (n-1)\overline { \mathbb C\mathbb P}\,\!^2, \omega _{\mu ,c_1,\ldots , c_{n-1}})$ is obtained from $(S^2 \times S^2, \mu \sigma \oplus \sigma )$ , by performing $n-1$ successive blowups of capacities $ c_{1}, \ldots , c_{n-1}$ . This can be naturally identified with $(\mathbb {X}_n, \omega )$ . One easy way to understand the equivalence is as follows: let $ \{ B, F, E_{1}, \ldots , E_{n-1} \} $ be the basis for $H_{2}({M}_{\mu ,c_1,\ldots ,c_{n-1}}; \mathbb {Z}) $ where the $ E_{i}$ represent the exceptional spheres arising from the blowups. We identify L with $ B+F-E_{1} $ , $ V_{1}$ with $ B - E_{1}$ , $ V_{2} $ with $ F - E_{1} $ , and $ V_{i} $ with $ E_{i-1} $ , with $3 \leq i \leq n$ . Then the uniqueness of symplectic blowups due to McDuff (see [Reference McDuff22, Corollary 1.3]) implies that the symplectomorphism type of a symplectic blowup of a rational ruled manifold along an embedded ball of capacity $ c \in (0,1) $ depends only on the capacity $ c $ and not on the particular embedding used in obtaining the blowup. Using this result and after rescaling, we conclude that, for parameters satisfying the relations

(2.4) $$ \begin{align} \mu = \dfrac{\nu - \delta_{2}}{\nu - \delta_{1}} , \quad c_{1}= \dfrac{\nu - \delta_{1} - \delta_{2}}{\nu - \delta_{1}}, \quad \mbox{and} \quad c_{i} = \dfrac{\delta_{i+1}}{\nu - \delta_{1}}, \quad 2 \leq i \leq n-1, \end{align} $$

there exists a symplectomorphism between two symplectic manifolds encoded by these parameters such that

$$ \begin{align*}\nu L - \delta_1 V_1 - \cdots - \delta_n V_n = \mu B + F -c_1 E_1 - \cdots -c_{n-1} E_{n-1}.\end{align*} $$

Summarizing the above discussion, we showed that

Lemma 2.3 Every symplectic form on $S^2 \times S^2 \#\, (n-1)\overline { \mathbb C\mathbb P}\,\!^2$ is, after rescaling, diffeomorphic to a form Poincaré dual to $\mu B + F -c_1 E_1 - \cdots -c_{n-1} E_{n-1}$ with

$$ \begin{align*}0 < c_{n-1} \leq \cdots \leq c_1 \leq 1 \leq \mu \quad \mbox{and} \quad c_i+c_j \leq 1.\end{align*} $$

Recall (see [Reference Li and Li16]) that the normalized reduced symplectic cone is defined as the space of reduced symplectic classes having area 1 on L, the line class. Note that cohomologous symplectic forms on a rational or ruled surface are diffeomorphic (cf. [Reference Lalonde and McDuff15, Reference Li and Liu20]). We represent such a class by $(1 | \delta _1, \ldots , \delta _n)$ , or $(\delta _1, \ldots , \delta _n) \in \mathbb {R}^n.$ For $3 \leq n \leq 8$ , such a cone is a n-simplex with one facet removed, where the monotone class is one of the vertices, namely $M_n = (\frac 13, \ldots , \frac 13)$ . We are interested in the case when the manifold is $\mathbb {X}_5$ where the normalized reduced cone is convexly generated by five half-closed intervals $\{MO,MA,MB,MC,MD\}$ , with vertices $M= (\frac 13,\frac 13,\frac 13,\frac 13,\frac 13)$ , which corresponds to the monotone case, $O=(0,0,0,0,0)$ , $A=(1,0,0,0,0)$ , $B=(\frac 12,\frac 12, 0,0,0)$ , $C=(\frac 13,\frac 13,\frac 13,0,0)$ , and $D=(\frac 13,\frac 13,\frac 13,\frac 13,0)$ (for more details, see [Reference Li and Li16]).

Let $N_\omega $ be the number of symplectic-2 spheres classes. Then Li, Li, and Wu proved the following.

Theorem 2.4 [Reference Li, Li and Wu18, Theorem 1.2]

Consider $\mathbb {X}_5$ with any symplectic form $\omega $ . Then the rank of the fundamental group of $\operatorname {\mathrm{Symp}}_h (\mathbb {X}_5, \omega )$ satisfies

$$ \begin{align*}\mathrm{rank}( \pi_1 (\operatorname{\mathrm{Symp}}_h (\mathbb{X}_5, \omega))) = N_\omega -5 + \mathrm{rank} ( \pi_0 (\operatorname{\mathrm{Symp}}_h(\mathbb{X}_5, \omega))),\end{align*} $$

where the rank of $\pi _0 (\operatorname {\mathrm{Symp}}_h(\mathbb {X}_5, \omega ))$ means the rank of its abelianization.

Moreover, along the edge $MA$ , when $\nu =1, \delta _1> \delta _2 =\delta _3=\delta _4=\delta _5$ , and $\delta _1+ \delta _2 +\delta _3 =1$ , or equivalently, when $\mu> 1$ and $c_1=c_2=c_3=c_4= 1/2$ , it follows from [Reference Li, Li and Wu18, Lemma 5.10] and its proof (in particular from sequence (37)) that $ \pi _1 (\operatorname {\mathrm{Symp}}_h (\mathbb {X}_5, \omega ))=\mathbb {Z}_5$ . This is the case we will study in detail in the forthcoming sections. In particular, we will show that a generating set of the fundamental group of $\operatorname {\mathrm{Symp}}_h(\mathbb {X}_5, \omega )$ can be realized by Hamiltonian circle actions except in some particular interval of values of $\mu $ .

4.1 Hamiltonian circle actions in ${M}_{\mu ,c_1,c_2,c_3,c_4}$

In this section, we list all equivalence classes of Hamiltonian circle actions on symplectic manifolds whose symplectic cohomology class belongs to the edge $MA$ of the reduced symplectic cone. Recall that, along this edge, we have $\mu> 1$ and $c_1=c_2=c_3=c_4= 1/2$ . Recall also that Karshon’s classification [Reference Karshon10, Section 6.2] implies that every compact four-dimensional Hamiltonian $S^1$ -space can be obtained from a minimal space, which can be $\mathbb {CP}^2$ , a Hirzebruch surface, or an irrational ruled manifold (see [Reference Karshon10, Section 6.3]), by a sequence of equivariant symplectic blowups at fixed points. It follows that the only possible Hamiltonian circle actions on the symplectic manifolds belonging to the edge $MA$ are the ones corresponding to the labeled graphs of Figure 4, where the values of a and b represent the symplectic area of the invariant spheres and depend on which sphere we perform the blowup. In our figures, we omit the genus label since, in our case, the invariant surfaces are always embedded spheres. Moreover, since the symplectic area of the spheres is positive, i.e., $a,b> 0$ , and $c_1=c_2=c_3=c_4= 1/2$ , then we can only have $a+b= 2\mu -2$ .

Figure 4:

Graphs representing Hamiltonian circle actions on symplectic manifolds belonging to the ray $MA$ .

4.2 Circle actions and homotopy classes of loops

A labeled graph only determines a circle action up to symplectomorphisms. Equivalently, a labeled graph defines a conjugacy class of circles in $\operatorname {\mathrm{Symp}}(M,\omega )$ . Consequently, any such graph defines an element of

where the action is by conjugation. The analysis of this action is done in two stages.

For any symplectic manifold $(M,\omega )$ belonging to the ray $MA$ , the action of $\operatorname {\mathrm{Symp}}(M,\omega )$ on homology induces a short exact sequence

$$\begin{align*}1\to \operatorname{\mathrm{Symp}}_{h}(M,\omega)\to\operatorname{\mathrm{Symp}}(M,\omega)\stackrel{f}{\longrightarrow}\operatorname{\mathrm{Aut}}_{c_1,[\omega]}\left(H_2(M,\mathbb{Z})\right)\to 1,\end{align*}$$

where $\operatorname {\mathrm {Aut}}_{c_1,[\omega ]}\left (H_2(M,\mathbb {Z})\right )$ is the group of automorphisms of the lattice $H_2(M,\mathbb {Z})$ preserving the intersection form and the classes dual to $c_1(M,\omega )$ and $[\omega ]$ . The fact that the map f is onto follows from three results in [Reference Li and Wu21] that we briefly recall. First, by [Reference Li and Wu21, Proposition 4.14], the group $\operatorname {\mathrm {Aut}}_{c_1,[\omega ]}\left (H_2(M,\mathbb {Z})\right )$ is generated by reflections about spherical homology classes A satisfying three conditions: $A\cdot A=-2$ , $c_1(A)=0$ , and $\omega (A)=0$ . Such a class is called a $(K, [\omega ])$ -null spherical class, where K denotes the symplectic canonical class. Second, a symplectic Dehn twist along a Lagrangian sphere L induces the reflection $R([L])$ in homology. Finally, the result follows from Proposition 5.6 in [Reference Li and Wu21] which proves existence of Lagrangian spheres representing $(K, [\omega ])$ -null spherical classes.

In our case, along $MA$ , the automorphism group is isomorphic to the Weyl group $D_4$ given by the trivalent Dynkin diagram (see [Reference Li, Li and Wu18]). It is easy to see that it fixes the classes $2B+2F-E_1-E_2-E_3-E_4$ and F, whereas it acts transitively on the eight exceptional classes ${E_1,\ldots ,E_4, F-E_1,\ldots ,F-E_4}$ . In particular, the only element of $\operatorname {\mathrm {Aut}}_{c_1,[\omega ]}$ that fixes the four exceptional classes $E_i$ is the identity.

In order to keep track of the action of on Hamiltonian circle actions, we consider extended graphs as defined in [Reference Karshon10, Section 5, p. 33] decorated with homology labels. Starting with the standard labelled graph of Figure 4, we add extra (dotted) edges that represent free invariant spheres connecting each interior fixed point to extrema of the moment map. Each such sphere is the closure of a free $\mathbb {C}^*$ -orbit, where the $\mathbb {C}^*$ action is defined from the choice of a generic $S^1$ -invariant almost-complex structure. Since the action of $\operatorname {\mathrm{Symp}}(M,\omega )$ preserves the genericity of almost-complex structures, an extended graph defines a configuration of invariant spheres that is well defined up to conjugation. We then label the edges of the extended graph with homology classes according to the sequence of blowups that is used to construct the Hamiltonian $S^1$ -manifold. Geometrically, this amounts to labeling invariant spheres with their homology class following a specific sequence of equivariant blowups performed on $\left (S^2\times S^2,\mu \sigma \otimes \sigma \right )$ , starting with the two fixed surfaces labeled $B=[S^2\times \operatorname {\mathrm {pt}}]$ and $F=[\operatorname {\mathrm {pt}}\times S^2]$ . The possible extended labeled graphs are shown in Figures 5–7. By construction, the group $\operatorname {\mathrm{Symp}}(M,\omega )$ acts on its corresponding extended labeled graph with kernel $\operatorname {\mathrm{Symp}}_h(M,\omega )$ . In our case, these extended labeled graphs classify $\operatorname {\mathrm{Symp}}_h$ -equivalence classes of $S^1$ -manifolds on the edge $MA$ endowed with a given framing $\phi :H_2(M,\mathbb {Z})\to \mathbb {Z}\langle B,F,E_1,E_2,E_3,E_4\rangle \simeq \mathbb {Z}^{1,5}$ .

Figure 5:

Family of graphs in the case $\mu>1$ .

Recall from [Reference Li, Li and Wu18] that, for $(M,\omega )$ belonging to the edge $MA$ , the symplectomorphism group $\operatorname {\mathrm{Symp}}_h(M,\omega )$ is not connected. Indeed, $\pi _0 (\operatorname {\mathrm{Symp}}_h(M,\omega ))= \pi _0(\operatorname {\mathrm {Diff}}^+(S^2,4))\simeq P_4(S^2) / \mathbb {Z}_2$ , where $P_4(S^2)$ is the pure braid group of four strings in $S^2.$ Since an extended labeled graph only determines an element in $\pi _1(\operatorname {\mathrm{Symp}}(M,\omega ))/\pi _0(\operatorname {\mathrm{Symp}}_h(M,\omega ))$ , we have to understand how $\pi _0(\operatorname {\mathrm{Symp}}_h(M,\omega ))$ acts on $\pi _1(\operatorname {\mathrm{Symp}}(M,\omega ))$ . We postpone this analysis to Section 4.4.

4.3 Extended labeled graphs along the edge $MA$

We now describe a finite set of one-parameter families of extended labeled graphs, parameterized by $\mu $ , that correspond to symplectic manifolds belonging to the edge $MA$ of the reduced symplectic cone. The number of elements in these families depends on the range of $\mu $ . As explained above, each such graph corresponds to a $\operatorname {\mathrm{Symp}}_h(M,\omega )$ -conjugacy class of Hamiltonian circle actions.

Notice that these actions only exist as long as the symplectic area of the classes corresponding to the fixed spheres is positive. Assuming $1 <\mu \leq \frac 32$ , we have four circle actions represented by the graphs in Figure 5. We can consider for example: $z_{0,12}, z_{0,13},z_{0,14}$ and $z_1$ . We do not consider flips of these graphs as they represent actions which are inverse to these ones.

Moreover, as we increase the value of $\mu $ , there are more classes that can be represented by the fixed spheres, as we see next. If we consider $ \mu> \frac 32 $ , then we can add the graphs in Figure 6 to the previous family. It should be clear that there are eight such graphs, because $i,j, \ell ,m=1,2,3,4$ are all distinct. More precisely, we have the graphs representing the following actions: $z_{0,123},z_{0,124},z_{0,134},z_{0,234}$ and $z_{1,1},z_{1,2}, z_{1,3},z_{1,4}$ .

Figure 6:

New family of graphs if $\mu>\frac 32$ .

Then there are no more possible classes for the fixed symplectic spheres unless we consider $\mu>2$ . In this case, eight new circle actions appear, where the following pairs of classes are represented by the fixed spheres: B and $B-E_1-E_2-E_3-E_4$ ; $B-2F$ and $B+2F-E_1-E_2-E_3-E_4$ ; and $B-F-E_i-E_j$ and $B+F -E_\ell -E_m$ with $i,j,\ell , m =1,2,3,4$ all distinct. If we restrict the range of values of $\mu $ further, it is easy to see that the number of circle actions keeps increasing. More precisely, when $\mu $ passes $k +\frac 12$ or $k+1$ , for $k \in \mathbb {Z}_{\geq 1}$ , the number of actions always increases by 8. Therefore, we obtain the following proposition.

Proposition 4.2 The Hamiltonian circle actions on the symplectic manifolds belonging to the edge $MA$ of the reduced symplectic cone are the ones represented by the labeled graphs in Figure 7. In particular, these actions satisfy the following existence conditions:

$$\begin{align*} & \bullet z_k \ \mbox{exists} \ \mbox{iff} \ \mu>k \ \mbox{and} \ \mu > 2-k, \\ & \bullet z_{k,i} \ \mbox{exists} \ \mbox{iff} \ \mu > k +\frac12 \ \mbox{and} \ \mu > \frac32-k, \\ & \bullet z_{k,ij} \ \mbox{exists} \ \mbox{iff}\ \mu > k +1,\\ & \bullet z_{k,ijl} \ \mbox{exists} \ \mbox{iff} \ \mu > k +\frac32,\\ & \bullet z_{k,1234} \ \mbox{exists} \ \mbox{iff} \ \mu > k +2,\end{align*}$$

where $k \in \mathbb {Z}_{\geq 0}$ and $i,j,\ell ,m =1,2,3,4$ are all distinct.

Figure 7:

Families of graphs of Hamiltonian $S^1$ -spaces encoded by the edge $MA$ .

4.4 The Seidel morphism along the edge $MA$

In this subsection, we prove our first main theorem, namely Theorem 1.3. The proof relies on the computation of the Seidel elements associated with the circle actions $z_{0,1i}, i=2,3,4$ , $z_1$ , and $z_{1,4}$ , and on the fact that they are linearly independent in the subgroup of invertible elements of the quantum homology of the manifold ${M}_{\mu ,c_i=1/2}$ .

4.4.1 Seidel elements and deformations

In order to describe the Seidel morphism

$$\begin{align*}\mathcal{S}:\pi_1(\operatorname{\mathrm{Symp}}({M}_{\mu,c_i=1/2}))\to {QH}_{2n}({M}_{\mu,c_i=1/2})^{\times},\end{align*}$$

our strategy is to combine the invariance property of $\mathcal {S}$ under the natural $\operatorname {\mathrm{Symp}}_{h}(M,\omega )$ action, the invariance of Gromov–Witten invariants with respect to symplectic deformations, and Theorem 2.2, which describes certain Seidel elements associated with subcircles of toric actions on NEF symplectic manifolds.

More precisely, we first observe that the Seidel morphism

$$\begin{align*}\mathcal{S}:\pi_1(\operatorname{\mathrm{Symp}}_0(M,\omega))\to {QH}_{2n}(M,\omega)^{\times}\end{align*}$$

defined in Section 2.2 is invariant under the action of $\pi _0(\operatorname {\mathrm{Symp}}_h(M,\omega ))$ on $\pi _1(\operatorname {\mathrm{Symp}}_0(M,\omega )$ . This follows from the definition of $\mathcal {S}$ and, in the case of Hamiltonian circle actions, can be seen directly from the formula (2.3) given by McDuff and Tolman. In particular, given a Hamiltonian circle action $\gamma :S^1\to \operatorname {\mathrm {Ham}}(M,\omega )$ , its image $\mathcal {S}(\gamma )$ is determined by the labeled extended graph associated with $\gamma $ .

Next, consider a Hamiltonian circle action $\rho $ on ${M}_{\mu ,c_i=1/2}$ and the corresponding loop $[\rho ]$ in $\operatorname {\mathrm{Symp}}({M}_{\mu ,c_i=1/2})$ . Let $\Omega (\rho )$ be the space of all symplectic forms that are invariant under this action and write $\Omega _0(\rho )$ for the connected component of $\omega _{\mu ,c_i=1/2}$ . A $\mathbb {Q}$ -generic symplectic form is a symplectic form whose cohomology class $ [\mu ;c_1,\ldots ,c_4]$ is given by coefficients that are linearly independent over $\mathbb {Q}$ . One can show that invariant $\mathbb {Q}$ -generic symplectic forms are dense in $\Omega _0(\rho )$ : let $\omega $ be any invariant symplectic form and $\delta _i$ be invariant closed two forms whose cohomology classes are a basis for $H^2(M,\mathbb {R})$ . Then, for sufficiently small $c_i$ , the two forms $ \omega + \sum _i c_i\delta _i$ are invariant and symplectic. The result follows readily.

The same argument shows that extended graphs whose moment map labels are small continuous perturbations of the labels associated with $\rho $ correspond to deformation equivalent symplectic forms invariant under the same circle action. Consequently, for any $S^1$ -manifold $(M,\omega ')$ associated with such an extended graph, there is no ambiguity as to what the Seidel element $\mathcal {S}([\rho ])\in {QH}_{2n}(M,\omega ')^{\times }$ is.Footnote ¹

We now observe that, due to deformation invariance of Gromov–Witten invariants, given any symplectic form $\omega '$ in $\Omega _0(\rho )$ , the quantum homology ring ${QH}(M,\omega ')$ is obtained from the quantum ring of a $\mathbb {Q}$ -generic class $\omega _{\mu ,c_1,c_2,c_3,c_4}$ by setting the values of the coefficients $\mu ,c_1,c_2,c_3,c_4$ equal to those of $\omega '$ . We thus have a natural specialization map ${QH}(M,\omega _{\mu ,c_1,c_2,c_3,c_4})\to {QH}(M,\omega ')$ , which sends the Seidel element of $[\rho ]$ computed relatively to the generic form $\omega _{\mu ,c_1,c_2,c_3,c_4}$ to the one computed relatively to the form $\omega '$ .

Finally, we can apply the previous discussion starting with a Hamiltonian circle actions $\rho $ on ${M}_{\mu ,c_i=1/2}$ . From the above remarks, we can find a deformation equivalent $\mathbb {Q}$ -generic form $\omega _{\mu ;c_1,c_2,c_3,c_4}$ such that the sizes of the blowups satisfy the inequalities $0 < c_4 < c_3 < c_2 < c_1 < c_i+c_j < 1 < \mu $ , with $i,j \in \{1,2,3,4\}$ distinct. Symplectic cohomology classes satisfying this condition are said to be reduced generic or, more simply, generic. By choosing the sizes carefully, we can embed the circle action $\rho $ into a toric action of a toric manifold that is NEF, and for which Theorem 2.2 applies. This allows us to compute the Seidel element of $\rho $ .

In what follows, we consider Hamiltonian actions fixing spheres in the same homology classes as the ones in Proposition 4.2 and we list in Figure 8 their graphs. Note that we use the same notation for the circle actions in the generic case as, for the circle actions along the edge $MA$ , we do not distinguish one case from the other with regard to notation.

Figure 8:

Graphs of the circle actions $z_k$ , $z_{k,i}$ , $z_{k,ij}$ , $z_{k,ijl}$ , and $z_{k,1234}$ in the generic case.

4.4.2 Computing Seidel elements from toric actions

First, consider the actions $z_{0,12}, z_{0,13}, z_{0,14}$ and the polygon of Figure 9. The action $z_{0,12}$ corresponds to the circle action whose moment map is the first component of the moment map associated with the toric action $T_{0,12}$ , represented in this figure. Moreover, it is clear that the homology classes of the fixed spheres are $B-E_1-E_2$ and $B-E_3-E_4$ . Then Theorem 2.2(2a) yields

$$ \begin{align*} \mathcal S (z_{0,12})= [B-E_3-E_4] \otimes q \, \frac{t^{\epsilon}}{1- t^{c_3+c_ 4-\mu}}, \end{align*} $$

where $\epsilon $ is the maximum of the momentum map of the action $z_{0,12}$ , $\phi _{\mathrm {max}}(F_{\mathrm {max}})$ , where $F_{\mathrm {max}}$ is the maximal 2-sphere whose momentum image is the edge in class $B-E_3-E_4$ , in the normalized polygon. In general, we obtain

$$ \begin{align*} \mathcal S (z_{0,1i})= [B-E_j-E_\ell] \otimes q \, \frac{t^{\epsilon}}{1- t^{c_j+c_ \ell-\mu}}, \quad \mbox{where} \quad j \neq \ell \neq i. \end{align*} $$

One can check that the normalized polygon yields

$$ \begin{align*}\epsilon = \frac{c_j^3+ 3c_1^2-c_1^3+c_\ell^3+3c_i^2-c_i^3-3\mu}{3(c_1^2+c_2^2+c_3^2+c_4^2-2\mu)}.\end{align*} $$

Hence, if $c_i=1/2$ for all i, we obtain $\epsilon = 1/2$ and

(4.1) $$ \begin{align} \mathcal S (z_{0,1i})= [B-E_j-E_\ell] \otimes q \, \frac{t^{\frac12}}{1- t^{1-\mu}}, \quad \mbox{where} \quad j \neq \ell \neq i. \end{align} $$

Note that the expression is well defined because $\mu>1$ .

Figure 9:

Toric action $T_{0,12}$ .

Consider now the polygon of Figure 10, which represents a toric action on ${M}_{\mu ,c_1,c_2,c_3,c_4}$ , and for which the homology classes of the edges are represented in the figure. The graph of the circle action obtained by the projection of the polygon onto the x-axis is also represented in Figure 10. Note that it becomes the action $z_1$ defined in Figure 5 if $c_i=1/2$ for all i.

Figure 10:

Toric action $T_1$ and its projection to the x-axis.

Now, Theorem 2.2(2a) gives

$$ \begin{align*} \mathcal S(z_1) = [B+F-E_1-E_2-E_3-E_4] \otimes q \, \frac{t^{1-\epsilon}}{1- t^{c_1+c_2+c_3+c_4-\mu-1}}, \end{align*} $$

where in this case the maximum of the momentum map on the invariant sphere is given by

$$ \begin{align*}\epsilon = \frac{-1 -c_1^3 + 3c_1^2 +3c_2^2-c_2^3 +3c_3^2 -c_3^3+3c_4^2-c_4^3+ 3c_1c_4^2-3c_3c_4^2-3\mu}{3(c_1^2+c_2^2+c_3^2+c_4^2-2\mu)},\end{align*} $$

which is simply equal to $1/2$ if $c_i=1/2$ for all i. Therefore, we obtain

(4.2) $$ \begin{align} \mathcal S(z_1)= [B+F-E_1-E_2-E_3-E_4] \otimes q \, \frac{t^{\frac12}}{1- t^{1-\mu}}. \end{align} $$

Finally, we compute the Seidel element of the circle action $z_{1,4}$ , seen as an element of the fundamental group of $\operatorname {\mathrm{Symp}}_h ({M}_{\mu ,c_1,c_2,c_3,c_4})$ . In order to do that, first consider the Delzant polygon of Figure 11. It represents a toric action on ${M}_{\mu ,c_1,c_2,c_3,c_4}$ and its projections onto the x-axis and the y-axis are represented in Figure 12. Note that the projection onto the x-axis corresponds to the graph of the action $z_{1,4}$ given in Figure 6. Let us denote the action whose graph is obtained by projection onto the y-axis by $s_{1,4}$ .

Figure 11:

Toric action $(z_{1,4}, s_{1,4})$ .

Figure 12:

Graphs of circle actions $z_{1,4}$ and $s_{1,4}$ , respectively.

Since $c_1(B-F-E_4)= -1 <0$ , the complex manifold corresponding to Figure 11 is not NEF and we cannot apply immediately Theorem 2.2 to compute the Seidel element of $z_{1,4}$ . Instead, we need to consider some auxiliary polygons for which the underlying complex manifolds are NEF and relate the circle actions represented on those polygons with the actions $z_{1,4}$ and $s_{1,4}$ . More precisely, consider the Delzant polygon, on the left in Figure 13 and apply the $GL(2, \mathbb {Z})$ transformation represented by the matrix

(4.3) $$ \begin{align} \left (\!\begin{array}{cc} 1 & 0 \\ 1 & 1 \end{array} \!\right) \end{align} $$

to this polygon as well as to the polygon of Figure 11. Then consider the projection onto the y-axis of the two transformed polygons and denote the action obtained this way from the polygon of Figure 13 by $t_{1,4}$ . It is easy to check that the two graphs coincide, which implies that as elements of $\pi _1(\operatorname {\mathrm{Symp}}_h ({M}_{\mu ,c_1,c_2,c_3,c_4}))$ the following identification holds

(4.4) $$ \begin{align} z_{1,4}+ s_{1,4} = t_{1,4}. \end{align} $$

Figure 13:

Polygon representing an NEF complex manifold and its transformation by the $GL(2, \mathbb {Z})$ matrix (4.3).

On the other hand, consider the polygon on the left in Figure 14, which represents a toric action, denoted by $(x_1,y_1)$ , on ${M}_{\mu ,c_1,c_2,c_3,c_4}$ and its transformation by the same $GL(2,\mathbb {Z})$ matrix (4.3). The projection of the transformed polygon onto the y-axis yields a graph that, up to translation, coincides with the graph of the action $s_{1,4}$ , which implies that $s_{1,4}=x_1+y_1.$

Figure 14:

Toric action $(x_1,y_1)$ and its transformation by the $GL(2, \mathbb {Z})$ matrix (4.3).

Note that, in Figure 14, we have $c_1(A) \geq 0$ for all homology classes A of the fixed spheres corresponding to edges of the polygon, so we can apply Theorem 2.2 to compute the Seidel element of $s_{1,4}=x_1+y_1$ . Moreover, it follows from equation (4.4) that the Seidel element of $z_{1,4}$ is given by

(4.5) $$ \begin{align} \mathcal S(z_{1,4})= \mathcal S(t_{1,4}) \mathcal S(s_{1,4})^{-1}. \end{align} $$

More precisely, Theorem 2.2(3a) yields the Seidel elements of $t_{1,4}$ and $-s_{1,4}$ , which are readable directly from the polygons on the right in Figures 13 and 14.

$$ \begin{align*} & \mathcal S(t_{1,4}) = E_2 \otimes q t^{\mu +1 -c_1-c_2 -\gamma} - (E_1-E_2) \otimes q \frac{t^{\mu + 1-2c_1-\gamma}}{1-t^{c_2-c_1}},\\ & \mathcal S(s_{1,4})^{-1} = E_3 \otimes q t^{\beta-c_3} - (F-E_3-E_4) \otimes q \frac{t^{\beta +c_4-1}}{1-t^{c_3+c_4-1}}, \end{align*} $$

where the exponents of the higher degree terms, namely of $E_2$ and $E_3$ , are the maximal values of the momentum map for the normalized polygons. Therefore, one can check that

$$ \begin{align*}\gamma = \frac{-1+ 3c_1^2 -3c_1^3 +3c_2^2 -3c_1c_2^2 -2c_2^3 +c_3^3 +c_4^3 +3c_1^2\mu+3c_2^2\mu -3\mu^2}{3(c_1^2+c_2^2+c_3^2+c_4^2-2\mu)}\end{align*} $$

and

$$ \begin{align*}\beta=\frac{3c_1^2 -2c_1^3 +3c_2^2 -3c_1c_2^2 -c_2^3 +2c_3^3 +3c_4^2-3\mu +3c_1^2\mu+3c_2^2\mu -3\mu^2}{3(c_1^2+c_2^2+c_3^2+c_4^2-2\mu)}.\end{align*} $$

Using [Reference Crauder, Miranda, Dijkgraaf, Faber and van der Geer7, Proposition 5.3], we can now compute the quantum product (4.5) (we leave the details to the interested reader since it is a long and boring computation) and finally obtain

$$ \begin{align*} \mathcal S(z_{1,4}) =((B+F -E_1- E_2-E_3)\otimes q + \nVdash\otimes t^{c_4 - \mu})t^{\beta- \gamma}, \end{align*} $$

where the identity $\nVdash $ is the homology class of the manifold, $[\mathbb C\mathbb P^2\#\, 5\overline { \mathbb C\mathbb P^2}] \in H_4(\mathbb C\mathbb P^2\#\, 5\overline { \mathbb C\mathbb P^2}, \mathbb {Z})$ . In what follows, we will suppress the identity from the expressions in order to simplify the notation.

If $c_i=1/2$ for all i, then

(4.6) $$ \begin{align} \mathcal S(z_{1,4}) = ((B+F -E_1- E_2-E_3)\otimes q + t^{\frac12 - \mu})t^{\frac{2-3\mu}{3(1-2\mu)}}. \end{align} $$

Note that this result agrees with McDuff–Tolman’s result as the leading term is given by the homology class of the edge where the action is maximal.

Remark 4.6 In a similar way, we can compute the Seidel element of the inverse of $z_{1,4}$ :

(4.7) $$ \begin{align} \begin{split} \mathcal S(z_{1,4})^{-1} = [(B-F -E_4)\otimes q & + [pt] \otimes q^2 \, t^{\frac32 - \mu} + (3F -E_1-E_2-E_3 +E_4)\otimes q\, t^{1 - \mu} \\ & + (B-E_4)\otimes q \, t^{2 - 2\mu} + t^{\frac12 - \mu}(1 + t^{2 - 2\mu})] \frac{t^{\frac{1-3\mu}{3(1-2\mu)}}}{(1-t^{1 - \mu})^4}. \end{split} \end{align} $$

Since all the possible Hamiltonian circle actions on ${M}_{\mu ,c_i=1/2}$ have semi-free maximal sets, the McDuff–Tolman result always applies. Observe that, when $\mu> \frac 32$ , the leading term is the homology class $B+F -E_4$ , corresponding to the homology class of the edge where the moment map is maximal, in accordance with the McDuff–Tolman formula. Otherwise, if $1 < \mu \leq \frac 32$ , the leading term is given by the class of a point, showing that the quantum homology class (4.7) is not the image of an effective Hamiltonian action under the Seidel homomorphism.

4.4.3 Injectivity of the Seidel morphism for $\mu>3/2$

Proposition 4.7 Consider the circle actions $z_{0,1i}, i=2,3,4$ , $z_1,$ and $z_{1,4}$ , defined in Figure 7. If $\mu> \frac 32$ and $c_1=c_2=c_3=c_4=1/2$ , then the Seidel elements of these five circle actions generate a free subgroup of rank 5 in the group of invertible elements of the quantum homology.▪

Proof Using the notation of Section 3 for the quantum homology ring, the Seidel elements of these five circle actions are given by the following expressions:

(4.8) $$ \begin{align} \mathcal S (z_{0,12}) = b_{34}, \quad \mathcal S (z_{0,13}) = b_{24}, \quad \mathcal S (z_{0,14}) = b_{23}, \end{align} $$

(4.9) $$ \begin{align} \quad \ \ \ \mathcal S (z_1) = b_{12} + f_{34}, \quad \mathcal S (z_{1,4}) = \left( b_{12} + f_{34}+e_4 + \frac{t^{1-\mu}}{1-t^{1-\mu}}\right) (1-t^{1-\mu})t^{\frac{1}{6(1-2\mu)}}. \hspace{-1pc} \end{align} $$

In order to show they are linearly independent, we first consider a simplification of the quantum algebra, namely, we set the $b_{ij},$ for all $i,j$ , equal to an element b, the $f_{ij}$ to f, and $e_j$ to e. Then the quantum homology algebra becomes isomorphic to the $\Pi ^{\mathrm {univ}}$ -algebra

(4.10) $$ \begin{align} \Pi^{\mathrm{univ}} [f,b,e]/ I', \end{align} $$

where $I'$ is the ideal generated by

$$ \begin{gather*} (1) \ f^2 = 0, \qquad (2) \ b^2 =1, \qquad (3) \ f(b+1)=0, \\ (4) \ f \left( e + \frac{t^{1-\mu}}{1-t^{1-\mu}} \right) = 0, \qquad (5) \ b \left( e + \frac{t^{1-\mu}}{1-t^{1-\mu}} \right) = f+ e + \frac{t^{1-\mu}}{1-t^{1-\mu}}, \\ (6) \ e^2 = 2(b+f+e) \frac{t^{1-\mu}}{1-t^{1-\mu}} + 2\frac{t^{1-\mu}}{(1-t^{1-\mu})^2} + \frac{t^{2-2\mu}}{(1-t^{1-\mu})^2}. \end{gather*} $$

Moreover, it is clear that the Seidel elements simplify to

$$ \begin{gather*} \mathcal S(z_{0,12}) = \mathcal S (z_{0,13}) = \mathcal S (z_{0,14}) = b, \quad \mathcal S(z_1) = b +f, \\ \mbox{and} \quad \mathcal S(z_{1,4}) = \left( b + f + e + \frac{t^{1-\mu}}{1-t^{1-\mu}}\right)(1-t^{1-\mu})\, t^{\frac{1}{6(1-2\mu)}}. \end{gather*} $$

We postpone the proof of the next two lemmas to Appendix A.

Lemma 4.8 Consider the elements b, $b+f$ , and $\left ( b + f + e + \frac {t^{1-\mu }}{1-t^{1-\mu }}\right )(1-t^{1-\mu })\, t^{\frac {1}{6(1-2\mu )}}$ contained in the subgroup of invertible elements of the algebra $ \Pi ^{\mathrm {univ}} [f,b,e]/ I'$ . They are linearly independent, that is, if

$$ \begin{align*}b^\alpha \, (b+f)^\beta \, \left( b + f + e + \frac{t^{1-\mu}}{1-t^{1-\mu}}\right)^\gamma(1-t^{1-\mu})^\gamma \, t^{\frac{\gamma}{6(1-2\mu)}}=1, \quad \mbox{where} \quad \alpha, \beta, \gamma \in \mathbb{Z},\end{align*} $$

then $\alpha =\beta =\gamma =0$ .

Lemma 4.9 The Seidel elements $b_{34}$ , $b_{24}$ , and $b_{23}$ are linearly independent in the subgroup of invertible elements of the quantum homology ${QH}_4^\times ({M}_{\mu ,c_i=1/2})$ .

Now, putting together the two lemmas, it is easy to conclude that the five Seidel elements $\mathcal S (z_{0,12}), \mathcal S (z_{0,13}),$ $\mathcal S (z_{0,14}), \mathcal S (z_1)$ , and $ \mathcal S (z_{1,4})$ are linearly independent. If

$$ \begin{gather*} \mathcal S (z_{0,12})^{\alpha_1}\, \mathcal S (z_{0,13})^{\alpha_2}\, \mathcal S (z_{0,14})^{\alpha_3}\, \mathcal S (z_1)^\beta\, \mathcal S (z_{1,4})^\gamma = 1, \end{gather*} $$

then

$$ \begin{gather*} b_{34}^{\alpha_1}\ b_{24}^{\alpha_2}\ b_{23}^{\alpha_3}\ (b_{12} + f_{34})^\beta\ \left( b_{12} + f_{34}+e_4 + \frac{t^{1-\mu}}{1-t^{1-\mu}}\right)^\gamma(1-t^{1-\mu})^\gamma\, t^{\frac{\gamma}{6(1-2\mu)}}=1. \end{gather*} $$

Using the simplification above of the quantum homology algebra, one obtains

$$ \begin{align*}b ^{\alpha_1+\alpha_2+\alpha_3}\, (b+f)^\beta \, \left( b + f + e + \frac{t^{1-\mu}}{1-t^{1-\mu}}\right)^\gamma(1-t^{1-\mu})^\gamma \, t^{\frac{\gamma}{6(1-2\mu)}}=1.\end{align*} $$

From Lemma 4.8, we conclude that $\alpha _1+\alpha _2+\alpha _3=\beta =\gamma =0$ . Therefore,

$$ \begin{align*}b_{34}^{\alpha_1}\ b_{24}^{\alpha_2}\ b_{23}^{\alpha_3}=1,\end{align*} $$

and Lemma 4.9 implies that $\alpha _1=\alpha _2=\alpha _3=0$ , so the five Seidel elements are linearly independent.

Next, we study the action of $\pi _0(\operatorname {\mathrm{Symp}}_h(M,\omega ))$ on $\pi _1(\operatorname {\mathrm{Symp}}_0(M,\omega ))$ .

Proposition 4.10 If $(M,\omega )$ belongs to the edge $MA$ of the reduced symplectic cone, and if $\mu>3/2$ , then the Seidel homomorphism is injective and the action of $\pi _0(\operatorname {\mathrm{Symp}}_h(M,\omega ))$ on $\pi _1(\operatorname {\mathrm{Symp}}_0(M,\omega ))$ is trivial.▪

Proof The Seidel homomorphism factors through

Consider arbitrary lifts in $\pi _1(\operatorname {\mathrm{Symp}}_0(M,\omega ))\simeq \mathbb {Z}^5$ of the five actions. By Proposition 4.7, these lifts generate a free subgroup of rank 5 in $\pi _1(\operatorname {\mathrm{Symp}}_0(M,\omega ))\simeq \mathbb {Z}^5$ on which the Seidel homomorphism $\mathcal {S}$ is injective. Consequently, $\mathcal {S}$ is injective on the whole group $\pi _1(\operatorname {\mathrm{Symp}}_0(M,\omega ))$ . Since

is also of rank $5$ , the triviality of the action follows readily.

Corollary 4.11 Each labeled extended graph in Figures 5–7 corresponds to a well-defined homotopy class in $\pi _1(\operatorname {\mathrm{Symp}}_0(M,\omega ))$ .

We now prove Theorem 1.3.

Proof Consider the extended labeled graph $z_{0,1i}$ , $i=2,3,4$ , $z_1$ , and $z_{1,4}$ of Figure 7. It follows from Corollary 4.11 that these graphs define five elements of $\pi _1 (\operatorname {\mathrm{Symp}}_h({M}_{\mu ,c_i=1/2}))$ . By Proposition 4.7, these five elements are linearly independent. This proves the second statement of the theorem. The first statement follows immediately from Remark 4.3.

Note that the proof of Proposition 4.7 does not depend on the value of $\mu $ , in addition to the condition $\mu>1$ . It follows that the five quantum homology classes $b_{34}$ , $b_{24}$ , $b_{23}$ , $b_{12}+f_{34}$ , and

(4.11) $$ \begin{align} \left( b_{12} + f_{34}+e_4 + \frac{t^{1-\mu}}{1-t^{1-\mu}}\right)(1-t^{1-\mu})\, t^{\frac{1}{6(1-2\mu)}} \end{align} $$

generate a free subgroup of rank 5 in the group of invertible elements of the quantum homology, for every $\mu>1$ . This observation proves the following result.

Proposition 4.12 When $ 1 < \mu \leq \frac 32$ , the quantum class ( 4.11 ) is not contained in the $\mathbb {Q}$ -subspace spanned by circle actions inside the $\mathbb {Q}$ -vector space formed by Seidel elements.▪

4.5 Relations between Hamiltonian circle actions on ${M}_{\mu ,c_1,c_2,c_3,c_4}$

In this section, we prove our second main result. The main step is to obtain a classification of all circle actions in $\operatorname {\mathrm{Symp}}_h({M}_{\mu ,c_i=1/2})$ , which also shows that although we have more and more circle actions on ${M}_{\mu ,c_i=1/2}$ as we increase the value of $\mu $ , they do not give new generators in the fundamental group of the symplectomorphism group $\operatorname {\mathrm{Symp}}_h({M}_{\mu ,c_i=1/2})$ . We show this by describing relations between the loops $z_k$ , $z_{k,i}$ , $z_{k,ij}$ , $z_{k,ijl}$ , and $z_{k,1234}$ , described in Section 4.3, that come from embedding pairs of loops inside torus actions. The tools we use are Delzant’s classification of toric actions and Karshon’s classification of Hamiltonian circle actions. We always consider the actions in the generic case, that is, when $0 < c_4 < c_3 < c_2 < c_1 < c_i+c_j < 1 < \mu $ , with $i,j \in \{1,2,3,4\}$ , in order to obtain the relations between the loops. As explained in Section 4.4.1, these relations between actions induce relations between Seidel elements in the quantum ring associated with the generic symplectic form. These relations map to similar relations in the quantum homology ring of ${M}_{\mu ,c_i=1/2}$ . By injectivity of the Seidel homomorphism when $\mu>3/2$ , we deduce that these relations hold in $\pi _1(\operatorname {\mathrm{Symp}}({M}_{\mu ,c_i=1/2}))$ as well.

Consider the manifold ${M}_{\mu ,c_1,c_2,c_3,c_4}$ endowed with a toric action, which we denote by $T_k$ , such that the momentum polygon is given in Figure 15. In addition to the homology classes indicated in the figure, it should be clear that the classes $E_3-E_4$ and $E_4$ are also represented in the bottom edges. Projecting onto the x-axis and the y-axis, we obtain the graphs in Figure 16 of the actions $z_k$ and $w_k$ , respectively, whose momentum maps are the first and second coordinates of the momentum map of the action $T_k$ .

Figure 15:

Toric action $T_k$ .

Figure 16:

Graphs of the circle actions $z_k$ and $w_k$ , respectively.

Performing the $GL(2, \mathbb {Z})$ transformation represented by the matrix

$$ \begin{align*}\left (\!\begin{array}{cc} 1 & 0 \\ j & -1 \end{array} \!\right)\end{align*} $$

to the polygon of Figure 15 yields a new polygon representing the same toric manifold. This new polygon has vertices

$$ \begin{align*} & (1, j- \mu), \ (0, k-\mu), \ (0,0), \ (1-c_1, (j+k)(1-c_1)), \ (1-c_2, (j+k)(1-c_2)-c_1+c_2), \\ & (1-c_3, (j+k)(1-c_3)-c_1-c_2+2c_3), \ (1-c_4, (j+k)(1-c_4)-c_1-c_2-c_3+3c_4), \ \mbox{and} \\ & (1, j+k-c_1-c_2-c_3-c_4). \end{align*} $$

Then perform the $GL(2, \mathbb {Z})$ transformation represented by the matrix

$$ \begin{align*}\left (\!\begin{array}{cc} 1 & 0 \\ k & -1 \end{array} \!\right)\end{align*} $$

to the polygon of the toric action $T_j$ . It should be clear, looking at the coordinates of the two transformed polygons, that projecting both polygons onto the y-axis, we obtain the same graph, which implies that

(4.12) $$ \begin{align} jz_k -w_k =kz_j-w_j, \quad j,k \geq 1. \end{align} $$

Now, consider the toric action $T_0$ on ${M}_{\mu ,c_1,c_2,c_3,c_4}$ represented in the Delzant polygon of Figure 17. Consider also its projections, in Figure 18, to the x-axis and the y-axis representing circle actions that we denote by $(z_0, y_0)$ .

Figure 17:

Toric action $T_0$ .

Figure 18:

Graphs of circle actions $z_0$ and $y_0$ , respectively.

Again, performing the $GL(2, \mathbb {Z})$ transformation represented by the matrix

$$ \begin{align*}\left (\!\begin{array}{cc} 1 & 0 \\ -k & 1 \end{array} \!\right)\end{align*} $$

to the polygon of Figure 17 yields a new polygon representing the same toric manifold. This new polygon has vertices

$$ \begin{align*} & (0, \mu), (1, \mu-k), (0,0), (1-c_1, -k(1-c_1)), (1-c_2, -k(1-c_2)+c_1-c_2), \\ & (1-c_3, -k(1-c_3)+c_1+c_2-2c_3), (1-c_4, -k(1-c_4)+c_1+c_2+c_3-3c_4), \ \mbox{and} \\ & (1, -k+c_1+c_2+c_3+c_4). \end{align*} $$

Therefore, projecting this new polygon onto the y-axis, it is easy to check that we obtain the graph of the circle action $w_k$ , which means we have the following identification:

(4.13) $$ \begin{align} w_k = -kz_0+y_0, \quad k \geq 1. \end{align} $$

Combining equations (4.12) and (4.13) yields

$$ \begin{align*}jz_k +kz_0= kz_j +jz_0.\end{align*} $$

Finally, setting $j=1$ implies that

(4.14) $$ \begin{align} z_k=kz_1 +(1-k)z_0, \quad k \geq 0. \end{align} $$

Next, we use a similar argument in order to obtain more relations between other circle actions listed in Proposition 4.2. For that, we need to consider the toric action $T_{k,4}$ on ${M}_{\mu ,c_1,c_2,c_3,c_4}$ , represented in the momentum polytope in Figure 19. Note that the two edges without homology labels represent the classes, $E_2-E_3$ and $E_3$ .

Figure 19:

Toric action $T_{k,4}$ .

Projecting onto the x- and y-axes, we obtain the graphs in Figure 20 of the actions $(z_{k,4},w_{k,4})$ , respectively, whose momentum maps are the first and second coordinates of the momentum map of the action $T_{k,4}$ .

Figure 20:

Graphs of the circle actions $z_{k,4}$ and $w_{k,4}$ .

Using the same argument as before, that is, performing the $GL(2, \mathbb {Z})$ transformation represented by the following matrix

$$ \begin{align*}\left (\!\begin{array}{cc} 1 & 0 \\ j & -1 \end{array} \!\right)\end{align*} $$

on the polygon of Figure 19 yields a new polygon with vertices

$$ \begin{align*} & (1, j+k-c_1-c_2-c_3), (1, j- \mu), (0, k-\mu), (0,-c_4), \ (c_4, (j+k)c_4), (1-c_1, (j+k)(1-c_1)), \\ & (1-c_2, (j+k)(1-c_2)-c_1+c_2), \ \mbox{and} \ (1-c_3, (j+k)(1-c_3)-c_1-c_2+2c_3). \end{align*} $$

Interchanging the role of k and j and then projecting onto the y-axis, it follows that

(4.15) $$ \begin{align} jz_{k,4} -w_{k,4} =kz_{j,4}-w_{j,4}, \quad j,k \geq 1. \end{align} $$

Now, consider the toric action on ${M}_{\mu ,c_1,c_2,c_3,c_4}$ represented in the polygon of Figure 21. Consider also its projections, in Figure 22, to the x- and y-axes representing circle actions that we denote by $(z_{0,4}, y_{0,4})$ .

Figure 21:

Toric action $T_{0,4}$ .

Figure 22:

Graphs of the circle actions $z_{0,4}$ and $y_{0,4}$ , respectively.

In this case, performing the $GL(2, \mathbb {Z})$ transformation represented by the matrix

$$ \begin{align*}\left (\!\begin{array}{cc} 1 & 0 \\ -k & 1 \end{array} \!\right)\end{align*} $$

to the polygon of Figure 21 yields a new polygon whose vertices are

$$ \begin{align*} & (0, \mu), (1, \mu-k), (0,c_4), (c_4, -kc_4), (1-c_1, -k(1-c_1)), (1-c_2, -k(1-c_2)+c_1-c_2), \\ & (1-c_3, -k(1-c_3)+c_1+c_2-2c_3), \ \mbox{and} \ (1, -k+c_1+c_2+c_3). \end{align*} $$

Therefore, projecting this new polygon onto the y-axis, it is clear that one obtains the graph of the circle action $w_{k,4}$ , which means we have the following relation:

(4.16) $$ \begin{align} w_k = -kz_{0,4}+y_{0,4}, \quad k \geq 1. \end{align} $$

Combining equations (4.15) and (4.16) yields

$$ \begin{align*}jz_{k,4} +kz_{0,4}= kz_{j,4} +jz_{0,4},\end{align*} $$

and setting $j=1$ implies that

$$ \begin{align*} z_{k,4}=kz_{1,4} +(1-k)z_{0,4}, \quad k \geq 0. \end{align*} $$

Therefore, it is clear that, in fact, we have the following identifications:

(4.17) $$ \begin{align} z_{k,i}=kz_{1,i} +(1-k)z_{0,i}, \quad k \geq 0,\quad i=1,2,3,4. \end{align} $$

Using a similar argument three more times applied to the appropriate toric actions on the manifold ${M}_{\mu ,c_1,c_2,c_3,c_4}$ , it follows that the following identifications hold:

(4.18) $$ \begin{align} z_{k,ij}=kz_{1,ij} +(1-k)z_{0,ij}, \quad k \geq 0,\quad i,j=1,2,3,4, \end{align} $$

(4.19) $$ \begin{align} z_{k,ij\ell}=kz_{1,ij\ell} +(k-1)z_{0,m}, \ \ k \geq 0,\ \ i,j,\ell, m=1,2,3,4, \ \ \mbox{and all indices are distinct}; \end{align} $$

and

(4.20) $$ \begin{align} z_{k,1234}=kz_{1,1234} +(k-1)z_{0}, \quad k \geq 0. \end{align} $$

In what follows, we will often use the next proposition. The proof follows from techniques similar to the ones used to obtain the previous relations, so we postpone it to Appendix B.

Proposition 4.13 Consider the circle actions $z_0$ , $z_{0,i}, \ i=1,2,3,4$ , $z_{0,1i}, i=2,3,4$ , $z_1$ , and $z_{1,4}$ , defined in Figure 7 or 8 through their graphs, as elements of $\pi _1 (\operatorname {\mathrm{Symp}}_h({M}_{\mu ,c_1,c_2,c_3,c_4}))$ . Then the following identifications hold:

$$ \begin{align*} z_{0,1}&= z_1 -z_{1,4} + z_{0,14},& z_{0,2}&= z_1 -z_{1,4} -z_{0,13}, \\ z_{0,3}&= z_1 -z_{1,4} - z_{0,12},& z_{0,4}&= z_1 -z_{1,4} - z_{0,12}-z_{0,13}+ z_{0,14}, \end{align*} $$

and $z_{0}= 2z_1 -2z_{1,4} - z_{0,12}-z_{0,13}+ z_{0,14}$ .▪

Putting together Proposition 4.13 with equations (4.14) and (4.17)–(4.20), we then obtain the following result.

Proposition 4.14 Consider the circle actions $z_{0,1i}, i=2,3,4$ , $z_1$ , and $z_{1,4}$ , defined in Figure 7 or 8 through their graphs, as elements of $\pi _1 (\operatorname {\mathrm{Symp}}_h({M}_{\mu ,c_1,c_2,c_3,c_4}))$ . Let $t= z_{0,12}+z_{0,13}- z_{0,14}$ . Then, for $k \in \mathbb {Z}_{\geq 0}$ and $i,j=1,2,3,4$ with $i \neq j$ , we have the following identifications:

$$ \begin{align*} & z_k= (2-k) z_1 + (k-1)(2z_{1,4} +t), & &z_{k,ij}=2kz_{1,4}-kz_1 +kt + z_{0,ij}, \\ & z_{k,1}= (2k-1)z_{1,4} + (1-k)z_1 +kt +z_{0,14}, & &z_{k,124}= (2k+1)z_{1,4}-(k+1)z_1 +kt + z_{0,12}, \\ & z_{k,2}= (2k-1)z_{1,4} + (1-k)z_1 +kt -z_{0,13}, & &z_{k,134}= (2k+1)z_{1,4}-(k+1)z_1 +kt + z_{0,13}, \\ & z_{k,3}= (2k-1)z_{1,4} + (1-k)z_1 +kt -z_{0,12}, & &z_{k,234}= (2k+1)z_{1,4}-(k+1)z_1 +kt - z_{0,14}, \\ & z_{k,4}= (2k-1)z_{1,4} + (1-k)z_1 +(k-1)t, & &z_{k,123}= (2k+1)z_{1,4}-(k+1)z_1 +(k+1)t, \end{align*} $$

and $z_{k,1234}=(2k+2)z_{1,4}-(k+2)z_1 +(k+1)t $ .▪

Remark 4.15 It is clear from the definition of the circle actions $z_{0,ij}$ that

$$ \begin{align*}z_{0,ij}=-z_{0,kl}.\end{align*} $$

Thus, we get

$$ \begin{align*}z_{23}= -z_{0,14}, \quad z_{24}= -z_{0,13}, \quad z_{34}= -z_{0,12},\end{align*} $$

which allows us to write $z_{k,ij}$ completely in terms of the chosen generators.

We now prove Theorem 1.5.