Vanishing of Rabinowitz Floer homology on negative line bundles

Abstract

Following Frauenfelder (Rabinowitz action functional on very negative line bundles, Habilitationsschrift, Munich/München, 2008), Albers and Frauenfelder (Bubbles and onis, 2014. arXiv:1412.4360) we construct Rabinowitz Floer homology for negative line bundles over symplectic manifolds and prove a vanishing result. Ritter (Adv Math 262:1035–1106, 2014) showed that symplectic homology of these spaces does not vanish, in general. Thus, the theorem $\mathrm {SH}=0\Leftrightarrow \mathrm {RFH}=0$ (Ritter in J Topol 6(2):391–489, 2013), does not extend beyond the symplectically aspherical situation. We give a conjectural explanation in terms of the Cieliebak–Frauenfelder–Oancea long exact sequence Cieliebak et al. (Ann Sci Éc Norm Supér (4) 43(6):957–1015, 2010).

1 Introduction

Negative line bundles give rise to a rather special class of contact manifolds which nevertheless contains many interesting examples. They arise at many places in modern contact and symplectic geometry such as Givental’s nonlinear Maslov index [17] and more generally contact rigidity [7, 8, 12, 25] etc.

Let us be more specific. We choose a closed connected symplectic manifold $(M,\omega )$ with integral symplectic form $[\omega ]\in \mathrm {H}^2(M,\mathbb {Z})$. We denote by $\wp :\Sigma \rightarrow M$ the principal $S^1$-bundle and by $\wp :E\rightarrow M$ the associated complex line bundle with first Chern class $c_1^E=-[\omega ]$. We refer to these bundles as negative line bundles. There exists an $S^1$-invariant 1-form $\alpha$ on $\Sigma$, and hence $E{\setminus } M$, with the property

$$\begin{aligned} d\alpha =\wp ^*\omega \end{aligned}$$

(1.1)

which is a contact form on $\Sigma$. For more details we refer to [16, Section 7.2]. If we denote by r the radial coordinate on E then the 2-form

$$\begin{aligned} \Omega :=d\big (\pi r^2\alpha \big )+\wp ^*\omega =2\pi rdr\wedge \alpha +\big (\pi r^2+1\big )\wp ^*\omega \end{aligned}$$

(1.2)

is a symplectic form on E. Throughout this article we make the assumption that $(E,\Omega )$ is semi-positive, see [20, Definition 6.4.1] and page 5 for an equivalent formulation.

Theorem 1.1

The Rabinowitz Floer homology $\mathrm {RFH}(\Sigma ,E)$ is well-defined.

In many situations we are able to prove the following vanishing result.

Theorem 1.2

We assume that one of the following is satisfied.

• (1) $(M,\omega )$ is symplectically aspherical: $\omega \big (\pi _2(M)\big )=0$.

• (2) There exists a constant $c\in \mathbb {Z}$ such that

  $$\begin{aligned} c_1^{TM}=c\omega :\pi _2(M)\rightarrow \mathbb {Z}\end{aligned}$$

  (1.3)

  and either $c\ge 1$ or $2c\nu \le -\mathrm{dim }\, M$ where $\nu \in \mathbb {Z}_{>0}$ is defined by $\omega \big (\pi _2(M)\big )=\nu \mathbb {Z}$.

Then $(E,\Omega )$ is semi-positive and Rabinowitz Floer homology vanishes

$$\begin{aligned} \mathrm {RFH}(\Sigma ,E)=0\;. \end{aligned}$$

(1.4)

Contact manifolds such as spheres, projective spaces etc. come from negative line bundles. In fact, the Boothby–Wang theorem [16, Theorem 7.2.5] characterizes these contact manifolds as those whose Reeb flow is periodic with all Reeb orbits having the same minimal period. On the sphere this corresponds to the Hopf fibration.

Remark 1.3

• (1) If $c=0$, i.e. $c_1^{TM}\big (\pi _2(M)\big )=0$, then $c_1^{TE}=-\wp ^*\omega :\pi _2(E)\rightarrow \mathbb {Z}$. Therefore, $(E,\Omega )$ is semi-positive if and only if $\nu \ge \tfrac{1}{2}\dim M-1$ or $\omega \big (\pi _2(M)\big )=0$, see Lemma 2.6. If $(E,\Omega )$ is semi-positive then $\mathrm {RFH}(\Sigma ,E)=0$ still holds.

• (2) It is worth pointing out that $\Sigma$ is not displaceable inside E since the zero-section $M\subset E$ is not even topologically displaceable. To our knowledge this is the first vanishing result for $\mathrm {RFH}$ result which is not due to a displaceability phenomenon, see also Ritter [23, Remark on p. 1044].

• (3) There are very few direct computations of $\mathrm {RFH}$. To our knowledge, the vanishing result in the displaceable case  [9], the computation for cotangent bundles  [5], and the computation for Brieskorn spheres  [13] are the only ones. The long exact sequence, [11] leads often to computational results if the symplectic homology and the connecting maps are known. The latter rarely happens, though.

• (4) Rabinowitz Floer homology, first constructed by Cieliebak and Frauenfelder in [9], is an invariant of contact type hypersurfaces in symplectic manifolds. It turned out to be an efficient tool for studying questions in symplectic topology and dynamics, see [2].

  In [15] Frauenfelder studied the Rabinowitz Floer homology of negative line bundles under the additional assumption of the line bundle being very negative. The implication of the latter is the generic absence of holomorphic spheres. Amongst many other things he established $C^\infty _{loc}$-compactness results, cf. [15, Theorem B]. Even though Rabinowitz Floer homology is not fully constructed in [15] all ingredients are basically contained therein, see also [3].

  The purpose of this article is to complete and extend the construction of Rabinowitz Floer homology to negative line bundles in the presence of holomorphic spheres under a semi-positivity assumption. In particular, we prove a transversality result made necessary due to the use of a rather restricted class of almost complex structures.

  It is worth pointing out that this is the first instance where Rabinowitz Floer homology is constructed in the presence of holomorphic spheres. Holomorphic spheres are a source for interesting symplectic topology and big technical problems at the same time. The latter is the reason we require semi-positivity.

• (5) The main new contribution is Theorem 1.2: Rabinowitz Floer homology vanishes in many cases. This should be contrasted with Ritter’s result that symplectic homology does not necessarily vanishes, see [23]. Thus, the theorem $\mathrm {SH}=0\Leftrightarrow \mathrm {RFH}=0$, [22], does not extend beyond the symplectically aspherical situation. We give a conjectural explanation of this in Sect.  4 below.

  For very negative line bundles Ritter in [23, Theorem 8] proved vanishing of symplectic homology. If we assume in addition that $c_1^{TM}=c\omega :\pi _2(M)\rightarrow \mathbb {Z}$ the Rabinowitz Floer homology vanishes according to Theorem 1.2 as well. The conjectural picture from Sect. 4 nicely relates these results.

2 Rabinowitz Floer homology and Hamiltonian Floer homology

2.1 Preliminaries

Let $\wp :E\rightarrow M$ be as described above. We denote by $\mathscr {L}(E)$ the component of contractible loops of the free loop space of E. Moreover, we denote by $\widetilde{\mathscr {L}}(E)$ the covering space of $\mathscr {L}(E)$ with deck transformations given by

$$\begin{aligned} \Gamma _E:=\frac{\pi _2(E)}{\ker \Omega \cap \ker c_1^{TE}}\;. \end{aligned}$$

(2.1)

We write elements in $\widetilde{\mathscr {L}}(E)$ as $[u,\bar{u}]$, where $u:S^1\rightarrow E$ and $\bar{u}:D^2\rightarrow E$ is a capping disk for u, i.e. $\bar{u}|_{S^1}=u$. Moreover, pairs $(u,\bar{u})$ and $(v,\bar{v})$ are equivalent if $u=v$ and $\Omega (-\bar{u}\#\bar{v})=c_1^{TE}(-\bar{u}\#\bar{v})=0$, where $-\bar{u}\#\bar{v}$ is the sphere formed by $\bar{u}$ with orientation reversed and $\bar{v}$. The expression $[u,\bar{u}]$ denotes the corresponding equivalence class. Analogously we define

$$\begin{aligned} \Gamma _M:=\frac{\pi _2(M)}{\ker \omega \cap \ker c_1^{TM}}\;, \end{aligned}$$

(2.2)

$\mathscr {L}(M)$ and $\widetilde{\mathscr {L}}(M)$. By definition of E we have

$$\begin{aligned} c_1^{TE}=\wp ^*\big (c_1^{TM}+c_1^E\big )\;. \end{aligned}$$

(2.3)

Remark 2.1

We point out that under assumption (1.3), i.e. $c_1^{TM}=c\omega :\pi _2(M)\rightarrow \mathbb {Z}$, we have $\ker \omega \cap \ker c_1^{TM}=\ker \omega$ and $\ker \Omega \cap \ker c_1^{TE}=\ker \Omega$. If we instead assume $\omega \big (\pi _2(M)\big )=0$ then $c_1^{TE}=\wp ^* c_1^{TM}:\pi _2(E)\rightarrow \mathbb {Z}$. In particular, since $\pi _2(E)\cong \pi _2(M)$ via $\wp _*$, we can identify in both cases $\Gamma _E\cong \Gamma _M$.

For $\tau >0$ we denote by $\mu _\tau :E\rightarrow \mathbb {R}$ the function $\mu _\tau =\pi r^2-\tau$ where as above r denotes the radial coordinate on E. We point out that along $\Sigma _\tau :=\{\mu _\tau =0\}$ the Hamiltonian vector field $X_{\mu _\tau }$ of $\mu _\tau$ agrees with the Reeb vector field R associated to the contact form $\alpha$. In particular, we use the convention $\Omega (X_{\mu _\tau },\cdot )=-d\mu _\tau$. The Rabinowitz action functional $\mathscr {A}^\tau$ is defined as

$$\begin{aligned} \begin{aligned} \mathscr {A}^\tau :\widetilde{\mathscr {L}}(E)\times \mathbb {R}&\rightarrow \mathbb {R}\\ \big ([u,\bar{u}],\eta \big )&\mapsto \int _{D^2}\bar{u}^*\Omega -\eta \int _0^1 \mu _\tau \big (u(t)\big ) dt . \end{aligned}\end{aligned}$$

(2.4)

In the article [9] Cieliebak and Frauenfelder developed a Floer theory for this functional in a slightly simpler set-up and for non-degenerate contact forms. The current set-up has been developed and studied Frauenfelder in [15] for very negative line bundles, see also [3].

In our setting the contact form is Morse–Bott non-degenerate. A general Morse–Bott approach to Rabinowitz Floer homology is currently not available in the literature. Instead of perturbing the contact form we choose the following perturbation. We fix $f:M\rightarrow \mathbb {R}$ and set

$$\begin{aligned} F:=\big (\pi r^2+1\big )f\circ \wp :E\rightarrow \mathbb {R}. \end{aligned}$$

(2.5)

The perturbed Rabinowitz action functional is

$$\begin{aligned} \begin{aligned} \mathscr {A}_f^\tau :\widetilde{\mathscr {L}}(E)\times \mathbb {R}&\rightarrow \mathbb {R}\\ \big ([u,\bar{u}],\eta \big )&\mapsto \int _{D^2}\bar{u}^*\Omega -\eta \int _0^1 \mu _\tau \big (u(t)\big ) dt-\int _0^1F\big (u(t)\big )dt \;. \end{aligned}\end{aligned}$$

(2.6)

Critical points of $\mathscr {A}_{f=0}^\tau$ correspond to capped Reeb orbits traversed in forward and backward direction and, in addition, to constant loops contained in $\Sigma _\tau$ together with cappings. In Lemma 2.3 we show that for $C^2$-small Morse functions f the critical points of $\mathscr {A}_f^\tau$ correspond to capped Reeb orbits which lie via $\wp :E\rightarrow M$ over $\mathrm {Crit}(f)$. The functional $\mathscr {A}_f^\tau$ is still Morse–Bott due to the remaining $S^1$-symmetry. This can be dealt with as in the article by Bourgeois-Oancea [6].

Remark 2.2

We split the tangent bundle $TE\cong V\oplus H$ in vertical resp. horizontal subspaces V resp. H according to $\alpha$. In particular, $\wp _*:(H,d\alpha )\mathop {\longrightarrow }\limits ^{\cong }(TM,\omega )$ is an isomorphism of symplectic vector bundles and V is spanned by the Reeb vector field R and the radial vector field $r\partial _r$.

Lemma 2.3

If $f:M\rightarrow \mathbb {R}$ is $C^2$-small then $\big ([u,\bar{u}],\eta \big )$ is a critical point of $\mathscr {A}_f^\tau$ if and only if the following equations are satisfied.

$$\begin{aligned} \left\{ \begin{aligned}&\pi r(u)^2=\tau \\&q:=\wp (u)\in \mathrm {Crit}(f)\\&\dot{u}=\big (\eta +f(q)\big )R(u)\\ \end{aligned} \right. \end{aligned}$$

(2.7)

In particular, necessarily $\eta +f(q)\in \mathbb {Z}$ and $u\subset \Sigma _\tau$ is a $(\eta +f(q))$-fold cover of the underlying simple periodic orbit.

Proof

The critical point equation for $\mathscr {A}_f^\tau$ is

$$\begin{aligned} \left\{ \begin{aligned} \;&\dot{u}=\eta X_{\mu _\tau }(u)+X_F(u)\\&\int _0^1\mu _\tau (u)dt=0\;. \end{aligned} \right. \end{aligned}$$

(2.8)

The Hamiltonian vector field of F is $X_F=(f\circ \wp ) X_{\mu _\tau }+X_f^h$ where $X_f^h$ is the horizontal lift of $X_f$, i.e. $X_f^h\in H$ and $\wp _*(X_f^h)=X_f$. Indeed,

$$\begin{aligned} \begin{aligned} \Omega (X_F,\cdot )&=\Big (2\pi rdr\wedge \alpha +\big (\pi r^2+1\big )\wp ^*\omega \Big )\Big ((f\circ \wp ) X_{\mu _\tau }+X_f^h,\cdot \Big )\\&=-2\pi r(f\circ \wp )+\big (\pi r^2+1\big )\wp ^*\big (\omega (X_f,\cdot )\big )\\&=-2\pi r(f\circ \wp )-\big (\pi r^2+1\big )\wp ^*df\\&=-d\big [\big (\pi r^2+1\big )f\circ \wp \big ]\\&=-dF\;. \end{aligned}\end{aligned}$$

(2.9)

We point out that $\Omega (X_{\mu _\tau },X_F)=dF(X_{\mu _\tau })=0$ since $\wp _*(X_{\mu _\tau })=0=dr(X_{\mu _\tau })$. Therefore the critical point Eq. (2.8) simplifies to

$$\begin{aligned} \left\{ \begin{aligned} \;&\dot{u}=\big (\eta +f(\wp (u))\big )X_{\mu _\tau }(u)+X_f^h(u)\\&\mu _\tau \big (u(t)\big )=0\quad \forall t\in S^1\;. \end{aligned} \right. \end{aligned}$$

(2.10)

The last equation translates into r(u(t)) being constant and $\pi r(u)^2=\tau$. The critical point equation together with $\wp _*(X_{\mu _\tau })=0$ implies that $\wp (u)$ is a 1-periodic solution of $X_f$ in M. Now, if the $C^2$-norm of f is sufficiently small the only 1-periodic solutions of $X_f$ are the critical points of f, see [19, p. 185]. Thus, $q:=\wp (u)\in \mathrm {Crit}(f)$ and u corresponds to a $(\eta +f(q))$-periodic orbit of R on $\Sigma _\tau$. This implies that $\eta +f(q)\in \mathbb {Z}$ due to our convention $S^1=\mathbb {R}/\mathbb {Z}$. $\square$

Remark 2.4

To summarize critical points of $\mathscr {A}_f^\tau$ correspond to all Reeb orbits over $\mathrm {Crit}(f)$ together with cappings. More precisely, all forward (i.e. $\eta +f(q)>0$) and backward (i.e. $\eta +f(q)<0$) iterations and also the “constants” (i.e. $\eta +f(q)=0$) together with cappings.

Convention 2.5

• From now on we assume that the Morse function $f:M\rightarrow \mathbb {R}$ is chosen $C^2$-small so that Lemma 2.3 applies.

• Every simple periodic Reeb orbit $v\subset \Sigma _\tau$ has a capping by its fiber disk $d_v\subset E$ and correspondingly the n-fold cover $v^n$ has $d_v^n$ as capping disk for $n\in \mathbb {Z}{\setminus }\{0\}$. Every non-constant critical point $\big ([u,\bar{u}],\eta \big )$ can be expressed in the form $u=v^n$ and $\bar{u}=d_v^n\#A$ for some $A\in \Gamma _E$. If u is a constant critical point, the capping disk $\bar{u}$ can be thought of as a sphere $A\in \Gamma _E$. We are going to adopt the notation $[u,\bar{u}]=[u,A]$.

Using Lemma 2.3 we compute the action value for a critical point $\big ([u,\bar{u}],\eta \big )=\big ([v^n,A],\eta \big )$ with $q=\wp (u)$.

$$\begin{aligned} \mathscr {A}_f^\tau \big ([v^n,A],\eta \big )= & {} \int _{D^2}\big (d_v^n\big )^*\Omega +\omega (A)-\eta \int _0^1 \underbrace{\mu _\tau (v^n)}_{=0} dt-\int _0^1F(v^n)dt\nonumber \\= & {} \int _{D^2}\big (d_v^n\big )^*\big [d\big (\pi r^2\alpha \big )\big ]-\int _0^1\big (\pi r^2+1\big )f\circ \wp (v^n)dt+\omega (A)\nonumber \\= & {} \int _{S^1}(v^n)^*\big (\pi r^2\alpha \big )-(\tau +1)f(q)+\omega (A)\\= & {} \int _{S^1}\tau \alpha \big ((\eta +f(q))R(v^n)\big )-(\tau +1)f(q)+\omega (A)\nonumber \\= & {} \tau \big (\eta +f(q)\big )-(\tau +1)f(q)+\omega (A)\nonumber \\= & {} \tau \eta +\omega (A)-f(q)\nonumber \\= & {} \tau n+\omega (A)-(\tau +1)f(q),\nonumber \end{aligned}$$

(2.11)

where we use $n=\eta +f(q)$ and $\Omega =\omega :\Gamma _E\cong \Gamma _M\rightarrow \mathbb {Z}$.

Next we explain how to define Floer homology for $\mathscr {A}_f^\tau$. This mainly follows the lines of Frauenfelder [15] and Albers and Frauenfelder [3]. We assume throughout that $(E,\Omega )$ is semi-positive. According to McDuff and Salamon [20, Exercise 6.4.3] the symplectic manifold $(E,\Omega )$ is semi-positive if and only if

• $(E,\Omega )$ is symplectically aspherical,

• $(E,\Omega )$ is monotone,

• $c_1^{TE}:\pi _2(E)\rightarrow \mathbb {Z}$ vanishes,

• the minimal Chern number $N_E$ of E satisfies $N_E\ge \tfrac{1}{2}\dim E-2$.

Since $\pi _2(E)\cong \pi _2(M)$ via $\wp _*$ the first condition is equivalent to $(M,\omega )$ being symplectically aspherical which is condition (1) in Theorem 1.2.

If we assume that there exists a constant $c\in \mathbb {Z}$ such that $c_1^{TM}=c\omega :\pi _2(M)\rightarrow \mathbb {Z}$ then

$$\begin{aligned} c_1^{TE}=\wp ^*\big (c_1^{TM}+c_1^E\big )=(c-1)\wp ^*\omega . \end{aligned}$$

(2.12)

Thus, if $c>1$ the symplectic manifold $(E,\Omega )$ is monotone and for $c=1$ we have $c_1^{TE}=0$ on $\pi _2(E)$. Furthermore, if we denote by $\nu \in \mathbb {Z}_{\ge 0}$ the generator of $\omega \big (\pi _2(M)\big )=\nu \mathbb {Z}$ then the minimal Chern number $N_E$ of E is

$$\begin{aligned} N_E=|c-1|\nu . \end{aligned}$$

(2.13)

Thus, we proved the following Lemma.

Lemma 2.6

The symplectic manifold $(E,\Omega )$ is semi-positive if

• $(M,\omega )$ is symplectically aspherical or

• $c_1^{TM}=c\omega :\pi _2(M)\rightarrow \mathbb {Z}$ with $c\ge 1$ or $N_E=|c-1|\nu \ge \tfrac{1}{2}\dim E-2$.

In the following Lemma we use the notation of Convention 2.5. For $\big ([u,\bar{u}],\eta \big )=\big ([u,A],\eta \big )\in \mathrm {Crit}(\mathscr {A}^\tau _f)$, we denote by $\mu _{\mathrm {CZ}}^E(u,\bar{u})\equiv \mu _{\mathrm {CZ}}^E(u,A)$ the Conley–Zehnder index of u with respect to the capping disk $\bar{u}$. We refer to [24] for a thorough discussion of the Conley–Zehnder index.

Lemma 2.7

The Conley–Zehnder index of a n-fold cover $v^n$ with its fiber disk $d_v^n$ is

$$\begin{aligned} \mu _{\mathrm {CZ}}^E(v^n,d_v^n)=2n. \end{aligned}$$

(2.14)

More generally, for any capping $\bar{u}=d_v^n\#A$ of $u=v^n$,

$$\begin{aligned} \mu _{\mathrm {CZ}}^E(v^n,d_v^n\#A)=2n+2c_1^{TE}(A). \end{aligned}$$

(2.15)

If $(M,\omega )$ is symplectically aspherical then all iterates $v^n$ are non-contractible inside $\Sigma$. Otherwise, the first iterate of v, which is contractible in $\Sigma$, is the orbit $v^{\nu }$.

If we assume $c_1^{TM}=c\omega$ then the Conley–Zehnder index of $v^n$ for $n\in \nu \mathbb {Z}{\setminus }\{0\}$ with respect to a capping disk $\bar{\nu }$ contained entirely in $\Sigma$ is

$$\begin{aligned} \mu _{\mathrm {CZ}}^E(u,\bar{\nu })=2cn. \end{aligned}$$

(2.16)

Proof

Since the linearized map of the Reeb flow is the identity in horizontal directions, the first assertion follows from the corresponding computation for $S^1\subset \mathbb {C}$. The relevant bit of the homotopy long exact sequence of the $S^1$-bundle $\Sigma \rightarrow M$ is

$$\begin{aligned} \cdots \longrightarrow \pi _2(M)\mathop {\longrightarrow }\limits ^{\delta }\pi _1(S^1)\mathop {\longrightarrow }\limits ^{i_*}\pi _1(\Sigma )\longrightarrow \cdots \;. \end{aligned}$$

(2.17)

If we identify $\pi _1(S^1)\cong \mathbb {Z}$ then $i_*(k)=[v^k]$ and $\delta (s)=-\omega (s)$ with respect to the homomorphism $\omega :\pi _2(M)\rightarrow \mathbb {Z}$. Thus, if $(M,\omega )$ is symplectically aspherical then all iterates $v^k$ are non-contractible in $\Sigma$. Otherwise $\omega \big (\pi _2(M)\big )=\nu \mathbb {Z}$ with $\nu >0$ and therefore the first iterate of v which is contractible in $\Sigma$ is the orbit $v^{\nu }$.

Now we assume that $c_1^{TM}=c\omega$ and recall that $\bar{\nu }$ is a capping which is entirely contained inside $\Sigma$.

$$\begin{aligned} \begin{aligned} \mu _{\mathrm {CZ}}^E(v^n,\bar{\nu })&\mathop {=}\limits ^{\phantom {(*)}}\mu _{\mathrm {CZ}}^E\Big (v^n,d_v^n\#\big (\underbrace{\bar{\nu }\#-d_v^n}_{\in \Gamma _E}\big )\Big )\\&\mathop {=}\limits ^{\phantom {(*)}}\mu _{\mathrm {CZ}}^E\big (v^n,d_v^n\big )+2c_1^{TE}\big (\bar{\nu }\#-d_v^n\big )\\&\mathop {=}\limits ^{\phantom {(*)}}2n+2(c-1)\wp ^*\omega \big (\bar{\nu }\#-d_v^n\big )\\&\mathop {=}\limits ^{\phantom {(*)}}2n+2(c-1)\bigg [\int _{\bar{\nu }}\wp ^*\omega -\underbrace{\int _{d_v^n}\wp ^*\omega }_{=0}\bigg ]\\&\mathop {=}\limits ^{(*)}2n+2(c-1)\int _{\bar{\nu }}d\alpha \\&\mathop {=}\limits ^{\phantom {(*)}}2n+2(c-1)\int _{v^n}\alpha \\&\mathop {=}\limits ^{\phantom {(*)}}2n+2(c-1)n\\&\mathop {=}\limits ^{\phantom {(*)}}2cn \end{aligned}\end{aligned}$$

(2.18)

where we used in $(*)$ that the disk $\bar{\nu }$ is contained inside $\Sigma$. $\square$

Definition 2.8

We point out that critical points of $\mathscr {A}_f^\tau$ are $S^1$-families, cf. Lemma 2.3. We choose a perfect Morse function $h:\mathrm {Crit}(\mathscr {A}_f^\tau )\rightarrow \mathbb {R}$ such that every critical manifold $S^1\cdot \big ([u,A],\eta \big )\subset \mathrm {Crit}(\mathscr {A}_f^\tau )$ gives rise to two critical points of h which we denote by $\big ([u,A]^\pm ,\eta \big )$ according to the maximum resp. minimum of h on $S^1\cdot \big ([u,A],\eta \big )$. We define the index of a critical point by

$$\begin{aligned} \mu \big ([u,A]^\pm ,\eta \big ):=\mu _{\mathrm {CZ}}^E(u,A)-\mu _{\mathrm {Morse}}(\wp (u),f)+\tfrac{1}{2} \dim M\pm \tfrac{1}{2} \in \tfrac{1}{2} +\mathbb {Z}\;, \end{aligned}$$

(2.19)

where $\mu _{\mathrm {Morse}}(\wp (u),f)$ is the Morse index of $\wp (u)\in \mathrm {Crit}(f)$. In case that u is a constant critical point we define $\mu _{\mathrm {CZ}}^E(u,A=0):=0$. We set

$$\begin{aligned} \mathfrak {C}:=\mathrm {Crit}(h)=\left\{ \big ([u,A]^\pm ,\eta \big )\right\} \subset \mathrm {Crit}(\mathscr {A}_f^\tau ) \end{aligned}$$

(2.20)

and

$$\begin{aligned} \mathfrak {C}_k:=\left\{ \big ([u,A]^\bullet ,\eta \big )\in \mathfrak {C}\,\bigr |\, \mu \big ([u,A]^\bullet ,\eta )\big )=k\right\} \;. \end{aligned}$$

(2.21)

Here, $\bullet$ indicates some choice of ±.

In order to define Rabinowitz Floer homology and to prove the vanishing result we rely on a fairly special class of almost complex structures which we describe next. In the next subsection we prove that this class is big enough to prove the necessary transversality results. We recall that we split the tangent bundle $TE\cong V\oplus H$ in vertical resp. horizontal subspaces V resp. H, see Remark 2.2. Let us abbreviate by

$$\begin{aligned} \mathfrak {j}:=\big \{j\in \Gamma (S^1\times M, \mathrm {Aut}(TM))\mid j_t:=j(t,\cdot ) \text { is an } \omega \text {-compatible almost complex structure}\big \} \end{aligned}$$

the space of $S^1$-families of compatible almost complex structures on $(M,\omega )$. Next we fix disjoint open balls around each point in $\mathrm {Crit}(f)$. The union of these balls is denoted by $\mathcal {U}$. For a fixed $j\in \mathfrak {j}$ we denote by $\mathcal {B}(j)$ the set of $B\in \Gamma _0(S^1\times E, \mathrm {L}(H,V))$ where $B_t:=B(t,\cdot )$ satisfies

$$\begin{aligned} iB_t+B_tj_t=0\;\forall t\in S^1\quad \text { and }\quad B_t(e)=0\;\forall e\in \wp ^{-1}(\mathcal {U})\;. \end{aligned}$$

(2.22)

Here the subscript 0 indicates compact support and $\mathrm {L}(H,V)$ is the space of linear maps. To describe the Floer equation we will choose a $S^1$-family $J_t$ of almost complex structures on E of the form

$$\begin{aligned} J_t=\begin{pmatrix} i &{}\quad B_t\\ 0 &{}\quad j_t \end{pmatrix}\;. \end{aligned}$$

(2.23)

The matrix representation refers to the splitting $TE\cong V\oplus H$. Moreover, $j\in \mathfrak {j}$ and $B\in \mathcal {B}(j)$ and i is the standard complex structure on $V_e\cong \mathbb {C}$, $e\in E$. We point out that $J_t$ is not $\Omega$-compatible. But, since $\begin{pmatrix} i &{} 0\\ 0 &{} j_t \end{pmatrix}$ is tame (even compatible) and B has compact support, the almost complex structure $J_t$ is $\Omega$-tame for sufficiently small $B_t$. We denote by $\mathcal {B}^T(j)\subset \mathcal {B}(j)$ the non-empty open convex subset consisting of those $B\in \mathcal {B}(j)$ for which the corresponding $J_t$ is tame.

We use $J_t$ to introduce a bilinear form $\mathfrak {m}$ on $T\big (\widetilde{\mathscr {L}}_E\times \mathbb {R}\big )$ as follows. For $(\hat{u}_1,\hat{\eta }_1),(\hat{u}_2,\hat{\eta }_2)\in T_{([u,A],\eta )}\big (\widetilde{\mathscr {L}}_E\times \mathbb {R}\big )=\Gamma (S^1,u^*TE)\times \mathbb {R}$ we set

$$\begin{aligned} \mathfrak {m}\big ((\hat{u}_1,\hat{\eta }_1),(\hat{u}_2,\hat{\eta }_2)\big ):=-\int _0^1\Omega \Big (J_t\big (u(t)\big )\hat{u}_1(t),\hat{u}_2(t)\Big )dt+\hat{\eta }_1\hat{\eta }_2\;. \end{aligned}$$

(2.24)

The bilinear form $\mathfrak {m}$ is not symmetric but positive definite since J is tame. Therefore we can define the vector field $\nabla \mathscr {A}_f^\tau (w)$ at $w=\big ([u,A],\eta \big )$ implicitly by

$$\begin{aligned} d\mathscr {A}_f^\tau (w)\hat{w}=\mathfrak {m}\big (\nabla \mathscr {A}_f^\tau (w),\hat{w}\big )\quad \forall \hat{w}\in T_w\big (\widetilde{\mathscr {L}}_E\times \mathbb {R}\big )\;. \end{aligned}$$

(2.25)

An explicit expression is

$$\begin{aligned} \nabla \mathscr {A}_f^\tau (w)= \begin{pmatrix} -J_t(u)\big (\partial _t u-\eta X_{\mu _\tau }(u)- X_F(u)\big )\\ \displaystyle -\int _0^1\mu _\tau (u)dt \end{pmatrix}\;. \end{aligned}$$

(2.26)

$\nabla \mathscr {A}_f^\tau$ is a gradient-like vector field for $\mathscr {A}_f^\tau$ since J is tame and B vanishes near critical points of $\mathscr {A}_f^\tau$: $B_t(e)=0$ for all $e\in \wp ^{-1}(\mathcal {U})$. Indeed, $d\mathscr {A}_f^\tau (w)\nabla \mathscr {A}_f^\tau (w)=\mathfrak {m}\big (\nabla \mathscr {A}_f^\tau (w),\nabla \mathscr {A}_f^\tau (w)\big )\ge 0$ with equality if and only if $w\in \mathrm {Crit}(\mathscr {A}_f^\tau )$. Moreover, $\mathfrak {m}$ is an inner product near critical points.

To construct Floer homology for $\mathscr {A}_f^\tau$ we study solutions $w=(u,\eta )\in C^\infty (\mathbb {R}\times S^1,E)\times C^\infty (\mathbb {R},\mathbb {R})$ to the Floer equations corresponding to positive gradient flow of $\mathscr {A}_f^\tau$

$$\begin{aligned} \left\{ \begin{aligned}&\partial _su+J_t(u)\big (\partial _tu-\eta X_{\mu _\tau }(u) - X_F(u)\big )=0\\&\partial _s\eta +\int _0^1\mu _\tau (u)dt=0\;. \end{aligned} \right. \end{aligned}$$

(2.27)

Due to the assumption that B vanishes near critical points the Floer equation thought of as a differential operator is Fredholm. The main ingredients for defining Floer homology are transversality and compactness for solution spaces of the Floer equation. This needs some attention in our framework due the restriction of the class of almost complex structures we consider and due to potential bubbling-off of holomorphic spheres.

The projection $\wp$ maps critical points of the Rabinowitz Floer action functional $\mathscr {A}_f^\tau$ to those of the action functional of classical mechanics $\mathfrak {a}_f$ on $(M,\omega )$

$$\begin{aligned} \begin{aligned} \mathfrak {a}_f:\widetilde{\mathscr {L}}(M)&\longrightarrow \mathbb {R}\\ \mathfrak {a}_f\big ([q,\bar{q}]\big )&:=\int _{D^2}\bar{q}^*\omega -\int _0^1f\big (q(t)\big )dt\;, \end{aligned}\end{aligned}$$

(2.28)

see Lemma 2.9. We recall that we chose the Morse function f in a $C^2$-small fashion, see Convention 2.5. This implies that all critical points of $\mathfrak {a}_f$ are critical points of f with some capping, i.e.

$$\begin{aligned} \mathrm {Crit}(\mathfrak {a}_f)\cong \mathrm {Crit}(f)\times \Gamma _M\;. \end{aligned}$$

(2.29)

We use the following convention for the Conley–Zehnder index for $(x,A)\in \mathrm {Crit}(\mathfrak {a}_f)\cong \mathrm {Crit}(f)\times \Gamma _M$

$$\begin{aligned} \mu _{\mathrm {CZ}}^M(x,A)=-\mu _{\mathrm {Morse}}(x,f)+\tfrac{1}{2} \dim M+2c_1^{TM}(A)\;. \end{aligned}$$

(2.30)

Lemma 2.9

The projection $\wp$ induces the map

$$\begin{aligned} \begin{aligned} \Pi :\mathrm {Crit}(\mathscr {A}_f^\tau )&\longrightarrow \mathrm {Crit}(\mathfrak {a}_f)\cong \mathrm {Crit}(f)\times \Gamma _M\\ \big ([u,A],\eta \big )&\longmapsto [\wp (u),A]\;. \end{aligned}\end{aligned}$$

(2.31)

Proof

This follows directly from the definition (2.6) of the action functional $\mathscr {A}_f^\tau$, see also Remark 2.1. $\square$

After a choice of $j\in \mathfrak {j}$ the action functional $\mathfrak {a}_f$ gives rise to the following Floer equation for $q:\mathbb {R}\times S^1\rightarrow M$

$$\begin{aligned} \partial _sq+j_t(q)\big (\partial _tq - X_f(q)\big )=0\;. \end{aligned}$$

(2.32)

We recall that solutions of either Floer equation is of finite energy if and only if it converges at $\pm \infty$ to critical points of $\mathscr {A}_f^\tau$ resp. $\mathfrak {a}_f$. That is, a solution $w=(u,\eta )$ of the Floer equation (2.27) has finite energy

$$\begin{aligned} \int _\mathbb {R}\int _{S^1}\Big (|\partial _su|^2+|\partial _s\eta |^2\Big )dtds<\infty \end{aligned}$$

(2.33)

if and only if there exists $(u_\pm ,\eta _\pm )\in \mathscr {L}(E)\times \mathbb {R}$ satisfying (2.7) such that

$$\begin{aligned} \lim _{s\rightarrow \pm \infty }\big (u(s,\cdot ),\eta (s)\big )=(u_\pm ,\eta _\pm ) \end{aligned}$$

(2.34)

and similarly for $\mathfrak {a}_f$. Following the usual Morse–Bott ideas we denote for $w_\pm \in \mathfrak {C}\subset \mathrm {Crit}(\mathscr {A}_f^\tau )$

$$\begin{aligned} \widehat{\mathcal {M}}(w_-,w_+):=\big \{w \text { solves }(2.27)\text { with}\lim _{s\rightarrow \pm \infty }w(s)\in W^{\pm }(w_\pm , h) \big \} \end{aligned}$$

(2.35)

the moduli space of finite energy solutions of the Floer equation of $\mathscr {A}_f^\tau$. Here $h:\mathrm {Crit}(\mathscr {A}_f^\tau )\rightarrow \mathbb {R}$ is the perfect Morse function from Definition 2.8 and $W^{+}(w_+, h)$ resp. $W^{-}(w_-, h)$ denotes the stable resp. unstable manifold of h on $\mathrm {Crit}(\mathscr {A}_f^\tau )$. Similarly, for $q_\pm \in \mathrm {Crit}(\mathfrak {a}_f)$ let

$$\begin{aligned} \widehat{\mathcal {N}}(q_-,q_+):=\big \{w \text { solves }(2.32)\text { with }\lim _{s\rightarrow \pm \infty }q(s)=q_\pm \big \}\;. \end{aligned}$$

(2.36)

Here, we abuse notation in the following sense. If $w_\pm =\big ([u_\pm ,\bar{u}_\pm ],\eta _\pm \big )\in \mathrm {Crit}(\mathscr {A}_f^\tau )$ is given and $w=(u,\eta )$ is a finite energy solution of (2.27) then by $\lim _{s\rightarrow \pm \infty }w(s)=w_\pm$ we mean that

$$\begin{aligned} \lim _{s\rightarrow \pm \infty }\big (u(s,\cdot ),\eta (s)\big )=(u_\pm ,\eta _\pm ) \end{aligned}$$

(2.37)

and

$$\begin{aligned} \big [(-\bar{u}_-)\#u\#\bar{u}_+\big ]=0\in \Gamma _E\;. \end{aligned}$$

(2.38)

The same remark applies to $\mathfrak {a}_f$. Unless $w_-=w_+$ the moduli space $\widehat{\mathcal {M}}(w_-,w_+)$ carries a free $\mathbb {R}$-action by shifts. We denote the quotient by

$$\begin{aligned} \mathcal {M}(w_-,w_+):=\widehat{\mathcal {M}}(w_-,w_+)/\mathbb {R}\end{aligned}$$

(2.39)

and similarly

$$\begin{aligned} \mathcal {N}(q_-,q_+):=\widehat{\mathcal {N}}(q_-,q_+)/\mathbb {R}\;. \end{aligned}$$

(2.40)

All moduli spaces depend on additional data, e.g. an almost complex structure, which we suppress in the notation.

Lemma 2.10

The projection $\wp$ induces the maps

$$\begin{aligned} \begin{aligned} \Pi :\widehat{\mathcal {M}}(w_-,w_+)&\longrightarrow \widehat{\mathcal {N}}\big (\Pi (w_-),\Pi (w_+)\big )\\ w=(u,\eta )&\longmapsto \Pi (w):=\wp (u)\;,\\ \Pi :\mathcal {M}(w_-,w_+)&\longrightarrow \mathcal {N}\big (\Pi (w_-),\Pi (w_+)\big )\\ [w]&\longmapsto [\Pi (w)]\;. \end{aligned}\end{aligned}$$

(2.41)

Proof

This follows immediately from the fact that $\wp _*(X_F)=X_f$, $\wp _*(X_{\mu _\tau })=0$ and the specific form of $J_t=\begin{pmatrix} i &{} B_t\\ 0 &{} j_t \end{pmatrix}$, i.e. $\wp _*\circ J=j\circ \wp _*$ $\square$

2.2 Transversality

We recall that $\mathfrak {j}$ is the space of $S^1$-families of compatible almost complex structures on $(M,\omega )$. We denote by

$$\begin{aligned} \mathfrak {j}_{\mathrm {reg}}(f)\subset \mathfrak {j}\end{aligned}$$

(2.42)

the subset of $j\in \mathfrak {j}$ with the following two properties.

• All finite energy solutions of the Floer equation for $\mathfrak {a}_f$ with respect to j are regular, i.e. the operator obtained by linearizing the Floer equation is a surjective Fredholm operator for all finite energy solutions.

• For every $t\in S^1$ all simple $j_t$-holomorphic spheres are regular, i.e. the operator obtained by linearizing the holomorphic sphere equation is a surjective Fredholm operator.

According to Flore et al. [14] and McDuff and Salamon [20, Chapter 2] the subset $\mathfrak {j}_{\mathrm {reg}}$ is of second category. For every $j\in \mathfrak {j}_{\mathrm {reg}}(f)$ the moduli space $\widehat{\mathcal {N}}(q_-,q_+)$ is a smooth manifold of dimension

$$\begin{aligned} \dim \widehat{\mathcal {N}}(q_-,q_+)=\mu _{\mathrm {CZ}}^M(q_+)-\mu _{\mathrm {CZ}}^M(q_-)\;. \end{aligned}$$

(2.43)

For $j\in \mathfrak {j}_{\mathrm {reg}}(f)$ we denote by

$$\begin{aligned} \mathcal {B}_{\mathrm {reg}}(j)\subset \mathcal {B}(j) \end{aligned}$$

(2.44)

the subset of $B\in \mathcal {B}(j)$ with the following two properties

• All finite energy solutions of the Floer equation for $\mathscr {A}^\tau _f$ with respect to the corresponding J are regular.

• For every $t\in S^1$ all simple $J_t$-holomorphic spheres are regular.

We refer to [4] for details on the linearization of the Rabinowitz Floer equations. For $B\in \mathcal {B}_{\mathrm {reg}}(j)$ the moduli space $\widehat{\mathcal {M}}(w_-,w_+)$ is a smooth manifold of dimension

$$\begin{aligned} \dim \widehat{\mathcal {M}}(w_-,w_+)=\mu (w_+)-\mu (w_-)\;. \end{aligned}$$

(2.45)

The next proposition shows that this class of almost complex structures is sufficiently large.

Proposition 2.11

For all $j\in \mathfrak {j}_{\mathrm {reg}}(f)$ the set $\mathcal {B}_{\mathrm {reg}}(j)\subset \mathcal {B}(j)$ is of second category.

Proof

We recall the splitting $TE\cong V\oplus H$, cf. Remark 2.2. Thus, we may consider the linearization of the Floer equation (2.27) in vertical resp. horizontal directions V resp. H. Since the projection $\wp$ induces an isomorphism $\wp _*:(H,d\alpha )\rightarrow (TM,\omega )$ and $j\in \mathfrak {j}_{\mathrm {reg}}$, it follows from Lemma 2.10 that the linearization is already surjective in horizontal directions. To show that it is for generic choice of B also surjective in vertical directions we distinguish two cases for $w\in \widehat{\mathcal {M}}(w_-,w_+)$.

Case 1 $\Pi (w)$ is non-constant. We claim that $\Pi (w)$ necessarily leaves the neighborhood $\mathcal {U}$. We recall that $\mathcal {U}$ is the union of disjoint neighborhoods of all critical points of f where each such neighborhood contracts onto a critical point, see the discussion before Eq. (2.22). If $\Pi (w)$ is contained in $\mathcal {U}$ then it has to be a gradient trajectory connecting the same critical point of f with cappings A and $A\#\Pi (w)$. Since $\Pi (w)$ is contained in $\mathcal {U}$ the two cappings are homotopic to each other: $A=A\#\Pi (w)\in \Gamma _M$. Thus, $\Pi (w)$ is a gradient trajectory from a critical point of $\mathfrak {a}_f$ to itself (including cappings) and therefore $\Pi (w)$ is constant which is a contradiction. Therefore $\Pi (w)$ necessarily leaves the neighborhood $\mathcal {U}$. By [14, Theorem 4.3] the set of regular points for $\Pi (w)$ is open and dense. Since $\Pi (w)$ leaves $\mathcal {U}$ we may apply Lemma 2.15 below. Thus a standard argument, see for instance [14, Section 5] or [20, Chapter 3] establishes that for generic $B \in \mathcal {B}(j)$ the linearization of the gradient flow equation is also vertically surjective.

Case 2 $\Pi (w)$ is constant. But then w is a vortex and vortices are by Albers and Frauenfelder [3, Proposition A.1] always transverse. In fact vortices are independent of the perturbation $B\in \mathcal {B}(j)$.

It remains to prove that generically all simple $J_t$-holomorphic spheres, $t\in S^1$, are regular. If we were not to restrict to upper triangular J this is a standard result which relies on the fact that simple curves are somewhere injective, see [20, Chapter 2] for details.

We argue again as above. Due to the definition of $\mathfrak {j}_\mathrm {reg}$ the linearization of a simple $J_t$-holomorphic sphere is already surjective in horizontal directions. For vertical directions we use the notion of somewhere horizontally injective points, see Definition 2.13 below. The important observation is that horizontally injective points still form a dense subset, see Lemma 2.14. Therefore, we can apply again Lemma 2.15 to conclude that for generic $B\in \mathcal {B}(j)$ all simple $J_t$-holomorphic curves are regular. $\square$

Remark 2.12

We recall that $\mathcal {B}^T(j)\subset \mathcal {B}(j)$ denotes the non-empty open convex subset consisting of those $B\in \mathcal {B}(j)$ for which the corresponding J is tame. From now on we always choose $j\in \mathfrak {j}_{\mathrm {reg}}$ and $B\in \mathcal {B}^T_{\mathrm {reg}}(j):=\mathcal {B}_{\mathrm {reg}}(j)\cap \mathcal {B}^T(j)$.

We recall the following notions and Lemmas considered in [3, 15].

Definition 2.13

A $J_t$-holomorphic curve $u:S^2 \rightarrow E$ is called somewhere horizontally injective if there exists $z \in S^2$ such that

$$\begin{aligned} d^hu(z):=\wp _*\circ du(z) \ne 0, \quad u^{-1}(u(z))=\{z\}\;. \end{aligned}$$

(2.46)

It remains to prove the following two Lemmas.

Lemma 2.14

Assume that $u:S^2 \rightarrow E$ is a simple $J_t$-holomorphic curve. Then u is horizontally injective on a dense set.

Proof

We denote by $I(u)\subset S^2$ the subset of injective points of u, by $R(\wp (u)) \subset S^2$ the subset of nonsingular points of $\wp (u)$ and by $S(u)\subset S^2$ the subset of horizontally injective points of u. Then

$$\begin{aligned} S(u)=I(u) \cap R(\wp (u)). \end{aligned}$$

(2.47)

We first observe that $\wp (u)$ is $j_t$-holomorphic. We claim that $\wp (u):S^2\rightarrow M$ is not constant, since otherwise u would lie in one fiber and hence itself must be constant, contradicting the assumption that it is simple. Therefore, it follows from [20, Lemma 2.4.1] that the complement of $R(\wp (u))$ is finite. Moreover, it follows from [20, Proposition 2.5.1] that the complement of I(u) is countable. Hence by (2.47) the complement of S(u) is countable. In particular, S(u) is dense in $S^2$. $\square$

Lemma 2.15

We fix $e\in E{\setminus }\wp ^{-1}(\mathcal {U})$, $(v,h)\in T_eE=V_e\oplus H_e$ with $h\ne 0$ and $t_0\in S^1$. Moreover, we fix $j\in \mathfrak {j}$ and $B\in \mathcal {B}(j)$. Then there exist $\widehat{B}\in \Gamma _0(S^1\times E, \mathrm {L}(H,V))$ and $\widehat{j}\in T_j\mathfrak {j}$ with

$$\begin{aligned} \left\{ \begin{aligned}&\widehat{B}(t_0,e)h=v\\&i\widehat{B}+B\widehat{j}+\widehat{B} j=0\;. \end{aligned} \right. \end{aligned}$$

(2.48)

Remark 2.16

The second equation asserts that the pair $(\widehat{j},\widehat{B})$ corresponds to a tangent vector of the space of almost complex structures we are considering.

Proof of Lemma 2.15

First we extend h resp. v to sections also denoted by h resp. v supported in a small neighborhood of e. Then we define $\widehat{j}$ by

$$\begin{aligned} \widehat{j} h:=v,\quad \widehat{j} jh:=-jv, \quad \widehat{j}|_{\text {span}\{h,j h\}^\perp }:=0, \end{aligned}$$

(2.49)

where $\perp$ refers to the metric $\omega (\cdot ,j\cdot )$ on H. We point out that $\text {span}\{h,j h\}^\perp$ is j-invariant. Then $\widehat{j}$ satisfies the equation

$$\begin{aligned} \widehat{j} j+j \widehat{j}=0 \end{aligned}$$

(2.50)

that is, $\widehat{j}\in T_j\mathfrak {j}$. Next we define $\widehat{B}$ by

$$\begin{aligned} \begin{aligned} \widehat{B} h&:=v\\ \widehat{B} jh&:=-\left( i\widehat{B}+B\widehat{j}\right) h\\ \widehat{B}|_{\text {span}\{h,j h\}^\perp }&:=0. \end{aligned}\end{aligned}$$

(2.51)

The first equation in (2.48) holds by construction. We show the second equation. We have

$$\begin{aligned} \begin{aligned} (i\widehat{B}+B\widehat{j}+\widehat{B} j)h=0 \end{aligned}\end{aligned}$$

(2.52)

by the very definition of $\widehat{B} jh$. Moreover,

$$\begin{aligned} \begin{aligned} (i\widehat{B}+B\widehat{j}+\widehat{B} j)jh&=i \widehat{B} jh + B\widehat{j} jh - \widehat{B}h\\&=-i \left( i\widehat{B}+B\widehat{j}\right) h - Bj\widehat{j}h - \widehat{B}h\\&=\widehat{B}h - iB\widehat{j}h - Bj\widehat{j}h - \widehat{B}h\\&=- iB\widehat{j}h - Bj\widehat{j}h \\&=- (iB + Bj)\widehat{j}h \\&=0 \end{aligned}\end{aligned}$$

(2.53)

where we used $jj=-1$, $\widehat{B} jh=-(i\widehat{B}+B\widehat{j})h$, $\widehat{j} j+j \widehat{j}=0$, $ii=-1$ and finally $iB + Bj=0$, see (2.22). Finally,

$$\begin{aligned} \begin{aligned} \left( i\widehat{B}+B\widehat{j}+\widehat{B} j\right) |_{\text {span}\{h,j h\}^\perp }=0 \end{aligned}\end{aligned}$$

(2.54)

since $\text {span}\{h,j h\}^\perp$ is j-invariant and $\widehat{B}$ and $\widehat{j}$ vanish on it. $\square$

This completes the discussion on transversality.

2.3 Compactness

In this subsection we discuss the appropriate compactness results for the moduli spaces $\mathcal {M}(w_-,w_+)$ of unparametrized gradient flow trajectories. This follows the usual scheme of Rabinowitz Floer homology, that is, we need to establish the following for a sequence $(u_\nu ,\eta _\nu )\in \mathcal {M}(w_-,w_+)$, $\nu \in \mathbb {N}$.

• (i) A uniform $C^0$-bound for the loops $u_\nu$.

• (ii) A uniform $C^0$-bound on the Lagrange multipliers $\eta _\nu$.

• (iii) A uniform bound on the derivatives of the loops $u_\nu$.

The first two are proved in [15, Proposition 6.2 & 6.4]. We point out that the set-up in [15] is the same as ours except for the following. Frauenfelder’s assumption of $(E,\Omega )$ being very negative is replaced by our assumption of semi-positivity. Moreover, the almost complex structures used are of the form $J= \begin{pmatrix} i&{}0\\ 0&{}j \end{pmatrix}$, i.e. $B=0$. The uniform $C^0$-bound for the loops is based on a maximum principle which continues to hold since in our setting B has compact support. The uniform $C^0$-bound on the Lagrange multipliers relies on a “fundamental lemma” which continues to hold verbatim.

To prove a uniform bound on the derivatives of the loops we argue by contradiction, i.e. by bubbling-off analysis. Indeed, since we already established uniform $C^0$-bounds for the loops and the Lagrange multipliers a blow-up of derivatives of $u_\nu$ leads to $J_t$-holomorphic spheres inside E. We claim that since we assume that E is semi-positive we can apply the results of Hofer-Salamon [18] and ensure that for generic $S^1$-family almost complex structure J of the form $J_t= \begin{pmatrix} i &{} B_t \\ 0 &{} j_t \end{pmatrix}$ with $B_t\in \mathcal {B}^T_{\mathrm {reg}}(j)$, $t\in S^1$ the moduli spaces $\widehat{\mathcal {M}}(w_-,w_+)$ are compact up to breaking as long as $\mu _{\mathrm {CZ}}(w_+)-\mu _{\mathrm {CZ}}(w_-)\le 2$. In [18] Hofer-Salamon argue that bubbling-off of $J_t$-holomorphic spheres of Chern number at least 2 never occurs for index reasons. Moreover, they rule out bubbling-off of $J_t$-holomorphic sphere with Chern number less than 2 by carefully studying moduli spaces of $J_t$-holomorphic spheres. The crucial input is that for simple holomorphic spheres the linearized operator is a surjective Fredholm operator, see [18, Theorem 2.2]. We establishes the corresponding result for our restricted class of almost complex structures in Proposition 2.11. Therefore, the results in [18] apply to the Floer equation for $\mathscr {A}_f^\tau$ and we conclude that the moduli spaces $\widehat{\mathcal {M}}(w_-,w_+)$ are compact up to breaking as long as $\mu (w_+)-\mu (w_-)\le 2$.

2.4 Rabinowitz Floer homology

We define Rabinowitz Floer homology with the help of Novikov rings. Alternative approaches are via mixed direct/inverse limits. How these relate has been studied in [10]. The current approach is as in the original article [9].

The spaces $\mathfrak {C}_k$ and $\mathfrak {C}$ of critical point of $\mathscr {A}_f^\tau$ were defined in Definition 2.8. The vector space $\mathrm {RFC}_*(\mathscr {A}_f^\tau )$, graded by $\mu$ (see (2.19)), is the set of all formal linear combinations

$$\begin{aligned} \xi =\sum _{w\in \mathfrak {C}} a_ww,\quad a_w\in \mathbb {Z}/2, \end{aligned}$$

(2.55)

subject to the Novikov condition

$$\begin{aligned} \forall \kappa \in \mathbb {R}:\; \#\big \{w\in \mathfrak {C}\mid a_w\ne 0,\;\mathscr {A}_f^\tau (w)\ge \kappa \big \}<\infty \;. \end{aligned}$$

(2.56)

It is a module over the Novikov ring

$$\begin{aligned} \Lambda _E:=\bigg \{\sum _{A\in \Gamma _E} n_A e^A\mid n_A\in \mathbb {Z}/2,\;\forall \kappa \in \mathbb {R}:\#\big \{A\in \Gamma _E\mid n_A\ne 0,\;\Omega (A)\ge \kappa \big \}<\infty \bigg \}\;.\nonumber \\ \end{aligned}$$

(2.57)

The multiplicative structure on $\Lambda _E$ is given by

$$\begin{aligned}\begin{aligned} \left( \sum _{A\in \Gamma _E}n_Ae^A\right) \cdot \left( \sum _{B\in \Gamma _E}m_Be^B\right)&:=\sum _A\sum _B(n_A\cdot m_B)\,e^{A+B}=\sum _C\left( \sum _An_A\cdot m_{C-A}\right) e^{C} \end{aligned}\end{aligned}$$

and the action of $\Lambda _E$ on $\mathrm {RFC}_k(\mathscr {A}_f^\tau )$ by

$$\begin{aligned} \left( \sum _{A\in \Gamma _E} n_A e^A\right) \cdot \left( \sum _{w\in \mathfrak {C}} a_ww\right) :=\sum _w \left( \sum _A n_A\cdot a_{w\#-A}\right) w\;, \end{aligned}$$

(2.58)

where we use the following notation. If $w=([u,B],\eta )$ then $w\#-A=([u,B-A],\eta )$. The differential $\partial$ on $\mathrm {RFC}_*(\mathscr {A}_f^\tau )$ is defined by

$$\begin{aligned} \begin{aligned} \partial :\mathrm {RFC}_k(\mathscr {A}_f^\tau )&\longrightarrow \mathrm {RFC}_{k-1}(\mathscr {A}_f^\tau )\\ \partial \, w&:=\sum _{z\in \mathfrak {C}_{k-1}} \#_2\mathcal {M}(z,w)\,z\;. \end{aligned}\end{aligned}$$

(2.59)

The compactness results described in Sect. 2.3 imply that $\mathcal {M}(z,w)$ is a finite set and $\#_2\mathcal {M}(z,w)\in \mathbb {Z}/2$ denotes its parity. Moreover, compactness up to breaking implies $\partial \circ \partial =0$. The Rabinowitz Floer homology is then defined by

$$\begin{aligned} \mathrm {RFH}_k(\mathscr {A}_f^\tau ):=\mathrm {H}_k\big (\mathrm {RFC}_*(\mathscr {A}_f^\tau ),\partial \big ),\quad k\in \tfrac{1}{2}+\mathbb {Z}\;. \end{aligned}$$

(2.60)

Remark 2.17

To define $\mathrm {RFH}_*(\mathscr {A}_f^\tau )$ we made auxiliary choices, notably $\tau$ and f. The assumption that f is $C^2$-small is not necessary for defining $\mathrm {RFH}_*(\mathscr {A}_f^\tau )$, see [3] for more details. Nevertheless, we decided to make this assumption throughout this article. The choices of $\tau$ and f become relevant in the proof of Theorem 1.2. The methods of Cieliebak and Frauenfelder [9] show that $\mathrm {RFH}_*(\mathscr {A}_f^\tau )$ is independent of all these choices.

Remark 2.18

We recall that we restrict ourselves to the class of almost complex structures J of the form $J= \begin{pmatrix} i &{} B \\ 0 &{} j \end{pmatrix}$ with $B\in \mathcal {B}^T_{\mathrm {reg}}(j)$. It is unclear to us whether it is possible to extend the definition of $\mathrm {RFH}_*(\mathscr {A}_f^\tau )$ beyond this class of almost complex structures. We crucially rely on Frauenfelder’s result, namely that the fact that the projection of the Floer equation of $\mathscr {A}_f^\tau$ gives the Floer equation of $\mathfrak {a}_f$ on M can be used to obtain uniform $C^0$-bounds for the Lagrange multiplier. For this $\wp$ needs to be J-j-holomorphic.

3 A filtration and the proof of vanishing

We use the fact that $\mathrm {RFC}_*(\mathscr {A}_f^\tau )$ admits a filtration. For $l\in \mathbb {Z}$ we set

$$\begin{aligned} \mathrm {RFC}_k^l(\mathscr {A}_f^\tau ):=\left\{ \sum _{w} a_ww\in \mathrm {RFC}_k(\mathscr {A}_f^\tau )\mid \mu _{\mathrm {CZ}}^M\big (\Pi (w)\big )= l\right\} \end{aligned}$$

(3.1)

and

$$\begin{aligned} \mathrm {RFC}_k^{\le l}(\mathscr {A}_f^\tau ):=\left\{ \sum _w a_ww\in \mathrm {RFC}_k(\mathscr {A}_f^\tau )\mid \mu _{\mathrm {CZ}}^M\big (\Pi (w)\big )\le l\right\} , \end{aligned}$$

(3.2)

where we recall that $A\in \Gamma _E\cong \Gamma _M$, see Remark 2.1.

Lemma 3.1

$$\begin{aligned} \partial \Big (\mathrm {RFC}_k^{\le l}(\mathscr {A}_f^\tau )\Big )\subset \mathrm {RFC}_{k-1}^{\le l}(\mathscr {A}_f^\tau ) \end{aligned}$$

(3.3)

hence we can decompose

$$\begin{aligned} \partial =\sum _{i\ge 0} \partial _i=\partial _0+\partial _1+\cdots \end{aligned}$$

(3.4)

with

$$\begin{aligned} \begin{aligned} \partial _i:\mathrm {RFC}_k^l(\mathscr {A}_f^\tau )&\longrightarrow \mathrm {RFC}_{k-1}^{l-i}(\mathscr {A}_f^\tau )\\ \partial _iw&:=\sum _{\begin{array}{c} z\in \mathfrak {C}_{k-1}\\ \mu _{\mathrm {CZ}}^M(\Pi (z))=\mu _{\mathrm {CZ}}^M(\Pi (w))-i \end{array}} \#_2\mathcal {M}(z,w)z\;. \end{aligned}\end{aligned}$$

(3.5)

Proof

This is a direct consequence of Lemma 2.10 together with (2.43). $\square$

Remark 3.2

From $\partial ^2=0$ and the filtration we derive for every $i\ge 0$ the equation

$$\begin{aligned} \sum _{j=0}^{i}\partial _j\partial _{i-j}=0. \end{aligned}$$

(3.6)

E.g. $\partial _0\partial _0=0$, $\partial _0\partial _1+\partial _1\partial _0=0$ etc. In particular, $\partial _0$ is a differential.

The main idea for proving Theorem 1.2 is that $\partial _0$ counts solutions of the Floer equation (2.27) which are entirely contained inside fibers of $\wp$ over critical points of f. Thus the homology of $\partial _0$ is the sum of $\mathrm {Crit}(f)\times \Gamma _M$-many copies of the Rabinowitz Floer homology of $\big (\Sigma _\tau \cap \wp ^{-1}(q),\wp ^{-1}(q)\big )\cong (S^1,\mathbb {C})$, $q\in \mathrm {Crit}(f)$, each of which vanishes.

Proposition 3.3

The differential $\partial _0$ counts precisely the solutions $w=(u,\eta )$ of the Floer equation (2.27) with image contained entirely in a fiber over some critical point of f. That is, there exists $q\in \mathrm {Crit}(f)$ such that $u(\mathbb {R}\times S^1)\subset \wp ^{-1}(q)$. Moreover, if $w_\pm =([u_\pm ,A_\pm ],\eta _\pm )\in \mathrm {Crit}(\mathscr {A}_f^\tau )$ are the asymptotic limits of w then

$$\begin{aligned} A_-=A_+\in \Gamma _E\;. \end{aligned}$$

(3.7)

Proof

Let $w=(u,\eta )$ be a gradient flow line from $w_-=([u_-,A_-],\eta _-)$ to $w_+=([u_+,A_+],\eta _+)$ with

$$\begin{aligned} \mu _{\mathrm {CZ}}^M\big (\Pi (w_+)\big )=\mu _{\mathrm {CZ}}^M\big (\Pi (w_-)\big )\;. \end{aligned}$$

(3.8)

Using Lemma 2.10 we see that

$$\begin{aligned} \wp (u)\in \widehat{\mathcal {N}}\big (\Pi (w_+),\Pi (w_-)\big )\;. \end{aligned}$$

(3.9)

According to (2.43), equation (3.8) implies that that

$$\begin{aligned} \dim \widehat{\mathcal {N}}\big (\Pi (w_+),\Pi (w_-)\big )=0\;, \end{aligned}$$

(3.10)

which in turn implies that $\wp (u)$ is s-independent, i.e. constant $\wp (u)=q\in \mathrm {Crit}(f)$, see Lemma 2.9. In other words, $u(\mathbb {R}\times S^1)\subset \wp ^{-1}(q)$. Moreover, in view of (2.38), we have

$$\begin{aligned} A_-=A_+\in \Gamma _M\cong \Gamma _E\;. \end{aligned}$$

(3.11)

This finishes the proof. $\square$

Corollary 3.4

$$\begin{aligned} \mathrm {H}_k\big (\mathrm {RFC}_*(\mathscr {A}_f^\tau ),\partial _0\big )=0\quad \forall k\in \tfrac{1}{2}+\mathbb {Z}\;. \end{aligned}$$

(3.12)

Proof

For $q\in \mathrm {Crit}(f)$ we fix an identification

$$\begin{aligned} \big (\Sigma _\tau \cap \wp ^{-1}(q),\wp ^{-1}(q)\big )\cong (S^1_\tau ,\mathbb {C}) \end{aligned}$$

(3.13)

together with the symplectic form, its primitive and the complex structure i. Here $S^1_\tau$ is the circle bounding a disk of area $\pi \tau ^2$. For $A\in \Gamma _M$ we denote by

$$\begin{aligned} \mathrm {RFC}_*(q,A) \end{aligned}$$

(3.14)

the vector space generated over $\mathbb {Z}/2$ by critical points of the form $\big ([u,A]^\pm ,\eta \big )\in \mathfrak {C}\subset \mathrm {Crit}(\mathscr {A}_f^\tau )$ with $\wp (u)=q$. Proposition 3.3 implies that $\mathrm {RFC}_*(q,A)$ is a $\partial _0$-subcomplex of $\mathrm {RFC}_*(\mathscr {A}_f^\tau )$. With the above identification we see that

$$\begin{aligned} \big (\mathrm {RFC}_{k+2c_1^{TE}(A)+\frac{1}{2}\dim M}(q,A),\partial _0\big )=\big (\mathrm {RFC}_k(S^1,\mathbb {C}),\partial \big )\;. \end{aligned}$$

(3.15)

Let v be the primitive Reeb orbit over q then all generators are of the form $\big ([v^n,A]^\pm , \eta =n-f(q)\big )$. Since $\mathrm {RFH}_*(S^1,\mathbb {C})=0$ due to [1, 9] and $\mu \big ([v^{n-1},A]^\pm , n-1-f(q)\big )+2=\mu \big ([v^n,A]^\pm , n-f(q)\big )$ from Lemma 2.7, we know that

$$\begin{aligned} \begin{aligned} \partial _0\big ([v^n,A]^-, n-f(q)\big )&=\big ([v^{n-1},A]^+,n-1-f(q)\big )\\ \partial _0\big ([v^n,A]^+,n-f(q)\big )&=0\;. \end{aligned}\end{aligned}$$

(3.16)

Let $\xi =\sum _w a_ww\in \mathrm {RFC}_k(\mathscr {A}_f^\tau )$ with $\partial _0\xi =0$, i.e. $\sum _w a_w\partial _0w=0$. If $a_w\ne 0$ then w is of the form $\big ([v^n,A]^+,n-f(q)\big )$ and we let $w'$ be the corresponding element $\big ([v^{n+1},A]^-,n+1-f(q)\big )$. Then

$$\begin{aligned} \xi ':=\sum _w a_w w' \end{aligned}$$

(3.17)

satisfies the Novikov condition, i.e. $\xi '\in \mathrm {RFC}_{k+1}(\mathscr {A}_f^\tau )$, since $\mathscr {A}_f^\tau (w')=\mathscr {A}_f^\tau (w)+\tau$ due to (2.11). From (3.16), $\partial _0\xi '=\xi$ and this completes the proof. $\square$

Lemma 3.5

We assume now that $c_1^{TM}=c\omega :\pi _2(M)\rightarrow \mathbb {Z}$.

• In case $c=0$ we have for all $\tau >0$

  $$\begin{aligned} \mathrm {RFC}_k(\mathscr {A}_f^\tau )=\bigoplus _{l=-\frac{1}{2}\dim M}^{\frac{1}{2}\dim M}\mathrm {RFC}_k^{l}(\mathscr {A}_f^\tau ) \end{aligned}$$

  (3.18)

  and

  $$\begin{aligned} \partial _n=0\quad \forall n\ge \dim M+1\;. \end{aligned}$$

  (3.19)

• In case $c\ge 1$ we assume $(c-1)\tau <1$. Then a formal sum $\xi =\sum _w a_w w$, $a_w\in \mathbb {Z}/2$, $w\in \mathfrak {C}_k$ satisfies the Novikov condition

  $$\begin{aligned} \forall \kappa \in \mathbb {R}:\; \#\big \{w\in \mathfrak {C}_k\mid a_w\ne 0,\;\mathscr {A}_f^\tau (w)\ge \kappa \big \}<\infty \end{aligned}$$

  (3.20)

  if and only if

  $$\begin{aligned} \forall \kappa \in \mathbb {R}:\; \#\big \{w\in \mathfrak {C}_k\mid a_w\ne 0,\;\mu _{\mathrm {CZ}}^M\big (\wp (w)\big )\ge \kappa \big \}<\infty . \end{aligned}$$

  (3.21)

  In particular, for all $\xi \in \mathrm {RFC}_k(\mathscr {A}_f^\tau )$ there exists $l(\xi )\in \mathbb {Z}$ with

  $$\begin{aligned} \xi \in \mathrm {RFC}_k^{\le l(\xi )}(\mathscr {A}_f^\tau )\;. \end{aligned}$$

  (3.22)

Proof

If we write $w=\big ([v^{n},A]^\pm ,\eta \big )\in \mathfrak {C}_k$ then according to (2.11) and Definition 2.8

$$\begin{aligned} \begin{aligned} \mathscr {A}_f^\tau (w)&=n\tau +\omega (A)-(\tau +1)f(\wp (v))\\ k=\mu (w)&=2n+2c_1^{TE}(A)-\mu _{\mathrm {Morse}}(\wp (v);f)+\tfrac{1}{2} \dim M\big ({\pm }\tfrac{1}{2} )\\&=2n+2(c-1)\omega (A)-\mu _{\mathrm {Morse}}(\wp (v);f)+\tfrac{1}{2} \dim M\big ({\pm }\tfrac{1}{2} )\;. \end{aligned}\end{aligned}$$

(3.23)

We solve the second equation for n

$$\begin{aligned} \begin{aligned} n&=\tfrac{1}{2} k-(c-1)\omega (A)+\tfrac{1}{2} \big [\mu _{\mathrm {Morse}}(\wp (v);f)-\tfrac{1}{2} \dim M\big ({\pm }\tfrac{1}{2} )\big ] \end{aligned}\end{aligned}$$

(3.24)

and abbreviate $e:=\mu _{\mathrm {Morse}}(\wp (v);f)-\tfrac{1}{2} \dim M\big ({\pm }\tfrac{1}{2} )$. In particular, $|e|\le \tfrac{1}{2} \dim M+\tfrac{1}{2}$. Thus, we can rewrite the action value as

$$\begin{aligned} \begin{aligned} \mathscr {A}_f^\tau (w)&=n\tau +\omega (A)-(\tau +1)f(\wp (v))\\&=\big (\tfrac{1}{2} k-(c-1)\omega (A)+\tfrac{1}{2} e\big )\tau +\omega (A)-(\tau +1)f(\wp (v))\\&=\big (1-(c-1)\tau \big )\omega (A)+\tfrac{1}{2} \big (k+e\big )\tau -(\tau +1)f(\wp (v))\;. \end{aligned}\end{aligned}$$

(3.25)

Next we observe that

$$\begin{aligned} \begin{aligned} \mu _{\mathrm {CZ}}^M(\Pi (w))&=-\mu _{\mathrm {Morse}}(\wp (v);f)+\tfrac{1}{2} \dim M+2c_1^{TM}(A)\\&=-\mu _{\mathrm {Morse}}(\wp (v);f)+\tfrac{1}{2} \dim M+2c\omega (A). \end{aligned}\end{aligned}$$

(3.26)

In case $c\ge 1$ and $(c-1)\tau <1$ equations (3.25) and (3.26) imply the if-and-only-if statement of the Lemma. The statement (3.22) follows from the if-part of the if-and-only-if statement since $\xi \in \mathrm {RFC}_k(\mathscr {A}_f^\tau )$ satisfies the Novikov condition by the very definition of $\mathrm {RFC}$.

The case $c=0$ follows immediately from equation (3.26). $\square$

We are now in the position to prove Theorem 1.2. We treat the symplectically aspherical case last and assume now that $c_1^{TM}=c\omega$. We first consider the case $c\ge 0$. If $c=0$ we assume that $(E,\Omega )$ is semi-positive.

Proof of Theorem 1.2 for $c\ge 0$ We fix $\xi \in \mathrm {RFC}_k(\mathscr {A}_f^\tau )$ with

$$\begin{aligned} \partial \xi =0\;. \end{aligned}$$

(3.27)

Our aim is to construct $\theta \in \mathrm {RFC}_{k+1}(\mathscr {A}_f^\tau )$ with $\partial \theta =\xi$. We split $\xi$ as follows.

$$\begin{aligned} \xi =\sum _{l=-\infty }^{l(\xi )} \xi _l \quad \text {with}\quad \xi _l\in \mathrm {RFC}_k^l(\mathscr {A}_f^\tau ) \;, \end{aligned}$$

(3.28)

where $l(\xi )\in \mathbb {Z}$ is taken from Lemma 3.5. If $c=0$ we set $l(\xi ):=\tfrac{1}{2} \dim M$. We expand $\partial \xi =0$ according to $\partial =\sum _{i\ge 0}\partial _i$ and collect terms in $\mathrm {RFC}_{k-1}^{l(\xi )-I}(\mathscr {A}_f^\tau )$ for all $I\ge 0$. We recall from Lemma 3.1 that $\partial _i$ drops the upper degree by i. This leads to

$$\begin{aligned} \sum _{i=0}^{I}\partial _i\xi _{l(\xi )+i-I}=0\;, \end{aligned}$$

(3.29)

since $\partial _i\xi _m\in \mathrm {RFC}_{k-1}^{l(\xi )-I}(\mathscr {A}^\tau _f)$ if and only if $m-i=l(\xi )-I$.

Claim 1

For all $l\le l(\xi )$ there exists $\theta _l\in \mathrm {RFC}_{k+1}^l(\mathscr {A}_f^\tau )$ such that

$$\begin{aligned} \sum _{i=0}^{I}\partial _i\theta _{l(\xi )+i-I}=\xi _{l(\xi )-I} \end{aligned}$$

(3.30)

holds for $I\ge 0$.

Proof of Claim 1

We inductively construct $\theta _l$. For $I=0$ equation (3.29) reduces to

$$\begin{aligned} \partial _0 \xi _{l(\xi )}=0. \end{aligned}$$

(3.31)

Corollary 3.4 implies that there exists $\theta _{l(\xi )}\in \mathrm {RFC}_{k+1}^{l(\xi )}(\mathscr {A}_f^\tau )$ with

$$\begin{aligned} \partial _0\theta _{l(\xi )}=\xi _{l(\xi )}. \end{aligned}$$

(3.32)

Now assume that we already constructed $\theta _{l(\xi )},\ldots , \theta _{l(\xi )-(I-1)}$ satisfying equation (3.30). Then we compute

$$\begin{aligned} \begin{aligned} \partial _0\bigg (\xi _{l(\xi )-I}-\sum _{i=1}^I\partial _i\theta _{l(\xi )+i-I}\bigg )&\mathop {=}\limits ^{\phantom {*(*)*}}\partial _0\xi _{l(\xi )-I}-\sum _{i=1}^I\partial _0\partial _i\theta _{l(\xi )+i-I}\\&\mathop {=}\limits ^{\phantom {*}(*)\phantom {*}}\partial _0\xi _{l(\xi )-I}+\sum _{i=1}^I\sum _{j=1}^i\partial _j\partial _{i-j}\theta _{l(\xi )+i-I}\\&\,\mathop {=}\limits ^{(**)}\partial _0\xi _{l(\xi )-I}+\sum _{j=1}^I\partial _j\left( \sum _{i=j}^I\partial _{i-j}\theta _{l(\xi )+i-I}\right) \\&\mathop {=}\limits ^{\phantom {*(*)*}}\partial _0\xi _{l(\xi )-I}+\sum _{j=1}^I\partial _j\left( \sum _{i=0}^{I-j}\partial _{i}\theta _{l(\xi )+i-(I-j)}\right) \\&\mathop {=}\limits ^{(***)}\partial _0\xi _{l(\xi )-I}+\sum _{j=1}^I\partial _j\xi _{l(\xi )+j-I}\\&\mathop {=}\limits ^{\phantom {*(*)*}}\sum _{j=0}^I\partial _j\xi _{l(\xi )+j-I}\\&\mathop {=}\limits ^{\phantom {*(*)*}}0\;. \end{aligned}\end{aligned}$$

(3.33)

Here we used equation (3.6) in $(*)$, the usual relabeling $\sum \nolimits _{i=1}^I\sum \nolimits _{j=1}^i=\sum \nolimits _{j=1}^I\sum \nolimits _{i=j}^I$ in $(**)$, the induction hypothesis (3.30) in $(***)$ and (3.29) at the end. Using again Corollary 3.4 we find $\theta _{l(\xi )-I}\in \mathrm {RFC}_{k+1}^{l(\xi )-I}(\mathscr {A}_f^\tau )$ with

$$\begin{aligned} \partial _0\theta _{l(\xi )-I}=\xi _{l(\xi )-I}-\sum _{i=1}^I\partial _i\theta _{l(\xi )+i-I}, \end{aligned}$$

(3.34)

in other words

$$\begin{aligned} \sum _{i=0}^I\partial _i\theta _{l(\xi )+i-I}=\xi _{l(\xi )-I}. \end{aligned}$$

(3.35)

This proves Claim 1. $\square$

Now we consider

$$\begin{aligned} \theta :=\sum _{l=-\infty }^{l(\xi )}\theta _l\;. \end{aligned}$$

(3.36)

We will show that $\theta$ satisfies the Novikov condition and $\partial \theta =\xi$. We need slightly different arguments for the cases $c=0$ and $c\ge 1$.

If $c=0$ then $\xi =\sum _{l=-\frac{1}{2}\dim M}^{\frac{1}{2}\dim M} \xi _l$, see Lemma 3.5, is the sum of finitely many non-zero $\xi _l$ each of which satisfies the Novikov condition. Since $\partial =\partial _0+\cdots \partial _{\dim M}$ we obtain only finitely many non-zero $\theta _l$ each of which satisfies the Novikov condition by (the proof of) Corollary 3.4. Thus, $\theta =\sum \theta _l$ satisfies the Novikov condition, too.

If $c\ge 1$ then Lemma 3.5 implies that each $\theta _i\in \mathrm {RFC}_{k+1}^i(\mathscr {A}_f^\tau )$ is a finite sum of elements in $\mathfrak {C}$ (as opposed to a general Novikov sum.) Thus, using again Lemma 3.5 we see that $\theta =\sum _{l=-\infty }^{l(\xi )}\theta _l$ satisfies the Novikov condition.

In both cases the equation $\partial \theta =\xi$ holds by construction. Indeed, the part of

$$\begin{aligned} \partial \theta =\sum _{l=-\infty }^{l(\xi )}\partial \theta _l=\sum _{l=-\infty }^{l(\xi )}\sum _{i=0}^\infty \underbrace{\partial _i\theta _l}_{\in \mathrm {RFC}_k^{l-i}}\in \mathrm {RFC}_k(\mathscr {A}_f^\tau ) \end{aligned}$$

(3.37)

in $\mathrm {RFC}_k^r(\mathscr {A}_f^\tau )$ is $\displaystyle \sum _{i=0}^{l(\xi )-r}\partial _i\theta _{i+r}$. By relabeling $I=l(\xi )-r$ we compute

$$\begin{aligned} \begin{aligned} \partial \theta&=\sum _{r=-\infty }^{l(\xi )}\sum _{i=0}^{l(\xi )-r}\partial _i\theta _{i+r}\\&=\sum _{I=0}^\infty \sum _{i=0}^{I}\partial _i\theta _{l(\xi )+i-I}\\&=\sum _{I=0}^\infty \xi _{l(\xi )-I}\\&=\sum _{l=-\infty }^{l(\xi )}\xi _l\\&=\xi , \end{aligned}\end{aligned}$$

(3.38)

where we used Eq. (3.35) in the third equality. Thus, for every $\xi \in \mathrm {RFC}_k(\mathscr {A}^\tau _f)$ with $\partial \xi =0$ we constructed $\theta \in \mathrm {RFC}_{k+1}(\mathscr {A}^\tau _f)$ with $\partial \theta =\xi$. This finishes the proof.$\square$

Proof of Theorem 1.2 for $2c\nu \le -\dim M$

In this proof we make the assumption that the Morse function $f:M\rightarrow \mathbb {R}$ additionally satisfies $f(M)\subset (0,1)$. We fix $\xi \in \mathrm {RFC}_k(\mathscr {A}_f^\tau )$ with

$$\begin{aligned} \partial \xi =0\;. \end{aligned}$$

(3.39)

We will again construct $\theta \in \mathrm {RFC}_{k+1}(\mathscr {A}_f^\tau )$ with $\partial \theta =\xi$. This time we split $\xi$ as follows.

$$\begin{aligned} \xi =\sum _{A\in \Gamma _M}\xi _A\quad \text {with}\quad \xi _A=\sum _{\begin{array}{c} {w\in \mathfrak {C}_k} \\ {[\Pi (w)]\in \mathrm {Crit}(f)\times \{A\}} \end{array}}a_ww\;. \end{aligned}$$

(3.40)

Claim 2

If $\mathcal {M}\Big (\big ([v,B],\hat{\eta }\big ),\big ([u,A],\eta \big )\Big )\ne \emptyset$ then $A=B\in \Gamma _M\cong \Gamma _E$.

Proof of Claim 2

We compare action and Conley–Zehnder index of $\Pi \big ([u,A],\eta \big )$ and $\Pi \big ([v,B],\hat{\eta }\big )$. Using that the moduli space is non-empty we conclude

$$\begin{aligned} \mathfrak {a}_f\big (\Pi \big ([u,A],\eta \big )\big )=\omega (A)-f(\wp (u))\ge \omega (B)-f(\wp (v))=\mathfrak {a}_f\big (\Pi \big ([v,B],\hat{\eta }\big )\big )\nonumber \\ \end{aligned}$$

(3.41)

and

$$\begin{aligned} \begin{aligned} \mu _{\mathrm {CZ}}^M\big (\Pi \big ([u,A],\eta \big )\big )&=-\mu _{\mathrm {Morse}}\big (\wp (u),f\big )+\tfrac{1}{2}\dim M+2c_1^{TM}(A)\\&\ge -\mu _{\mathrm {Morse}}\big (\wp (v),f\big )+\tfrac{1}{2}\dim M+2c_1^{TM}(B)\\&=\mu _{\mathrm {CZ}}^M\big (\Pi \big ([v,B],\hat{\eta }\big )\big )\;. \end{aligned}\end{aligned}$$

(3.42)

We assume now that $A\ne B$. We recall that $f(M)\subset (0,1)$. Thus, the first inequality simplifies to

$$\begin{aligned} \omega (A)>\omega (B) \end{aligned}$$

(3.43)

since $\omega \big (\pi _2(M)\big )=\nu \mathbb {Z}$. From $c_1^{TM}=c\omega$ with $c<0$ we conclude then $c_1^{TM}(A)<c_1^{TM}(B)$. The minimal Chern number of M equals $-c\nu$. Thus, we have

$$\begin{aligned} c_1^{TM}(A)\le c_1^{TM}(B)+c\nu \end{aligned}$$

(3.44)

and from (3.42)

$$\begin{aligned} \begin{aligned} 2c_1^{TM}(A)&\ge -\mu _{\mathrm {Morse}}\big (\wp (v),f\big )+\mu _{\mathrm {Morse}}\big (\wp (u),f\big )+2c_1^{TM}(B)\\&\ge -\dim M+2c_1^{TM}(B). \end{aligned}\end{aligned}$$

(3.45)

We conclude that

$$\begin{aligned} 2c\nu \ge -\dim M. \end{aligned}$$

(3.46)

In case $2c\nu <-\dim M$ we arrive at a contradiction. It remains to treat the case $2c\nu =-\dim M$. In this case we claim that the inequality (3.44) necessarily becomes the equality

$$\begin{aligned} c_1^{TM}(A)=c_1^{TM}(B)+c\nu . \end{aligned}$$

(3.47)

Otherwise (3.44) is actually of the form $c_1^{TM}(A)\le c_1^{TM}(B)+2 c\nu$ since $-c\nu$ is the minimal Chern number of M. As above this implies then that $4cv\ge -\dim M$, i.e. $2\dim M\le \dim M$, and thus $\dim M=0$. I.e. we are left with the case $\Sigma =S^1\subset \mathbb {C}=E$ in which Theorem 1.2 is true: $\mathrm {RFH}_*(S^1,\mathbb {C})=0$, [1, 9]

We combine $c_1^{TM}(A)=c_1^{TM}(B)+c\nu$ with (3.42) and arrive at

$$\begin{aligned} \mu _{\mathrm {Morse}}\big (\wp (v),f\big )-\mu _{\mathrm {Morse}}\big (\wp (u),f\big )\ge -2c\nu =\dim M \end{aligned}$$

(3.48)

which turns the inequality (3.42) into an equality:

$$\begin{aligned} \mu _{\mathrm {CZ}}^M\big (\Pi \big ([u,A],\eta \big )\big )=\mu _{\mathrm {CZ}}^M\big (\Pi \big ([v,B],\hat{\eta }\big )\big ). \end{aligned}$$

(3.49)

Now we proceed as in the proof of Proposition 3.3 in order to conclude that all element in $\mathcal {M}\Big (\big ([v,B],\hat{\eta }\big ),\big ([u,A],\eta \big )\Big )$ are actually differentials which are entirely contained in fibers of E and thus $A=B$, again by Proposition 3.3. $\square$

We recall that we split the cycle $\xi$ as

$$\begin{aligned} \xi =\sum _{A\in \Gamma _M}\xi _A\;. \end{aligned}$$

(3.50)

It follows from the Claim 2 and $\partial \xi =0$ that

$$\begin{aligned} \partial \xi _A=0\quad \forall A\in \Gamma _M\;. \end{aligned}$$

(3.51)

Observe that for every A the sum

$$\begin{aligned} \xi _A=\sum _{\begin{array}{c} {w\in \mathfrak {C}_k} \\ {[\Pi (w)]\in \mathrm {Crit}(f)\times \{A\}} \end{array}}a_ww \end{aligned}$$

(3.52)

is finite. Indeed, we know that w is of the form $w=([u,A],\eta )$ with fixed A and u being an l-fold cover of a simple Reeb orbit over a critical point of f. Moreover, the index of w is fixed: $\mu (w)=k$. Therefore, Definition 2.8 of the index $\mu$ and the index formula Lemma 2.7 allow only for finitely many combinations. The number of possibilities is bounded by $\tfrac{1}{2}\dim M$. In particular, the number of possibilities does not depend on A.

Now, we apply again the inductive procedure (3.30) from the proof in case $c\ge 0$ to obtain $\theta _A\in \mathrm {RFC}_{k+1}(\mathscr {A}_f^\tau )$ with

$$\begin{aligned} \partial \theta _A=\xi _A. \end{aligned}$$

(3.53)

If we set

$$\begin{aligned} \theta :=\sum _{A\in \Gamma _M}\theta _A \end{aligned}$$

(3.54)

then Claim 2 implies $\partial \theta =\xi$. As above it remains to check that $\theta$ satisfies the Novikov condition (2.56). For this we express for some A

$$\begin{aligned} \theta _A=\sum _{\begin{array}{c} {z\in \mathfrak {C}_{k+1}} \\ {[\Pi (z)]\in \mathrm {Crit}(f)\times \{A\}} \end{array}}b_zz\;. \end{aligned}$$

(3.55)

The same argument we used to conclude that each $\xi _A$ is a finite sum gives the same for $\theta _A$. Moreover, since $\mu (\theta _A)=k+1$, $\partial \theta _A=\xi _A$ again the claim, the index formula Lemma 2.7 and the computation of the action (2.11) implies that exists $C>0$ such that

$$\begin{aligned} |\mathscr {A}_f^\tau (z)-\mathscr {A}_f^\tau (w)|\le C \end{aligned}$$

(3.56)

whenever $\mathcal {M}(w,z)\ne \emptyset$ for some w appearing in $\xi _A$ and z in $\theta _A$. The constant C does not depend on A, indeed we may choose $C=\tfrac{\tau }{2}\dim M+\max f-\min f$.

Thus, $\xi$ satisfying the Novikov condition implies that $\theta$ satisfies the Novikov condition since their actions are of bounded distance. This completes the proof.

Proof of Theorem 1.2 for $\omega \big (\pi _2(M)\big )=0$

We follow the proof of the case $2c\nu \le -\dim M$. We first establish Claim 2, i.e. that $\mathcal {M}\big (\big ([v,B],\hat{\eta }\big ),\big ([u,A],\eta \big )\big )\ne \emptyset$ implies $A=B\in \Gamma _M$ holds without assuming $c_1^{TM}=c\omega$ under the assumption that the Morse function $f:M\rightarrow \mathbb {R}$ is sufficiently small.

We assume otherwise. Then we find a sequence $\epsilon _n\rightarrow 0$ and a sequence of elements $w_n\in \mathcal {M}\big (\big ([v_n,B_n],\hat{\eta }_n\big ),\big ([u_n,A_n],\eta _n\big )\big )$ where $w_n$ satisfies the Floer equation for $\mathscr {A}_{f_n}^\tau$ with $f_n:=\epsilon _nf$. By definition of $\mathcal {M}$ we have $c_1^{TE}(-A_n\#w_n\#B_n)=0$. Since $\omega \big (\pi _2(M)\big )=0$ we can identify $c_1^{TE}=c_1^{TM}:\pi _2(E)\cong \pi _2(M)\rightarrow \mathbb {Z}$. We recall that $\Pi (w_n)$ are solutions of the Floer equation of $\mathfrak {a}_{f_n}$ with

$$\begin{aligned} c_1^{TM}(-A_n\#\Pi (w_n)\#B_n)=c_1^{TE}(-A_n\#w_n\#B_n)=0\;. \end{aligned}$$

(3.57)

We point out that the Floer cylinders $\Pi (w_n)$ topologically form spheres since their asymptotic limits lie in $\mathrm {Crit}(f_n)$. Therefore, we can rewrite

$$\begin{aligned} c_1^{TM}(-A_n\#\Pi (w_n)\#B_n)=-c_1^{TM}(A_n)+c_1^{TM}(\Pi (w_n))+c_1^{TM}(B_n)=0\;. \end{aligned}$$

(3.58)

Moreover, they have uniformly bounded energy since their energy is given by the action difference of $\mathfrak {a}_{f_n}$ which, in turn, is bounded by $\max f_n-\min f_n$ thanks to our assumption $\omega \big (\pi _2(M)\big )=0$. Therefore, we can take the Floer-Gromov limit of $\Pi (w_n)$. Floer-Gromov compactness implies that we find a bubble tree of holomorphic spheres in $(M,\omega )$. Since we assume that $\omega \big (\pi _2(M)\big )=0$ all holomorphic spheres are constant and therefore

$$\begin{aligned} c_1^{TM}(\Pi (w_n))=0 \end{aligned}$$

(3.59)

for sufficiently large n from which we conclude

$$\begin{aligned} c_1^{TM}(A_n)=c_1^{TM}(B_n) \end{aligned}$$

(3.60)

for sufficiently large n. That is, the above claim indeed holds for sufficiently small $f:M\rightarrow \mathbb {R}$. We now can proceed as in the proof of the case $2c\nu \le -\dim M$.

Remark 3.6

In the latter two cases of the proof of Theorem 1.2 we assume that the auxiliary Morse function f is very small. This is an echo of the ‘true’ proof of Theorem 1.2 in the full Morse–Bott setting, i.e. the case of $\mathscr {A}_{f=0}^\tau$. Indeed, in both cases $2c\nu \le -\dim M$ and $\omega \big (\pi _2(M)\big )=0$ the Morse–Bott differential is of the form $\partial =\partial _0+$ auxiliary Morse trajectories which immediately implies the Theorem.

4 A conjectural explanation

Let V be a Liouville domain, i.e. a compact exact symplectic manifold with contact type boundary. We recall one of the main theorems by Cieliebak–Frauenfelder–Oancea in [11]. There is a long exact sequence between symplectic (co-)homology $\mathrm {SH}$ and Rabinowitz Floer homology $\mathrm {RFH}$ as follows.

$$\begin{aligned} \cdots \longrightarrow \mathrm {SH}^{-*}(V)\longrightarrow \mathrm {SH}_*(V)\longrightarrow \mathrm {RFH}_*(\partial V,V)\longrightarrow \mathrm {SH}_{-*-1}(V)\longrightarrow \cdots \;. \end{aligned}$$

(4.1)

Moreover, the map $\mathrm {SH}^{-*}(V)\rightarrow \mathrm {SH}_*(V)$ splits as

(4.2)

where $\mathrm {PD}$ denotes Poincare duality and $d=\tfrac{1}{2}\dim V$. As observed by Ritter in [22] this long exact sequence together with the fact that $\mathrm {SH}$ is a ring with unity leads to the statement

$$\begin{aligned} \mathrm {SH}^*(V)=0\quad \Longleftrightarrow \quad \mathrm {SH}_*(V)=0\quad \Longleftrightarrow \quad \mathrm {RFH}_*(\partial V,V)=0. \end{aligned}$$

(4.3)

Note that our (co-)homology and grading conventions match with the ones in [11].

In [21] Oancea proves $\mathrm {SH}_*(E)=0$ for negative line bundles $\wp :E\rightarrow M$ under the condition that $(E,\Omega )$ is symplectically aspherical. We point out that even in the symplectically aspherical case E is not a Liouville manifold since $[\Omega ]\ne 0\in \mathrm {H}^2(E)$. Ritter computes in [23, Theorem 1] for more general negative line bundles

$$\begin{aligned} \mathrm {SH}_*(E)\cong \mathrm {QH}_{*+d}(E,\Sigma )/\ker r^k \end{aligned}$$

(4.4)

where $d=\tfrac{1}{2}\dim E$, $\mathrm {QH}_*(E,\Sigma )$ is the relative quantum homology of the disk bundle inside E with boundary $\Sigma$ and $r:\mathrm {QH}_*(E)\rightarrow \mathrm {QH}_{*-2}(E)$ is the map given by quantum intersection product with $\mathrm {PD}(\wp ^*c_1^E)\in \mathrm {QH}_{2d-2}(E)$. Finally (4.4) holds for any $k\ge \dim \mathrm {H}_*(M)$. In particular,

$$\begin{aligned} \mathrm {SH}_*(E)=0\quad \Longleftrightarrow \quad \wp ^*c_1^E \text { is nilpotent in }\mathrm {QH}^*(E)\; \end{aligned}$$

(4.5)

which generalizes Oancea’s computation. Considering for instance the bundle $\wp :\mathcal {O}(-n)\rightarrow \mathbb {C}\mathrm {P}^m$ it follows that

$$\begin{aligned} \mathrm {SH}_*(\mathcal {O}(-n))\ne 0, \end{aligned}$$

(4.6)

for $n\le m$, see [23, Section 1.5]. On the other hand, Theorem 1.2 applies since $(\mathbb {C}\mathrm {P}^m,n\omega _\mathrm {FS})$ is monotone with $c=\tfrac{m+1}{n}$ with corresponding negative line bundle $\mathcal {O}(-n)$, i.e. we conclude

$$\begin{aligned} \mathrm {RFH}_*(\Sigma ,\mathcal {O}(-n))=0. \end{aligned}$$

(4.7)

In this case $\Sigma$ is a Lens space. This is, of course, no contradiction to (4.3) since the space $\mathcal {O}(-n)$ is not a Liouville manifold. Also, $(\mathbb {C}\mathrm {P}^m,n\omega _\mathrm {FS})$ is not symplectically aspherical.

We offer the following conjectural explanation of Ritter’s result (4.4) in terms of the long exact sequence (4.1) from [11] and Theorem 1.2. We claim that the long exact sequence (4.1) remains valid for negative line bundles E (and probably even more generally) but the splitting of the map $\mathrm {SH}^{-*}(V)\rightarrow \mathrm {SH}_*(V)$ needs to be corrected as follows.

(4.8)

Here, as in [23], we identify $\mathrm {QH}_*(E,\Sigma )$ as Floer homology of a Hamiltonian with very small slope at infinity or equivalently as symplectic homology in the action window $(-\varepsilon ,\varepsilon )$. Then $c_*$ is just a continuation homomorphism induced by a canonical inclusion map. We refer to [23] for details. In particular, if $\mathrm {RFH}_*(\Sigma ,E)=0$ then the map $c_*$ is surjective and

$$\begin{aligned} \mathrm {SH}_*(E)\cong \mathrm {QH}_{*+d}(E,\Sigma )/\ker c_*. \end{aligned}$$

(4.9)

Ritter’s important observation in [23] is that $c_*$ is indeed surjective and can be identified with $r^k$ for large k under his assumptions.

As mentioned above Ritter’s and the present result holds for the bundle $\mathcal {O}(-n)\rightarrow \mathbb {C}\mathrm {P}^m$. In fact, from inspection of Ritter’s article [23] it seems that Theorem 1.2 applies to all examples Ritter considers.