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|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
101942634cd545c4cdd5085c20eb1e894cea2a1c | subsection | 94 | 126 | Proof of Proposition | Furthermore, these
evaluation maps are invariant under the domain and Reeb rotations used
to obtain transversality for \sbox
{\operatorname{ev}}\widetilde{\usebox {
}}_Y and for
\sbox
{\operatorname{ev}}\widetilde{\usebox {
}}_{W,Y}
in the vertical directions, so the transversality follows immediately. | {
"cite_spans": []
} | 1804.08013 | Morse-Bott Split Symplectic Homology | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
-0.07513494789600372,
0.031189244240522385,
-0.01513685379177332,
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0.002050418173894286,
0.005458880681544542,
0.03... | |
6a993ba9650ebe987ec34e02042155a0bc5e52ea | subsection | 95 | 126 | Monotonicity and the differential | The results of the previous section show that the moduli spaces of
Floer cylinders with cascades that project to simple chains of pearls are
transverse.We now impose monotonicity conditions on (X, \omega ) and on (\Sigma ,
\omega _\Sigma ) in order to show that these moduli spaces are sufficient for
the purposes of def... | {
"cite_spans": []
} | 1804.08013 | Morse-Bott Split Symplectic Homology | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
-0.027718745172023773,
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-0.014340177178382874,
0.0021407236345112324,
-0... | |
90d8dad86e4defaf3b5acadfdec1ae48f9f4d173 | subsection | 96 | 126 | Index inequalities from monotonicity and transversality | First, we consider the Fredholm index contributions of a plane in W that could
appear as an augmentation plane, to obtain some bounds on the possible indices.Lemma 6.1
If v \colon W is a J_W holomorphic plane asymptotic to a
given closed Reeb orbit \gamma in Y,
the Fredholm index for the deformations of v
(as an unpar... | {
"cite_spans": []
} | 1804.08013 | Morse-Bott Split Symplectic Homology | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
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0.012533742934465408,
0.016828628256917,
0... | |
49157ce9645ab5976da4e446ce1a6a8738e53927 | subsection | 97 | 126 | Index inequalities from monotonicity and transversality | See Figure REF .
A cylinder with one augmentation puncture and whose projection
to \Sigma is trivial. The positive puncture converges to an orbit p_{k_+}
and the negative puncture converges to an orbit \widehat{q}_{k_-}. The augmentation
plane has index 0.
If B \in H_2(X;\mathbb {Z}) is the class represented by the au... | {
"cite_spans": []
} | 1804.08013 | Morse-Bott Split Symplectic Homology | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
-0.03117370791733265,
0.0024719240609556437,
-0.02340698428452015,
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0.0406494177877903,
-0.014892579056322575,
0.005443573463708162,
-0.013954163528978825,
0.... | |
9716efc3dcea71c784cae7d47a3c84b1bda12ea0 | subsection | 98 | 126 | Index inequalities from monotonicity and transversality | \end{aligned}By Lemma REF , we have that for each j=1, \dots , k,
| \gamma _j|_0 \ge 0.Consider the chain of pearls in \Sigma obtained by projecting the upper level
of this split Floer trajectory to \Sigma .
By Proposition REF , if this is a simple chain of pearls, it has Fredholm indexI_\Sigma M(p) + 2 \langle c_1(T\S... | {
"cite_spans": []
} | 1804.08013 | Morse-Bott Split Symplectic Homology | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
-0.0444713830947876,
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0.03394109755754471,
-0.020465383306145668,
0.015429159626364708,
-0.02635623887181282,
0.00928... | |
4194448e5944a5b55dd6ec16e2ce1129d47a3959 | subsection | 99 | 126 | Index inequalities from monotonicity and transversality | Then, either i(\sbox
{p}\widetilde{\usebox {
}}) = i(\sbox
{q}\widetilde{\usebox {
}})
or \sbox
{p}\widetilde{\usebox {
}} = \check{p} and \sbox
{q}\widetilde{\usebox {
}} = \hat{q}. Since N =0,
this is a pure Morse differential term.
N_1 = 1, N_0 = k = 0 and \sbox
{p}\widetilde{\usebox {
}} = p,
\sbox
{q}\widet... | {
"cite_spans": []
} | 1804.08013 | Morse-Bott Split Symplectic Homology | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
-0.005887405946850777,
-0.008119506761431694,
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0.019276197999715805,
0.009409165941178799,
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0.00795162282884121,
0.009981499053537846,
-0.01836046390235424,
0.00... | |
76f2f9d7b10a234845c0818b350468f2809e341b | subsection | 100 | 126 | Index inequalities from monotonicity and transversality | Let \gamma _j, j=1, \dots ,
k, be the corresponding Reeb orbits with multiplicities k_j = B_j \bullet \Sigma = K \omega (B_j).Let B \in H_2(X) be the spherical homology class in X represented by the
lower level v_0 in W, connecting to the critical point x.
Let k_- = B \bullet \Sigma be the multiplicity of the orbit
to ... | {
"cite_spans": []
} | 1804.08013 | Morse-Bott Split Symplectic Homology | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
-0.021131351590156555,
0.02991955354809761,
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-0.013106015510857105,
-0.021695872768759727,
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-0.02869896963238716,
0.012274493463337421,
-0.004760275594890118,... | |
771ea86b9eb5bd409134bacda3f9cc346658371b | subsection | 101 | 126 | Index inequalities from monotonicity and transversality | By passing to a simple underlying chain of pearls as necessary, and applying
monotonicity and Proposition REF (to \mathcal {M}^*_{k,(X,\Sigma )}((B;A_1,\ldots ,A_N);x,p,J_W)), we obtainI_X &M(p) + 2 \langle c_1(T\Sigma ), A \rangle + 2 \left( \langle c_1(TX), B \rangle - B \bullet \Sigma \right) + M(x) - 2n +1 + N + 2k... | {
"cite_spans": []
} | 1804.08013 | Morse-Bott Split Symplectic Homology | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
-0.019056683406233788,
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-0.02006368152797222,
-0.007556302938610315,
... | |
dd19000527468cd2858d2abc762b7a4f25a8b803 | subsection | 102 | 126 | Index inequalities from monotonicity and transversality | We will now adapt an argument originally due
to Biran and Khanevsky to show that
if \overline{W} is a Weinstein domain (or equivalently, if W is a Weinstein
manifold of finite-type), and \Sigma has minimal Chern number at least 2,
then there can only be rigid augmentation planes
if the isotropic skeleton has codimensio... | {
"cite_spans": []
} | 1804.08013 | Morse-Bott Split Symplectic Homology | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
-0.04799896478652954,
0.018903598189353943,
0.005069185048341751,
-0.002668092492967844,
0.011076684109866619,
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0.010733398608863354,
0.03728845342993736,
0.015974227339029312,
-0.021451540291309357,
-... | |
c90663583445ff75a06018da2db786ecebe5ab92 | subsection | 103 | 126 | Index inequalities from monotonicity and transversality | Thus, the augmentation plane can only exist if there is a spherical class B with (\tau _X - K ) \,\omega (B) = 1.By applying Lemma REF ,
we have B = \imath _*A, where A \in \pi _2(\Sigma ) is a spherical class in \Sigma .Now observe that \langle c_1(T\Sigma ), A \rangle + \langle c_1(N\Sigma ), A
\rangle = \langle c_1(... | {
"cite_spans": []
} | 1804.08013 | Morse-Bott Split Symplectic Homology | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
-0.03613597899675369,
0.024370422586798668,
-0.016236774623394012,
-0.002216533524915576,
-0.0036834131460636854,
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0.016236774623394012,
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0.020585911348462105,
-0.00939260795712471,
0.00029804935911670327,
0.02290545031428337,
-0.02101319469511509,
... | |
ff2784d4d39837fe54d4cd3b31653824058f9242 | subsection | 104 | 126 | Orientations | In order to orient our moduli spaces, we will take the
point of view of coherent orientations, which is implemented in the Morse–Bott setting in
,.
Some authors
,
have used the alternative approach of canonical orientations. We
find it more straightforward to use coherent orientations in our
computations, especially si... | {
"cite_spans": []
} | 1804.08013 | Morse-Bott Split Symplectic Homology | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
-0.02568386308848858,
-0.005119985435158014,
-0.04419516772031784,
0.03339054808020592,
-0.021242979913949966,
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0.004906334914267063,
0.0026210357900708914,
0.06781882792711258,
0.03339054808020592,
-0.024295132607221603,
-0.0056006996892392635,
0.0253176037222147,
0.0... | |
44ca60ce384b7830757e5701ace70bac513394f2 | subsection | 105 | 126 | Orienting the moduli spaces of curves | We now explain the first part of this method: how to orient the moduli spaces
of Floer punctured cylinders, but without considering their constraints coming from evaluation maps.
We begin by sketching the situation for the non-degenerate case and then
discuss the modifications needed for the Morse–Bott situation.First,... | {
"cite_spans": []
} | 1804.08013 | Morse-Bott Split Symplectic Homology | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
-0.00804209429770708,
-0.01490915846079588,
-0.044132329523563385,
0.04590250551700592,
-0.017686504870653152,
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0.053990382701158524,
-0.019761882722377777,
0.0024130099918693304,
0.04556678235530853,
0.02... | |
116c2df3d6bc14159685b23a97afcbc405f6618c | subsection | 106 | 126 | Orienting the moduli spaces of curves | Indeed, given two such operators&D \colon W^{1,p}(\dot{S}, E) \rightarrow L^{p}(\dot{S}, \Lambda ^{0,1}T^*\dot{S}
\otimes E) \\
{and}
&D^{\prime } \colon W^{1,p}(\dot{S}^{\prime }, E^{\prime }) \rightarrow L^{p}(\dot{S}^{\prime }, \Lambda ^{0,1}T^*\dot{S}^{\prime }
\otimes E^{\prime })that have a matching asymptotic op... | {
"cite_spans": []
} | 1804.08013 | Morse-Bott Split Symplectic Homology | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
-0.05083676427602768,
0.052728649228811264,
-0.040950141847133636,
0.002521241083741188,
0.0011910098837688565,
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0.007266208995133638,
0.015577532351016998,
0.045191626995801926,
-0.022260155528783798,
-0.041835054755210876,
0.057214245200157166,
... | |
1c8cb411b66780f03f8f21dfd5302806761634da | subsection | 107 | 126 | Orienting the moduli spaces of curves | Recall that for
fixed \delta , this conjugation is not unique, but depends on a contractible family of choices (of cut-off functions), so
the orientation of the determinant bundle does not depend on the choices.From
*Section 1.8, a coherent orientation of the determinant
bundle over non-degenerate Cauchy–Riemann opera... | {
"cite_spans": []
} | 1804.08013 | Morse-Bott Split Symplectic Homology | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
-0.02732948586344719,
-0.0082705644890666,
-0.02191241830587387,
0.05261421948671341,
-0.0003433352685533464,
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0.04446573182940483,
0.010139835067093372,
0.017639802768826485,
0.060732193291187286,
-0.023713022470474243,
-0.010078797116875648,
0.033204335719347,
0.0395... | |
325d413028369684b508754068ff0e5fbaa5ca6e | subsection | 108 | 126 | Orienting the moduli spaces of curves | We may identify the kernel (and cokernel) of this operator with those ofD^{\tilde{v}} \colon W^{1,p, -\delta }(\mathbb {R}\times S^1, \rightarrow L^{p, -\delta }({\operatorname{Hom}}^{0,1}(T(\mathbb {R}\times S^1), ).At \pm \infty , the -\delta -perturbed asymptotic operators (see Definition REF )
associated to D^{\til... | {
"cite_spans": []
} | 1804.08013 | Morse-Bott Split Symplectic Homology | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
-0.01530718244612217,
-0.02138121798634529,
-0.018237370997667313,
0.01424651499837637,
0.0036093206144869328,
0.003014126094058156,
0.04380021244287491,
0.020923376083374023,
0.03204893693327904,
0.046516742557287216,
-0.03607794642448425,
-0.017764266580343246,
0.021594878286123276,
0.00... | |
4b64943f0c71ad783e2ff308a5a4b516d632a3a0 | subsection | 109 | 126 | Orienting the moduli spaces of curves | For each k>0,\Psi _k \colon W^{1,p}(\mathbb {R}\times S^1, \rightarrow L^{p}({\operatorname{Hom}}^{0,1}(T(\mathbb {R}\times S^1),)is an operator given by\Psi _k (F)(\partial _s) = F_s + i F_t +
\begin{pmatrix} a(s) - \delta &0 \\ 0 & -\delta \end{pmatrix} Fwhere the function a\colon \mathbb {R}\rightarrow \mathbb {R} i... | {
"cite_spans": []
} | 1804.08013 | Morse-Bott Split Symplectic Homology | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
-0.04595218598842621,
-0.008863951079547405,
-0.027842875570058823,
0.010084460489451885,
-0.00492399325594306,
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0.04564705863595009,
0.020092640072107315,
0.02061135694384575,
0.03829348832368851,
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-0.02886505238711834,
0.007818889804184437,
0.0... | |
a7cf28e70cbb24c89c0fbce48ac27aea91c5f491 | subsection | 110 | 126 | Orienting the moduli spaces of curves | For these choices of capping operators, D^{\tilde{v}}
are oriented in the direction of the Reeb flow, as wanted.We now analyze how, for \delta > 0 and small, a coherent orientation scheme relates
the orientations of D \colon W^{1,p,-\delta }(\dot{S}, \rightarrow L^{1,p,-\delta }(\dot{S}, T^*\dot{S} \otimes
and of D \c... | {
"cite_spans": []
} | 1804.08013 | Morse-Bott Split Symplectic Homology | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
-0.05541601777076721,
0.03564202040433884,
-0.01791255548596382,
0.02300863340497017,
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0.05373767018318176,
-0.01274019107222557,
-0.01102369837462902,
0.0205673985183239,
0.02438182... | |
8acfc88024684ac87228c9a7da4cc3ebb6e9f860 | subsection | 111 | 126 | Orienting the moduli spaces of curves | Assume the
asymptotic operators at the punctures in \Gamma are complex linear.Let \mathbf {\delta } and \mathbf {\delta ^{\prime }} be vectors of sufficiently small
weights so that the differential operator induces a Fredholm operator on W^{1,p,\mathbf {\delta }}
and on W^{1,p,\mathbf {\delta ^{\prime }}}, and \delta _... | {
"cite_spans": []
} | 1804.08013 | Morse-Bott Split Symplectic Homology | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
-0.022928936406970024,
0.02233397401869297,
-0.018535368144512177,
0.02842089720070362,
-0.02341710962355137,
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0.039755694568157196,
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0.04363057762384415,
-0.03713175654411316,
0.0020289744716137648,
0.023279812186956406,
-0... | |
67bc6813f6adaeefee8a05e17b412091e90b37d6 | subsection | 112 | 126 | Orienting the moduli spaces of curves | By the coherent orientation, this operator is oriented and hence induces an
orientation on the space of sections V_{z_0}.The result now follows by the gluing property of coherent orientations.In particular, if z_0 is a positive puncture, this identifies
the determinant bundle ofD_\delta \colon W^{1,p, \mathbf {\delta }... | {
"cite_spans": []
} | 1804.08013 | Morse-Bott Split Symplectic Homology | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
-0.04423234239220619,
0.0035753713455051184,
-0.02236037887632847,
0.05042914301156998,
-0.00940204318612814,
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0.013225439004600048,
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0.057908039540052414,
-0.02434457652270794,
-0.015301215462386608,
0.039714474231004715,
... | |
fbce4f6f83de7ced28ab019a2a531aeaf46fde36 | subsection | 113 | 126 | Orientations with constraints | We have now explained how to orient all of the moduli spaces of punctured
cylinders with ends free to move in the corresponding Morse–Bott families of
orbits. This is not yet sufficient to orient our moduli spaces of cascades.
The additional ingredient
necessary is to orient moduli spaces of holomorphic curves with
con... | {
"cite_spans": []
} | 1804.08013 | Morse-Bott Split Symplectic Homology | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
-0.013993927277624607,
-0.0397690013051033,
-0.009988031350076199,
0.023745419457554817,
0.020800134167075157,
0.005749413277953863,
0.009049507789313793,
-0.009896468371152878,
0.046605728566646576,
0.04773500934243202,
-0.026049762964248657,
-0.018511051312088966,
0.018801001831889153,
0... | |
4e2fc19193f1260981b36ef816eee53c1e07205f | subsection | 114 | 126 | Orientations with constraints | In our problem, the
asymptotic operators of D^{\tilde{v}} are constant on each Morse–Bott
family of orbits, dramatically simplifying the problem to consider.We also notice another key feature regarding cascades: suppose that
\tilde{v}_- and \tilde{v}_+ are two (punctured) cylinders so the
asymptotic limit of \tilde{v}_... | {
"cite_spans": []
} | 1804.08013 | Morse-Bott Split Symplectic Homology | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
-0.04773777723312378,
0.004814983811229467,
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0.03220163285732269,
0.030522875487804413,
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0.02008405327796936,
0.012094689533114433,
0.07636823505163193,
0.042213138192892075,
-0.014223660342395306,
-0.03430771082639694,
0.021137092262506485,
0.004... | |
b25743df0b403e1a50b1e55ef3c0a9512c202cca | subsection | 115 | 126 | Orientations with constraints | After the conjugation described in Definition REF ,
we obtain non-degenerate operators \hat{D}_{\tilde{v}_-} and \hat{D}_{\tilde{v}_+} that have
asymptotic operators \mathbf {A}+\delta and \mathbf {A}-\delta respectively. If we consider
now a \delta –perturbed Cauchy–Riemann operator coming from the
linearization at a ... | {
"cite_spans": []
} | 1804.08013 | Morse-Bott Split Symplectic Homology | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
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0.0470980666577816,
0.0134762... | |
a2bc20ea70dcd918f91909ea14fb66ee6ffd45b1 | subsection | 116 | 126 | A calculation of signs | Having now explained the general framework of our orientations, let us now give
an explicit description of the signs associated to a Floer cylinder with
cascades contributing to the differential.
By Propositions
REF and REF , there are
four types of contributions to the differential, referred to as
Cases 0 through 3. W... | {
"cite_spans": []
} | 1804.08013 | Morse-Bott Split Symplectic Homology | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
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-0... | |
46abf5b9291fcdbd5a0207f254fdc09e1005472a | subsection | 117 | 126 | A calculation of signs | For all p\in \operatorname{Crit}(f_\Sigma ), we will assume that the
orientations on critical submanifolds of \Sigma and Y
are such that the
restrictions of \pi _\Sigma \colon Y \rightarrow \Sigma toW^u_Y(p) \rightarrow W^u_\Sigma (p) \qquad \text{and} \qquad W^s_Y(\widehat{p}) \rightarrow W^s_\Sigma (p)are orientation... | {
"cite_spans": []
} | 1804.08013 | Morse-Bott Split Symplectic Homology | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
-0.02840597927570343,
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5880ad3d5f83d1b3ee56b58049b51fc65b201390 | subsection | 118 | 126 | A calculation of signs | These spaces are unions of fibre productsW^s_{Y}(\widehat{q}) \times _{\operatorname{ev}} {\mathcal {M}}^*_{H,0,\mathbb {R}\times Y;k_-,k_+}(A;J_Y) \times _{\operatorname{ev}} W_{Y}^u(p)defined with respect to the inclusion mapsW^s_{Y}(\widehat{q}), W_{Y}^u(p) &\rightarrow Yand the evaluation maps from (REF )\tilde{\op... | {
"cite_spans": []
} | 1804.08013 | Morse-Bott Split Symplectic Homology | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
-0.0033892584033310413,
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b5c70cc3fec4006eeb705296c0588f88e0f3f4d3 | subsection | 119 | 126 | A calculation of signs | The sign of such a contribution to the differential
is obtained by comparing this orientation with the one induced
by s-translation on the domain of the punctured Floer
cylinder in {\mathcal {M}}^*_{H,1,\mathbb {R}\times Y;k_-,k_+}(0;J_Y).Finally, Case 3 Floer cylinders with cascades that contribute to the differential... | {
"cite_spans": []
} | 1804.08013 | Morse-Bott Split Symplectic Homology | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
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-0.011778352782130241,
... | |
a7d7606503bd093531d62251568fa9f9339b69d4 | subsection | 120 | 126 | A calculation of signs | We have evaluation maps(\operatorname{ev}^1_-,\operatorname{ev}^1_+) \colon \mathcal {M}^*_{H}(B;J_W) \rightarrow W \times Y \qquad \text{and} \qquad (\operatorname{ev}^2_-,\operatorname{ev}^2_+) \colon \mathcal {M}^*_{H,k_+}(0;J_Y) \rightarrow Y \times Yand can write{\mathcal {M}}^*_{H,0,W;k_+}((B,0);J_W) = \mathcal {... | {
"cite_spans": []
} | 1804.08013 | Morse-Bott Split Symplectic Homology | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
-0.02555297315120697,
0.00046576966997236013,
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... | |
8ebf2ed6098455d96cce5d4291c2002087eb3a18 | subsection | 121 | 126 | A calculation of signs | Helv.,
volume=80,
number=4,
pages=771793,AbreuMacarinidynamicallyconvexellipticarticle
author=Abreu, Miguel,
author=Macarini, Leonardo,
title=Multiplicity of periodic orbits for dynamically convex contact
forms,
date=2016,
ISSN=1661-7746,
journal=Journal of Fixed Point Theory and Applications,
pages=130,
url=http://dx.... | {
"cite_spans": []
} | 1804.08013 | Morse-Bott Split Symplectic Homology | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
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... | |
148efb0cea8abd7741d04a7ef045b01b7bee0282 | subsection | 122 | 126 | A calculation of signs | J.,
volume=146,
number=1,
pages=71174,
url=http://dx.doi.org/10.1215/00127094-2008-062,
review=2475400 (2010e:53147),BourgeoisThesisbook
author=Bourgeois, Frédéric,
title=A Morse-Bott approach to contact homology,
publisher=ProQuest LLC, Ann Arbor, MI,
date=2002,
ISBN=978-0493-62828-8,
url=http://gateway.proquest.com/o... | {
"cite_spans": []
} | 1804.08013 | Morse-Bott Split Symplectic Homology | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
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... | |
6d4de4151e21139ca2f06b83cbc407797d89957d | subsection | 123 | 126 | A calculation of signs | Z.,
volume=212,
number=1,
pages=1338,
url=http://dx.doi.org/10.1007/BF02571639,
review=1200162,FOOObook
author=Fukaya, Kenji,
author=Oh, Yong-Geun,
author=Ohta, Hiroshi,
author=Ono, Kaoru,
title=Lagrangian intersection floer theory: Anomaly and obstruction,
publisher=American Mathematical Society,
date=2009,Frauenfelde... | {
"cite_spans": []
} | 1804.08013 | Morse-Bott Split Symplectic Homology | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
-0.040769487619400024,
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0.007995213381946087,
... | |
e451949a67f1ed5eaecff59259121f0ca36be490 | subsection | 124 | 126 | A calculation of signs | Nonlinear Differential Equations Appl.,
volume=35,
publisher=Birkhäuser,
address=Basel,
pages=381475,JoyceCornersincollection
author=Joyce, Dominic,
title=On manifolds with corners,
date=2012,
booktitle=Advances in geometric analysis,
series=Adv. Lect. Math. (ALM),
volume=21,
publisher=Int. Press, Somerville, MA,
pages... | {
"cite_spans": []
} | 1804.08013 | Morse-Bott Split Symplectic Homology | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
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... | |
b77a1c77cb3bf653d5a1e355dfafe0594bfed194 | subsection | 125 | 126 | A calculation of signs | Soc.,
volume=27,
number=1,
pages=133,
url=https://doi-org.umiss.idm.oclc.org/10.1112/blms/27.1.1,
review=1331677,SchmaeschkeOrientationsunpublished
author=Schmäschke, Felix,
title=Floer homology of Lagrangians in clean intersection,
date=2016,
note=arXiv:1606.05327,SchwarzThesisthesis
author=Schwarz, Matthias,
title=Co... | {
"cite_spans": []
} | 1804.08013 | Morse-Bott Split Symplectic Homology | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
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28ce18bc5e149be694658b3f6f730dce3bca605e | abstract | 0 | 4 | Abstract | This paper describes a new algorithm for exact Bayesian inference that is
based on a recently proposed compositional semantics of Bayesian networks in
terms of channels. The paper concentrates on the ideas behind this algorithm,
involving a linearisation (`stretching') of the Bayesian network, followed by a
combination... | {
"cite_spans": []
} | 1804.08032 | A Channel-based Exact Inference Algorithm for Bayesian Networks | [
"Bart Jacobs"
] | [
"cs.AI"
] | 2,018 | en | Computer Science | [
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eafedf93c85da27e208ea78b67c06f5cc89a6e59 | subsection | 1 | 4 | Introduction | In general, inference is about answering probabilistic queries of the
form: given this-and-this as evidence, what is the likelihood of that?
The focus in this paper is on exact inference, where precise
answers are sought and not approximations. In general, inference is
computationally very expensive. In probabilistic g... | {
"cite_spans": []
} | 1804.08032 | A Channel-based Exact Inference Algorithm for Bayesian Networks | [
"Bart Jacobs"
] | [
"cs.AI"
] | 2,018 | en | Computer Science | [
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0... | |
7d82df3d9dd8716f3563c20ab7a05fd85c967187 | subsection | 2 | 4 | Introduction | The implementation
builds on the EfProb library for channel-based
probabilistic computations. This paper concentrates on the methodology
to use channel-based compositional semantics for Bayesian inference
— the main intellectual contribution — and not on this prototype
implementation. Nevertheless, a brief comparison ... | {
"cite_spans": []
} | 1804.08032 | A Channel-based Exact Inference Algorithm for Bayesian Networks | [
"Bart Jacobs"
] | [
"cs.AI"
] | 2,018 | en | Computer Science | [
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0.0039... | |
d7e1120bf11cf8659c7b4dd77ee44c9cf3efec73 | subsection | 3 | 4 | The channel-based approach to Bayesian networks | This section explains in a concrete way how Bayesian networks can be
described conveniently in terms of states \omega , channels c, and
forward transformation c \gg \omega of a state along a channel. For
a more elaborate introduction we refer to , and
to for a more abstract account of the underlying
category theory. W... | {
"cite_spans": []
} | 1804.08032 | A Channel-based Exact Inference Algorithm for Bayesian Networks | [
"Bart Jacobs"
] | [
"cs.AI"
] | 2,018 | en | Computer Science | [
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0.0... | |
528ae8ba9e12dba3696c5d62668d475740f807c7 | abstract | 0 | 279 | Abstract | If $(X, \omega)$ is a symplectic manifold, and $\Sigma$ is a smooth
symplectic submanifold Poincar\'e dual to a positive multiple of $\omega$, $X
\setminus \Sigma$ admits a compactification as a Liouville domain, which we
then complete to $(W, d\lambda)$.
Under monotonicity assumptions on $X$ and on $\Sigma$, we const... | {
"cite_spans": []
} | 10.1112/topo.12105 | 1804.08014 | Symplectic Homology of complements of smooth divisors | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
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c0664b655609104bcecb321426d978f3be8fdfa7 | subsection | 1 | 279 | Introduction | Symplectic homology
is a version of Hamiltonian Floer homology for a class of open symplectic
manifolds with contact boundary, including Liouville manifolds SymplecticHomology,ViterboSH.
Some of the applications of symplectic homology include special cases of
the Weinstein conjecture , and a proof that there are
infini... | {
"cite_spans": []
} | 10.1112/topo.12105 | 1804.08014 | Symplectic Homology of complements of smooth divisors | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
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0.... |
d13c79459b380c54dfb936392928aa9d7db41242 | subsection | 2 | 279 | Introduction | We call a Morse function perfect
if the corresponding Morse differential vanishes.Theorem 1.1 If \Sigma and X \setminus \Sigma admit perfect Morse functions, then
there is a chain complex computing the symplectic homology of X\setminus \Sigma , whose differential is expressed in terms of Gromov–Witten invariants of
(\S... | {
"cite_spans": []
} | 10.1112/topo.12105 | 1804.08014 | Symplectic Homology of complements of smooth divisors | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
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... |
ac8483ea7712a156c6a34204a194de52c5709e20 | subsection | 3 | 279 | Introduction | As studied in our paper , this also allows us to achieve
transversality by geometric arguments involving monotonicity and automatic
transversality.
For more on the relation between this work and , see Remark REF .Remark 1.4 The contact boundaries Y of the Liouville domains X\setminus \Sigma are such that the Reeb flow ... | {
"cite_spans": []
} | 10.1112/topo.12105 | 1804.08014 | Symplectic Homology of complements of smooth divisors | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
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... |
9977db0dc69bd419f9e9f9a8568de8adc4a69dbf | subsection | 4 | 279 | Almost complex structures and neck-stretching | In this section, we consider a local model of the divisor, and identify a class
of almost complex structures that are well-behaved in this local model. | {
"cite_spans": []
} | 10.1112/topo.12105 | 1804.08014 | Symplectic Homology of complements of smooth divisors | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
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0... |
2d71b6368689c57904983030a99332201d30bd19 | subsection | 5 | 279 | Symplectic hyperplane sections | We adopt some of our terminology from *Section 2.2,
with minor modifications.
A variant on this construction is also explicitly described in
*Section 3.1.Let (X,\omega ) be a closed symplectic manifold such that
[K\omega ] admits a lift to H^2(X;\mathbb {Z}) for some K>0.
We refer to a closed codimension 2 symplectic s... | {
"cite_spans": []
} | 10.1112/topo.12105 | 1804.08014 | Symplectic Homology of complements of smooth divisors | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
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0.02150551788508892,
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0.01... |
19a84282b5af99d724fc15aa906930b5cae72cb1 | subsection | 6 | 279 | Symplectic hyperplane sections | This contact form is\alpha _{\rho _0} = -\frac{f(\rho _0^2)}{K} \Theta |_Y.Observe that by following the flow of the Liouville vector field, we obtain that
E\setminus \Sigma
equipped with the symplectic form \omega _E is then symplectomorphic
to a piece of symplectization \left( (-\infty , -\ln f(\rho _0^2)) \times Y,... | {
"cite_spans": []
} | 10.1112/topo.12105 | 1804.08014 | Symplectic Homology of complements of smooth divisors | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
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a946e4c757785ba61fbf17df848768c128fba3ad | subsection | 7 | 279 | Symplectic hyperplane sections | See also
*Proposition 5.Lemma 2.2Let \Sigma \subset X be a symplectic hyperplane section, as in
Definition REF , and let E \rightarrow \Sigma be the normal bundle
to \Sigma , endowed with a Hermitian metric and symplectic form
\omega _E as in Equation (REF ).Then,the symplectic form \omega |_{X \setminus \Sigma } is ex... | {
"cite_spans": []
} | 10.1112/topo.12105 | 1804.08014 | Symplectic Homology of complements of smooth divisors | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
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0141bd76c793378883b48f7071aad42b9459a74a | subsection | 8 | 279 | Symplectic hyperplane sections | The Gysin sequence applied to the circle bundle Y \rightarrow \Sigma gives0 \rightarrow H^1(\Sigma ; \mathbb {R}) \xrightarrow{} H^1(Y; \mathbb {R}) \xrightarrow{}
H^0(\Sigma ; \mathbb {R}) \xrightarrow{} H^2(\Sigma ; \mathbb {R}).The last map is an injection from H^0(\Sigma ; \mathbb {R})
to H^2(\Sigma ; \mathbb {R}) ... | {
"cite_spans": []
} | 10.1112/topo.12105 | 1804.08014 | Symplectic Homology of complements of smooth divisors | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
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5268856c87c17a70239c23ef158cb8f798b390c4 | subsection | 9 | 279 | Symplectic hyperplane sections | Notice that this flow preserves the
levels \rho = \text{constant}.Also notice that \pi ^*\mu (\tilde{V}) =
\mu (V) = \omega _\Sigma (V, V) = 0 and \omega _E(\tilde{V}, \cdot ) =
f(\rho ^2) \pi ^*\omega _\Sigma (V, \cdot ) = f(\rho ^2)\pi ^*\mu .Finally, we calculate:\frac{d}{dt} (\phi ^t)^*\left( \lambda _E - t\pi ^*\m... | {
"cite_spans": []
} | 10.1112/topo.12105 | 1804.08014 | Symplectic Homology of complements of smooth divisors | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
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0.0... |
8786af94634fa81c20ec4d59d35639397d9825bb | subsection | 10 | 279 | Symplectic hyperplane sections | We may now compactify by gluing in
\partial \overline{W} = \lbrace 0 \rbrace \times Y.Note that this implies that the vector field dual to \lambda
points outwards at the boundary, and is thus a Liouville vector field
and \lambda is a Liouville form for \omega .
This fact was also pointed out to us by McLean, using a d... | {
"cite_spans": []
} | 10.1112/topo.12105 | 1804.08014 | Symplectic Homology of complements of smooth divisors | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
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a869bd288efc7b0c3e73ea2b629508648ecb2e4b | subsection | 11 | 279 | Almost complex structures | Since the Hermitian connection on the complex line bundle E above defines a horizontal distribution,
any almost complex structure J_\Sigma on
(\Sigma , \omega _\Sigma ) may be lifted to an almost complex structure J_E
by requiring that J_E preserve the horizontal and vertical subspaces,
is i on the vertical subspaces a... | {
"cite_spans": []
} | 10.1112/topo.12105 | 1804.08014 | Symplectic Homology of complements of smooth divisors | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
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0... |
dc213bad9fd5c51293700f8434f51a8c4ec5bb1f | subsection | 12 | 279 | Almost complex structures | Recall that the induced contact form on Y,
seen as the \rho _0-circle bundle, is \alpha _{\rho _0} = \operatorname{e}^{-\rho _0^2} \alpha .As observed earlier, there is an exact symplectomorphism\psi _1 \colon (E \setminus \Sigma , \omega _E) &\rightarrow \left( (-\infty , 0) \times Y, d(\operatorname{e}^{r} \alpha ) \... | {
"cite_spans": []
} | 10.1112/topo.12105 | 1804.08014 | Symplectic Homology of complements of smooth divisors | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
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df88a3a1decf3edfd99e6542237e146c4078e642 | subsection | 13 | 279 | Almost complex structures | This verifies that J_Y is a cylindrical almost complex structure on \mathbb {R}\times Y,
adapted to the contact form \alpha .
Notice that
J_Y is both \mathbb {R}–translation
invariant and Reeb-flow invariant since these two actions correspond to
the * action on
the Hermitian line bundle E. Even though \psi _2 is not a ... | {
"cite_spans": []
} | 10.1112/topo.12105 | 1804.08014 | Symplectic Homology of complements of smooth divisors | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
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0... |
adc2ba1a6d5e48ca7093935d61a912f8a6479e4a | subsection | 14 | 279 | Almost complex structures | Define the almost complex structure J_W on W as follows:J_W :=
{\left\lbrace \begin{array}{ll}
J_X & \text{ on } W\setminus \big ([-\epsilon /2,\infty )\times Y\big ) \\
G_*J_1 & \text{ on } [-\epsilon /2,-\epsilon /4)\times Y \\
J_Y & \text{ on } [-\epsilon /4,\infty )\times Y
\end{array}\right.}Notice that we may use... | {
"cite_spans": []
} | 10.1112/topo.12105 | 1804.08014 | Symplectic Homology of complements of smooth divisors | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
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0.... |
517d619910dc97dddd5999999ecf780415614e40 | subsection | 15 | 279 | Almost complex structures | Let J_W be the almost complex structure corresponding to
J_X by the construction above, for fixed diffeomorphism g \colon (-\infty , 0)
\rightarrow \mathbb {R}.For each \kappa \ge \epsilon /4 take a diffeomorphismf_\kappa : [-\epsilon /4,\epsilon /4] \rightarrow [-\kappa ,\kappa ]such that f_\kappa ^{\prime } \equiv 1 ... | {
"cite_spans": []
} | 10.1112/topo.12105 | 1804.08014 | Symplectic Homology of complements of smooth divisors | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
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63d8c4b038a140874ae7abb819a7ac55e3830467 | subsection | 16 | 279 | The chain complex | We will describe two chain complexes associated to W whose generators are (essentially) the same, but for which the differentials are a priori different.
We first define the group underlying these chain complexes.
[Figure: An admissible Hamiltonian and the graphical procedure for computing the action of a periodic orbi... | {
"cite_spans": []
} | 10.1112/topo.12105 | 1804.08014 | Symplectic Homology of complements of smooth divisors | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
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ec917d5616a2cd5f260b65a72de5e7d4c7c69ce4 | subsection | 17 | 279 | The chain complex | Fix throughout a Morse function f_\Sigma \colon \Sigma \rightarrow \mathbb {R} and a gradient-like vector field Z_\Sigma \in \mathfrak {X}(\Sigma ), which means that
\frac{1}{c} |df_\Sigma |^2 \le df_\Sigma (Z_\Sigma )\le c |df_\Sigma |^2 for some constant c>0.
Denote the time-t flow
of Z_\Sigma by \varphi ^t_{Z_\Sigma... | {
"cite_spans": []
} | 10.1112/topo.12105 | 1804.08014 | Symplectic Homology of complements of smooth divisors | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
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... |
6156114cacf22110af59a048f8efc4f0f865bd25 | subsection | 18 | 279 | The chain complex | In the following, given a critical point for f_\Sigma , p \in \Sigma , we denote the two critical points in the fibre above p by \widehat{p} and p, the fibrewise maximum and fibrewise minimum of f_Y, respectively.We will denote by M(p) the Morse index of a critical point p \in \Sigma of f_\Sigma , and by \tilde{M} (\ti... | {
"cite_spans": []
} | 10.1112/topo.12105 | 1804.08014 | Symplectic Homology of complements of smooth divisors | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
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127e907645e775add59b2ae27c307925e727c052 | subsection | 19 | 279 | The chain complex | The complex is then given by:SC_*(W,H) = \left(\bigoplus _{k>0} \,
\bigoplus _{p\in \text{Crit}(f_\Sigma )}\mathbb {Z}\langle p_k, \widehat{p}_k \rangle \right)
\oplus \left(\bigoplus _{x\in \text{Crit}(f_W)}\mathbb {Z}\langle x \rangle \right)Recall that \lambda is a Liouville form on W.Definition 3.3
The Hamiltonian... | {
"cite_spans": []
} | 10.1112/topo.12105 | 1804.08014 | Symplectic Homology of complements of smooth divisors | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
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... |
1c5dcf3187b95b4d4a073f878063c9b5e6f046fe | subsection | 20 | 279 | The chain complex | We first define the group underlying these chain complexes.
[Figure: An admissible Hamiltonian and the graphical procedure for computing the action of a periodic orbit]Definition 3.1
Consider a function h \colon (0, +\infty ) \rightarrow \mathbb {R} with the following properties:h(\rho ) = 0 for \rho \le 2;
h^{\prime... | {
"cite_spans": []
} | 10.1112/topo.12105 | 1804.08014 | Symplectic Homology of complements of smooth divisors | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
-0.010846838355064392,
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1859cc11ac1f9a7fb4b55c84f8d7a51365897de2 | subsection | 21 | 279 | The chain complex | Fix throughout a Morse function f_\Sigma \colon \Sigma \rightarrow \mathbb {R} and a gradient-like vector field Z_\Sigma \in \mathfrak {X}(\Sigma ), which means that
\frac{1}{c} |df_\Sigma |^2 \le df_\Sigma (Z_\Sigma )\le c |df_\Sigma |^2 for some constant c>0.
Denote the time-t flow
of Z_\Sigma by \varphi ^t_{Z_\Sigma... | {
"cite_spans": []
} | 10.1112/topo.12105 | 1804.08014 | Symplectic Homology of complements of smooth divisors | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
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... |
13c85bae5315e378e74c5c4cf71bf564cc25a440 | subsection | 22 | 279 | The chain complex | In the following, given a critical point for f_\Sigma , p \in \Sigma , we denote the two critical points in the fibre above p by \widehat{p} and p, the fibrewise maximum and fibrewise minimum of f_Y, respectively.We will denote by M(p) the Morse index of a critical point p \in \Sigma of f_\Sigma , and by \tilde{M} (\ti... | {
"cite_spans": []
} | 10.1112/topo.12105 | 1804.08014 | Symplectic Homology of complements of smooth divisors | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
0.008347146213054657,
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ae29d351a62e479bf09cd72a1f4ee0957773084c | subsection | 23 | 279 | The chain complex | The complex is then given by:SC_*(W,H) = \left(\bigoplus _{k>0} \,
\bigoplus _{p\in \text{Crit}(f_\Sigma )}\mathbb {Z}\langle p_k, \widehat{p}_k \rangle \right)
\oplus \left(\bigoplus _{x\in \text{Crit}(f_W)}\mathbb {Z}\langle x \rangle \right)Recall that \lambda is a Liouville form on W.Definition 3.3
The Hamiltonian... | {
"cite_spans": []
} | 10.1112/topo.12105 | 1804.08014 | Symplectic Homology of complements of smooth divisors | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
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... |
029d706b207e694e4529a1ea21c569fb6a2a44ed | subsection | 24 | 279 | Floer moduli spaces before and after splitting | In this section, we discuss the differential in the Morse–Bott symplectic homology of W, before and after stretching the neck.In this section, we discuss the differential in the Morse–Bott symplectic homology of W, before and after stretching the neck. | {
"cite_spans": []
} | 10.1112/topo.12105 | 1804.08014 | Symplectic Homology of complements of smooth divisors | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
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... |
6a9a8061a8eade60128adc379a9c528d14fc6c19 | subsection | 25 | 279 | Morse–Bott symplectic homology | Consider H \colon W \rightarrow \mathbb {R} as defined in Section , together with the auxiliary data of (f_\Sigma , Z_\Sigma ), (f_Y, Z_Y) and (f_W, Z_W).
Fix an almost complex structure J_\kappa as in (REF ) for a large \kappa >0.Definition 4.1
A map \tilde{v} \colon \mathbb {R}\times S^1 \rightarrow W is a Floer cyl... | {
"cite_spans": []
} | 10.1112/topo.12105 | 1804.08014 | Symplectic Homology of complements of smooth divisors | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
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40345e1d330b46a6ef2cc537c8c057ec8af349c6 | subsection | 26 | 279 | Morse–Bott symplectic homology | Then we define\mathcal {M}(S_-, S_+) = \bigcup _{x_- \in S_-, x_+ \in S_+} \mathcal {M}(x_-, x_+).Given a Floer cylinder \tilde{v}, we denote its asymptotic limits by \tilde{v}(+\infty ) = x_+ and \tilde{v}(-\infty ) = x_-.Note that \mathcal {A}(\tilde{v}(s_0,.)) \le \mathcal {A}(\tilde{v}(s_1,.)) if s_0 \le s_1.Defini... | {
"cite_spans": []
} | 10.1112/topo.12105 | 1804.08014 | Symplectic Homology of complements of smooth divisors | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
-0.013492598198354244,
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0.015254491940140724,
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0.0006645238026976585,
0.0... |
098f2b01628c9bcf0b405c9d5c518ca7f4e485d0 | subsection | 27 | 279 | Morse–Bott symplectic homology | Then, for any sequence s_k \rightarrow +\infty [resp., s_k \rightarrow -\infty ] there is a subsequence we also denote (s_k)_{k=1}^\infty and a 1-periodic orbit \gamma (t) of the Hamiltonian vector field so that \tilde{v}(s_k, t) \rightarrow \gamma (t) in C^\infty .If \gamma is a Morse–Bott non-degenerate orbit, then t... | {
"cite_spans": []
} | 10.1112/topo.12105 | 1804.08014 | Symplectic Homology of complements of smooth divisors | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
-0.031861335039138794,
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0.0195013340562582,
-... |
0ba812af0b79ea723a26b3070b6d73faf2bed706 | subsection | 28 | 279 | Morse–Bott symplectic homology | Lemma REF
implies that there are subsequences (denoted also by s_k^\pm ) such that
\tilde{v}(s_k^\pm ,.) converge in C^1 (actually C^\infty ) to 1-periodic
X_H-orbits x_\pm , with x_+\in W_0. Then,0 \le E(\tilde{v}) &= \lim _{k\rightarrow \infty } E(\tilde{v}|_{[s_k^-,s_k^+]\times S^1}) = \lim _{k\rightarrow \infty } ... | {
"cite_spans": []
} | 10.1112/topo.12105 | 1804.08014 | Symplectic Homology of complements of smooth divisors | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
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0.02... |
0932b71f80eb65e21c8a0f72e37778a50116fdd9 | subsection | 29 | 279 | Morse–Bott symplectic homology | Let (f_i, Z_i), i=0, \dots , N+1 be the pair of Morse function and
gradient-like vector field of f_i = f_Y, Z_i = Z_Y if S_i = Y_k for some
k, and f_i = -f_W, Z_i=-Z_W if S_i = W_0.Let x be a critical point of f_0 and y a critical point of f_N (so x and y are generators of the chain complex (REF )).A Floer cylinder wit... | {
"cite_spans": []
} | 10.1112/topo.12105 | 1804.08014 | Symplectic Homology of complements of smooth divisors | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
-0.016177205368876457,
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... |
b328189af63a25e1eaad5c50c23b7d791d1e2a00 | subsection | 30 | 279 | Morse–Bott symplectic homology | For the case s\rightarrow -\infty , observe that \nu _{i-1} is a negative flow line of
Z_W, which agrees with \partial _r on [-\epsilon /4,\infty ) \times Y (in
particular on \partial W_0).
If i>0, then the sublevel \tilde{v}_{i-1} must be such that
\lim _{s\rightarrow +\infty } \tilde{v}_{i-1}(s,.) \in W_0, which as w... | {
"cite_spans": []
} | 10.1112/topo.12105 | 1804.08014 | Symplectic Homology of complements of smooth divisors | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
-0.0250825397670269,
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0.00... |
0e75ab884f12b22d2d415901103889787db4c4a5 | subsection | 31 | 279 | Morse–Bott symplectic homology | Given generators x, y, denote by\mathcal {M}_{H,N}(x,y;J_\kappa )the space of Floer cylinders with N cascades from x to
y (i.e. with negative end at x and positive end at y).We then define\partial _{\text{pre}} \, y = \sum _{|x|=|y| - 1} \# \left( \mathcal {M}_{H, 0}(x,y; J_\kappa ) / \mathbb {R}\right) x
+ \sum _{|x| ... | {
"cite_spans": []
} | 10.1112/topo.12105 | 1804.08014 | Symplectic Homology of complements of smooth divisors | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
-0.012385145761072636,
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0.0059979818761348724... |
405cb31ace9e7362ec9660c8010763b1a1c8cdc4 | subsection | 32 | 279 | Morse–Bott symplectic homology | Then we define\mathcal {M}(S_-, S_+) = \bigcup _{x_- \in S_-, x_+ \in S_+} \mathcal {M}(x_-, x_+).Given a Floer cylinder \tilde{v}, we denote its asymptotic limits by \tilde{v}(+\infty ) = x_+ and \tilde{v}(-\infty ) = x_-.Note that \mathcal {A}(\tilde{v}(s_0,.)) \le \mathcal {A}(\tilde{v}(s_1,.)) if s_0 \le s_1.Defini... | {
"cite_spans": []
} | 10.1112/topo.12105 | 1804.08014 | Symplectic Homology of complements of smooth divisors | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
-0.013492598198354244,
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0.00928998552262783,
0.0006645238026976585,
0.0... |
c14262beb5f5cf086edf59aa4c854c71cc23048f | subsection | 33 | 279 | Morse–Bott symplectic homology | Then, for any sequence s_k \rightarrow +\infty [resp., s_k \rightarrow -\infty ] there is a subsequence we also denote (s_k)_{k=1}^\infty and a 1-periodic orbit \gamma (t) of the Hamiltonian vector field so that \tilde{v}(s_k, t) \rightarrow \gamma (t) in C^\infty .If \gamma is a Morse–Bott non-degenerate orbit, then t... | {
"cite_spans": []
} | 10.1112/topo.12105 | 1804.08014 | Symplectic Homology of complements of smooth divisors | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
-0.031861335039138794,
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0.0195013340562582,
-... |
0f84d76956034ca9fcf8f2a6e366176417e0e693 | subsection | 34 | 279 | Morse–Bott symplectic homology | Lemma REF
implies that there are subsequences (denoted also by s_k^\pm ) such that
\tilde{v}(s_k^\pm ,.) converge in C^1 (actually C^\infty ) to 1-periodic
X_H-orbits x_\pm , with x_+\in W_0. Then,0 \le E(\tilde{v}) &= \lim _{k\rightarrow \infty } E(\tilde{v}|_{[s_k^-,s_k^+]\times S^1}) = \lim _{k\rightarrow \infty } ... | {
"cite_spans": []
} | 10.1112/topo.12105 | 1804.08014 | Symplectic Homology of complements of smooth divisors | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
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0.007250070571899414,
0.02... |
9b81a3e069e13f8c577ea78cef49d8e920f0b121 | subsection | 35 | 279 | Morse–Bott symplectic homology | Let (f_i, Z_i), i=0, \dots , N+1 be the pair of Morse function and
gradient-like vector field of f_i = f_Y, Z_i = Z_Y if S_i = Y_k for some
k, and f_i = -f_W, Z_i=-Z_W if S_i = W_0.Let x be a critical point of f_0 and y a critical point of f_N (so x and y are generators of the chain complex (REF )).A Floer cylinder wit... | {
"cite_spans": []
} | 10.1112/topo.12105 | 1804.08014 | Symplectic Homology of complements of smooth divisors | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
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... |
c5a9a70cf9c4420326f97954f9f2a73ff0a117ac | subsection | 36 | 279 | Morse–Bott symplectic homology | For the case s\rightarrow -\infty , observe that \nu _{i-1} is a negative flow line of
Z_W, which agrees with \partial _r on [-\epsilon /4,\infty ) \times Y (in
particular on \partial W_0).
If i>0, then the sublevel \tilde{v}_{i-1} must be such that
\lim _{s\rightarrow +\infty } \tilde{v}_{i-1}(s,.) \in W_0, which as w... | {
"cite_spans": []
} | 10.1112/topo.12105 | 1804.08014 | Symplectic Homology of complements of smooth divisors | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
-0.0250825397670269,
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0.00... |
04a2311c01ec437e6b17383bd46f13f948368e71 | subsection | 37 | 279 | Morse–Bott symplectic homology | Given generators x, y, denote by\mathcal {M}_{H,N}(x,y;J_\kappa )the space of Floer cylinders with N cascades from x to
y (i.e. with negative end at x and positive end at y).We then define\partial _{\text{pre}} \, y = \sum _{|x|=|y| - 1} \# \left( \mathcal {M}_{H, 0}(x,y; J_\kappa ) / \mathbb {R}\right) x
+ \sum _{|x| ... | {
"cite_spans": []
} | 10.1112/topo.12105 | 1804.08014 | Symplectic Homology of complements of smooth divisors | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
-0.026537680998444557,
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-0.003158884355798363... |
d336451571b89930369ba038f5a77b9b30a2f397 | subsection | 38 | 279 | Morse–Bott split symplectic homology | Recall the definition of the almost complex structures J_\kappa , J_Y
and J_W in Section REF . Given a sequence
\tilde{v}_{\kappa _n} of finite energy J_{\kappa _n}-Floer cylinders in
W, with \kappa _n\rightarrow \infty , SFT compactness
implies that there is a subsequence converging to an SFT building.
Observe that t... | {
"cite_spans": []
} | 10.1112/topo.12105 | 1804.08014 | Symplectic Homology of complements of smooth divisors | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
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0.0045579285360872746,
-0.02363257296383381,
... |
d601c7a66693b9b56b5c533d6addb207ad158017 | subsection | 39 | 279 | Morse–Bott split symplectic homology | This solves the Floer equation\partial _s \tilde{v} + J_Y (\partial _t \tilde{v} - X_H(\tilde{v})) = 0,and has finite hybrid energy, given
byE( \tilde{v} ) = \sup \left\lbrace
\int _{\mathbb {R}\times S^1} \tilde{v} {}^{*}\left( d( \eta \alpha ) - dH \wedge dt
\right) \, | \, \eta \colon \mathbb {R}\rightarrow [0, \in... | {
"cite_spans": []
} | 10.1112/topo.12105 | 1804.08014 | Symplectic Homology of complements of smooth divisors | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
-0.0021248154807835817,
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... |
b132162f0aa4c9e8ef72454ffb5ffd4704f0624a | subsection | 40 | 279 | Morse–Bott split symplectic homology | Writing A \in H_2(\Sigma ; \mathbb {Z}) for the (spherical) homology class
represented by the projection of \tilde{v} to \Sigma , these satisfy the
conditions
&k_+ - k_- = K \omega (A) > 0 \\
&\gamma _-(0) = \lim _{s \rightarrow -\infty } \tilde{v}(s, 0) \\
&\lim _{s \rightarrow +\infty } \tilde{v}(s, 0) = \gamma _+(0... | {
"cite_spans": []
} | 10.1112/topo.12105 | 1804.08014 | Symplectic Homology of complements of smooth divisors | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
-0.05990760028362274,
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0.020630430430173874,
0.03936872258782387,
0.00... |
b79348be3da02f21667de7c732e003b189c394a0 | subsection | 41 | 279 | Morse–Bott split symplectic homology | These satisfy
asymptotic matching conditions as follows: \gamma (0) = \tilde{v}_0(0), the
plane \tilde{v}_0 converges at its positive puncture to the same Reeb orbit
as \tilde{v}_1 converges to at its negative puncture, which has
multiplicity k_+, and \gamma _+(0) = \lim _{s \rightarrow \infty } \tilde{v}_1(s,
0).
See ... | {
"cite_spans": []
} | 10.1112/topo.12105 | 1804.08014 | Symplectic Homology of complements of smooth divisors | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
-0.045721184462308884,
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0.0016328995116055012,
0... |
ce9b328020c4e64c84a3464abd1f2f937f8dbaac | subsection | 42 | 279 | Morse–Bott split symplectic homology | We also define
\mathcal {M}^*_{H,k_+}(0;J_Y) as the space of Floer cylinders \tilde{v} \colon \mathbb {R}\times S^1 \rightarrow \mathbb {R}\times Y such that
\lim _{s\rightarrow \infty }\tilde{v}(s,.) is a Hamiltonian orbit of multiplicity k_+, \lim _{s\rightarrow -\infty }\tilde{v}(s,.)
is the corresponding Reeb orbit... | {
"cite_spans": []
} | 10.1112/topo.12105 | 1804.08014 | Symplectic Homology of complements of smooth divisors | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
-0.018916158005595207,
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0.010579319670796394,
-0.00047528755385428667,
0... |
c0228141e5d016fd6c25dad1ddbff19bf5a35817 | subsection | 43 | 279 | Morse–Bott split symplectic homology | If we identify the images of the simple Reeb orbits underlying x_\pm (t) = \lim _{s\rightarrow \pm \infty } v(s,t) with S^1, then the following map keeps track of the asymptotic effect of the action:\operatorname{rot}_{\tilde{v}}\colon S^1 \times S^1 &\rightarrow S^1 \times S^1 \\
(\theta _1,\theta _2) &\mapsto \left(\... | {
"cite_spans": []
} | 10.1112/topo.12105 | 1804.08014 | Symplectic Homology of complements of smooth divisors | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
-0.04198567941784859,
0.035394906997680664,
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-0.00023754729772917926,
... |
53fcf85fca0fb0c97cb081de74441169ee4d2c5c | subsection | 44 | 279 | Morse–Bott split symplectic homology | The non-constant orbits
again form manifolds Y_k, where k\ge 1. The critical points of the
Morse function f_W \colon W\rightarrow \mathbb {R} are below the neck-stretching region.This then gives a different description of the differential on the
chain complex (REF ) (defined above in terms of Floer
cylinders with casca... | {
"cite_spans": []
} | 10.1112/topo.12105 | 1804.08014 | Symplectic Homology of complements of smooth divisors | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
-0.012331102043390274,
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0.005093447398394346,
-0.02254088781774044,
0.... |
928f91b13160ffef19df11aaba7b055f278b5034 | subsection | 45 | 279 | Morse–Bott split symplectic homology | It states
that under our monotonicity assumptions, the cascades contributing to the split Floer differential can be of only four simple types.
[Figure: Case 1 in Proposition][Figure: Case 2 in Proposition][Figure: Case 3 in Proposition]Proposition 4.13
In what follows, q,p\in \operatorname{Crit}(f_\Sigma ) and x\in \o... | {
"cite_spans": []
} | 10.1112/topo.12105 | 1804.08014 | Symplectic Homology of complements of smooth divisors | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
-0.04603965952992439,
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0.0030967695638537407,
0.010114080272614956,
-0.... |
713a37dd6f448c6879bc0105b8798c628ed9bfd9 | subsection | 46 | 279 | Morse–Bott split symplectic homology | Writing B \in H_2(X; \mathbb {Z}) for the homology class represented by the plane U,
these satisfy
&k_+-k_-= K \omega (B) > 0 \\
&\gamma _-(0) = \lim _{s \rightarrow -\infty } \tilde{v}(s, 0) \\
&\lim _{s \rightarrow +\infty } \tilde{v}(s, 0) = \gamma _+(0),
and the Reeb orbit to which \tilde{v} converges at P is the... | {
"cite_spans": []
} | 10.1112/topo.12105 | 1804.08014 | Symplectic Homology of complements of smooth divisors | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
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0.04069025442004204,
0.0233... |
38052284451964183b97313b0daca4ca0c06b61a | subsection | 47 | 279 | Morse–Bott split symplectic homology | Using the notation from ,
we write\mathcal {M}^*_{H,l,\mathbb {R}\times Y;k_-,k_+}(A;J_Y)for the space of simple Floer cylinders \tilde{v} \colon \mathbb {R}\times S^1\setminus \Gamma \rightarrow \mathbb {R}\times Y,
where \Gamma is a set of l augmentation punctures, \lim _{s\rightarrow \pm \infty } \tilde{v}(s,.) are ... | {
"cite_spans": []
} | 10.1112/topo.12105 | 1804.08014 | Symplectic Homology of complements of smooth divisors | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
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0.... |
7f407da88a363f50e15b2837c8c7ef1621fb5c49 | subsection | 48 | 279 | Morse–Bott split symplectic homology | We also define
\mathcal {M}^*_{H,k_+}(0;J_Y) as the space of Floer cylinders \tilde{v} \colon \mathbb {R}\times S^1 \rightarrow \mathbb {R}\times Y such that
\lim _{s\rightarrow \infty }\tilde{v}(s,.) is a Hamiltonian orbit of multiplicity k_+, \lim _{s\rightarrow -\infty }\tilde{v}(s,.)
is the corresponding Reeb orbit... | {
"cite_spans": []
} | 10.1112/topo.12105 | 1804.08014 | Symplectic Homology of complements of smooth divisors | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
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0... |
164575e3217393d09a1ffbcca50c0c9cf588d396 | subsection | 49 | 279 | Morse–Bott split symplectic homology | If we identify the images of the simple Reeb orbits underlying x_\pm (t) = \lim _{s\rightarrow \pm \infty } v(s,t) with S^1, then the following map keeps track of the asymptotic effect of the action:\operatorname{rot}_{\tilde{v}}\colon S^1 \times S^1 &\rightarrow S^1 \times S^1 \\
(\theta _1,\theta _2) &\mapsto \left(\... | {
"cite_spans": []
} | 10.1112/topo.12105 | 1804.08014 | Symplectic Homology of complements of smooth divisors | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
-0.055875226855278015,
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-0... |
7f98943fde548100de36689814a2bb4cf89be1b0 | subsection | 50 | 279 | Sketch of isomorphism between the two Floer complexes | Proposition 5.1 For a generic choice of almost complex structures of the cylindrical type J_\kappa , J_W and J_Y described in Section REF , and for R>0 large enough,
the presplit and split chain complexes are well-defined and are chain isomorphic.
Furthermore, these complexes compute the symplectic homology of W.We wil... | {
"cite_spans": []
} | 10.1112/topo.12105 | 1804.08014 | Symplectic Homology of complements of smooth divisors | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
-0.031952857971191406,
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-0.002632219810038805,
0.... |
ed5c65b72b7b2cbb2c186acfa714ea8f2bd54c16 | subsection | 51 | 279 | Sketch of isomorphism between the two Floer complexes | We address this difficulty by means of the Abouzaid–Seidel
Lemma REF below.
A usual action filtration argument
gives that this is a chain isomorphism.
[Figure: By perturbing J in (i.e.~away from the cylindrical end of W), the curve on the left canbe made transverse. Such a perturbation only makes the plane in Wtransve... | {
"cite_spans": []
} | 10.1112/topo.12105 | 1804.08014 | Symplectic Homology of complements of smooth divisors | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
-0.050163764506578445,
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0.011183101683855057,
0.009... |
16139f582e689b3806ded706533a95943202aec6 | subsection | 52 | 279 | Sketch of isomorphism between the two Floer complexes | Furthermore,
any such configuration in \Phi \circ \partial _{pre} or in \tilde{\partial }\circ \Phi can be glued, showing that all such configurations arise as
boundary components of these 1-dimensional moduli spaces.The presence of degenerate orbits of X_H along \partial \operatorname{supp}dH
introduces two main diffi... | {
"cite_spans": []
} | 10.1112/topo.12105 | 1804.08014 | Symplectic Homology of complements of smooth divisors | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
-0.04719918221235275,
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-0.00004808921949006617,
... |
390fea375345e0071f5372af2a10589aa3d93499 | subsection | 53 | 279 | Sketch of isomorphism between the two Floer complexes | Let i denote the complex structure on S with i \partial _\sigma =
\partial _\tau .Let J be an S-dependent family of almost complex structures on [r_0, r_0 + \delta ] \times Y for small \delta >0
and let H \colon S \times [r_0, r_0 + \delta ] \times Y \rightarrow \mathbb {R} be an S-dependent family of Hamiltonians
with... | {
"cite_spans": []
} | 10.1112/topo.12105 | 1804.08014 | Symplectic Homology of complements of smooth divisors | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
-0.037863556295633316,
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-0.023630399256944656,
0.0... |
e92035d869beb0cabdb0ee48ddf6a20744af31fa | subsection | 54 | 279 | Sketch of isomorphism between the two Floer complexes | Furthermore, since the image of the annulus S is not below r=r_0,
we have dr( d\tilde{v} \circ i)(\frac{\partial }{\partial \tau }) \ge 0 along \lbrace 0\rbrace \times S^1, hence\tilde{v}^*\lambda (\frac{\partial }{\partial \tau }) \le \lambda (X_H) \beta (\frac{\partial }{\partial \tau }).The result now follows.Follow... | {
"cite_spans": []
} | 10.1112/topo.12105 | 1804.08014 | Symplectic Homology of complements of smooth divisors | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
-0.01962069608271122,
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0... |
e564ccac9125cdbe1289521ec76ac289cbe0e477 | subsection | 55 | 279 | Sketch of isomorphism between the two Floer complexes | Observe that they are equal if H is z-independent and \beta = dt.Lemma 5.4
Let (W, d\lambda ) be an exact symplectic manifold with cylindrical end ((r_0, +\infty ) \times Y, d( \operatorname{e}^r \alpha )), r_0 < \log 2.Let (\beta ,H, J) be a monotone triple on the cylinder (\mathbb {R}\times S^1, i) with 1-form \beta... | {
"cite_spans": []
} | 10.1112/topo.12105 | 1804.08014 | Symplectic Homology of complements of smooth divisors | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
-0.033873312175273895,
0.020690185949206352,
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0.028212502598762512,
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0.004455408547073603,
... |
ba4cade2d6810ac421ba5356c286361892d94a67 | subsection | 56 | 279 | Sketch of isomorphism between the two Floer complexes | Take a regular value r_1-c of \tilde{r} \circ \tilde{v} \colon \mathbb {R}\times S^1 \rightarrow \mathbb {R} in the interval (r_1-\delta /2, r_1).By hypothesis, there exists a k_0 sufficiently large so that for each k \ge k_0, \min _{t \in S^1} \tilde{r}( \tilde{v}(s_{k}, t) ) > r_1-c/2. Fix such a k.Define the subdoma... | {
"cite_spans": []
} | 10.1112/topo.12105 | 1804.08014 | Symplectic Homology of complements of smooth divisors | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
-0.07812514156103134,
0.05288705602288246,
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0.01565554551780224,
-0.022201577201485634,
0.0230... |
42fbd029e48f0c0cedc05fc07ee8914391e3e55b | subsection | 57 | 279 | Sketch of isomorphism between the two Floer complexes | Combining Stokes's Theorem with Lemma REF (and implicitly using biholomorphisms between neighborhoods of the connected components of \Gamma in S_k and annuli of the form (-\epsilon ,0]\times S^1), we obtainE_{\mathrm {topo}} ( \tilde{v}|_{S^k} ) &= \int _{ \lbrace s_k \rbrace \times S^1 } \tilde{v}^*\lambda - H(s,t,\ti... | {
"cite_spans": []
} | 10.1112/topo.12105 | 1804.08014 | Symplectic Homology of complements of smooth divisors | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
0.006448494270443916,
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-0.... |
10ac86d0000e5674ee41855c191612701af987d4 | subsection | 58 | 279 | Sketch of isomorphism between the two Floer complexes | As S_k has non-empty interior, we conclude that \partial _s\tilde{v} = 0, which contradicts the fact that \tilde{v}(s,t) \rightarrow \gamma _- as s \rightarrow -\infty .We now explain how the previous result completes the argument that none of the problematic configurations described earlier occur, and thus \Phi is a c... | {
"cite_spans": []
} | 10.1112/topo.12105 | 1804.08014 | Symplectic Homology of complements of smooth divisors | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
-0.055203475058078766,
0.0016306997276842594,
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0.034... |
0de87330f63c4d7a19adca50ce1da536206fbbcd | subsection | 59 | 279 | Sketch of isomorphism between the two Floer complexes | See Figure
\ref {fig:transversality split presplit}.)
For these complexes to be well-defined, we also need to establish
that there are no curves counted either in \partial or in the proof of \partial ^2=0 that
are asymptotic to the degenerate constant orbits at
\partial \operatorname{supp}dH. As we pointed out in Rem... | {
"cite_spans": []
} | 10.1112/topo.12105 | 1804.08014 | Symplectic Homology of complements of smooth divisors | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
-0.058504458516836166,
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0.0016718556871637702,
0.0... |
e83c0359521ff1652c24f698ba02ac110180a34f | subsection | 60 | 279 | Sketch of isomorphism between the two Floer complexes | In addition,
\tilde{H} is a small time-dependent perturbation of H near the non-constant periodic orbits of X_H, using auxiliary Morse functions on the manifolds of orbits in a manner similar to *page 73.
Picking a generic almost complex structure on W, we get a chain complex SC_*(W,\tilde{H}) that computes the symplec... | {
"cite_spans": []
} | 10.1112/topo.12105 | 1804.08014 | Symplectic Homology of complements of smooth divisors | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
-0.023722046986222267,
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0.02... |
c5dcfc60634bd3e7edebaf30d8d2e6cdbc0c5c1a | subsection | 61 | 279 | Sketch of isomorphism between the two Floer complexes | (We may arrange for
such convergence to happen at the +\infty puncture by changing our
subsequence and shifts.) We will again show that this kind of behaviour does
not occur for energy reasons.The rest of this section will now prove these two claims.
In order to rule both of these problem configurations out, we will
co... | {
"cite_spans": []
} | 10.1112/topo.12105 | 1804.08014 | Symplectic Homology of complements of smooth divisors | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
-0.0686754360795021,
0.015825873240828514,
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... |
993fe4486cffc64d5dbf9818237e4d6e71967f21 | subsection | 62 | 279 | Sketch of isomorphism between the two Floer complexes | Let i denote the complex structure on S with i \partial _\sigma =
\partial _\tau .Let J be an S-dependent family of almost complex structures on [r_0, r_0 + \delta ] \times Y for small \delta >0
and let H \colon S \times [r_0, r_0 + \delta ] \times Y \rightarrow \mathbb {R} be an S-dependent family of Hamiltonians
with... | {
"cite_spans": []
} | 10.1112/topo.12105 | 1804.08014 | Symplectic Homology of complements of smooth divisors | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
-0.037863556295633316,
0.05375954136252403,
-0.000518202141392976,
0.03789407014846802,
0.01905382052063942,
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0.019084330648183823,
-0.023630399256944656,
0.0... |
ce7b85eab6eb54c0849ed1cae978350962c356ef | subsection | 63 | 279 | Sketch of isomorphism between the two Floer complexes | Furthermore, since the image of the annulus S is not below r=r_0,
we have dr( d\tilde{v} \circ i)(\frac{\partial }{\partial \tau }) \ge 0 along \lbrace 0\rbrace \times S^1, hence\tilde{v}^*\lambda (\frac{\partial }{\partial \tau }) \le \lambda (X_H) \beta (\frac{\partial }{\partial \tau }).The result now follows.Follow... | {
"cite_spans": []
} | 10.1112/topo.12105 | 1804.08014 | Symplectic Homology of complements of smooth divisors | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
-0.01962069608271122,
0.047022536396980286,
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0.010924119502305984,
0... |
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