3. Compactness and additional properties 63
3.4. A Z -grading for RF H
Now, we associate to each contractible periodic Reeb trajectory v a Conley-Zehnder indexµCZ(v), which allows us to define aZ-grading for Rabinowitz-Floer homology. For simplicity, let us make the following assumptions:
(A) The map i∗ :π1(Σ)→π1(W) induced by the inclusion is injective.
(B) The integralIc1 :π2(W)→Z of the first Chern classc1(T W)vanishes on spheres.
Remark.
• Assumption (A) is automatically satisfied if Σ is simply connected.
• One can associate a Conley-Zehnder index to every closed Reeb trajectoryv, but it will depend on the trivialization ofv∗ξ. In order to fix the trivialization, we restrict ourself to contractible trajectories. These have, due to (B), a unique Conley-Zehnder index (see below). Note that the two ends of a solution of the Rabinowitz-Floer equation are either both contractible or not. Hence, the contractible closed Reeb trajectories generate a subcomplex ofRF C(W,Σ). Its homology is by abuse of notation also denoted by RF H(W,Σ). If Σ is simply connected this version of Rabinowitz-Floer homology coincides with the original one.
To obtain the (transversal) Conley-Zehnder index of a closed contractible Reeb trajectory v choose a mapufrom the unit discD ⊂Cto Σ such thatu(e2πit) = v(t). The existence of such maps is guaranteed by assumption (A). Now choose a symplectic trivialization Φ :D×R2n−2 →u∗ξ of the pullback bundle (u∗ξ, u∗dα). Such trivializations exist and are homotopically unique as D is contractible. The linearization of the Reeb flow ψt alongv with respect to Φ defines a path Ψ in the group Sp(2n−2) starting at 1 by
Ψ(t) := Φ(v(t))−1 ◦dψt(v(0))◦Φ(v(0)).
The Conley-Zehnder index of this path is the (transverse) Conley-Zehnder indexµCZ(v).
It is independent from the choice ofudue to assumption (B). Indeed, if we choose another disc u′ :D→Σ with u′|∂D =v and another trivialization Φ′ ofu′∗ξ, then we can glue u and u′ together to a mapw:S2 →Σ.
Now, (w∗ξ, w∗dα) is trivial. Indeed,
w∗c1(ξ) =
w∗c1(T V) = 0 and hence [w∗c1(ξ)] = 0∈H2(S2), as the complement ofξinT V is trivialized by the Reeb and Liouville vector field. Therefore, there exists a trivialization Φ′′ ofw∗ξand both Φ and Φ′ are homotopic to Φ′′ along v (see [47, Lem.5.2]).
The transversal Conley-Zehnder indexµCZ allows us to grade RF H as follows.
Proposition 61(Frauenfelder & Cieliebak, [14]). Ifπ1(Σ)→π1(V)is injective and Ic1 vanishes, then we have a Z-grading of the Rabinowitz-Floer homology RF H(Σ, V), which is independent of V and given by the index
µ(c) := µCZ(c) +indh(c)− 1
2dimc
crit(AH) +1
2, wherec∈crit(h)and dimc
crit AH
is the real dimension of the connected component of crit
AH
which containsc.
Proof: Recall that the boundary operator ∂F counted points in the zero-dimensional manifoldsM(c−, c+, m), which where given as quotientsM(c −, c+, m) /Rm ifm̸= 0 and M(c −, c+,0) /R if m = 0. The Global Transversality Theorem 38 gave the following dimension formula near a flow line (v, η) withm cascades by
dim(v,t) M(c −, c+, m)m̸=0=
µCZ(c+,¯c+) +indh(c+)− 1
2dimc+(crit AH
)
−
µCZ(c−,¯c−) +indh(c−)− 1
2dimc−(crit AH
) +m−1 +
m
k=1
2c1(¯v−k#vk#¯vk+) dim(v,t) M(c −, c+, m)m=0= indh(c+)−indh(c−).
Using condition (B), we get a dimension formula for the moduli space by dim(v,t) M(c−, c+, m) =
µCZ(c+,c¯+) +indh(c+)−1
2dimc+(crit AH
)
−
µCZ(c−,¯c−) +indh(c−)− 1
2dimc−(crit AH
)
−1
=µ(c+)−µ(c−)−1.
Note that the formula holds also for m = 0, as then µCZ(c+,c¯+) = µCZ(c−,c¯−) and dimc+
crit AH
= dimc− crit
AH
, since (v, η) is simply a Morse trajectory on one connected component of crit
AH
. The moduli space is hence zero-dimensional if and only ifµ(c+)−µ(c−) = 1. It follows that ∂F reduces the index µexactly by 1 so that µ provides a well-defined grading for the homology.
Discussion 62. The term 12 in the definition of µ, which does not appear in [14], has no influence on the relative grading given by the other terms. It normalizesµsuch that it takes values in Z and fits with the grading of symplectic (co)homology (see [16]).
Note that this convention differs also from the one used previously by the author in [23], where 12dim Σ was added instead of 12. So all indices in this work are shifted by
−12dim Σ + 12 =−n+ 1 =−(n−2)−1 in comparison to the indices in [23].
The gradingµ allows us to define more refined invariants for contact structure:
Definition 63. Let ck(W) := dim2RF Hk(W,Σ) denote the kth Betti-number of the Rabinowitz-Floer homology of the filling W of Σ. We define
Ck(Σ) :={ck(W)|W Liouville domain, ∂W = Σ} ⊂[0,∞]
to be the set of all Betti numbers ck(W) for varying fillings W of Σ.
Observe that Ck(Σ) is (trivially) independent from W. It is therefore an invariant of (Σ, ξ), asRF Hk(W,Σ) was apart from ξ only dependent onW. The most simplest case is, when |Ck(Σ)|= 1, i.e. if the group RF Hk(W,Σ) does not depend on W at all and is itself an invariant of the contact structure.
Proposition 64. Assume that (W,Σ) is a Liouville domain satisfying (A) and (B).
Then RF Hk(W,Σ)is independent ofW, ifΣadmits a contact form for which the closed contractible Reeb orbits are Morse-Bott (MB) and for all c∗ = (v∗, η∗) ∈ RF C∗(W,Σ) with ∗ ∈ {k−1, k, k+ 1} holds that ηk−1 ≥ηk≥ηk+1.
Proof: At first, observe that RF H(W,Σ) does not depend on the particular contact form. Moreover, the grading µ of the chain complex and the chain complex itself do not depend on W. The chain groups RF Ck(W,Σ) are hence independent of W. Now consider (v∗, η∗)∈RF C∗(W,Σ) with∗ ∈ {k−1, k, k+ 1}.
If ηk−1 ≥ ηk ≥ ηk+1, the Lemmas 15 and 16 tell us that if there are flow lines with cascades fromck−1 to ck or fromck tock+1 then they must have zero cascades, i.e. they are h-Morse flow lines on Ck, as each cascade would reduce the action. Since h-Morse flow lines are independent of the fillingW, so are the boundary operators
∂∗F :RF C∗+1(W,Σ)→RF C∗(W,Σ), ∗ ∈ {k−1, k}
and hence the quotient
RF Hk(W,Σ) = ker∂k−1F im∂kF .
In [16, Cor.1.15], Cieliebak, Frauenfelder and Oancea proved another criterion under which RF Hk(W,Σ) does not depend on W, i.e. where|Ck(Σ)|= 1 for allk.
Theorem 65. Let (W,Σ) be a Liouville domain, dimW = 2n, satisfying assumption (A) and (B).RF Hk(W,Σ)is independent of W for all k if Σ admits a contact form for which all contractible closed Reeb orbits v are Morse-Bott (MB) and satisfy
µCZ(v)>3−n.
Proof: (Sketch)
The theorem is shown by proving that all AH-gradient trajectories lie entirely in the symplectization of Σ and are hence independent of the filling. This follows from a Gromov compactness argument, as trajectories leaving the symplectization would lead to the bubbling-off of holomorphic planes. The latter cannot happen due to the condition µCZ(v)>3−n.
We finish the section with a short discussion of what happens with (A) and (B) under handle attachment. Let (W,Σ) be a Liouville domain satisfying (A) and (B). Assume that (W′,Σ′) is obtained from (W,Σ) by attaching ak-handleHk2nas described in Section 5. In general, one cannot (topologically) expect that (A) still holds for (W′,Σ′) as is shown by attaching a 1-handle to the unit ball inR3. The result is diffeomorphic to the full 2-torus, which violates (A). However, we have the following lemma.
Lemma 66. Assume that (W,Σ) satisfies (A), dimW = 2n ≥ 4 and that (W′,Σ′) is obtained by attaching ak-handle Hk2n, k ≤n−2. Then (W′,Σ′) satisfies (A) if
1. k≥3 or
2. k= 1, W has 2 components (W1,Σ1), (W2,Σ2) and Hk2n is glued to W so that Σ′ is the connected sum Σ1#Σ2 resp. W′ is the boundary connected sum W1#W2.
Proof:
@1. The first part of this proof follows Milnor, [39, Lem.2]. Set l = 2n−k and note that k < l. Let X denote the space which is obtained by gluing the handle Hk2n ∼= Dk ×Dl to Σ. It is formed from the topological sum Σ + (Dk×Dl) by identifyingSk−1×Dl with the attaching region in Σ. The subset Σ∪(Dk×0) is a deformation retract ofX. This subset is formed from Σ by attaching a k-cell. It follows thus that the map π1(Σ)→π1(X) induced by inclusion is an isomorphism ask≥3. But Σ′ is also embedded topologically in X and similar arguments show that π1(Σ′)→π1(X) is also an isomorphism as l > k ≥3.
Analogously, we consider the space Y which is obtained by gluing the handle Hk2n to W. Note that Y = W′ and the same reasonings as for Σ shows that π1(W) → π1(W′) is also an isomorphism. Combining all these spaces, we obtain the following diagram:
π1(Σ)
(a) //π1(X)
(b)
π1(Σ′)
(c)
oo
π1(W) //π1(W′) π1(W′) .
As all the maps are induced by inclusions, the above diagram commutes. Moreover, all horizontal maps are isomorphisms. As (a) is injective by assumption, it follows from the commutativity of the left square that (b) is also injective. Then it follows from the commutativity of the right square that (c) is also injective.
@2. Without loss of generality we assume thatW =W1∪W2and Σ = Σ1∪Σ2, such that W′ =W1#W2 and Σ′ = Σ1#Σ2. Now we apply the van Kampen Theorem in Σ′ to the two sets Σ′1 ∼= Σ1\ {pt.} and Σ′2 ∼= Σ2\ {pt.}coming from Σ1 and Σ2 plus the connecting cylinder. This gives a mapπ1(Σ1\ {pt.})∗π1(Σ2\ {pt.})→π1(Σ1#Σ2) induced by inclusion.
Analogously, we obtain a map π1(W1\ {pt.})∗π1(W2\ {pt.})→π1(W1#W2). As the intersections Σ′1 ∩Σ′2 ∼= D1×S2n−2 and W1′ ∩W2′ ∼= D1×D2n−1 are simply connected (as n ≥ 2), both maps are isomorphisms. Moreover, they fit into the following diagram:
π1(Σ1)∗π1(Σ2)
(a)
π1(Σ1\ {pt.})∗π1(Σ2\ {pt.})
(b)
oo //π1(Σ1#Σ2)
(c)
π1(W1)∗π1(W2)oo π1(W1\ {pt.})∗π1(W2\ {pt.}) //π(W1#W2) .
Again all the maps are induced by inclusion, so this diagram also commutes. The two horizontal maps on the left are isomorphisms as dim Σ,dimW ≥3. Therefore, all horizontal maps are isomorphisms. As (a) is injective by assumption, it follows as above that (c) is also injective.
A similar result holds for the behaviour of (B) under handle attachment.
Lemma 67. Let(W,Σ)be as above, satisfying (B). Let (W′,Σ′)be obtained by attaching a k-handle. Then (W′,Σ′) also satisfies (B) if k = 1 or k ≥3.
Proof: Assume thatIc1 :π2(W)→Zvanishes. AsW′is obtained fromW by attaching a handle, W ⊂W′ is an open subset and c1(T W′)|V =c1(T W). As in the proof above, there is a retraction ρ : W′ = W ∪Hk2n → W ∪Dk, where ∂Dk is identified with an embedded sphereSk−1 in Σ. Let w:S2 →W′ be any smooth sphere.
• Ifk≥3, we may assume that there is a point qin the interior of the attached disc Dk such thatq ̸∈im(ρ◦w). Note that there exists a retraction R:W′\ {q} →W and that R ◦w is well-defined. It follows that w and R◦w are homotopic and hence
S2
w∗c1(T W′) =
S2
(R◦w)∗c1(T W′) =
S2
(R◦w)∗c1(T W) = 0.
• If k = 1, then D1 is a line. Hence we may split ρ◦w into a finite number of spheres ˜wi such that no ˜wi passes over the line. It may well hit its centre but D1 ⊈ image( ˜wi). Now, we may homotope each ˜wi into W to obtain spheres wi :S2 →W. Then
S2
w∗c1(T W′) =
S2
(ρ◦w)∗c1(T W′) =
i
S2
˜
wi∗c1(T W′)
=
i
S2
wi∗c1(T W) = 0.