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d83accfbb128b12553028518b4f6cb93d9a37367 | subsection | 18 | 24 | Convergence of SC Paths to Dubins Paths | As previously mentioned, Dubins paths are proven to be optimal in terms of length.
SC paths are, however, longer than the Dubins path.
This happens because SC turns have longer turning radii than a circular arc with radius \kappa _{\max }^{-1}.Figure REF shows the Dubins path for a given start and goal configurations \... | {
"cite_spans": []
} | 1801.08995 | Trajectory Generation using Sharpness Continuous Dubins-like Paths with
Applications in Control of Heavy Duty Vehicles | [
"Rui Oliveira",
"Pedro F. Lima",
"Marcello Cirillo",
"Jonas Mårtensson",
"Bo Wahlberg"
] | [
"cs.SY",
"cs.RO"
] | 2,018 | en | Computer Science | [
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0.0... | |
7872c8a980477caa334913345b928f095319b1de | subsection | 19 | 24 | Notes on Computational Cost | As previously mentioned, given two configurations \mathbf {q}_{\mathrm {s}} and \mathbf {q}_{\mathrm {g}}, the SC method computes all possible 16 SC turns and how they can be connected.
The connection process, as detailed in section REF , is computationally cheap.
The bulk of processing comes from finding all 16 possib... | {
"cite_spans": []
} | 1801.08995 | Trajectory Generation using Sharpness Continuous Dubins-like Paths with
Applications in Control of Heavy Duty Vehicles | [
"Rui Oliveira",
"Pedro F. Lima",
"Marcello Cirillo",
"Jonas Mårtensson",
"Bo Wahlberg"
] | [
"cs.SY",
"cs.RO"
] | 2,018 | en | Computer Science | [
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ec4bb36634bedc6ce43bf6c45034da575eb01d53 | subsection | 20 | 24 | Precomputation of Cubic Curvature Paths | As previously stated, the SC method computation speed is limited by the generation of the cubic curvature paths.
Depending on the application, some assumptions can be made that greatly improve the computation speed, by allowing the precomputation of the cubic curvature paths to be used.If one assumes that the start and... | {
"cite_spans": []
} | 1801.08995 | Trajectory Generation using Sharpness Continuous Dubins-like Paths with
Applications in Control of Heavy Duty Vehicles | [
"Rui Oliveira",
"Pedro F. Lima",
"Marcello Cirillo",
"Jonas Mårtensson",
"Bo Wahlberg"
] | [
"cs.SY",
"cs.RO"
] | 2,018 | en | Computer Science | [
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0.0... | |
50be65b1abba1f309af1759a8934bd3db7e7ba0f | subsection | 21 | 24 | Timing Evaluation | We test the steering method, measuring its computational speed for several problem instances. The method is implemented in C++ and running on a Linux Mint distribution. The computer used is equipped with an Intel Core i7-6820HQ Processor running at 2.70 GHz, and with 16,0 GB of RAM.We generate 1000 random pairs of star... | {
"cite_spans": []
} | 1801.08995 | Trajectory Generation using Sharpness Continuous Dubins-like Paths with
Applications in Control of Heavy Duty Vehicles | [
"Rui Oliveira",
"Pedro F. Lima",
"Marcello Cirillo",
"Jonas Mårtensson",
"Bo Wahlberg"
] | [
"cs.SY",
"cs.RO"
] | 2,018 | en | Computer Science | [
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0.01... | |
8580fbf809e4216c7ac3c17bd385342145cdfb0d | subsection | 22 | 24 | Simulations | A simulation test is run in order to understand how the proposed paths affect the performance of a vehicle tracking them.
A kinematic vehicle model coupled with a detailed steering actuator model are used to simulate a vehicle.In the test, two paths consisting of a straight segment, a turn, and a straight segment are g... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 401,
"openalex_id": "",
"raw": "T. Fraichard and A. Scheuer, \"From Reeds and Shepp's to continuous-curvature paths\" IEEE Transactions on Robotics 20, no. 6 (2004): 1025-1035.",
"source_ref_id": "852c4e48cc57ab28f5ba70d844a... | 1801.08995 | Trajectory Generation using Sharpness Continuous Dubins-like Paths with
Applications in Control of Heavy Duty Vehicles | [
"Rui Oliveira",
"Pedro F. Lima",
"Marcello Cirillo",
"Jonas Mårtensson",
"Bo Wahlberg"
] | [
"cs.SY",
"cs.RO"
] | 2,018 | en | Computer Science | [
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7b488fdc981b0b83c78accae97eb55e2a9c07be7 | subsection | 23 | 24 | Conclusions | This paper presented the concept of SC paths.
SC paths respect not only the maximum steering angle constraints, but also maximum steering rate and acceleration constraints.
These properties ease the low-level controller task and introduce an higher degree of smoothness, improving the driving comfort and reducing actuat... | {
"cite_spans": []
} | 1801.08995 | Trajectory Generation using Sharpness Continuous Dubins-like Paths with
Applications in Control of Heavy Duty Vehicles | [
"Rui Oliveira",
"Pedro F. Lima",
"Marcello Cirillo",
"Jonas Mårtensson",
"Bo Wahlberg"
] | [
"cs.SY",
"cs.RO"
] | 2,018 | en | Computer Science | [
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9c6544d60ce0db5e737dc91c94c6efbb65d2174e | abstract | 0 | 126 | Abstract | We introduce a chain complex associated to a Liouville domain $(\overline{W},
d\lambda)$ whose boundary $Y$ admits a Boothby--Wang contact form (i.e. is a
prequantization space). The differential counts cascades of Floer solutions in
the completion $W$ of $\overline{W}$, in the spirit of Morse--Bott homology (as
in wor... | {
"cite_spans": []
} | 1804.08013 | Morse-Bott Split Symplectic Homology | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
-0.03403608500957489,
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... | |
5f098fa8ea4fcc4e2b3745293e3709b2d8a0408e | subsection | 1 | 126 | Introduction and statement of main results | In this paper we define Morse–Bott split symplectic homology theory for
Liouville manifolds W of finite-type whose boundary Y = \partial W is a
prequantization space.
This is inspired by the construction of Bourgeois and Oancea for positive symplectic homology, SH^+
.
Our main result is that we obtain transversality fo... | {
"cite_spans": []
} | 1804.08013 | Morse-Bott Split Symplectic Homology | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
-0.03137967362999916,
0.007539668586105108,
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0.027625100687146187,
0.0420023649930954,
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0.01214130874723196,
0.02509152889251709,
0.002... | |
2f306cb536b8bfa1978d0c814ff0367a663c02e8 | subsection | 2 | 126 | Introduction and statement of main results | The differential will be obtained from moduli spaces of Floer cylinders with
cascades
together with asymptotic boundary conditions given in terms of the auxiliary Morse functions.We then prove three main results.
The first is a transversality theorem for “simple cascades”.
These are elements of the relevant moduli spac... | {
"cite_spans": []
} | 1804.08013 | Morse-Bott Split Symplectic Homology | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
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0.0058403159491717815,
0.0... | |
3970c0246a8cd9c701400910a4b267f0f437a6bc | subsection | 3 | 126 | Set–up | We now provide details of the classes of Liouville manifolds for which we prove
transversality in split Floer homology.We begin by summarizing some constructions from , specificallyProposition 2.1 (*Lemma 2.2)
Let (\overline{W}, d\lambda ) be a Liouville domain
with boundary Y = \partial \overline{W}. Assume that
\alp... | {
"cite_spans": []
} | 1804.08013 | Morse-Bott Split Symplectic Homology | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
-0.05497979745268822,
0.023020120337605476,
-0.007803042884916067,
0.02480498142540455,
0.03420219197869301,
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0.03682608902454376,
0.012768615037202835,
0.007326317485421896,
0.03905335068702698,
-0.01453059259802103,
0.0030968086794018745,
0.013699183240532875,
0.01515... | |
d92f85138579be11197007416a12e1e8a837698f | subsection | 4 | 126 | Set–up | We may then extend the distribution \xi _x smoothly to the 0-section by
definining \xi _x = T_x \Sigma if x \in \Sigma .Notice that this gives a splitting T_x N\Sigma = \ker d\pi \oplus \xi _x
and d\pi |_{\xi } \colon \xi _x \rightarrow T_{\pi (x)} \Sigma is a symplectic
isomorphism.Any almost complex structure J_\Sigm... | {
"cite_spans": []
} | 1804.08013 | Morse-Bott Split Symplectic Homology | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
-0.03413848578929901,
0.02166070230305195,
-0.00022821436868980527,
0.05268736928701401,
0.007402011193335056,
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0.048873867839574814,
0.029852109029889107,
0.014300639741122723,
0.03981298208236694,
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0.020638683810830116,
-0.019326837733387947,
0.... | |
d6aa76073f5a0a124b09151002ccb63ba36bdc4e | subsection | 5 | 126 | Set–up | These will also be relevant for the grading given
in Definition REF .Definition 2.4
Given a symplectic manifold (X^{2n},\omega ) and a
codimension-2 symplectic submanifold \Sigma ^{2n-2} \subset X, say that
(X, \Sigma , \omega ) is a monotone triple ifX is spherically monotone: there exists a constant \tau _X > 0 so
t... | {
"cite_spans": []
} | 1804.08013 | Morse-Bott Split Symplectic Homology | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
-0.029750028625130653,
0.011816101148724556,
-0.00923013687133789,
-0.027232717722654343,
0.0012233745073899627,
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0.056357234716415405,
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0.023189766332507133,
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-0.006541192065924406,
0.01968078874051571,
... | |
3aa94281e14e8b5bf7cf99ac8ebce527d4ecdf7d | subsection | 6 | 126 | Set–up | In particular, such almost complex structures are
cylindrical and Reeb-invariant on W \setminus \overline{W}.A compatible almost complex structure J_Y on the symplectization
\mathbb {R}\times Y is admissible if J_Y is cylindrical and
Reeb–invariant.In the following, we will identify W with X \setminus \Sigma by means o... | {
"cite_spans": []
} | 1804.08013 | Morse-Bott Split Symplectic Homology | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
-0.00853643286973238,
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0.04497838020324707,
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0.012846606783568859,
0.012709290720522404,
0.037... | |
4ed3f55ca651ab1147180dfd58ae689cf83bb5d6 | subsection | 7 | 126 | The chain complex | We will now describe the chain complex for the split symplectic homology
associated to W.Definition 3.1
Let h \colon (0, +\infty ) \rightarrow \mathbb {R} be a smooth function with the following properties:h(\rho ) = 0 for \rho \le 2;
h^{\prime }(\rho ) > 0 for \rho > 2;
h^{\prime }(\rho ) \rightarrow +\infty as \rh... | {
"cite_spans": []
} | 1804.08013 | Morse-Bott Split Symplectic Homology | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
-0.00029698992148041725,
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0.01598692499101162,
0.027778808027505875,
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0.029685644432902336,
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-0.002324434695765376,
0.03716044872999191,
... | |
24f73167e9d38230a1210b3aaf881cebd35afa91 | subsection | 8 | 126 | 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": []
} | 1804.08013 | Morse-Bott Split Symplectic Homology | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
-0.009124988690018654,
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0.031739089637994766,
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0.001132993958890438,
0.012512526474893093,
... | |
ebc32b04269ed1d5d3775fa2e98ef5927c9f704a | subsection | 9 | 126 | 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": []
} | 1804.08013 | Morse-Bott Split Symplectic Homology | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
0.005289755295962095,
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0.028710367158055305,
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0.021021723747253418,
0.01396617479622364,
0.0020518300589174032,
0.050159238278865814,
-0.025811871513724327,
0.0008333176374435425,
0.01298983953893184... | |
27211ebe217dc6fb051c76501c3dec3406140c42 | subsection | 10 | 126 | The chain complex | The action of \gamma _k is the negative of the
y-intercept of the tangent line to the graph of h at \operatorname{e}^{b_k}.
See
Figure REF . The convexity of h implies that
\mathcal {A}(\gamma _k) is monotone increasing in k. | {
"cite_spans": []
} | 1804.08013 | Morse-Bott Split Symplectic Homology | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
0.005246732849627733,
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0.04356123507022858,
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-0.0006501178722828627,
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... | |
dadaed5895fdd7bbc51f29a67cf19e8ed718ed2a | subsection | 11 | 126 | Gradings | We will now define the gradings of the generators.
For this, we will assume that (X, \Sigma , \omega ) is a monotone triple as in
Definition REF .Definition 3.4
For a critical point \sbox
{p}\widetilde{\usebox {
}} of f_Y, and a multiplicity k, we define| \sbox
{p}\widetilde{\usebox {
}}_k | = \tilde{M}( \sbox
{p}\... | {
"cite_spans": []
} | 1804.08013 | Morse-Bott Split Symplectic Homology | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
-0.02808237448334694,
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0.03474831581115723,
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-0.012538681738078594,
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... | |
285976104a73a4e908b75ef68e761eef9f00b447 | subsection | 12 | 126 | Gradings | Notice that this index does not explicitly
depend on the covering multiplicity k of the orbit.Notice also that Y may be capped off by the normal disk bundle over \Sigma ,
and each orbit bounds a disk fibre. The trivialization induced by the fibre
differs from the constant trivialization only in a k-fold winding in the ... | {
"cite_spans": []
} | 1804.08013 | Morse-Bott Split Symplectic Homology | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
-0.05176602676510811,
0.033330485224723816,
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0.012903353199362755,
0.016909420490264893,
0... | |
362c0cea9a23b634f7794647625078b283a5a0b9 | subsection | 13 | 126 | Gradings | Recall that c_1(L) = c_1(TX).Putting this together, we obtain that the Conley–Zehnder index of the orbit
with respect to the trivialization induced by the disk \dot{B} is given by\operatorname{CZ}_H^W( \sbox
{p}\widetilde{\usebox {
}}_k) = \tilde{M}( \sbox
{p}\widetilde{\usebox {
}}) +1 - n - 2k + 2 \langle c_1(TX), ... | {
"cite_spans": []
} | 1804.08013 | Morse-Bott Split Symplectic Homology | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
-0.04591885581612587,
0.03731479123234749,
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0.03328735753893852,
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0.03588078171014786,
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0.007082335185259581,
-0.0... | |
7fa8240892910484a4b3e5a6b66cff437324e305 | subsection | 14 | 126 | Gradings | This gives a a Conley–Zehnder index, which we denote by
\operatorname{CZ}_{L^{\otimes N}}(\gamma ).
Notice that this Conley–Zehnder index
depends only on the trivialization of
L^{\otimes N} and not on the further trivialization of TW^N.
The Seidel–McLean grading is
then defined to be\operatorname{SM}(\gamma ) = \frac{1... | {
"cite_spans": []
} | 1804.08013 | Morse-Bott Split Symplectic Homology | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
-0.08272446691989899,
0.04368218407034874,
-0.006933134980499744,
-0.016850151121616364,
0.004655159078538418,
-0.02350473962724209,
0.02176477760076523,
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0.03824862092733383,
0.02634362317621708,
-0.021139001473784447,
-0.010783180594444275,
0.008364023640751839,
-0... | |
ce2c354cad839d85ad7db5216df62348d9e85dc8 | subsection | 15 | 126 | Gradings | From this, we obtainN \operatorname{CZ}_0( \widehat{p}_m) = \operatorname{CZ}_{L^{\otimes N}}(\gamma _m) + 2mwNow, it follows from this that\operatorname{CZ}_H^W(\sbox
{p}\widetilde{\usebox {
}}_m) &= \operatorname{CZ}_0(\sbox
{p}\widetilde{\usebox {
}}_m) +2 \left( \frac{\tau _X -K}{K} \right) m
\\
&= \operatorname{... | {
"cite_spans": []
} | 1804.08013 | Morse-Bott Split Symplectic Homology | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
-0.040393322706222534,
0.0030889909248799086,
-0.02645091339945793,
0.012027996592223644,
0.009045785292983055,
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0.021554291248321533,
-0.03389500081539154,
-0.009648329578340054,
-0.007558493409305811,... | |
d0bd53f13091971df1c65029eb6868b8cef57ab6 | subsection | 16 | 126 | Split symplectic homology moduli spaces | In this section, we describe the moduli spaces of cascades that contribute to the differential in the Morse–Bott split symplectic homology of W.We also define auxiliary moduli spaces of spherical “chains of pearls” in \Sigma and in X. (These are familiar objects, reminiscent of ones considered in the literature for Flo... | {
"cite_spans": []
} | 1804.08013 | Morse-Bott Split Symplectic Homology | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
-0.020243225619196892,
0.00491207093000412,
-0.024362042546272278,
-0.009046143852174282,
-0.007886772975325584,
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0.02305012196302414,
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0.041432250291109085,
0.014774350449442863,
-0.010670787654817104,
-0.002593329409137368,
-0.018961815163493156,
... | |
76318a7f4a9ec674ad74f2a4c3112e032681a01c | subsection | 17 | 126 | Split Floer cylinders with cascades | We now identify the moduli spaces of split Floer cylinders with cascades
we use to define the differential on the chain complex (REF ).First, we define the basic building blocks: split Floer cylinders. We consider
two types of basic split Floer cylinders: one connecting two non-constant
1-periodic Hamiltonian orbits an... | {
"cite_spans": []
} | 1804.08013 | Morse-Bott Split Symplectic Homology | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
-0.01967361941933632,
-0.013538015075027943,
-0.05021428316831589,
0.017277376726269722,
0.02797652967274189,
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0.027274444699287415,
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0.05277841538190842,
-0.025107141584157944,
-0.021001474931836128,
-0.009722339920699596,
0... | |
c309a36bcbca6dd5badce959dbeedb1e838f811c | subsection | 18 | 126 | Split Floer cylinders with cascades | A split Floer cylinder from x_- to x_+ consists
of \tilde{v}_1=(b,v) \colon \mathbb {R}\times S^1\setminus \Gamma \rightarrow \mathbb {R}\times Y (where
\Gamma = \lbrace z_1,\ldots ,z_k\rbrace \subset \mathbb {R}\times S^1 is a (possibly empty) finite subset),
\tilde{v}_0 \colon \mathbb {R}\times S^1 \rightarrow W, and... | {
"cite_spans": []
} | 1804.08013 | Morse-Bott Split Symplectic Homology | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
-0.01658940687775612,
0.0039756582118570805,
-0.05384308099746704,
0.02176310494542122,
0.017016733065247536,
0.00637936731800437,
0.03183579072356224,
0.016665715724229813,
0.014674071222543716,
0.027791455388069153,
-0.004570862744003534,
0.01088155247271061,
-0.02043534256517887,
0.0196... | |
3e19086f21cce44badaea245e017b8a281048e6e | subsection | 19 | 126 | Split Floer cylinders with cascades | Recall that the Hofer energy of a punctured pseudoholomorphic curve \tilde{u} in the symplectization of Y with contact form \alpha is given by:\sup \lbrace \int \tilde{u} {}^{*}d(\psi \alpha ) \, | \, \psi \colon \mathbb {R}\rightarrow [0,1] \text{ smooth and nondecreasing} \rbrace .In a symplectic manifold either comp... | {
"cite_spans": []
} | 1804.08013 | Morse-Bott Split Symplectic Homology | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
-0.012596276588737965,
0.00773629592731595,
-0.0035515243653208017,
0.0018320296658203006,
0.009163009002804756,
-0.07092367857694626,
0.02296474389731884,
0.019165262579917908,
0.044952914118766785,
0.028656339272856712,
0.005230010952800512,
0.004020737484097481,
-0.01902793161571026,
0.... | |
d9020674840f6cf855f365a75cfa990e438cb431 | subsection | 20 | 126 | Split Floer cylinders with cascades | Then, \tilde{v} has finite
hybrid energy if and only if \tilde{v}|_{S_0} has finite Floer energy
and \tilde{v}|_{S_1} has finite Hofer energy.Equivalently, \tilde{v} has finite hybrid
energy if and only if the punctures \lbrace \pm \infty \rbrace \cup \Gamma can be partitioned into
\Gamma _F and \Gamma _C (with +\infty... | {
"cite_spans": []
} | 1804.08013 | Morse-Bott Split Symplectic Homology | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
-0.024607598781585693,
0.016705095767974854,
0.012791984714567661,
0.017986582592129707,
-0.003871158231049776,
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0.007456270046532154,
0.012273287400603294,
0.04601147770881653,
0.04335696995258331,
0.02109876461327076,
-0.010946033522486687,
-0.012074962258338928,
0.0... | |
5c97e14845a2255b48dda58b6dd5c1bd05d9fcd3 | subsection | 21 | 126 | Split Floer cylinders with cascades | Let (f_i, Z_i), i=0, \dots , N 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_i}
and f_i = -f_W, Z_i=-Z_W if S_i = W.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 with 0 cascades... | {
"cite_spans": []
} | 1804.08013 | Morse-Bott Split Symplectic Homology | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
-0.012104276567697525,
0.0016579502262175083,
-0.0459764190018177,
0.0005625018384307623,
0.03837978467345238,
0.013225466012954712,
0.020242437720298767,
0.002717169700190425,
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0.03880690410733223,
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0.001969709759578109,
-0.013927163556218147,
0... | |
f5d6790a10835fd863e034d05521f2020f75f1d1 | subsection | 22 | 126 | Split Floer cylinders with cascades | Since we require that the sublevels are non-trivial,
it follows that any such cascade with
collections of orbits S_i = Y_{k_i}, i=1, \dots , N and S_0 = Y_{k_0}, or,
if S_0 = W, with k_0 = 0, we must have that the sequence of multiplicities
is monotone increasing k_0 < k_1 < \dots < k_N.By a standard SFT-type compactne... | {
"cite_spans": []
} | 1804.08013 | Morse-Bott Split Symplectic Homology | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
-0.007279652636498213,
-0.02565428987145424,
-0.042914848774671555,
0.00030236502061598003,
0.012079339474439621,
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-0.0023159063421189785,
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0.018267806619405746,
0.03189617022871971,
-0.025440631434321404,
-0.007428450509905815,
-0.0097825098782777... | |
606999b95a35edb931084554108a9d133137393a | subsection | 23 | 126 | Transversality for the Floer and holomorphic moduli spaces | In this section, we will build the transversality theory needed
for the Floer cylinders with cascades that appear in the split Floer differential
as in Equation (REF ).
In the process, we will also discuss transversality for pseudoholomorphic
curves in X and in \Sigma , which will be necessary for the proof of our
main... | {
"cite_spans": []
} | 1804.08013 | Morse-Bott Split Symplectic Homology | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
-0.051899924874305725,
0.02448541484773159,
0.0011060390388593078,
0.028406132012605667,
-0.0049085249193012714,
0.018413642421364784,
-0.009923839010298252,
-0.002644958673045039,
-0.007162556052207947,
0.02364635095000267,
-0.011121412739157677,
0.0012843400472775102,
-0.009107658639550209... | |
e1bfd18878a00ba0c93c3dc2972862a3919bef41 | subsection | 24 | 126 | Statements of transversality results | Before stating the main result of this section, we will introduce some
definitions allowing us to relate transversality for split Floer cylinders
with cascades to transversality problems for spheres in \Sigma and in X
with various constraints.Lemma 5.1
Let \tilde{v} \colon \mathbb {R}\times S^1 \setminus \Gamma \right... | {
"cite_spans": []
} | 1804.08013 | Morse-Bott Split Symplectic Homology | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
-0.005403159186244011,
-0.0034819087013602257,
-0.021963689476251602,
0.004136316943913698,
0.015080003067851067,
-0.0017648030770942569,
0.012508160434663296,
-0.012760003097355366,
0.0074865808710455894,
0.0422484315931797,
-0.006528817117214203,
0.009676843881607056,
-0.013645266182720661... | |
4cb3b15aaea1f51d1e577bc55137ab1904e11fb3 | subsection | 25 | 126 | Statements of transversality results | If such a chain of
pearls is the projection to \Sigma of the components in \mathbb {R}\times Y is a split Floer cylinder, then the additional marked points in
the pseudoholomorphic spheres correspond to augmentation punctures in the Floer
cylinders, where they converge to cylinders over Reeb orbits that
are capped by p... | {
"cite_spans": []
} | 1804.08013 | Morse-Bott Split Symplectic Homology | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
0.004324977286159992,
-0.008146517910063267,
-0.0389019139111042,
0.017605021595954895,
0.011113742366433144,
-0.015713321045041084,
0.02189186029136181,
0.0021968139335513115,
-0.005404314491897821,
0.002759366063401103,
-0.004214373882859945,
0.013180883601307869,
-0.006777323316782713,
... | |
2f78071174faaa8f4a5830c463d459dd9a6ef99c | subsection | 26 | 126 | Statements of transversality results | Additionally, again by Lemma REF ,
if any of the sublevels have augmentation planes,
then those correspond to spheres in X passing through \Sigma at the
images of the corresponding marked points in the chain of pearls.Observe that we allow the sphere w_1 to be unstable in the definition
of a chain of pearls in \Sigma w... | {
"cite_spans": []
} | 1804.08013 | Morse-Bott Split Symplectic Homology | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
0.003798979800194502,
-0.00308381044305861,
-0.05013432726264,
-0.01224369928240776,
0.0320701003074646,
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0.017103036865592003,
0.06261450797319412,
0.019010156393051147,
0.0016963817179203033,
-0.004577083978801966,
0.0022046288941055536,
0.0007371012470684946,
0.009... | |
28f9fc7601992b90a8ea99e72ec0eec58ba56723 | subsection | 27 | 126 | Statements of transversality results | In , there is no such
condition on constant spheres.Definition 5.8
Given a finite hybrid energy Floer cylinder with N cascades, we obtain an
augmented chain of pearls (possibly with a sphere in X) by the following
construction:cylinders in \mathbb {R}\times Y are projected to \Sigma : by
Lemma REF these form holomorph... | {
"cite_spans": []
} | 1804.08013 | Morse-Bott Split Symplectic Homology | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
0.01684591919183731,
0.004859986715018749,
-0.023178398609161377,
-0.0016021934570744634,
0.0026836739853024483,
-0.03164713457226753,
-0.01139846257865429,
0.029709244146943092,
0.00511175999417901,
0.029419323429465294,
0.01211563404649496,
0.008567919954657555,
-0.016296597197651863,
0.... | |
275575aff1dfb6251abf245ec279dfd08ad520e5 | subsection | 28 | 126 | Statements of transversality results | Note that when both x, y are generators in \mathbb {R}\times Y,
the moduli space \mathcal {M}_{H,N}^*(x,y; J_W) will depend on J_W only insofar as
augmentation planes appear, otherwise it depends only on J_Y.Proposition 5.9
There exists a residual set \mathcal {J}^{reg}_W \subset \mathcal {J}_W of
almost complex struc... | {
"cite_spans": []
} | 1804.08013 | Morse-Bott Split Symplectic Homology | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
-0.026793450117111206,
0.028136173263192177,
-0.01028404664248228,
-0.0038469808641821146,
-0.010459516197443008,
-0.06390146166086197,
0.018859170377254486,
0.03515496104955673,
0.013602711260318756,
0.01908804476261139,
-0.017348606139421463,
-0.01248886063694954,
-0.0029124144930392504,
... | |
43c4902f64ab79db88049fe864cf75e7a9e8c6ea | subsection | 29 | 126 | Statements of transversality results | In the case of a Floer cascade that
descends to W, there are therefore N+1 cylinders in the cascade.Remark 5.10
These index formulas justify that the moduli spaces are rigid (modulo
their \mathbb {R}, \mathbb {R}^{N} and \mathbb {R}^{N+1} actions) when the index difference is
1, which then justifies the definition of ... | {
"cite_spans": []
} | 1804.08013 | Morse-Bott Split Symplectic Homology | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
-0.00132238760124892,
0.0053086187690496445,
-0.017710650339722633,
0.031043216586112976,
0.03780102729797363,
0.002333961660042405,
0.017298774793744087,
0.024621007964015007,
0.019800538197159767,
0.07566307485103607,
-0.008428195491433144,
-0.015613135881721973,
0.007795127108693123,
0.... | |
a10322662bdc06c6decd69ecf5b20229dbacd6b4 | subsection | 30 | 126 | A Fredholm theory for Morse–Bott asymptotics | In this section, we collect some facts about Cauchy–Riemann operators on
Hermitian vector bundles over punctured Riemann surfaces, specifically in the
context of degenerate asymptotic operators.
These facts can mostly be found in the literature, but not in a unified way.
The main reference for these results is
. Additi... | {
"cite_spans": []
} | 1804.08013 | Morse-Bott Split Symplectic Homology | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
-0.01762622594833374,
0.015718625858426094,
-0.049689169973134995,
0.05564088374376297,
0.004314991645514965,
-0.05200881510972977,
0.07007760554552078,
0.014459609985351562,
0.014154394157230854,
0.05521358177065849,
-0.019930606707930565,
0.014741934835910797,
0.02956017293035984,
0.0484... | |
e0ff708aa990d6323287cb4de0bcc5881d5186d5 | subsection | 31 | 126 | A Fredholm theory for Morse–Bott asymptotics | We take the push-forward of the standard complex structure in \mathbb {R}\times S^1 by the final map of the isotopy, to produce a family of complex
structures on \mathbb {R}\times S^1, which can be assumed standard near \Gamma
and outside of a compact set.For each z \in \Gamma , let \beta _{z} \colon \dot{S} \rightarr... | {
"cite_spans": []
} | 1804.08013 | Morse-Bott Split Symplectic Homology | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
-0.06299303472042084,
0.01927330531179905,
-0.04739737510681152,
0.018510308116674423,
0.02249315194785595,
-0.03616606071591377,
-0.003906544763594866,
0.015870338305830956,
0.00529519934207201,
0.006481659598648548,
-0.008011468686163425,
0.03171015530824661,
-0.003460191423073411,
0.035... | |
54861d347af7a606b463ab1099efa0ea1a60a223 | subsection | 32 | 126 | A Fredholm theory for Morse–Bott asymptotics | By a similar construction, we may define W^{m,p,
\mathbf {\delta }} as well.We will say that a differential operator D \colon \Gamma (E) \rightarrow \Gamma (\Lambda ^{0,1} T^*\dot{S} \otimes E) is a Cauchy–Riemann operator if it is
a real linear Cauchy–Riemann operator *Definition
C.1.5 such that, near \pm \infty , it ... | {
"cite_spans": []
} | 1804.08013 | Morse-Bott Split Symplectic Homology | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
-0.013295101001858711,
0.036063820123672485,
-0.0538211315870285,
0.00625853706151247,
0.003993107005953789,
0.005392792169004679,
0.03710118681192398,
0.03777242824435234,
-0.0035297235008329153,
0.05257018655538559,
-0.031121062114834785,
0.007948075421154499,
0.038809794932603836,
0.013... | |
6e59f0e84d51914300324c466ceb43a941ed6210 | subsection | 33 | 126 | A Fredholm theory for Morse–Bott asymptotics | This will consist of a discrete set of eigenvalues. If an
asymptotic operator \mathbf {A}_z does not have 0 in its spectrum,
we say the asymptotic operator is non-degenerate. If all the asymptotic
operators are non-degenerate, we say D itself is non-degenerate.Note that we obtain a path of symplectic matrices associate... | {
"cite_spans": []
} | 1804.08013 | Morse-Bott Split Symplectic Homology | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
-0.030690811574459076,
0.023212693631649017,
-0.03974086418747902,
0.020099353045225143,
0.022113867104053497,
-0.05561278760433197,
0.04569283500313759,
-0.016512908041477203,
0.03971033915877342,
0.037604257464408875,
-0.023655274882912636,
-0.00183614541310817,
0.013628491200506687,
0.0... | |
fdb0ebb316f9866a778f438c8da2e35ad319ef4c | subsection | 34 | 126 | A Fredholm theory for Morse–Bott asymptotics | It
can often be combined with Proposition REF to compute
Conley–Zehnder indices of interest.Lemma 5.13
Given a constant C\ge 0, the spectrum \sigma (A_C) of the operator\mathbf {A}_C:= - i \frac{d}{dt} - \begin{pmatrix} C & 0 \\
0 & 0
\end{pmatrix} \colon W^{1,p}(S^1, \rightarrow L^p(S^1,is the set\left\lbrace \frac{1... | {
"cite_spans": []
} | 1804.08013 | Morse-Bott Split Symplectic Homology | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
-0.03330431878566742,
0.013118727132678032,
-0.03397589921951294,
-0.0016207618173211813,
0.012866884469985962,
-0.0015234589809551835,
0.026252716779708862,
0.014614521525800228,
0.04887279123067856,
0.034678008407354355,
-0.016881108283996582,
0.010325560346245766,
0.0012382277054712176,
... | |
967949f96b0d2e150f57263e6a0ae1624f7b0d92 | subsection | 35 | 126 | A Fredholm theory for Morse–Bott asymptotics | Then\operatorname{CZ}(\mathbf {A}+ \delta ) = {\left\lbrace \begin{array}{ll}
0 & \text{ if } C > 0 \\
-1 & \text{ if } C = 0
\end{array}\right.}
\qquad \text{and} \qquad \operatorname{CZ}(\mathbf {A}_C - \delta ) = 1.For any n\ge 0, taking-i\frac{d}{dt}\colon W^{1,p}(S^1,n) \rightarrow L^p(S^1,n),we have\operatorname{... | {
"cite_spans": []
} | 1804.08013 | Morse-Bott Split Symplectic Homology | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
-0.032935675233602524,
0.011957877315580845,
-0.047526270151138306,
0.015414750203490257,
-0.0006376708042807877,
-0.022389542311429977,
0.04053621366620064,
0.015765778720378876,
-0.004773230757564306,
0.02367156185209751,
-0.0376058854162693,
0.008462850004434586,
0.022542163729667664,
0... | |
7e2c682637ad31a28f7919f241a037384485c4ec | subsection | 36 | 126 | A Fredholm theory for Morse–Bott asymptotics | A negative weight \delta _z < 0 always corresponds to
allowing exponential growth.Theorem 5.16
Let \mathbf {\delta } \colon \Gamma \cup \lbrace \pm \infty \rbrace \rightarrow \mathbb {R} such that
-\delta _z \notin \sigma ( \mathbf {A}_z ) for positive punctures z \in \Gamma _+ and such that +\delta _z \notin \sigma (... | {
"cite_spans": []
} | 1804.08013 | Morse-Bott Split Symplectic Homology | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
-0.07038795202970505,
0.003975728992372751,
-0.035468690097332,
0.025746850296854973,
0.0010492566507309675,
-0.031653210520744324,
0.022160300984978676,
0.009958399459719658,
0.041207168251276016,
0.023686490952968597,
-0.03644545376300812,
0.00987445842474699,
-0.0017322272760793567,
-0.... | |
f13878d42a6600ac3cd08e9431bda3e66a468a92 | subsection | 37 | 126 | A Fredholm theory for Morse–Bott asymptotics | We then define\begin{split}
W^{1, p, \mathbf {\delta }}_{\mathbf {V}} (\dot{S},E)
= &\lbrace u \in W^{1,p}_\text{loc}(\dot{S},E)\,
| \, \exists \, c_z \in V_z, z \in \Gamma , c_- \in V_-, c_+ \in V_+ \\
&\qquad \text{such that } u - \sum c_z \mu _z - c_- \mu _- - c_+ \mu _+
\in W^{1,p, \mathbf {\delta }}(\dot{S},E) \rb... | {
"cite_spans": []
} | 1804.08013 | Morse-Bott Split Symplectic Homology | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
-0.03978480398654938,
0.04448331892490387,
-0.033560801297426224,
0.037099942564964294,
0.014736243523657322,
0.0005405958509072661,
0.038442373275756836,
0.009946200996637344,
0.040517039597034454,
0.04106621816754341,
-0.009107181802392006,
0.010670810006558895,
0.05360575392842293,
-0.0... | |
ad9f7ff1924992418ab10f1b0b976c05ebe6c0aa | subsection | 38 | 126 | A Fredholm theory for Morse–Bott asymptotics | Combining Theorem REF with Lemma REF , we have:Theorem 5.18
Let \delta > 0 be sufficiently small that
for z \in \Gamma _+,
[-\delta , 0) \cap \sigma ( \mathbf {A}_z) = \emptyset and such that for z \in \Gamma _-, (0, \delta ] \cap \sigma (\mathbf {A}_z) = \emptyset .For each z \in \Gamma , fix the subspace V_z \subset... | {
"cite_spans": []
} | 1804.08013 | Morse-Bott Split Symplectic Homology | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
-0.05982653424143791,
0.02852443791925907,
-0.015063466504216194,
-0.010378072038292885,
0.009958370588719845,
0.004368710797280073,
0.03510230407118797,
-0.0056316303089261055,
0.012774186208844185,
0.03107316978275776,
-0.043832093477249146,
0.0006839226116426289,
0.01385777909308672,
0.... | |
f15a0ecdbcea50b8db58fe64195c21c6ea0d6485 | subsection | 39 | 126 | A Fredholm theory for Morse–Bott asymptotics | The former indices are used in *Proposition 4.22, whereas
the latter can be computed using Theorem REF . Additionally,
the latter arises naturally in the linearization of the non-linear problem.We first learned this result from Wendl . We give a
proof of this formulation since it is slightly stronger than what we
have ... | {
"cite_spans": []
} | 1804.08013 | Morse-Bott Split Symplectic Homology | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
-0.016220856457948685,
0.017289021983742714,
-0.013603851199150085,
0.01706012897193432,
-0.00268758088350296,
0.008377470076084137,
0.04480191320180893,
0.02150064706802368,
0.022889262065291405,
0.039339009672403336,
-0.051149871200323105,
-0.003917878493666649,
0.007526752073317766,
0.0... | |
10ade16eda4cd0ea0bc1c9126aa130cf71e2b2c8 | subsection | 40 | 126 | A Fredholm theory for Morse–Bott asymptotics | See also the very closely related *Proposition 3.15.Note that W^{1,p, \mathbf {\delta }}_{\mathbf {V}}( \dot{S}, E) is a subspace of W^{1,p, \mathbf {\delta }^{\prime }}(\dot{S}, E), and thus the kernel of D_\delta is contained in the kernel of D_{\delta ^{\prime }}.Now, by a linear version of the analysis done in , an... | {
"cite_spans": []
} | 1804.08013 | Morse-Bott Split Symplectic Homology | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
-0.002661544131115079,
0.02698557637631893,
-0.022238679230213165,
0.005418484099209309,
-0.03066403977572918,
-0.005231508053839207,
0.031244046986103058,
0.0036880034022033215,
0.04047836363315582,
0.025886615738272667,
-0.019567595794796944,
-0.03081667423248291,
0.022498156875371933,
-... | |
56b46de9b6a89bc3bb0cb35c928f61bce8bbbc87 | subsection | 41 | 126 | The linearization at a Floer solution | The first step in the proof of Proposition REF is to
set up the appropriate Fredholm problem.
Given a Floer solution \tilde{v} \colon \mathbb {R}\times S^1 \setminus \Gamma \rightarrow \mathbb {R}\times Y, we consider exponentially weighted Sobolev spaces of sections
of the pull-back bundle \tilde{v}^*T( \mathbb {R}\ti... | {
"cite_spans": []
} | 1804.08013 | Morse-Bott Split Symplectic Homology | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
-0.025854989886283875,
0.011218073777854443,
-0.03217374160885811,
0.010134423151612282,
0.0039034318178892136,
0.006898733787238598,
0.01522452849894762,
0.008218956179916859,
0.048138510435819626,
0.04157555475831032,
-0.005528907757252455,
0.031074827536940575,
-0.0009086067439056933,
0... | |
54b725b2ad0a0131be483467a639d2bfb0741061 | subsection | 42 | 126 | The linearization at a Floer solution | Since the relevant moduli spaces in the differential involve connecting orbits
of bounded multiplicities, for any given moduli space, we may choose \delta
sufficiently small.We now adapt an observation first used in
,
to show that the linearization of the Floer operator is upper triangular with
respect to the splitt... | {
"cite_spans": []
} | 1804.08013 | Morse-Bott Split Symplectic Homology | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
-0.05126098915934563,
-0.00570202199742198,
-0.03402142971754074,
0.07585405558347702,
0.01786506548523903,
0.02384551241993904,
0.018704159185290337,
0.017498914152383804,
0.01585124060511589,
0.04485336318612099,
-0.014882468618452549,
0.007872222922742367,
0.03621833026409149,
0.0541596... | |
a274569adb00dd2b9613aba6bec80a3fc64353ae | subsection | 43 | 126 | The linearization at a Floer solution | The linearized projection d\pi _\Sigma induces an isomorphism of complex vector bundles\tilde{v}^*\big (T (\mathbb {R}\times Y) \big ) \cong ( \mathbb {R}\oplus \mathbb {R}R) \oplus w^*T\Sigma .To see this, note that for each point
p \in Y, d\pi _\Sigma induces a symplectic isomorphism (\xi _p, d\alpha ) \cong (T_{\pi ... | {
"cite_spans": []
} | 1804.08013 | Morse-Bott Split Symplectic Homology | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
-0.04245821386575699,
0.0064404625445604324,
-0.009843834675848484,
0.03183602914214134,
0.0080963633954525,
-0.011637091636657715,
0.027547474950551987,
0.0011713400017470121,
0.052225738763809204,
0.02202271856367588,
-0.025090331211686134,
0.009599646553397179,
0.03128660470247269,
0.03... | |
04757ca0b59cca0d503784413ad157051e0627c1 | subsection | 44 | 126 | The linearization at a Floer solution | Let \mathbf {V_\Sigma } be the kernels of the asymptotic operators of
\dot{D}^\Sigma _w
at each of the punctures, \pm \infty and \Gamma .
(These are explicitly given by
V_\Sigma (- \infty ) = T_{w(0)} \Sigma , V_\Sigma (+\infty ) = T_{w(\infty )}
\Sigma , V_\Sigma (z) = T_{w(z)} \Sigma for each marked point z \in \Gamm... | {
"cite_spans": []
} | 1804.08013 | Morse-Bott Split Symplectic Homology | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
-0.011536377482116222,
-0.006580770947039127,
-0.006035234313458204,
0.0008521621348336339,
0.010590272955596447,
-0.004108690191060305,
0.034090299159288406,
0.007645139005035162,
0.0381799153983593,
0.04379550740122795,
-0.023500028997659683,
0.0005617498536594212,
0.016236383467912674,
... | |
8a59e183384a68d3a88eebb6cfc2279e1ea41a27 | subsection | 45 | 126 | The linearization at a Floer solution | Notice that again these are chosen so that they precisely give the kernels of the
corresponding asymptotic operators of Dv.Lemma 5.22
The isomorphism \tilde{v}^*T( \mathbb {R}\times Y) \cong (\mathbb {R}\oplus \mathbb {R}R) \oplus w^*T\Sigma
induces a decomposition:D_{\tilde{v}} = \begin{pmatrix}
D^{ \tilde{v}} & M \... | {
"cite_spans": []
} | 1804.08013 | Morse-Bott Split Symplectic Homology | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
-0.032314982265233994,
-0.009085728786885738,
-0.024945702403783798,
0.012945828028023243,
0.014326614327728748,
0.025815369561314583,
0.014311357401311398,
0.019163182005286217,
0.043880023062229156,
0.06389761716127396,
-0.0324065275490284,
-0.004245346412062645,
0.023084310814738274,
0.... | |
78c65cd7225f949ccfc869c0b532ec84bab4349a | subsection | 46 | 126 | The linearization at a Floer solution | Let \nabla
be the Levi-Civita connection on T \Sigma for the metric \omega _\Sigma (\cdot ,
J_\Sigma \cdot ).Then, it follows that the linearization D_{\tilde{v}} applied to a section \zeta of \tilde{v}^* T(\mathbb {R}\times Y) satisfies\begin{aligned}D_{\tilde{v}} \zeta \left(\partial _s \right)
&= \sbox
{\nabla }\w... | {
"cite_spans": []
} | 1804.08013 | Morse-Bott Split Symplectic Homology | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
-0.004184816963970661,
-0.004882922396063805,
-0.006260058842599392,
0.03781212866306305,
0.04126069322228432,
0.012283600866794586,
0.02163744904100895,
0.046998124569654465,
0.03476030379533768,
0.029663752764463425,
-0.022339370101690292,
-0.010116804391145706,
0.0023670727387070656,
0.... | |
419b6ff7417cb4be506182fa5ce7c463ecf309e6 | subsection | 47 | 126 | The linearization at a Floer solution | For any two vector fields V and W in T\Sigma ,
since d\alpha ( \tilde{V}, \tilde{W}) = K\omega _\Sigma (V, W),
we have the following[\tilde{V}, \tilde{W}] = \sbox
{[V, W]}\widetilde{\usebox {
}} - K\omega _\Sigma (V, W) R.From this, it follows that the Levi-Civita connection \sbox
{\nabla }\widetilde{\usebox {
}} sat... | {
"cite_spans": []
} | 1804.08013 | Morse-Bott Split Symplectic Homology | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
-0.014703366905450821,
0.015458755195140839,
-0.0210593119263649,
0.04251539334654808,
0.023546751588582993,
0.01494753360748291,
0.05695170536637306,
0.021211914718151093,
0.03589239344000816,
0.03854769840836525,
-0.009682705625891685,
-0.0013677107635885477,
-0.015054355375468731,
0.015... | |
f0744a8d288739ca768facfb288da785a7e1c0e6 | subsection | 48 | 126 | The linearization at a Floer solution | We compute\sbox
{\nabla }\widetilde{\usebox {
}}_s R &= \sbox
{\nabla }\widetilde{\usebox {
}}_{\pi _\xi v_s} R = -\frac{1}{2} J_Y \pi _\xi v_s \\
\sbox
{\nabla }\widetilde{\usebox {
}}_s \zeta &= \sbox
{ \nabla _{w_s} \eta }\widetilde{\usebox {
}} - \frac{K}{2} \omega _\Sigma (w_s, \eta )R
-\frac{1}{2} \alpha (v_s... | {
"cite_spans": []
} | 1804.08013 | Morse-Bott Split Symplectic Homology | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
-0.04030722752213478,
0.06215429678559303,
-0.02059604786336422,
0.06645657867193222,
-0.012472051195800304,
-0.006487755104899406,
0.02617986500263214,
0.021892836317420006,
0.027903830632567406,
0.009619117714464664,
-0.01716337352991104,
-0.0004381427715998143,
-0.0050689163617789745,
0... | |
d5231bc1b443baf53a45ea23d75beb536044ea73 | subsection | 49 | 126 | The linearization at a Floer solution | We then obtain the following covariant derivatives of J_Y, where \tilde{W} is a
section of \tilde{v}^*\xi :(\sbox
{\nabla }\widetilde{\usebox {
}}_{\zeta }J_Y) \partial _r &= \sbox
{\nabla }\widetilde{\usebox {
}}_\zeta R - J_Y \sbox
{\nabla }\widetilde{\usebox {
}}_\zeta \partial _r = -\frac{1}{2} J_Y \zeta \\
(\sb... | {
"cite_spans": []
} | 1804.08013 | Morse-Bott Split Symplectic Homology | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
-0.031107360497117043,
0.03566896170377731,
-0.019512660801410675,
-0.008047637529671192,
0.014378953725099564,
0.00027508806670084596,
0.04344961419701576,
0.038354046642780304,
0.027857793495059013,
0.03868968412280083,
-0.01284570712596178,
-0.022167235612869263,
-0.015233300626277924,
... | |
26b2c000f408658e9c141594290665c1daed4d1e | subsection | 50 | 126 | The linearization at a Floer solution | Observe that in particular, D_{ab} is a pointwise linear map from \tilde{v}^*\xi |_p to \mathbb {R}\partial _r \oplus \mathbb {R}R. The map is surjective except at
critical points of the pseudoholomorphic map w, of which there are finitely many
if w is non-constant.
The decay claim follows since w converges to a point,... | {
"cite_spans": []
} | 1804.08013 | Morse-Bott Split Symplectic Homology | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
-0.005249959882348776,
0.017474429681897163,
-0.03015674464404583,
0.004872237332165241,
0.0030599329620599747,
0.006768479943275452,
0.03244597092270851,
0.006608234252780676,
0.029012132436037064,
0.06122918054461479,
-0.03589507192373276,
-0.005967250559478998,
0.018237505108118057,
0.0... | |
e53b7d2de5930af28bea3021869abe06e4832a96 | subsection | 51 | 126 | The linearization at a Floer solution | For this, we need to compute the Conley–Zehnder indices of the
appropriately perturbed asymptotic operators.
We will first identify the (Morse–Bott degenerate) asymptotic operators at each of the
punctures, and then apply Corollary REF to obtain the
Conley–Zehnder indices of the \pm \delta -perturbed operators.Recall f... | {
"cite_spans": []
} | 1804.08013 | Morse-Bott Split Symplectic Homology | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
-0.022824402898550034,
-0.001745967660099268,
-0.046655766665935516,
0.024807807058095932,
-0.0015514414990320802,
0.009001602418720722,
0.03643360733985901,
-0.0009654791210778058,
0.044184137135744095,
0.04006475955247879,
-0.05031743273139,
-0.011854653246700764,
-0.0064231776632368565,
... | |
65013239e04cb0f2747a46941854f6b5aff40105 | subsection | 52 | 126 | The linearization at a Floer solution | In the case of \delta –exponential growth, the
relevant asymptotic operators are \mathbf {A}_+ - \delta and \mathbf {A}_- + \delta ,
respectively.For the case of exponential decay, Corollary REF then gives the Conley–Zehnder
index of 0 for \mathbf {A}_+ + \delta and of 1 for \mathbf {A}_- - \delta .In the case of expon... | {
"cite_spans": []
} | 1804.08013 | Morse-Bott Split Symplectic Homology | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
-0.06427042931318283,
0.008277946151793003,
-0.039673104882240295,
0.035736311227083206,
-0.012687764130532742,
0.0022907403763383627,
0.011489941738545895,
0.04541044682264328,
0.03527854382991791,
0.03475974500179291,
-0.035675276070833206,
0.014511200599372387,
-0.018127556890249252,
0.... | |
a8a9763cbed185cb037bd1b87838cfa17ffeeade | subsection | 53 | 126 | The linearization at a Floer solution | This adjusted Chern number then becomesc_1(E,l,\textbf {A}_\Gamma )
&= \frac{1}{2}( \operatorname{Ind}(D^C_{\tilde{v}}|_{W^{1,p,\delta }}) -2 +\# \Gamma _0 ) \\
&= \frac{1}{2}( -c-1-2\#\Gamma -2+(1-c) ) = -\# \Gamma -1-c < 0.as necessary to apply *Proposition 2.2.Now, applying
Theorem REF , we compute that the Fredholm... | {
"cite_spans": []
} | 1804.08013 | Morse-Bott Split Symplectic Homology | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
-0.029963307082653046,
-0.002561145694926381,
-0.03310609981417656,
0.023036962375044823,
0.041100382804870605,
-0.0028567358385771513,
0.005377833731472492,
0.009901316836476326,
0.029963307082653046,
0.0498269647359848,
-0.04320574551820755,
-0.017651500180363655,
-0.017971882596611977,
... | |
05c350756209c6efc019d0755740ea0ca118c329 | subsection | 54 | 126 | The linearization at a Floer solution | It thus remains to study transversality for D^\Sigma _w, specifically with
respect to the evaluation maps that will allow us to define the moduli spaces of
chains of pearls in \Sigma (see Section REF ).
Additionally, we need to consider transversality for moduli spaces of planes in
W asymptotic to Reeb orbits in Y, or ... | {
"cite_spans": []
} | 1804.08013 | Morse-Bott Split Symplectic Homology | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
-0.03713217005133629,
0.0349658727645874,
-0.016384530812501907,
-0.012013797648251057,
-0.02085442654788494,
-0.011800218373537064,
0.01211295835673809,
-0.031945258378982544,
0.0025972684379667044,
0.03182321414351463,
-0.0032170279882848263,
-0.005343279335647821,
0.010007684119045734,
... | |
7f70ed0a337a8a1b50728805d790dcc281ab1120 | subsection | 55 | 126 | Transversality for chains of pearls in | In this section and the next, we show that for generic almost complex structure (in a sense
to be made precise), the moduli spaces of chains of pearls and moduli spaces of
chains of pearls with spheres in X (possibly augmented as well) are
transverse. We begin with the definition of several moduli spaces that will be u... | {
"cite_spans": []
} | 1804.08013 | Morse-Bott Split Symplectic Homology | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
-0.020824262872338295,
0.014760054647922516,
-0.03426468372344971,
-0.019344443455338478,
0.015423684380948544,
-0.03569873794913292,
0.014165075495839119,
-0.00034897803561761975,
0.024760277941823006,
0.008665334433317184,
-0.017040807753801346,
0.02048863284289837,
0.001869797590188682,
... | |
cfc0cf8495dc5c1ee1a28f980cd73749c4555467 | subsection | 56 | 126 | Transversality for chains of pearls in | In the following, we will writel = B \bullet \Sigma = K \omega (B)which is the order of contact of v with \Sigma .Let{\mathcal {M}^*_{k, (X, \Sigma )}}( (B; A_1, \dots , A_N); J_W)denote the moduli
space of N parametrized J_\Sigma -holomorphic spheres in \Sigma , representing the
classes A_i, and of a J_X-holomorphic s... | {
"cite_spans": []
} | 1804.08013 | Morse-Bott Split Symplectic Homology | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
0.008429420180618763,
0.017865793779492378,
-0.03139863535761833,
-0.03161223232746124,
-0.0037169544957578182,
-0.002158846938982606,
0.04354310408234596,
-0.027019914239645004,
0.01942199282348156,
0.008635387755930424,
0.005393302999436855,
0.027935326099395752,
0.006407884880900383,
0.... | |
419996c114b0b05ddd97628544ee0cac271d3578 | subsection | 57 | 126 | Transversality for chains of pearls in | We impose r \ge 2 and in general will require
r to be sufficiently large that the Sard–Smale theorem holds (this will
depend on the Fredholm indices associated to the collection of homology classes
and will also depend on the order of contact to \Sigma for the spheres in X).For each of these moduli spaces, we also cons... | {
"cite_spans": []
} | 1804.08013 | Morse-Bott Split Symplectic Homology | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
-0.04977006837725639,
0.03420738875865936,
-0.030393004417419434,
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0.016859572380781174,
-0.02496132254600525,
0.03149154782295227,
-0.0015867833280935884,
0.... | |
1c66dd72e6ca652c7aa46e6142bfbf4ed15a25eb | subsection | 58 | 126 | Transversality for chains of pearls in | Their dimensions are\dim &~{\mathcal {M}^*_{k, \Sigma }}((A_1,\ldots ,A_N); q,p;J_\Sigma ) =~
M(p) + \sum _{i=1}^N 2 \, \langle c_1(T\Sigma ), A_i\rangle - M(q) + N-1 + 2k, \\
\dim &~{\mathcal {M}^*_{k, (X, \Sigma )}}((B; A_1, \dots , A_N); x, p, J_W) \\
=&~ M(p) + \sum _{i=1}^N 2 \, \langle c_1(T\Sigma ), A_i \rangle ... | {
"cite_spans": []
} | 1804.08013 | Morse-Bott Split Symplectic Homology | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
-0.033759500831365585,
0.009973703883588314,
-0.031683869659900665,
-0.051158156245946884,
-0.01505594328045845,
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0.025762222707271576,
0.017673373222351074,
-0.005341693293303251,
0.03681189566850662,
-0.004258092492818832,
... | |
8571d31976d12ea22cfcca7ce37ca68d0a2ba22d | subsection | 59 | 126 | Transversality for chains of pearls in | Letr \ge \max _i B_i
\bullet \Sigma + 2.The universal moduli space
\mathcal {M}^*_X( (B_1, \dots , B_k); J^r_W)
is a Banach manifold and the evaluation maps\operatorname{ev}^a_\Sigma \colon &\mathcal {M}^*_X( (B_1, \dots , B_k); J^r_W)
\rightarrow \Sigma ^k: (f_1, f_2, \dots , f_k) \mapsto ( f_1(\infty ), \dots , f_k(\... | {
"cite_spans": []
} | 1804.08013 | Morse-Bott Split Symplectic Homology | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
-0.05124137923121452,
-0.007351260632276535,
-0.030625803396105766,
-0.020218828693032265,
0.02143958769738674,
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0.0473044291138649,
0.005607863888144493,
0.04147530347108841,
0.023957403376698494,
-0.021302253007888794,
0.014015844091773033,
0.01991363801062107,
-0.0... | |
a12d2ebe5a835f45a0c1cf0180e0f732dfcf6539 | subsection | 60 | 126 | Transversality for chains of pearls in | For this,
we will need the following lemma about evaluation maps
intersecting with the flow diagonals.Lemma 5.30
Suppose f_0 \colon \mathcal {M}_0 \rightarrow \Sigma and f_1 \colon \mathcal {M}_1 \rightarrow \Sigma are
submersions.Then,F \colon \mathcal {M}_0 \times \mathcal {M}_1 &\rightarrow \Sigma ^3 \\
(m_0, m_1) ... | {
"cite_spans": []
} | 1804.08013 | Morse-Bott Split Symplectic Homology | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
-0.05421843379735947,
-0.013752930797636509,
-0.015316630713641644,
-0.010381441563367844,
0.026285413652658463,
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0.011113710701465607,
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0.033043649047613144,
0.0009620568016543984,
-0.02184603177011013,
0.02416488341987133,
-... | |
fce13356c230d78d92a81785f94c1b3c97f0ef9e | subsection | 61 | 126 | Transversality for chains of pearls in | Suppose also that \mathcal {M}_1 is a manifold with a submersion e\colon \mathcal {M}_1 \rightarrow \Sigma .Then the map\hat{\operatorname{ev}}\colon \mathcal {M}_0 \times \mathcal {M}_1 &\rightarrow B\times \Sigma ^3 \\
(m, n) &\mapsto (\operatorname{ev}_-(m), \operatorname{ev}_+(m), e(n), e(n))is transverse to A \tim... | {
"cite_spans": []
} | 1804.08013 | Morse-Bott Split Symplectic Homology | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
-0.04918908327817917,
-0.025586256757378578,
0.008803380653262138,
-0.02334345318377018,
0.01701173186302185,
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0.014936759136617184,
0.02856140211224556,
-0.012114184908568859,
0.021390536800026894,
-0.02401476912200451,
-0.... | |
e324ba02aa9fbdc20e8ec90e3684e340b8f3fce4 | subsection | 62 | 126 | Transversality for chains of pearls in | Let \Delta \subset \Sigma ^2 denote the diagonal.Then
if \sum _{i=1}^N A_i \ne 0,
the universal evaluation map\operatorname{ev}_\Sigma \colon \mathcal {M}_{k, \Sigma }^*((A_1, \dots , A_N); J^r_\Sigma ) \rightarrow \Sigma ^{2N}is transverse to the submanifold
\lbrace x\rbrace \times S \times \lbrace y \rbrace for all x... | {
"cite_spans": []
} | 1804.08013 | Morse-Bott Split Symplectic Homology | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
-0.07005869597196579,
0.004561748821288347,
-0.014448080211877823,
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0.022442582994699478,
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0.022534122690558434,
0.008620026521384716,
0.016934920102357864,
0.026577144861221313,
-... | |
4afb13a674f601ce8029c5f63c071f58317d0e0e | subsection | 63 | 126 | Transversality for chains of pearls in | Then, at least one of A_1, \dots , A_{N-1} or A_2, \dots , A_N is a
collection of spheres satisfying the hypotheses of the lemma.
For simplicity of notation,
let us assume that A_1+ \dots + A_{N-1} \ne 0.
Let S_0 = S_1 \times S_2 \times \dots \times S_{N-2}.
Let k=k_0+k_N where k_N is
the number of marked points we con... | {
"cite_spans": []
} | 1804.08013 | Morse-Bott Split Symplectic Homology | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
-0.014270465821027756,
-0.009790913201868534,
-0.009081205353140831,
0.006524731870740652,
0.015521993860602379,
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0.025366326794028282,
0.005120579153299332,
0.021565955132246017,
0.006246190518140793,
0.007272596005350351,
0.015735669061541557,
-0.00786020327359438,
... | |
5b22d182ca67330adb92613df8fa438e886c4c5d | subsection | 64 | 126 | Transversality for chains of pearls in | Suppose that S \subset \Sigma ^{2N-2} is obtained by taking the product of
some number of copies of \Delta _{f_\Sigma } \subset \Sigma ^2 and of the
complementary number of copies of
\lbrace (p, p) \, | \, p \in \operatorname{Crit}(f_\Sigma ) \rbrace \subset \Sigma ^2,
in arbitrary order.Let \Delta \subset \Sigma \time... | {
"cite_spans": []
} | 1804.08013 | Morse-Bott Split Symplectic Homology | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
-0.0355578288435936,
0.019457610324025154,
-0.0059288484044373035,
-0.0006834008963778615,
-0.025088872760534286,
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0.02095317654311657,
0.015131154097616673,
0.013254066929221153,
0.04099069908261299,
0.003782788524404168,
0.02911774255335331,
-0.029789220541715622,
0... | |
0bd4d55476028ea4c7259a1d67709a90df8267cb | subsection | 65 | 126 | Transversality for chains of pearls in | It suffices therefore to prove that\operatorname{ev}_\Sigma &\colon \mathcal {M}_{k, \Sigma }^*((A_1, \dots , A_N); J^r_\Sigma ) \rightarrow \Sigma ^{2N}is transverse to
W^{s}_\Sigma (q) \times S \times W^u_{\Sigma }(p) .The proposition follows immediately if at least one of
the A_i, i=1, \dots , N is non-zero, or if ... | {
"cite_spans": []
} | 1804.08013 | Morse-Bott Split Symplectic Homology | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
-0.029592782258987427,
0.02968435361981392,
-0.008187998086214066,
-0.03195837140083313,
0.006142905913293362,
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0.03168366104364395,
-0.01071383897215128,
-0.001313475426286459,
0.024815814569592476,
... | |
94d7dcadd3b9d33026df033843fb8861b7bc4082 | subsection | 66 | 126 | Transversality for chains of pearls in | The case of N \ge 2 is similar, using the description of the tangent space to the flow diagonal at
(x, y) \in \Delta _{f_\Sigma }, such that \varphi ^t_{Z_\Sigma }(x) = y for some t > 0, asT_{(x,y)} \Delta _{f_\Sigma } = \lbrace (v, d\varphi ^t_{Z_\Sigma }(x)v + c Z_\Sigma (y)) \, | \, v \in T_x \Sigma , c \in \mathbb ... | {
"cite_spans": []
} | 1804.08013 | Morse-Bott Split Symplectic Homology | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
-0.031181536614894867,
0.036019787192344666,
0.00643701059743762,
0.009447562508285046,
0.027854284271597862,
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0.027167467400431633,
0.012568769045174122,
0.028510577976703644,
0.060043152421712875,
0.006566742900758982,
0.021428721025586128,
0.009920704178512096,
0.0... | |
7b69f0effb7ea617e4d7ce751e8a44e3ed10b0d3 | subsection | 67 | 126 | Transversality for spheres in | We will now consider transversality for a chain of pearls with a sphere in X.
We will extend the results from
Section 6 in . In that paper, Cieliebak and Mohnke prove that the moduli space of simple
curves not contained in \Sigma ,
with a condition on the order of contact with \Sigma ,
can be made transverse by a pertu... | {
"cite_spans": []
} | 1804.08013 | Morse-Bott Split Symplectic Homology | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
-0.04221678525209427,
0.049695536494255066,
-0.01825425587594509,
0.007951895706355572,
0.034707508981227875,
-0.021108392626047134,
0.01843740977346897,
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0.02852609194815159,
0.026435093954205513,
0.007520723156630993,
-0.014583563432097435,
0.014781979843974113,
-0.0... | |
35db4a41b5a3ef75b9d55e6a4ce6f8c83726fc7d | subsection | 68 | 126 | Transversality for spheres in | (This is well-defined, by *Lemma 6.4.)Define the space of simple pseudoholomorphic maps into X that have
order of contact l at \infty to a point in \Sigma to be\begin{split}
\mathcal {M}^*_{\infty , l, (X, \Sigma )}(J_W)
\lbrace (f,J_X) \in \, & W^{m,p}(\mathbb {CP}^1, X) \times J_W
\, | \,
\overline{\partial }_{J_X} f... | {
"cite_spans": []
} | 1804.08013 | Morse-Bott Split Symplectic Homology | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
-0.017949003726243973,
0.01205757912248373,
-0.007516908459365368,
-0.017567435279488564,
0.021474700421094894,
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0.03464646264910698,
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0.035531699657440186,
0.006841531489044428,
-0.004720008000731468,
0.002167311729863286,
0.01602589525282383,
-0.... | |
781002278663e4c3361c87f64e64451f72ed46a8 | subsection | 69 | 126 | Transversality for spheres in | We will obtain transversality
of the evaluation map at 0 by varying J_X freely in the complement
of our chosen neighbourhood of the divisor, X \setminus \varphi (\overline{\mathcal {U}}).The linearized Cauchy–Riemann operator at f with respect to a torsion-free connection is(D_f \xi )(z) = \nabla _s \xi (z) + J_X(f(z))... | {
"cite_spans": []
} | 1804.08013 | Morse-Bott Split Symplectic Homology | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
-0.07692070305347443,
0.04575560986995697,
-0.015918312594294548,
0.001980250235646963,
0.017993951216340065,
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0.05759895592927933,
0.009897436015307903,
0.016711939126253128,
0.033668071031570435,
-0.027502205222845078,
0.00544854998588562,
0.010721586644649506,
0.010... | |
bf5413eefd6314f46fcff6fa99aea8d3afe08e53 | subsection | 70 | 126 | Transversality for spheres in | For the induction step, note that D_f\xi = 0 combined with the product rule implies thatD^{(k-i,i)}\xi (z) = J_X(f(z)) \Big (D^{(k-i+1,i-1)} \xi (z) + \sum _{\alpha ,\beta }
D^\alpha (J_X(f(z)) D^{\beta }\xi (z) + \sum _{\alpha ^{\prime },\beta ^{\prime }} D^{\alpha ^{\prime }}A(z)
D^{\beta ^{\prime }}\xi (z) \Big ).He... | {
"cite_spans": []
} | 1804.08013 | Morse-Bott Split Symplectic Homology | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
-0.06544943153858185,
0.009959365241229534,
-0.03589950501918793,
0.019720304757356644,
0.011928342282772064,
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0.011707023717463017,
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0.021887706592679024,
-0.012409139424562454,
-0.020819269120693207,
0.017766591161489487,
-... | |
d9afc06a5284e68f91b78f9e0ffe031a29ae206a | subsection | 71 | 126 | Transversality for spheres in | By Lemma 6.5 in ,T_{(f,J_X)}
\mathcal {M}^*_{\infty , k, (X, \Sigma )}(J^r()
= \lbrace (\xi ,Y) \in &T_f W^{m,p}(\mathbb {CP}^1, X)\times T_{J_X}J^r( \, | \, \\
&D_f\xi + \frac{1}{2}Y(f) \circ df \circ j = 0,\\
&\xi (0) \in T_{f(0)}\Sigma , d^k \xi (0) \in T_{f(0)}\Sigma \rbrace .We argue by induction on k. The case k=... | {
"cite_spans": []
} | 1804.08013 | Morse-Bott Split Symplectic Homology | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
-0.06316245347261429,
0.045647844672203064,
-0.019818365573883057,
-0.0060301837511360645,
0.027797583490610123,
0.009253147058188915,
0.013441093266010284,
0.012868968769907951,
0.032130468636751175,
0.026851672679185867,
-0.013425836339592934,
-0.033503562211990356,
-0.03115404210984707,
... | |
6ef6dc1b0939bc0d455bba82d140d89c6e77b65f | subsection | 72 | 126 | Transversality for spheres in | We have
D_f \xi _2 + \frac{1}{2} Y_2(f) \circ df \circ j = 0
as well as \xi _2(0) = v, d^{k-1}(\xi _2)(0) \in T_{f(0)}\Sigma and \pi _(\frac{\partial ^{k}}{\partial s^k} \xi _2(0) = 0. Lemma \ref {bootstrap} implies that d^{k}(\xi _2)(0) \in T_{f(0)}\Sigma , hence (\xi _2,Y_2)\in T_{(f,J_X)} \mathcal {M}^*_{\infty , ... | {
"cite_spans": []
} | 1804.08013 | Morse-Bott Split Symplectic Homology | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
-0.11622090637683868,
0.021852459758520126,
0.00011904068378498778,
-0.00790473073720932,
0.009186578914523125,
0.03726515918970108,
0.026735691353678703,
0.0016166166169568896,
0.020967375487089157,
0.02144043706357479,
-0.024156734347343445,
-0.01748807169497013,
0.005138838198035955,
0.... | |
29d46d01d4c981c39d24f346d5546964feda733a | subsection | 73 | 126 | Proof of Proposition | We are now ready to complete the proof of Proposition . To this end, we will show that the transversality problem for a
cascade reduces to the already solved transversality problem for chains of
pearls. The two key ingredients of this are the splitting of the linearized
operator given by Lemma REF and a careful study o... | {
"cite_spans": []
} | 1804.08013 | Morse-Bott Split Symplectic Homology | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
-0.01108737476170063,
0.01108737476170063,
-0.012483788654208183,
0.005295690149068832,
0.032476164400577545,
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0.005268983077257872,
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0.06080123782157898,
-0.012079362757503986,
-0.00362075655721128,
0.01599390059709549,
0.02... | |
c3e893b5217f6fdb935ad6b3c6663c458bf7493c | subsection | 74 | 126 | Proof of Proposition | Let J_Y \in J_Y.Define {\mathcal {M}}_{H, k, \mathbb {R}\times Y; k_-, k_+}^*((A_1, \dots , A_N); J_Y )
to be a set of tuples of
punctured cylinders (\tilde{v}_1, \ldots , \tilde{v}_N) with the
following properties:There is a partition of \Gamma = \Gamma _1 \cup \dots \cup \Gamma _N of
k augmentation marked points
with... | {
"cite_spans": []
} | 1804.08013 | Morse-Bott Split Symplectic Homology | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
-0.037562787532806396,
-0.013777090236544609,
-0.011023198254406452,
0.024731630459427834,
0.0027386352885514498,
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0.0036616851575672626,
0.016859007999300957,
-0.004218566231429577,
0.04708316922187805,
-0.00952801015228033,
0.028149204328656197,
-0.02907988242805004,
... | |
9bbe2529f20a109e1ea8400ca1e60966120508e8 | subsection | 75 | 126 | Proof of Proposition | Let J_W be an almost complex structure on W as given by Lemma , matching J_Y on the cylindrical end.Definition 5.38 Define the moduli space{\mathcal {M}}_{H, k, W; k_+}^*\left((B; A_1,\ldots ,A_N);J_W\right)to consist of
tuples(\tilde{v}_0, \tilde{v}_1, \ldots , \tilde{v}_N)with the propertiesThe map \tilde{v}_0 \colon... | {
"cite_spans": []
} | 1804.08013 | Morse-Bott Split Symplectic Homology | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
-0.0291427094489336,
0.0178365595638752,
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0.016829533502459526,
0.00570266367867589,
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0.016829533502459526,
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0.024473773315548897,
0.04443119093775749,
-0.013297314755618572,
-0.002241013338789344,
0.01319050882011652,
0.02... | |
abf2f365f7d11d80a5bd64255840541cbe8a8792 | subsection | 76 | 126 | Proof of Proposition | For each i \ge 2, the cylinder
\tilde{v}_i has multiplicities k_{i} and k_{i-1} at \pm \infty :
\tilde{v}_i(+\infty , \cdot ) \in Y_{k_i}, \tilde{v}_i(-\infty , \cdot ) \in Y_{k_{i -1}}.
The cylinder \tilde{v}_0 converges at +\infty to a Reeb orbit of multiplicity
k_0.Observe that these moduli spaces are non-empty onl... | {
"cite_spans": []
} | 1804.08013 | Morse-Bott Split Symplectic Homology | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
-0.03603506460785866,
0.003495569108054042,
-0.025523565709590912,
0.014851878397166729,
-0.013905995525419712,
0.0020119070541113615,
0.04171035811305046,
-0.022640151903033257,
0.02578292228281498,
0.02140440233051777,
0.0015904554165899754,
-0.018765084445476532,
0.013890739530324936,
0... | |
1d5120df67f918a94227a6d7aba06ad2c7121906 | subsection | 77 | 126 | Proof of Proposition | In
order to impose these conditions, we will need to study these evaluation maps
and establish their transversality.Proposition 5.39 For J_Y \in J_Y^{reg},
{\mathcal {M}}_{H, k, \mathbb {R}\times Y}^*\left((A_1,\ldots ,A_N);J_Y\right)
is a manifold of dimensionN(2n-1) + \sum _{i=i}^N 2 \, \langle c_1(T\Sigma ), A_i\ran... | {
"cite_spans": []
} | 1804.08013 | Morse-Bott Split Symplectic Homology | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
-0.06648193299770355,
0.04273402690887451,
-0.011240575462579727,
0.020420759916305542,
0.04276455193758011,
0.030295372009277344,
-0.013018615543842316,
-0.005154791753739119,
0.022603247314691544,
0.05207446217536926,
-0.013888558372855186,
-0.018604563549160957,
0.013155975379049778,
-0... | |
9aadb60711d92c2def415353af60775eb619973d | subsection | 78 | 126 | Proof of Proposition | Since the Fredholm index is the sum of these, each component \tilde{v}_i contributes an index of
1 + 2n-2 + 2 \, \langle c_1(T\Sigma ), A_i \rangle + 2k_i = 2n-1 + 2 \, \langle c_1(T\Sigma ), A_i \rangle + 2k_i, where k_i is the number of punctures.We now consider the case of a collection(\tilde{v}_0, \tilde{v}_1, \dot... | {
"cite_spans": []
} | 1804.08013 | Morse-Bott Split Symplectic Homology | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
-0.014568333514034748,
0.0027935351245105267,
-0.022165223956108093,
0.039601460099220276,
0.031180810183286667,
0.0067311800085008144,
0.00044167411397211254,
0.020853310823440552,
0.006654906086623669,
0.05796823650598526,
-0.029685840010643005,
-0.0018372498452663422,
0.010624967515468597... | |
f036518d3521a06d01b05174fc50d15113124304 | subsection | 79 | 126 | Proof of Proposition | We then have asymptotic evaluation maps\begin{aligned}\sbox
{\operatorname{ev}}\widetilde{\usebox {
}}_{Y} &\colon {\mathcal {M}}_{H, k, \mathbb {R}\times Y; k_-, k_+}^*\left((A_1,\ldots ,A_N);J_Y\right) \rightarrow Y^{2N} \\
&(\tilde{v}_1, \ldots , \tilde{v}_N) \mapsto \left(\lim _{s\rightarrow - \infty } v_1(s, 1), ... | {
"cite_spans": []
} | 1804.08013 | Morse-Bott Split Symplectic Homology | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
-0.040137991309165955,
0.018771760165691376,
-0.015299747698009014,
0.027699792757630348,
0.0055208816193044186,
-0.03082841821014881,
0.018649667501449585,
0.02872231975197792,
0.014994516037404537,
0.05732254683971405,
0.008386245928704739,
0.008592276833951473,
-0.024540642276406288,
0.... | |
0509fcbcd5f5bbce749dd31b6be479a4d7cd00f0 | subsection | 80 | 126 | Proof of Proposition | Combining all of these evaluation maps over all punctures in \Gamma , we obtain\sbox
{\operatorname{ev}}\widetilde{\usebox {
}}^a_{\Sigma } &\colon {\mathcal {M}}_{H, k, \mathbb {R}\times Y; k_-, k_+}^*\left((A_1,\ldots ,A_N);J_Y\right) \rightarrow \Sigma ^k \\
\sbox
{\operatorname{ev}}\widetilde{\usebox {
}}^a_{\Sig... | {
"cite_spans": []
} | 1804.08013 | Morse-Bott Split Symplectic Homology | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
0.005874714348465204,
-0.015106407925486565,
-0.03009074367582798,
-0.012176680378615856,
0.03857474774122238,
-0.0355534665286541,
0.004100855905562639,
0.019928252324461937,
0.04263364151120186,
0.06420986354351044,
0.010162493214011192,
0.01564047299325466,
0.01147476676851511,
0.031235... | |
6cfc03140ba912437c901ef76d0479cd9476d1d9 | subsection | 81 | 126 | Proof of Proposition | \end{aligned}Write \Delta _{\Sigma ^k} \subset \Sigma ^k \times \Sigma ^k to denote the diagonal \Sigma ^k.
Then, define{\mathcal {M}}_{H}^*(\tilde{q}_{k_-},\tilde{p}_{k_+}; (A_1,\ldots ,A_N), (B_1, \dots , B_k) ; J_W)
= \sbox
{\operatorname{ev}}\widetilde{\usebox {
}}^{-1} \left(
W_Y^s(\tilde{q}) \times \left( \tilde... | {
"cite_spans": []
} | 1804.08013 | Morse-Bott Split Symplectic Homology | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
-0.03304058313369751,
-0.0098465820774436,
0.014361823908984661,
-0.03877615928649902,
-0.009678785689175129,
0.02635924704372883,
0.02822025865316391,
0.032979566603899,
0.032186347991228104,
0.04335241764783859,
0.002812491962686181,
0.025611791759729385,
-0.00884743221104145,
0.00801607... | |
4ccfcf4c26edeb417a7e970b7e420335b3409b07 | subsection | 82 | 126 | Proof of Proposition | \end{split}Similarly, if x \in W is a critical point of f_W,
and letting W_W^u(x) be the descending manifold of x in W
for the gradient-like vector field -Z_W, we define\sbox
{\operatorname{ev}}\widetilde{\usebox {
}} \colon {\mathcal {M}}_{H, k, W; k_+}^*\left( (B; A_1,\ldots ,A_N);J_W \right)
&\times \mathcal {M}^*_... | {
"cite_spans": []
} | 1804.08013 | Morse-Bott Split Symplectic Homology | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
-0.033328622579574585,
0.016618529334664345,
-0.0050130276940763,
0.0055242497473955154,
0.024584436789155006,
-0.023226264864206314,
0.020830389112234116,
0.00028684703283943236,
0.01719842292368412,
0.052892401814460754,
-0.010895895771682262,
-0.026461461558938026,
0.0071151419542729855,
... | |
00d6f17d5f7bb955fabf51dbe443b7e298e4e45f | subsection | 83 | 126 | Proof of Proposition | The following result will provide the final step in the proof of Proposition REF .Proposition 5.40
Let J_W \in J_W^{reg} and let J_Y \in J_Y^{reg} be the induced almost complex structure on \mathbb {R}\times Y.Let \tilde{q}, \tilde{p} denote critical points of f_Y, and let x be a critical point of f_W in W. | {
"cite_spans": []
} | 1804.08013 | Morse-Bott Split Symplectic Homology | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
0.015886569395661354,
0.038243748247623444,
0.019076092168688774,
-0.0026077015791088343,
0.0037866041529923677,
-0.017305830493569374,
0.0177636556327343,
0.006638479884713888,
-0.011033611372113228,
0.03418435528874397,
0.020678482949733734,
0.0045591858215630054,
0.004448544699698687,
0... | |
b6cadf81ea4701a684ec2c6dd643677e3dfe5d90 | subsection | 84 | 126 | Proof of Proposition | Let k_+ and k_- be non-negative multiplicities, k_+ > k_-.Let A_1, \dots , A_N be spherical homology classes in \Sigma ,
let B, B_1, \dots , B_k be spherical homology classes in X, k \ge 0.Let \tilde{\Delta }\subset Y \times Y and
\Delta _{\Sigma ^k} \subset \Sigma ^k \times \Sigma ^k be the diagonals.Then,the evaluati... | {
"cite_spans": []
} | 1804.08013 | Morse-Bott Split Symplectic Homology | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
-0.06321758031845093,
-0.018016401678323746,
-0.0014034791383892298,
-0.017909614369273186,
0.008954807184636593,
-0.017467213794589043,
0.02692544274032116,
0.006834333296865225,
0.04677246883511543,
0.044209592044353485,
0.0019793633837252855,
0.01063288003206253,
0.027062740176916122,
-... | |
495341dcac581968d59c5ef47f5cab37ce275ad5 | subsection | 85 | 126 | Proof of Proposition | The
fibres have a locally free (S^1)^N torus action by constant rotation by the
action of the Reeb vector field.We will study the case of\pi _\Sigma ^\mathcal {M}\colon {\mathcal {M}}_{H, k, \mathbb {R}\times Y}^*\left((A_1,\ldots ,A_N);J_Y\right) \rightarrow \mathcal {M}_{k, \Sigma }^*((A_1, \dots , A_N); J_\Sigma )in... | {
"cite_spans": []
} | 1804.08013 | Morse-Bott Split Symplectic Homology | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
-0.02099023386836052,
0.040210939943790436,
-0.008664573542773724,
0.007177256513386965,
0.028068335726857185,
-0.002814460778608918,
0.030585333704948425,
0.037770215421915054,
0.04808228090405464,
0.05961470305919647,
-0.018457980826497078,
-0.009427299723029137,
-0.00038660698919557035,
... | |
73d9ac842d313bbef41e6f3c6125e12bc9e83f1c | subsection | 86 | 126 | Proof of Proposition | Then, k_+ > k_-.Denote by w^*Y the pullback under w
of the S^1-bundle Y\rightarrow \Sigma .
The map \tilde{v} gives a section s of w^*Y, defined in
the complement of \Gamma \cup \lbrace 0,\infty \rbrace . By
*Theorem 11.16, the Euler number \int _{\mathbb {CP}^1}e(w^*Y) (where e is
the Euler class) is the sum of the lo... | {
"cite_spans": []
} | 1804.08013 | Morse-Bott Split Symplectic Homology | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
-0.013309644535183907,
0.02050219103693962,
-0.025231119245290756,
0.021051356568932533,
-0.015758007764816284,
-0.03127194195985794,
0.02358362078666687,
-0.005346739664673805,
0.054123345762491226,
0.05415385216474533,
-0.022759873420000076,
0.022149689495563507,
-0.006731095258146524,
-... | |
2e14382c94690cf844d2bdf022fbfff9edea35fb | subsection | 87 | 126 | Proof of Proposition | If A=0, we get an equality, but the assumptions of the Lemma imply that \sum _{i=1}^m
k_i > 0. In either case, we get k_+ > k_-, as wanted.Recall that the gradient-like vector field Z_Y has the property that d\pi _\Sigma Z_Y = Z_\Sigma .
Also recall that we may use the contact form \alpha as a connection to lift vector... | {
"cite_spans": []
} | 1804.08013 | Morse-Bott Split Symplectic Homology | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
-0.02406330034136772,
-0.004406009800732136,
-0.031524907797575,
0.020279090851545334,
0.014831659384071827,
0.003061318537220359,
0.007495939265936613,
0.014572258107364178,
0.03366115316748619,
0.04360996186733246,
-0.022354301065206528,
0.02096574194729328,
0.012878519482910633,
0.01130... | |
6b8e95aba0777a0e336348c4f753da1b902df677 | subsection | 88 | 126 | Proof of Proposition | Then, there exists a positive g = g(\tilde{x}, \tilde{y}) > 0 so that\begin{aligned}T_{(\tilde{x}, \tilde{y})} \sbox
{\Delta }\widetilde{\usebox {
}}_{f_Y}
&= \mathbb {R}(R, gR) \oplus H
\end{aligned}where the subspace H is such that d\pi _\Sigma |_H \colon H \rightarrow T\Delta _{f_\Sigma } induces a linear
isomorphi... | {
"cite_spans": []
} | 1804.08013 | Morse-Bott Split Symplectic Homology | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
-0.03961402550339699,
-0.006504421588033438,
-0.000019491806597216055,
-0.004364257212728262,
0.03195368871092796,
0.019684936851263046,
0.0022469819523394108,
0.009102375246584415,
0.026292361319065094,
0.0391867570579052,
-0.03283874690532684,
-0.020692072808742523,
-0.0029622777365148067,... | |
b4a573c2ee828866b35d64ff0c58f8901b69d69c | subsection | 89 | 126 | Proof of Proposition | From this, it now follows that \pi _\Sigma ( \sbox
{\Delta }\widetilde{\usebox {
}}_{f_Y}) \subset \Delta _{f_\Sigma } \cup \lbrace (p, p)\, | \, p \in \operatorname{Crit}(f_\Sigma ) \rbrace .We now consider the consequences of Equation
(REF ) in this case of x=y. Any v \in T_xY may be written as v_0 +
aR where \alpha... | {
"cite_spans": []
} | 1804.08013 | Morse-Bott Split Symplectic Homology | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
-0.015994174405932426,
0.002638351870700717,
-0.009324847720563412,
0.020221641287207603,
0.029058117419481277,
0.022404052317142487,
0.007585023529827595,
0.018893880769610405,
0.0496918223798275,
0.01594838872551918,
-0.03803194686770439,
-0.0016415775753557682,
-0.016894608736038208,
0.... | |
15a7c804cf008d1642ca494539d7b5d08397c565 | subsection | 90 | 126 | Proof of Proposition | For each i=1, \dots N,
let \tilde{y}_{i} = \tilde{v}_i(-\infty , 0) \in Y and \tilde{x}_i = \tilde{v}_i(+\infty , 0) \in Y, with&\tilde{y}_1 \in W^s_Y(\tilde{q}), \tilde{x}_N \in W^u_Y(\tilde{p})\\
&(\tilde{x}_i, \tilde{y}_{i+1}) \in \sbox
{\Delta }\widetilde{\usebox {
}}_{f_Y} \quad \text{ for } 1 \le i \le N-1.Let w... | {
"cite_spans": []
} | 1804.08013 | Morse-Bott Split Symplectic Homology | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
-0.06391315907239914,
0.025577472522854805,
0.005970874801278114,
-0.00005716921805287711,
0.01968671754002571,
0.037481069564819336,
0.008149385452270508,
0.021579084917902946,
0.04010596498847008,
0.0562826469540596,
-0.03427625447511673,
-0.01771804504096508,
0.0010778096038848162,
0.00... | |
e2e6026c7c33454df8adfbee2daf4300909405b4 | subsection | 91 | 126 | Proof of Proposition | Notice
that by rotating by the action of the Reeb vector field on \tilde{v}_i, we
obtain that the image of d\tilde{\operatorname{ev}} contains the subspace\lbrace (a_1R, a_1R, a_2R, a_2R, \dots , a_{N}R, a_{N}R )\, | \, (a_1, \dots , a_N)
\in \mathbb {R}^{N} \rbrace \subset (TY)^{2N}.In the case of the chain of pearls ... | {
"cite_spans": []
} | 1804.08013 | Morse-Bott Split Symplectic Homology | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
-0.062268033623695374,
0.0004657327081076801,
-0.022179745137691498,
0.02278991788625717,
0.02983740158379078,
0.03420013189315796,
-0.022423814982175827,
0.008252574130892754,
-0.004885188303887844,
0.03133232146501541,
-0.012790726497769356,
0.0022347543854266405,
-0.02516958862543106,
0... | |
72c0cda1ed5312d1f24e96930dbcc81ad0ebbe9b | subsection | 92 | 126 | Proof of Proposition | This establishes that \tilde{\operatorname{ev}}_Y defined on
{\mathcal {M}}_{H, k, \mathbb {R}\times Y; k_-, k_+}^*\left((A_1,\ldots ,A_N);J_Y\right) is transverse to
W^s_Y(\tilde{q}) \times \sbox
{\Delta }\widetilde{\usebox {
}}_{f_Y}^N \times W^u_Y(\tilde{p}).We now consider
the case of\sbox
{\operatorname{ev}}\wid... | {
"cite_spans": []
} | 1804.08013 | Morse-Bott Split Symplectic Homology | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
-0.04516511783003807,
0.04025188460946083,
0.004253302700817585,
0.0009927551727741957,
0.035887956619262695,
0.03140196204185486,
0.003770753275603056,
0.016524938866496086,
0.0363762304186821,
0.062437720596790314,
-0.019973358139395714,
-0.007842861115932465,
0.016326580196619034,
0.024... | |
47c6a55e938b26f04baee9df2f9bc5d485470bbb | subsection | 93 | 126 | Proof of Proposition | Observe that domain rotation on the plane \tilde{v}_1 then gives that (0, lR, 0, \dots ,
0) \in TW \oplus TY \oplus TY^{2N} is in the image of d\sbox
{\operatorname{ev}}\widetilde{\usebox {
}}_{W,Y}.As before, the Reeb rotation on each of the punctured cylinders \tilde{v}_1,
\dots , \tilde{v}_N gives that the followin... | {
"cite_spans": []
} | 1804.08013 | Morse-Bott Split Symplectic Homology | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
-0.07836196571588516,
0.0065339780412614346,
0.0009583549690432847,
0.004821335896849632,
0.015371822752058506,
0.03619053587317467,
0.035580240190029144,
0.018354643136262894,
0.0006145868101157248,
0.037716273218393326,
-0.007338805589824915,
-0.00952061265707016,
-0.014265662059187889,
... |
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