Transfer morphism and handle attaching

In the following, we construct a map π∗(W, V) :SH∗(V)→SH∗(W) for an exactly em-bedded Liouville subdomainW ⊂V, as first suggested by Viterbo in [52]. Analogously, we construct a mapπ^∗(W, V) :SH^∗(W)→SH^∗(V) in cohomology. Then we will show that π∗(W, V) and π^∗(W, V) are isomorphisms if V is obtained from W by attaching a subcritical handle H_k²ⁿ as described in Section 5.

As shown above, we have always maps SH_∗(V)→SH_∗^>0(V, ∂W) and SH_>0^∗ (V, ∂W)→ SH^∗(V). The maps π∗(W, V) and π^∗(W, V) are obtained by showing the identities SH_∗^>0(V, ∂W) = SH∗(W) and SH_>0^∗ (V, ∂W) = SH^∗(W). This is done by giving an explicit cofinal sequence (H_n)⊂Ad(V, ∂W).

The following proposition is based on ideas by Viterbo, [52]. Its proof is taken from McLean, [37]. We include it here for completeness and to add a missing argument for the homotopy case. See also Cieliebak, [12], for a slightly different approach.

Proposition 92. There exists a cofinal sequence (H_n)⊂Ad(V, ∂W) and a sequence of monotone decreasing admissible homotopies (H_n,n+1) between them such that

1. K_n := H_n|_W, K_n,n+1 := H_n,n+1|_W are sequences of admissible Hamiltonians / homotopies on (W, ω).

2. all 1-periodic orbits of X_Hn in W have positive action and all 1-periodic orbits of X_Hn in V \W have negative action.

3. all A^H-gradient trajectories of Hn or Hn,n+1 connecting 1-periodic orbits in W are entirely contained inW.

Proof: It will be convenient to use z = e^r rather than r for the second coordinate in the completion (W ,ω) = 

W ∪(∂W ×[1,∞), d(zα)

. Note that we can embed W into V using the flow ofY_λ, whereλ=zα. The cylindrical end ∂W ×[1,∞) is then a subset ofV. The first coordinates will be denotedz_W for∂W ×(0,∞) andz_V for∂V ×(0,∞).

To begin, assume thatSpec(∂W, λ) and Spec(∂V, λ) are discrete and let k :N→R\

Spec(∂W, λ)∪4·Spec(∂V, λ)

be an increasing function such thatk(n)→ ∞. Let µ:N→R be defined by µ(n) = dist

k(n), Spec(∂W, λ)

= min

a∈Spec(∂W,λ)|k(n)−a|.

Choose an increasing sequence A=A(n) with A > 2k

µ >1 and A(n+ 1) >2A(n)

which satisfies additionally the conditions (⊕) and (⊕⊕) below. Note that we can always achieve ^2k_µ >1, as we may choose k arbitrarily large whilst making µ arbitrarily small.

Let alsoε(n)>0 be a sequence tending to zero.

We assume thatH_n|_W is aC²-small negative Morse function insideW\

∂W×[1−ε,1) and for 1−ε ≤z_W ≤A of the formH_n=g(z) withg(1) =−ε, g^′ ≥0 andg^′ ≡k(n) for 1≤z_W ≤A−ε. For A ≤z_W ≤ 2A we assume that H_n ≡ B is constant with B being arbitrarily close to k·(A−1).

Now we describeH_n on∂V ×[1,∞): We keep H_n constant until we reach z_V = 2A+P, whereP is some constant such that{z_W ≤1} ⊂ {z_V ≤P}, which implies{z_W ≤2A} ⊂ {z_V ≤ 2A+P}. Then let H_n = f(z_V) for z_V ≥ 2A+P with 0 ≤ f^′ ≤ ¹₄k(n) and f^′ ≡ ¹₄k(n) for z_V ≥2A+P +ε. Figure 7 gives a schematic illustration of H_n.

zW = 1 zW =A zV = 2A+P B

slopek

slope ¹₄k

Fig. 7: The Hamiltonian Hn

As the action of an XH-orbit on level z = const. is h^′(z)·z−h(z), we distinguish five types of 1-periodic orbits of X_H:

• critical points inside W of action>0 (as H is negative and C²-small inside W)

• non-constant orbits nearz_W = 1 of action ≈1·g^′(z)>0

• non-constant orbits on z_W = a for a near A of action ≈ g^′(a) · a − B <

(k−µ)·A−B ≈ −µ·A+k <−k → −∞

• critical points inA < z_W ; z_V <2A+P of action−B <0

• non-constant orbits on z_V = a for a near 2A +P of action ≈ f^′(a)·a −B ≤

1

4k·(2A+P +ε)−B ≈ −¹₂kA+k·(¹₄P +¹₄ε+ 1)<0 for Asufficiently large (this is condition (⊕)).

Obviously, (H_n) satisfies 1. and 2. of the proposition’s claims. It only remains to show thatA^H-gradient trajectories connecting two orbits of non-negative action are contained entirely inside z_W ≤ 1. By Gromov’s Monotonicity Lemma (see [49], Prop. 4.3.1 and [42], Lem. 1) there exists a K > 0 such that any J-holomorphic curve which intersects zW = A and zW = 2A has area greater than KA. Note that inside A ≤ zW ≤ 2A the equation (54) reduces to an ordinary J-holomorphic curve equation, as X_H ≡ 0 there. Any A^H-gradient trajectory connecting two orbits of non-negative action which intersects zW = A and zW = 2A has therefore area greater than KA – in other words the action difference between its ends is greater than KA.

Fork(n) fixed, the maximal action difference of two 1-periodic orbits in W is bounded from above. So forA(n) sufficiently large (this is condition (⊕⊕)) no such A^H-gradient trajectory can touchz_W = 2A. It follows then from the Maximum Principle that in fact all theseA^H-gradient trajectories have to remain inside z_W ≤1.

For the construction of the homotopies Hn,n+1 we have to sharpen this argument. As A(n+ 1)>2A(n), we can take for H_n,n+1 the following interpolations:

At first, in time s ∈ [0,1/2], decrease H_n+1 in the area z_W ≤ 2A(n) to H_n and keep it unchanged in z_W ≥A(n+ 1). Then decrease in time s ∈[1/2,1] the remaining part to H_n (see Figures 8 and 9).

Fors∈[−∞,1/2] the homotopyH_n,n+1is then constantB(n+ 1) in the areaA(n+ 1)≤ z_W ≤2A(n+1) so that noA^H-gradient trajectory can leavez_W ≤1 in this time interval.

Fors∈[1/2,∞] the homotopy H_n,n+1 is constant B(n) in the areaA(n)≤z_W ≤2A(n) so that again noA^H-gradient trajectory can leave z_W ≤1 in this time interval.

Hn

Hn+1

zW = 1

A(n) 2A(n)

A(n+ 1) 2A(n+ 1) B(n+1)

B(n)

Fig. 8: Two Hamiltonian Hn and Hn+1

zW = 1

A(n) 2A(n)

A(n+ 1) 2A(n+ 1) B(n+1)

B(n)

Fixed for times≥¹₂ Fixed for times≤¹₂

Fig. 9: The homotopyHn,n+1 at time s= ¹₂

Corollary 93. SH_∗^>0(V, ∂W)≃SH_∗(W) and SH_>0^∗ (V, ∂W)≃SH^∗(W).

Proof: We only prove the corollary for homology, cohomology being completely analog.

Take the sequence of Hamiltonians (H_n) constructed in Proposition 92. Clearly it is cofinal and (H_n) ⊂ Ad^>0(V, ∂W), as 1-periodic orbits with positive action are either isolated critical points inside W (as H is Morse and C²-small there) or isolated Reeb-orbits nearz_W = 1 – in both cases non-degenerate. Hence we have

SH_∗^>0(V, ∂W) = lim

−→F H_∗^>0(H_n).

Let H˜_n ∈ Ad(W) be the linear extension of H_n|_W with slope k(n). Due to k(n) ̸∈ Spec(∂W, λ), we have obviously F C_∗^>0(H_n) = F C_∗( ˜H_n). As any A^H-gradient trajectory connecting 1-periodic orbits inW stays inW, the two boundary operators∂^Hn and∂^H˜n coincide and we have F H_∗^>0(H_n) = F H∗( ˜H_n). As the A^H-gradient trajectories for the homotopiesH_n,n+1 stay inside W, the continuation maps

σ(Hn+1, Hn) :F H_∗^>0(Hn)→F H_∗^>0(Hn+1) coincide with the continuation maps

σ( ˜H_n+1,H˜_n) :F H∗( ˜H_n)→F H∗( ˜H_n+1).

Hence we have SH_∗^>0(V, ∂W) = lim

−→F H_∗^>0(H_n) = lim

−→ F H_∗( ˜H_n) = SH_∗(W).

We finish this section with the following Invariance Theorem due to Cieliebak, [12].

Theorem 94 (Invariance of SH under subcritical surgery).

Let W be an exact Liouville domain and let V be obtained from W by attaching to

∂W ×[0,1] a subcritical exact symplectic handle H_k²ⁿ, k < n, as described in Section 5.

Moreover, assume that the Conley-Zehnder index is well-defined on W. Then it holds that

SH_∗(V)∼=SH_∗(W) and SH^∗(V)∼=SH^∗(W).

Proof: The idea of the proof is to construct yet another cofinal sequence of Hamilto-nians (H_n)⊂Ad^w(V)∩Ad(V, ∂W) for which we can directly show that

SH∗(W)

(∗)∼= SH_∗^>0(V, ∂W) ⁽¹⁾= lim

n→∞F H_∗^>0(H_n)⁽²⁾≃ lim

n→∞F H∗(H_n)⁽³⁾= SH∗(V) SH^∗(W)

(∗)∼= SH_>0^∗ (V, ∂W) ⁽⁴⁾= lim

n→∞F H_>0^∗ (Hn)⁽⁵⁾≃ lim

n→∞F H^∗(Hn)⁽⁶⁾= SH^∗(V).

Note that the isomorphisms (∗) have been shown in Corollary 93.

To start, fix sequences k(n) ̸∈Spec(∂W, λ), k(n)→ ∞ and ε(n) →0. Then choose an increasing sequence of non-degenerate HamiltoniansH_nonW that is on∂W×(−ε(n),0]

of the form

H_n|_{∂W×(−ε(n),0]} =k(n)·e^r−

1 +ε(n)

and extend H_n over the handle by a function ψ with α = k(n) and β = −1−ε(n) as described in Section 5.

For eachn choose the handle so thin such that each trajectory ofXHn which leaves and reenters the handle has length greater than 1. Thus we obtain a cofinal weakly admissible sequence (H_n), whose 1-periodic orbits having positive action are all contained in W. This shows already the identities (1),(3),(4) and (6). Now recall that we had the long exact sequences

· · · →F H_j+1^>0 (H_n)→F H_j^≤0(H_n)→F H_j(H_n)→F H_j^>0(H_n)→. . .

· · · →F H_≤0^j−1(H_n)→F H_>0^j (H_n)→F H^j(H_n)→F H_≤0^j (H_n)→. . .

Note that F H_j^>0(H_n) is generated by all 1-periodic orbits of H_n inside W, while F H_j^≤0(H_n) is generated by all other orbits. These are finitely many, lying on the handle, and are explicitly given in (51). Observe that H_n is on the handle time-independent.

The orbits there are therefore of Morse-Bott type. We can now use either the definition of SH with Morse-Bott techniques, as described in [7], or perturb H_n near these orbits to make it non-degenerate, as described in [13]. In both cases we obtain for each orbit γ two generators in the chain complex whose indices are µ_CZ(γ) and µ_CZ(γ) + 1. We have shown in Section 5.4 that the possible values ofµ_CZ(γ) increase to ∞as the slope α=k(n) tends to∞. Therefore,F H_j^≤0(H_n) becomes eventually zero fornlarge enough, as well asF H_j+1^≤0(Hn). This implies for n large enough that

F H_j(H_n)→F H_j^>0(H_n)

is an isomorphism. As the direct limit is an exact functor, these maps converge to an isomorphism in the limit, proving (2). In the cohomology case, the line of arguments is the same. Even though taking inverse limits is not exact, it still takes the isomorphism

F H_>0^j (Hn)→F H^j(Hn)

to an isomorphism in the limit, as it is left exact (see Theorem 73). This proves (5).

Im Dokument Rabinowitz-Floer homology on Brieskorn manifolds (Seite 135-140)