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337bc8e3d0018d329a230c69d384906470209102 | subsection | 164 | 279 | Orientations | In the special case when g_2 is the inclusion of a submanifold, this agrees with the orientation on the preimage, as defined in .
This is the reason for the name used for this orientation convention, and for the sign in Part (1) above.Lemma 8.5
The identification of \ker f with the fibre sum orientation (as in Definit... | {
"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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... |
325ad3e01f4cfcce9a4d65c646e2aff94b5d25f4 | subsection | 165 | 279 | Orientations | \\
&\hspace{142.26378pt} .(f_1,f_2)(H) \oplus \Delta \oplus (f_1,f_2)^{-1}(\Delta ) \cong \\
&\stackrel{(P2)}{\cong } (-1)^{\dim V_1 \dim W + \dim V_1 \dim V_2 + \dim V_2 \dim W} W \oplus W \oplus (f_1,f_2)^{-1}(\Delta ) \cong \\
&\stackrel{\varphi }{\cong } (-1)^{\dim V_1 \dim W + \dim V_1 \dim V_2 + \dim V_2 \dim 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 | [
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804bd1c5d48c2d2ea1de28a2e312ede9e32d58dc | subsection | 166 | 279 | Orientations | In *Section 7, these spaces are oriented using the fibre sum rule.
the spaces of cylinders are given a coherent orientation and the critical manifolds are oriented as follows.For any oriented manifold S with a Morse–Smale pair (f_S,Z_S), fixing an orientation on a critical manifold of a critical point p
also fixes an 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 | [
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3161445201da4e764189a8f6fd670d18d8892bec | subsection | 167 | 279 | Orientations | Under our conventions, these maps
are always orientation-preserving for W^u_Y(\widehat{p}), and preserve orientations for W^s_Y(p) iff M(p) is even.Each cascade in (REF ) projects to a chain of pearls in \Sigma from q to p, with exactly one sphere, contained in\mathcal {M}(q,p;A)W^s_{\Sigma }(q) \times _{\operatorname{... | {
"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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f47db8d09df36e0c12cfafea27dc3e06fbf1359c | subsection | 168 | 279 | Orientations | To compare the two signs, it will be useful to also consider the fibre product\overline{\mathcal {M}({q}_{k_-},\widehat{p}_{k_+};A)}\pi _\Sigma ^{-1}\left(W^s_{\Sigma }(q) \right) \times _{\operatorname{ev}} \mathcal {M}^*_{H,0,\mathbb {R}\times Y;k_-,k_+}(A;J_Y) \times _{\operatorname{ev}} \pi _\Sigma ^{-1}\left(W^u_{... | {
"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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26f49d2e8129d174979c70c24c5cf1d174fa2aa8 | subsection | 169 | 279 | Orientations | Recall that the left sides in (REF ) were given orientations in (REF ), which agree with
the fibre sum orientations on the right sides.Note that \mathcal {M}(\widehat{q}_{k_-},{p}_{k_+};A) is a codimension 2 submanifold of \overline{\mathcal {M}({q}_{k_-},\widehat{p}_{k_+};A)}.
Given an element (x,u,y) \in \mathcal {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.... |
3cebac58e0d83fb9f69182244068a215abdaf740 | subsection | 170 | 279 | Orientations | \\
&\qquad .\lbrace 1\rbrace {}{i\,}\times {}{{s}}{} \big ( \pi _\Sigma ^{-1}\left(W^s_{\Sigma }(q) \right) \times _{\operatorname{ev}} \mathcal {M}^*_{H;k_-,k_+}(A) \times _{\operatorname{ev}} \pi _\Sigma ^{-1}\left(W^u_{\Sigma }(p) \right) \big ) {}{\widehat{u}\,}\times {}{i} \lbrace 1\rbrace = \\
&= (-1)^{M(q)} \lbr... | {
"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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5d6028e107a8c69c49fd36290bb714e8165860a9 | subsection | 171 | 279 | Orientations | The last identity follows from the fact that the fibre sum
orientation on the zero dimensional vector space
0 {}{di\,}\oplus {}{d{s}}{} \, \mathbb {R}^2 {}{d\widehat{u}\,}\oplus {}{di} \,0
is negative, which is a simple calculation (the reason ultimately being the fact that the matrix in (REF ) has
negative determinant... | {
"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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9bedcaf76597bfc31171954cc4ccba7a21dd7833 | subsection | 172 | 279 | Orientations | This can be identified with T_{(a,b)} (A {}{f_1\,}\times {}{f_2}B), via the restriction of (\pi _A,\pi _B) to the fibre sum.
This gives an identificationT_{(\tilde{a}, \tilde{b})} (\tilde{A} {}{\tilde{f}_1\,}\times {}{\tilde{f}_2}{} \tilde{B}) \cong \left( T_{(a,b)} (A {}{f_1\,}\times {}{f_2}B) \right) \oplus \mathbb {... | {
"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... |
5b0bf0f064993c1ad0c9d9fb4a3a3d4d9bb173ba | subsection | 173 | 279 | Orientations | The commutative diagram
T AT B(d f) R [df]id T RTATB(d f) [df] T Y
defines the isomorphism \varphi .Denote by \psi the composition T A \oplus T B \oplus \mathbb {R}\oplus \mathbb {R}\rightarrow T A \oplus \mathbb {R}\oplus T B \oplus \mathbb {R}\rightarrow T \tilde{A} \oplus T\tilde{B}, where the first isomorphism... | {
"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... |
04f21f2e53da748e349bda239bbce408fee9f2b8 | subsection | 174 | 279 | Orientations | By Part (2) in Definition REF and the commutativity of the diagram, we conclude that the isomorphism \chi changes orientations by (-1)^{\dim B + 1 + \dim B + 1} = 1, as wanted.The space (REF ) is an S^1-bundle over (REF ). With respect to the connections discussed above (see (REF )), we have a decompositionT_{(x,u,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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4d3dc77db27a24e5c11103fba827eac327dfada7 | subsection | 175 | 279 | Orientations | Writing (REF ) as(W^s_{\Sigma }(q) \times _{\operatorname{ev}} \mathcal {M}^*_{0,\Sigma }(A;J_\Sigma )) \times _{\operatorname{ev}} W_{\Sigma }^u(p)we apply Lemma REF to W^s_{\Sigma }(q) \hookrightarrow \Sigma and \operatorname{ev}_\Sigma ^1\colon \mathcal {M}^*_{0,\Sigma }(A;J_\Sigma )\rightarrow \Sigma to get a sign(... | {
"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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02b3d9ee30f25e4c1898bf178d43d36f0d39804a | subsection | 176 | 279 | Orientations | We then apply Lemma REF to \operatorname{ev}^2\colon W^s_{\Sigma }(q) \times _{\operatorname{ev}} \mathcal {M}^*_{0,\Sigma }(A;J_\Sigma ) \rightarrow \Sigma and W_{\Sigma }^u(p) \hookrightarrow \Sigma to get the sign(-1&)^{\dim (W^s_{\Sigma }(q) \times _{\operatorname{ev}} \mathcal {M}^*_{0,\Sigma }(A;J_\Sigma ) + \dim... | {
"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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73c39e3b13048cec4fd2d09bf2fd18e8a72a263b | subsection | 177 | 279 | Orientations | This oriented zero-dimensional vector space coincides with the quotient of
\left( T_{(x,u,y)}\mathcal {M}(\widehat{q}_{k_-},{p}_{k_+};A) \right) \oplus \mathbb {R}_{\operatorname{domain}} by the infinitesimal
*-action on the domain of u.
On the other hand, by Definition REF ,
the Gromov–witten sign of (\pi _\Sigma (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 | [
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4c2036e285062faede02dcc3335e12e395571c6a | subsection | 178 | 279 | Orientations | The corresponding sign associated to an element in the zero-dimensional
\operatorname{ev}^{-1}(pt) is its Gromov–Witten sign.Observe that this is oriented diffeomeorphic to \mathcal {M}_X^*(B;J_W) \times _{\operatorname{ev}} \lbrace pt\rbrace ,
by Lemma REF . Analogously to what was pointed out in Remark REF ,
this ori... | {
"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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c947e8594bae8b205e24337a4b1f8608c50d12db | subsection | 179 | 279 | Orientations | We have oriented isomorphisms\big [&W^s_{Y}(\widehat{p}) \times _{\operatorname{ev}} \big ( \mathcal {M}_X^*(B;J_W) \times _{\tilde{\operatorname{ev}}} \mathcal {M}^*_{H,1,\mathbb {R}\times Y;k_-,k_+}(0;J_Y) \big ) \big ]\times _{\operatorname{ev}} W_{Y}^u({p}) \\
& \cong \big [\big (\mathcal {M}_X^*(B;J_W) \times _{\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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877c34ce17736c15da352381e3b4ec602dda6f96 | subsection | 180 | 279 | Orientations | The first isomorphism preserves orientations, by Lemma REF (\dim \big ( \mathcal {M}_X^*(B;J_W) \times _{\tilde{\operatorname{ev}}} \mathcal {M}^*_{H,1,\mathbb {R}\times Y;k_-,k_+}(0;J_Y) \big ) + \dim Y is even).
The second and third isomorphisms are orientation-preserving, by the associativity of the fibre sum orient... | {
"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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d224174c74e04415437da6fce08d8d83c144809e | subsection | 181 | 279 | Orientations | Denote by \operatorname{ev}\colon \mathcal {M}^*_{H}(B;J_W)/* \rightarrow X\times \Sigma the evaluation map at (0,\infty )
(where we quotiented by the space * of domain automorphisms). Write W^u_{X}(x) for \psi \big (W^u_{W}(x)\big ).Definition 8.18 The preimage orientation on \operatorname{ev}^{-1}\left(W^u_{X}(x) \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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0... |
3e0f7ed3cbc4ad9c7c6db32af2e9d67b96c4e2f3 | subsection | 182 | 279 | Orientations | The corresponding sign associated to an element in this
zero-dimensional manifold is its Gromov–Witten sign.If W^u_{X}(x) \subset X and W_{\Sigma }^u(p)\subset \Sigma represent homology classes, then the Gromov–Witten orientation determines the signs with which elements in \operatorname{ev}^{-1}\left(W^u_{X}(x) \times ... | {
"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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947d76bf6148ac471e0bd8d052bee46ed3372cd4 | subsection | 183 | 279 | Orientations | We used
the associativity of the fibre sum orientation, the oriented diffeomorphism \mathcal {M}^*_{H,k_+}(0;J_Y)\cong \mathbb {R}_2 \times Y
(see *Lemma 5.22), the fact that W and \mathcal {M}^*_{H}(B;J_W) are
even dimensional and *Proposition 7.5.(a)&(c).
Now, if we think of \mathcal {M}^*_{H}(B;J_W)
as a space of ps... | {
"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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0.049436550587415695,
-0.038176003843545914,
-0.0020598561968654394,
0.009116770699620247,
... |
fce4e5467ceca822ad7a3c68b9e05a1f525d973a | subsection | 184 | 279 | Orientations | Each of the
Cases 1 through 3 motivates one of the following definitions.Definition 8.20
Let q,p\in \operatorname{Crit}(f_\Sigma ) and A \in H_2(\Sigma ;\mathbb {Z}). Definen_A(q,p) := \# \mathcal {M}_A(q,p),where (using the notation of (REF ))\mathcal {M}_A(q,p) := \mathcal {M}(q,p;A) / *.Here, \mathcal {M}(q,p;A) is... | {
"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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0.0025274551007896... |
37a3c083bb9304902f787f56f5978c84c2e27d81 | subsection | 185 | 279 | Orientations | Similarly, the coefficients n_B can
only be non-trivial if\, \langle c_1(TX),B\rangle - B\bullet \Sigma = (1 - K/\tau _X) \, \langle c_1(TX),B\rangle = (\tau _X - K) \, \omega (B) = 1.The coefficients n_B(x,p) can only be non-zero if\, \langle c_1(TX),B\rangle - B\bullet \Sigma = (1 - K/\tau _X) \, \langle c_1(TX),B\ra... | {
"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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47f3c6cfa3821a16577014d1036d66c86f8386a6 | subsection | 186 | 279 | Orientations | Denoting also by f and g the appropriate isomorphisms in Part (1) of Definition REF , we have an isomorphismg \circ r \circ f^{-1} \colon W \rightarrow W,which is -\operatorname{Id}, hence it changes orientation by (-1)^{\dim W}.Denote also by r the induced map (V_1 \oplus V_2)/\ker f \rightarrow (V_2 \oplus V_1)/\ker ... | {
"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.... |
733ee334878d9ee69da89d170622abcdad18a47b | subsection | 187 | 279 | Orientations | In the special case when g_2 is the inclusion of a submanifold, this agrees with the orientation on the preimage, as defined in .
This is the reason for the name used for this orientation convention, and for the sign in Part (1) above.Lemma 8.5
The identification of \ker f with the fibre sum orientation (as in Definit... | {
"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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... |
ccf8007f107bdec8d49a45ff33c0f3afa685affe | subsection | 188 | 279 | Orientations | \\
&\hspace{142.26378pt} .(f_1,f_2)(H) \oplus \Delta \oplus (f_1,f_2)^{-1}(\Delta ) \cong \\
&\stackrel{(P2)}{\cong } (-1)^{\dim V_1 \dim W + \dim V_1 \dim V_2 + \dim V_2 \dim W} W \oplus W \oplus (f_1,f_2)^{-1}(\Delta ) \cong \\
&\stackrel{\varphi }{\cong } (-1)^{\dim V_1 \dim W + \dim V_1 \dim V_2 + \dim V_2 \dim 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 | [
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0... |
de81cc1e0f8c41c128b7e0809d8208fa25312966 | subsection | 189 | 279 | Orientations | In *Section 7, these spaces are oriented using the fibre sum rule.
the spaces of cylinders are given a coherent orientation and the critical manifolds are oriented as follows.For any oriented manifold S with a Morse–Smale pair (f_S,Z_S), fixing an orientation on a critical manifold of a critical point p
also fixes an 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 | [
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... |
ae64e89044ff0f76dee012d3893cede88894170b | subsection | 190 | 279 | Orientations | Under our conventions, these maps
are always orientation-preserving for W^u_Y(\widehat{p}), and preserve orientations for W^s_Y(p) iff M(p) is even.Each cascade in (REF ) projects to a chain of pearls in \Sigma from q to p, with exactly one sphere, contained in\mathcal {M}(q,p;A)W^s_{\Sigma }(q) \times _{\operatorname{... | {
"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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-... |
e29213b011122d783ae8ef168b9387e3eb539458 | subsection | 191 | 279 | Orientations | To compare the two signs, it will be useful to also consider the fibre product\overline{\mathcal {M}({q}_{k_-},\widehat{p}_{k_+};A)}\pi _\Sigma ^{-1}\left(W^s_{\Sigma }(q) \right) \times _{\operatorname{ev}} \mathcal {M}^*_{H,0,\mathbb {R}\times Y;k_-,k_+}(A;J_Y) \times _{\operatorname{ev}} \pi _\Sigma ^{-1}\left(W^u_{... | {
"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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7a680eef8ab2068a232f1b20d02af048227d9591 | subsection | 192 | 279 | Orientations | Recall that the left sides in (REF ) were given orientations in (REF ), which agree with
the fibre sum orientations on the right sides.Note that \mathcal {M}(\widehat{q}_{k_-},{p}_{k_+};A) is a codimension 2 submanifold of \overline{\mathcal {M}({q}_{k_-},\widehat{p}_{k_+};A)}.
Given an element (x,u,y) \in \mathcal {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.... |
70dd46eb9b8b5515172290d2f6cc87b21b580807 | subsection | 193 | 279 | Orientations | \\
&\qquad .\lbrace 1\rbrace {}{i\,}\times {}{{s}}{} \big ( \pi _\Sigma ^{-1}\left(W^s_{\Sigma }(q) \right) \times _{\operatorname{ev}} \mathcal {M}^*_{H;k_-,k_+}(A) \times _{\operatorname{ev}} \pi _\Sigma ^{-1}\left(W^u_{\Sigma }(p) \right) \big ) {}{\widehat{u}\,}\times {}{i} \lbrace 1\rbrace = \\
&= (-1)^{M(q)} \lbr... | {
"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.... |
b4adbb221573d1b63983089ead127d0c462c3288 | subsection | 194 | 279 | Orientations | The last identity follows from the fact that the fibre sum
orientation on the zero dimensional vector space
0 {}{di\,}\oplus {}{d{s}}{} \, \mathbb {R}^2 {}{d\widehat{u}\,}\oplus {}{di} \,0
is negative, which is a simple calculation (the reason ultimately being the fact that the matrix in (REF ) has
negative determinant... | {
"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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0.0038... |
260e479fdae55631e407e9dd2cd52f6780159a96 | subsection | 195 | 279 | Orientations | This can be identified with T_{(a,b)} (A {}{f_1\,}\times {}{f_2}B), via the restriction of (\pi _A,\pi _B) to the fibre sum.
This gives an identificationT_{(\tilde{a}, \tilde{b})} (\tilde{A} {}{\tilde{f}_1\,}\times {}{\tilde{f}_2}{} \tilde{B}) \cong \left( T_{(a,b)} (A {}{f_1\,}\times {}{f_2}B) \right) \oplus \mathbb {... | {
"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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cd17efbe30c05cd62e4157eb45810e322e78d218 | subsection | 196 | 279 | Orientations | The commutative diagram
T AT B(d f) R [df]id T RTATB(d f) [df] T Y
defines the isomorphism \varphi .Denote by \psi the composition T A \oplus T B \oplus \mathbb {R}\oplus \mathbb {R}\rightarrow T A \oplus \mathbb {R}\oplus T B \oplus \mathbb {R}\rightarrow T \tilde{A} \oplus T\tilde{B}, where the first isomorphism... | {
"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... |
b52e3bbd3a6e17b8c55de15a60ab474b96b69294 | subsection | 197 | 279 | Orientations | By Part (2) in Definition REF and the commutativity of the diagram, we conclude that the isomorphism \chi changes orientations by (-1)^{\dim B + 1 + \dim B + 1} = 1, as wanted.The space (REF ) is an S^1-bundle over (REF ). With respect to the connections discussed above (see (REF )), we have a decompositionT_{(x,u,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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531e9ce85423995658fb0477e45e67dffb2004ef | subsection | 198 | 279 | Orientations | Writing (REF ) as(W^s_{\Sigma }(q) \times _{\operatorname{ev}} \mathcal {M}^*_{0,\Sigma }(A;J_\Sigma )) \times _{\operatorname{ev}} W_{\Sigma }^u(p)we apply Lemma REF to W^s_{\Sigma }(q) \hookrightarrow \Sigma and \operatorname{ev}_\Sigma ^1\colon \mathcal {M}^*_{0,\Sigma }(A;J_\Sigma )\rightarrow \Sigma to get a sign(... | {
"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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0.01... |
370c872315de7ce2aaf5ed14dec35ecadbfd4cad | subsection | 199 | 279 | Orientations | We then apply Lemma REF to \operatorname{ev}^2\colon W^s_{\Sigma }(q) \times _{\operatorname{ev}} \mathcal {M}^*_{0,\Sigma }(A;J_\Sigma ) \rightarrow \Sigma and W_{\Sigma }^u(p) \hookrightarrow \Sigma to get the sign(-1&)^{\dim (W^s_{\Sigma }(q) \times _{\operatorname{ev}} \mathcal {M}^*_{0,\Sigma }(A;J_\Sigma ) + \dim... | {
"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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... |
33ebb567f9c137e8021861c3a1bda810e7c03dfe | subsection | 200 | 279 | Orientations | This oriented zero-dimensional vector space coincides with the quotient of
\left( T_{(x,u,y)}\mathcal {M}(\widehat{q}_{k_-},{p}_{k_+};A) \right) \oplus \mathbb {R}_{\operatorname{domain}} by the infinitesimal
*-action on the domain of u.
On the other hand, by Definition REF ,
the Gromov–witten sign of (\pi _\Sigma (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 | [
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... |
25bd0c6e7b714cdd93a1649230b9efb1eaf49a4c | subsection | 201 | 279 | Orientations | The corresponding sign associated to an element in the zero-dimensional
\operatorname{ev}^{-1}(pt) is its Gromov–Witten sign.Observe that this is oriented diffeomeorphic to \mathcal {M}_X^*(B;J_W) \times _{\operatorname{ev}} \lbrace pt\rbrace ,
by Lemma REF . Analogously to what was pointed out in Remark REF ,
this ori... | {
"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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36f045367f1cb20ea03e72b35ef11c00aa9427c4 | subsection | 202 | 279 | Orientations | We have oriented isomorphisms\big [&W^s_{Y}(\widehat{p}) \times _{\operatorname{ev}} \big ( \mathcal {M}_X^*(B;J_W) \times _{\tilde{\operatorname{ev}}} \mathcal {M}^*_{H,1,\mathbb {R}\times Y;k_-,k_+}(0;J_Y) \big ) \big ]\times _{\operatorname{ev}} W_{Y}^u({p}) \\
& \cong \big [\big (\mathcal {M}_X^*(B;J_W) \times _{\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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4dc843de5ce8057e6bd78ef061a6f8acbaded81b | subsection | 203 | 279 | Orientations | The first isomorphism preserves orientations, by Lemma REF (\dim \big ( \mathcal {M}_X^*(B;J_W) \times _{\tilde{\operatorname{ev}}} \mathcal {M}^*_{H,1,\mathbb {R}\times Y;k_-,k_+}(0;J_Y) \big ) + \dim Y is even).
The second and third isomorphisms are orientation-preserving, by the associativity of the fibre sum orient... | {
"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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0.012513378635048866,
... |
743a8c7742229f45b40451a6c34ec937206efc10 | subsection | 204 | 279 | Orientations | Denote by \operatorname{ev}\colon \mathcal {M}^*_{H}(B;J_W)/* \rightarrow X\times \Sigma the evaluation map at (0,\infty )
(where we quotiented by the space * of domain automorphisms). Write W^u_{X}(x) for \psi \big (W^u_{W}(x)\big ).Definition 8.18 The preimage orientation on \operatorname{ev}^{-1}\left(W^u_{X}(x) \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 | [
-0.0712776705622673,
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0.014440135098993778,
0... |
917b225e01fabc2ecce1ade86552f9d9720bd71c | subsection | 205 | 279 | Orientations | The corresponding sign associated to an element in this
zero-dimensional manifold is its Gromov–Witten sign.If W^u_{X}(x) \subset X and W_{\Sigma }^u(p)\subset \Sigma represent homology classes, then the Gromov–Witten orientation determines the signs with which elements in \operatorname{ev}^{-1}\left(W^u_{X}(x) \times ... | {
"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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0.014876813627779484,
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0.0... |
69b9cad948753f28f935327670904e9cc41162e3 | subsection | 206 | 279 | Orientations | We used
the associativity of the fibre sum orientation, the oriented diffeomorphism \mathcal {M}^*_{H,k_+}(0;J_Y)\cong \mathbb {R}_2 \times Y
(see *Lemma 5.22), the fact that W and \mathcal {M}^*_{H}(B;J_W) are
even dimensional and *Proposition 7.5.(a)&(c).
Now, if we think of \mathcal {M}^*_{H}(B;J_W)
as a space of ps... | {
"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 | [
-0.01922532543540001,
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... |
b330c69b0ade1cf1df52ddbc52479f93b0d93f21 | subsection | 207 | 279 | Orientations | Each of the
Cases 1 through 3 motivates one of the following definitions.Definition 8.20
Let q,p\in \operatorname{Crit}(f_\Sigma ) and A \in H_2(\Sigma ;\mathbb {Z}). Definen_A(q,p) := \# \mathcal {M}_A(q,p),where (using the notation of (REF ))\mathcal {M}_A(q,p) := \mathcal {M}(q,p;A) / *.Here, \mathcal {M}(q,p;A) is... | {
"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 | [
-0.022386575117707253,
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0.011338259093463421,
0.0025274551007896... |
51ef1b7a2147f4dae6d50e85e91de13590e3c2b8 | subsection | 208 | 279 | Orientations | Similarly, the coefficients n_B can
only be non-trivial if\, \langle c_1(TX),B\rangle - B\bullet \Sigma = (1 - K/\tau _X) \, \langle c_1(TX),B\rangle = (\tau _X - K) \, \omega (B) = 1.The coefficients n_B(x,p) can only be non-zero if\, \langle c_1(TX),B\rangle - B\bullet \Sigma = (1 - K/\tau _X) \, \langle c_1(TX),B\ra... | {
"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 | [
-0.01463727094233036,
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-0.0623953714966774,
0.012851493433117867,
-0.039... |
fe692a0c48e72d83b5451d54ae65a2abc40c4966 | subsection | 209 | 279 | A formula for the symplectic homology differential | Recall that the symplectic chain complex (REF ) isSC_*(W,H) = \left(\bigoplus _{k>0} \bigoplus _{p_k\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)as an abelian group, where f_\Sigma : \Sigma \rightarrow... | {
"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 | [
-0.03375885635614395,
0.019809672608971596,
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0.027303161099553108,
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0.019855458289384842,
-0.020465925335884094,
0.... |
b0848ecd8d4b366b861274d588c799a66ac3e74b | subsection | 210 | 279 | A formula for the symplectic homology differential | If W is Weinstein, this term is trivial if n\ge 3 and the minimal Chern number of spheres in \Sigma is at least 2.Given p\in \operatorname{Crit}(f_\Sigma ) and x\in \operatorname{Crit}(f_W),\langle \partial (p_{k_+}) , x \rangle = k_+ \sum _{B \in H_2(X;\mathbb {Z})} \delta _{k_+,(B\bullet \Sigma )} \, n_B(x,p)where 2(... | {
"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 | [
-0.01838868483901024,
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0.06152960658073425,
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0.032473959028720856,
-0.032229792326688766,
... |
c48f8572aa1b4bd03bcfdf5c6b8050d608d81025 | subsection | 211 | 279 | A formula for the symplectic homology differential | The comment about W Weinstein follows from Lemma *Lemma 6.6 (and, by Lemma REF below, it can be interpreted as saying that a certain relative Gromov–Witten invariant of the triple (X,\Sigma ,\omega ) vanishes).The contributions in \langle \partial (p_{k_+}) , x \rangle are the ones in Proposition REF ,
and consist of 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 | [
-0.0439930222928524,
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0.03474899381399155,
-0.012241469696164131,
0.02225... |
f7b86e5e27423bb7cc282711a33129a985a34458 | subsection | 212 | 279 | A formula for the symplectic homology differential | There are two flow lines of -Z_Y from \widehat{p}_{k_+}, which cancel one another.Remark 9.3 At least if there were no contributions \langle \partial (p_{k_+}) , x \rangle , it would be immediate from Theorem REF that \partial ^2 = 0.Recall that the symplectic chain complex (REF ) isSC_*(W,H) = \left(\bigoplus _{k>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 | [
-0.03667448088526726,
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0.0019832290709018707,
-0.007787988055497408,
0... |
389a75d5eb510315610aac8bcae87fcb5c256fab | subsection | 213 | 279 | A formula for the symplectic homology differential | If W is Weinstein, this term is trivial if n\ge 3 and the minimal Chern number of spheres in \Sigma is at least 2.Given p\in \operatorname{Crit}(f_\Sigma ) and x\in \operatorname{Crit}(f_W),\langle \partial (p_{k_+}) , x \rangle = k_+ \sum _{B \in H_2(X;\mathbb {Z})} \delta _{k_+,(B\bullet \Sigma )} \, n_B(x,p)where 2(... | {
"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 | [
-0.01838868483901024,
-0.007588193751871586,
-0.025515252724289894,
0.013841109350323677,
0.012429529801011086,
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0.06152960658073425,
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0.032473959028720856,
-0.032229792326688766,
... |
84f202c3f1eaff76621759410fad5d2571295d69 | subsection | 214 | 279 | A formula for the symplectic homology differential | The comment about W Weinstein follows from Lemma *Lemma 6.6 (and, by Lemma REF below, it can be interpreted as saying that a certain relative Gromov–Witten invariant of the triple (X,\Sigma ,\omega ) vanishes).The contributions in \langle \partial (p_{k_+}) , x \rangle are the ones in Proposition REF ,
and consist of 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 | [
-0.0439930222928524,
0.017282972112298012,
-0.0129965515807271,
0.005609720479696989,
0.014171122573316097,
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0.03749474510550499,
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0.03474899381399155,
-0.012241469696164131,
0.02225... |
6527e4106135fc714d98581be753aee26e8d104f | subsection | 215 | 279 | A formula for the symplectic homology differential | There are two flow lines of -Z_Y from \widehat{p}_{k_+}, which cancel one another.Remark 9.3 At least if there were no contributions \langle \partial (p_{k_+}) , x \rangle , it would be immediate from Theorem REF that \partial ^2 = 0.Recall that the symplectic chain complex (REF ) isSC_*(W,H) = \left(\bigoplus _{k>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 | [
-0.03667448088526726,
0.016430290415883064,
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0.02068660408258438,
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0.024698829278349876,
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0.0019832290709018707,
-0.007787988055497408,
0... |
5c1033bc03bdcc680ff32d4deda9fea76ebaf7da | subsection | 216 | 279 | A formula for the symplectic homology differential | If W is Weinstein, this term is trivial if n\ge 3 and the minimal Chern number of spheres in \Sigma is at least 2.Given p\in \operatorname{Crit}(f_\Sigma ) and x\in \operatorname{Crit}(f_W),\langle \partial (p_{k_+}) , x \rangle = k_+ \sum _{B \in H_2(X;\mathbb {Z})} \delta _{k_+,(B\bullet \Sigma )} \, n_B(x,p)where 2(... | {
"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 | [
-0.01838868483901024,
-0.007588193751871586,
-0.025515252724289894,
0.013841109350323677,
0.012429529801011086,
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0.06152960658073425,
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0.032473959028720856,
-0.032229792326688766,
... |
f7f737f77a9994cfed43a490ad0b2b00bcae4544 | subsection | 217 | 279 | A formula for the symplectic homology differential | The comment about W Weinstein follows from Lemma *Lemma 6.6 (and, by Lemma REF below, it can be interpreted as saying that a certain relative Gromov–Witten invariant of the triple (X,\Sigma ,\omega ) vanishes).The contributions in \langle \partial (p_{k_+}) , x \rangle are the ones in Proposition REF ,
and consist of 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 | [
-0.04127923399209976,
0.023507501929998398,
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0.008351951837539673,
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0.027580508962273598,
-0.014598754234611988,
0.0233... |
801581ee1e4eca2163da09ffe8209b0c2e7d598a | subsection | 218 | 279 | Relation to Gromov–Witten invariants | All the coefficients contributing to the differential in Theorem REF are either combinatorial or topological, except for the n_A(q,p), the n_B and the n_B(x,p). These are harder to determine, but can sometimes be related to Gromov–Witten invariants, absolute or relative.We denote by \operatorname{GW}^\Sigma _{0,k,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 | [
-0.04693432152271271,
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... |
a2705717319e68791aa7adff199d5d3863eda20d | subsection | 219 | 279 | Relation to Gromov–Witten invariants | Furthermore, it is lacunary if it does not have critical points of consecutive Morse index.Lemma 9.4
If the Morse function f_\Sigma is perfect, then W^s_\Sigma (q) and W^u_\Sigma (p) represent classes in H_*(\Sigma ;\mathbb {Z}) andn_A(q,p) = \operatorname{GW}^\Sigma _{0,2,A}\big ([W^s_\Sigma (q)],[W^u_\Sigma (p)]\big... | {
"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 | [
-0.050054192543029785,
0.008782374672591686,
0.013612298294901848,
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0.03915825113654137,
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0.02099834382534027,
0.031024442985653877,
-0.02... |
c08a74a91d6abb314df3934dcb83bd1f72ebff99 | subsection | 220 | 279 | Relation to Gromov–Witten invariants | By the description of the
Gysin sequence in Morse homology *(3.26), \langle \partial _{f_E}(p), \widehat{q}\rangle is the
Euler class of E\rightarrow S integrated over S, which is precisely what we wanted to show.The following result does not need any hypotheses on the auxiliary Morse functions.Lemma 9.5
The count of ... | {
"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 | [
-0.054557912051677704,
-0.0020710951648652554,
0.0058242399245500565,
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0.026775484904646873,
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0.031367748975753784,
-0.007307952735573053,
... |
f9c5d63baec86ae6d20d48f072bc96274dacca33 | subsection | 221 | 279 | Relation to Gromov–Witten invariants | Nevertheless, it can be computationally
quite challenging to use this fact to compute relative invariants of
(X,\Sigma ,\omega ).Depending on what part of symplectic homology one wants to compute, it may also
be possible to ignore the contributions coming from relative Gromov–Witten
invariants.
By Theorem REF , if
X is... | {
"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 | [
-0.057177651673555374,
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0.03710139915347099,
-0.014973282814025879,
0... |
a992f324dd49c81d3bd426e28c600f3a63b498dd | subsection | 222 | 279 | Relation to Gromov–Witten invariants | There is an induced long exact sequence\ldots \rightarrow H_*(\Sigma )[t]t[1] \rightarrow SH_{*}^+(W,H) \rightarrow H_{*}(\Sigma )[t]t \stackrel{[\Delta ]}{\rightarrow } H_{*}(\Sigma )[t]t[1] \rightarrow \ldotswith connecting homomorphism [\Delta ] (we are not being careful keeping track of the degrees * in the sequenc... | {
"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 | [
-0.05102119967341423,
-0.008460303768515587,
-0.011023569852113724,
-0.029386013746261597,
-0.009909769520163536,
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0.0102912075817585,
0.031644128262996674,
0.01861419342458248,
0.006076313555240631,
0.023786498233675957,
0.014548060484230518,
0.... |
8beb99b4ede62b257b48b029e08e432c7b7b7aff | subsection | 223 | 279 | Relation to Gromov–Witten invariants | We also denote denote by \operatorname{GW}^{(X,\Sigma )}_{0,k,(s_1,\ldots ,s_l),B}(C_1,\ldots ,C_k;D_1,\ldots ,D_l) the genus 0 relative Gromov–Witten invariant of (X,\Sigma ,\omega ), counting pseudohomolorphic spheres in X class B\in H_2(X;\mathbb {Z}) with k marked points constrained to go through representatives of... | {
"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 | [
-0.053369518369436264,
0.010916491970419884,
-0.009207691065967083,
-0.04442882537841797,
0.0002760590869002044,
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-0.004291073884814978,
0.012243865057826042,
0.014280694536864758,
0.04183511063456535,
-0.00324977352283895,
0.0008582148002460599,
0.02732555940747261,
-... |
79e0ead21e20d209b6c3ffc7e71d26ca4df21ae3 | subsection | 224 | 279 | Relation to Gromov–Witten invariants | The orientation conventions in Definition REF are precisely the ones used in Gromov–Witten theory (recall Remark REF ).
The second statement is proven along similar lines. Note that the W^u_W(x) define homology classes in W, but that the W^s_W(x) (which we do not consider) would define relative classes in H_*(W,Y;\math... | {
"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 | [
-0.04175694286823273,
0.01052317675203085,
-0.005788128823041916,
-0.00847043190151453,
0.023030418902635574,
-0.04227585345506668,
-0.01694086380302906,
0.001336382352747023,
0.024266643449664116,
0.037483569234609604,
-0.03400382399559021,
0.026067564263939857,
0.01244619395583868,
0.003... |
8bf40816ae14772073b15a2aa3e724439f5e838a | subsection | 225 | 279 | Relation to Gromov–Witten invariants | That is not a problem, though, since the numbers we are computing are invariants.Remark 9.7Computing Gromov–Witten invariants can be a very hard problem, both in
the absolute and relative case, but it has been done for a large class
of important examples, especially using tools from algebraic geometry.
We implicitly as... | {
"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 | [
-0.05124134570360184,
0.031709205359220505,
-0.008491902612149715,
0.0019665658473968506,
0.0267193540930748,
-0.0637236014008522,
0.00824775081127882,
0.01034592930227518,
0.029435543343424797,
0.014801698736846447,
-0.017975671216845512,
0.03772144392132759,
-0.0059054200537502766,
0.022... |
55091f63626d15c5c5f199acbee540352850b9ca | subsection | 226 | 279 | Relation to Gromov–Witten invariants | The variable t has degree 2\frac{\tau _X - K}{K}, by (REF ) (which should also force us to take a global shift of C_*(\Sigma ) by 1-n; we don't do this to avoid making the notation even heavier). The first summand contains the orbits decorated with {} \, and the second summand, with the degree shift up by 1, contains 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 | [
-0.07292686402797699,
0.000735657406039536,
-0.028789330273866653,
-0.02331218682229519,
0.004611327312886715,
-0.00596154248341918,
-0.017529910430312157,
0.01638565957546234,
0.043267905712127686,
0.025722740218043327,
0.013669973239302635,
0.03390031307935715,
-0.006487897597253323,
0.0... |
3803779cf33fb870d546039d1d1a323d7eddab3e | subsection | 227 | 279 | Relation to Gromov–Witten invariants | These are harder to determine, but can sometimes be related to Gromov–Witten invariants, absolute or relative.We denote by \operatorname{GW}^\Sigma _{0,k,A}(C_1,\ldots ,C_k) the genus 0 Gromov–Witten invariant of (\Sigma ,\omega _\Sigma ), counting pseudohomolorphic spheres in \Sigma in class A\in H_2(\Sigma ;\mathbb {... | {
"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 | [
-0.04919036105275154,
0.02363395318388939,
-0.019346581771969795,
-0.030103158205747604,
-0.005221898667514324,
-0.09337624162435532,
0.006015291437506676,
0.028424827381968498,
0.038540586829185486,
0.014372617937624454,
-0.0020330692641437054,
0.006045806687325239,
0.0025327543262392282,
... |
43f69ef242d5c050aee1efec56dd559953067c75 | subsection | 228 | 279 | Relation to Gromov–Witten invariants | Furthermore, it is lacunary if it does not have critical points of consecutive Morse index.Lemma 9.4
If the Morse function f_\Sigma is perfect, then W^s_\Sigma (q) and W^u_\Sigma (p) represent classes in H_*(\Sigma ;\mathbb {Z}) andn_A(q,p) = \operatorname{GW}^\Sigma _{0,2,A}\big ([W^s_\Sigma (q)],[W^u_\Sigma (p)]\big... | {
"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 | [
-0.050054192543029785,
0.008782374672591686,
0.013612298294901848,
-0.04480460658669472,
0.009774302132427692,
-0.07117462158203125,
-0.03027668222784996,
0.003569031832739711,
0.02235652133822441,
0.03915825113654137,
-0.020845739170908928,
0.02099834382534027,
0.031024442985653877,
-0.02... |
93d2877f8206c7755430d5bbbfa3d6e3ee3482f4 | subsection | 229 | 279 | Relation to Gromov–Witten invariants | By the description of the
Gysin sequence in Morse homology *(3.26), \langle \partial _{f_E}(p), \widehat{q}\rangle is the
Euler class of E\rightarrow S integrated over S, which is precisely what we wanted to show.The following result does not need any hypotheses on the auxiliary Morse functions.Lemma 9.5
The count of ... | {
"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 | [
-0.054557912051677704,
-0.0020710951648652554,
0.0058242399245500565,
-0.00136165926232934,
0.026775484904646873,
-0.050682712346315384,
0.005633531603962183,
0.012876643799245358,
0.014547250233590603,
0.02390722744166851,
-0.021725522354245186,
0.031367748975753784,
-0.007307952735573053,
... |
e96d9c484e1bae7f89b5317e063eacde058ec580 | subsection | 230 | 279 | Relation to Gromov–Witten invariants | Nevertheless, it can be computationally
quite challenging to use this fact to compute relative invariants of
(X,\Sigma ,\omega ).Depending on what part of symplectic homology one wants to compute, it may also
be possible to ignore the contributions coming from relative Gromov–Witten
invariants.
By Theorem REF , if
X is... | {
"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 | [
-0.057177651673555374,
0.02683444321155548,
-0.0040160128846764565,
-0.022120488807559013,
0.020198294892907143,
-0.04881763085722923,
-0.0023836735635995865,
0.02151026949286461,
0.03121277317404747,
0.005922951735556126,
0.005617841612547636,
0.03710139915347099,
-0.014973282814025879,
0... |
55c4e1e4f6c3198c9ab65c470bb17994af233bb2 | subsection | 231 | 279 | Relation to Gromov–Witten invariants | There is an induced long exact sequence\ldots \rightarrow H_*(\Sigma )[t]t[1] \rightarrow SH_{*}^+(W,H) \rightarrow H_{*}(\Sigma )[t]t \stackrel{[\Delta ]}{\rightarrow } H_{*}(\Sigma )[t]t[1] \rightarrow \ldotswith connecting homomorphism [\Delta ] (we are not being careful keeping track of the degrees * in the sequenc... | {
"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 | [
-0.04533027857542038,
-0.0007297488045878708,
-0.0159953311085701,
-0.015300876460969448,
-0.013248041272163391,
-0.03849257901310921,
-0.014346957206726074,
0.01475905068218708,
0.03134962543845177,
0.02434403821825981,
0.0115233538672328,
0.026007674634456635,
0.030449125915765762,
0.026... |
25316456e8358b75a4a92b64448435405437cccc | subsection | 232 | 279 | Example: | We now illustrate the results in this paper with the computation of the symplectic homology of the completion W of X\setminus \Sigma , where (X,\Sigma , \omega ) = (\mathbb {CP}^1 \times \mathbb {CP}^1,\Delta ,\omega _{FS}\oplus \omega _{FS}), where \Delta is the diagonal and \omega restricts to the Fubini-Study form 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 | [
-0.02982536144554615,
0.019298763945698738,
-0.02633175067603588,
-0.027796320617198944,
-0.007940716110169888,
-0.08268719166517258,
0.04268611967563629,
-0.0017401305958628654,
0.028162464499473572,
0.018108800053596497,
-0.017818937078118324,
-0.00986296497285366,
0.018474942073225975,
... |
247e5ab158b473e050183694eec1813d1c5f214f | subsection | 233 | 279 | The coefficients in the differential | The manifold \Sigma = \mathbb {CP}^1 admits a lacunary Morse function f_\Sigma : \Sigma \rightarrow \mathbb {R}, with one critical point of Morse index 0, denoted by m (for minimum),
and one critical point of index 2, denoted by M (for maximum).
Lemma REF implies that all the numbers n_A(q,p) (giving the Case 1 contrib... | {
"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 | [
-0.045922666788101196,
0.018277525901794434,
0.01891830749809742,
-0.05892137438058853,
0.010557635687291622,
-0.040033578872680664,
-0.017026476562023163,
0.023464802652597427,
0.02262568473815918,
0.012937680818140507,
-0.007323215715587139,
0.028346946462988853,
0.00892516877502203,
0.0... |
cc17a6bade827b8e005105fd725cf4d2c6cd2c15 | subsection | 234 | 279 | The coefficients in the differential | We will now compute these invariants.Proposition 10.1n_{L_i} = \operatorname{GW}_{L_i,0,(1)}^{\mathbb {CP}^1\times \mathbb {CP}^1,\Delta }\left(\emptyset ; pt \right) = 1,
for i=1,2 (the point constraint is in \Delta , not in \mathbb {CP}^1 \times \mathbb {CP}^1);
n_{L_i}(e,M) = \operatorname{GW}_{L_i,1,(1)}^{\mathbb ... | {
"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 | [
-0.055708903819322586,
0.04569656401872635,
-0.0064294179901480675,
-0.0053228712640702724,
0.020268883556127548,
-0.04099564999341965,
-0.010767844505608082,
-0.0032585894223302603,
0.0032356951851397753,
0.015102455392479897,
-0.022756706923246384,
-0.0014833449386060238,
-0.01259937044233... |
ce4a7860b1abb2f350228b3ca84402dbb2425efd | subsection | 235 | 279 | The coefficients in the differential | This satisfies the vanishing normal Nijenhuis
tensor condition *(4.6), and can thus be used
for computing relative Gromov–Witten invariants.[Proof of Proposition REF ]Given i=1,2,
\operatorname{GW}_{L_i,0,(1)}^{\mathbb {CP}^1\times \mathbb {CP}^1,\Delta }\left(\emptyset ; [pt] \right) = 1
is the fact that if one fixe... | {
"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 | [
-0.06971944868564606,
0.037057533860206604,
-0.013087659142911434,
-0.02351962961256504,
0.006639220751821995,
-0.003998795058578253,
-0.03464604914188385,
-0.004185761325061321,
0.0190324317663908,
-0.01843719184398651,
-0.000654859934002161,
0.015781503170728683,
0.0034607890993356705,
0... |
172b55e33f05b8f4ef6d22b32cf2d6f99e536018 | subsection | 236 | 279 | The coefficients in the differential | To prove that
\operatorname{GW}_{L_1+L_2,1,(2)}^{\mathbb {CP}^1 \times \mathbb {CP}^1, \Delta }\left(pt; pt \right) = 1
we show that, up to domain automorphism, there is a unique holomorphic map U: \mathbb {CP}^1 \rightarrow \mathbb {CP}^1 \times \mathbb {CP}^1, such that U_*[\mathbb {CP}^1] = L_1+L_2 \in H_2(\mathbb... | {
"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 | [
-0.080764040350914,
0.04068725183606148,
-0.03055359236896038,
-0.017077656462788582,
0.010271012783050537,
-0.00599015224725008,
-0.019061656668782234,
0.009439258836209774,
0.007039382588118315,
0.001106461277231574,
0.014368734322488308,
0.005497967824339867,
0.04700552672147751,
-0.004... |
4e9747cfb9e66cd7b1c80f3f89fdcce8a79728b1 | subsection | 237 | 279 | The coefficients in the differential | This invariant counts the
number of lines in \mathbb {CP}^1 through two generic points, which is 1 (the integral
complex structure on \mathbb {CP}^1 is regular).The manifold W = T^*S^2 also admits a lacunary Morse function f_W: W \rightarrow \mathbb {R}
growing at infinity, with one critical point of index 0, denoted b... | {
"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 | [
-0.05438971519470215,
0.04398180544376373,
0.012468124739825726,
-0.031315289437770844,
0.0072450945153832436,
-0.02441737987101078,
-0.013421929441392422,
0.021166816353797913,
0.007718181237578392,
0.020937904715538025,
-0.027774769812822342,
0.01284201629459858,
0.023089686408638954,
-0... |
686850e808bfb8cd4a8ee202736415655426614b | subsection | 238 | 279 | The coefficients in the differential | We will now compute these invariants.Proposition 10.1n_{L_i} = \operatorname{GW}_{L_i,0,(1)}^{\mathbb {CP}^1\times \mathbb {CP}^1,\Delta }\left(\emptyset ; pt \right) = 1,
for i=1,2 (the point constraint is in \Delta , not in \mathbb {CP}^1 \times \mathbb {CP}^1);
n_{L_i}(e,M) = \operatorname{GW}_{L_i,1,(1)}^{\mathbb ... | {
"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 | [
-0.055708903819322586,
0.04569656401872635,
-0.0064294179901480675,
-0.0053228712640702724,
0.020268883556127548,
-0.04099564999341965,
-0.010767844505608082,
-0.0032585894223302603,
0.0032356951851397753,
0.015102455392479897,
-0.022756706923246384,
-0.0014833449386060238,
-0.01259937044233... |
fb118d0a72679efeafbaa0771cf4a5b0d3c30e59 | subsection | 239 | 279 | The coefficients in the differential | This satisfies the vanishing normal Nijenhuis
tensor condition *(4.6), and can thus be used
for computing relative Gromov–Witten invariants.[Proof of Proposition REF ]Given i=1,2,
\operatorname{GW}_{L_i,0,(1)}^{\mathbb {CP}^1\times \mathbb {CP}^1,\Delta }\left(\emptyset ; [pt] \right) = 1
is the fact that if one fixe... | {
"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 | [
-0.06971944868564606,
0.037057533860206604,
-0.013087659142911434,
-0.02351962961256504,
0.006639220751821995,
-0.003998795058578253,
-0.03464604914188385,
-0.004185761325061321,
0.0190324317663908,
-0.01843719184398651,
-0.000654859934002161,
0.015781503170728683,
0.0034607890993356705,
0... |
baea13879ab382b71c99a77e1104f772bca8a837 | subsection | 240 | 279 | The coefficients in the differential | To prove that
\operatorname{GW}_{L_1+L_2,1,(2)}^{\mathbb {CP}^1 \times \mathbb {CP}^1, \Delta }\left(pt; pt \right) = 1
we show that, up to domain automorphism, there is a unique holomorphic map U: \mathbb {CP}^1 \rightarrow \mathbb {CP}^1 \times \mathbb {CP}^1, such that U_*[\mathbb {CP}^1] = L_1+L_2 \in H_2(\mathbb... | {
"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 | [
-0.080764040350914,
0.04068725183606148,
-0.03055359236896038,
-0.017077656462788582,
0.010271012783050537,
-0.00599015224725008,
-0.019061656668782234,
0.009439258836209774,
0.007039382588118315,
0.001106461277231574,
0.014368734322488308,
0.005497967824339867,
0.04700552672147751,
-0.004... |
9d21332008776286223472d159e8c128e5c490a8 | subsection | 241 | 279 | The coefficients in the differential | This invariant counts the
number of lines in \mathbb {CP}^1 through two generic points, which is 1 (the integral
complex structure on \mathbb {CP}^1 is regular).The manifold W = T^*S^2 also admits a lacunary Morse function f_W: W \rightarrow \mathbb {R}
growing at infinity, with one critical point of index 0, denoted b... | {
"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 | [
-0.05438971519470215,
0.04398180544376373,
0.012468124739825726,
-0.031315289437770844,
0.0072450945153832436,
-0.02441737987101078,
-0.013421929441392422,
0.021166816353797913,
0.007718181237578392,
0.020937904715538025,
-0.027774769812822342,
0.01284201629459858,
0.023089686408638954,
-0... |
4a63248fb1e8e55a8fcf3ad636f7e760665d9f1b | subsection | 242 | 279 | The coefficients in the differential | We will now compute these invariants.Proposition 10.1n_{L_i} = \operatorname{GW}_{L_i,0,(1)}^{\mathbb {CP}^1\times \mathbb {CP}^1,\Delta }\left(\emptyset ; pt \right) = 1,
for i=1,2 (the point constraint is in \Delta , not in \mathbb {CP}^1 \times \mathbb {CP}^1);
n_{L_i}(e,M) = \operatorname{GW}_{L_i,1,(1)}^{\mathbb ... | {
"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 | [
-0.055708903819322586,
0.04569656401872635,
-0.0064294179901480675,
-0.0053228712640702724,
0.020268883556127548,
-0.04099564999341965,
-0.010767844505608082,
-0.0032585894223302603,
0.0032356951851397753,
0.015102455392479897,
-0.022756706923246384,
-0.0014833449386060238,
-0.01259937044233... |
4ae37e6cc39a263d0d5c168d56e3ceddca8f28b4 | subsection | 243 | 279 | The coefficients in the differential | This satisfies the vanishing normal Nijenhuis
tensor condition *(4.6), and can thus be used
for computing relative Gromov–Witten invariants.[Proof of Proposition REF ]Given i=1,2,
\operatorname{GW}_{L_i,0,(1)}^{\mathbb {CP}^1\times \mathbb {CP}^1,\Delta }\left(\emptyset ; [pt] \right) = 1
is the fact that if one fixe... | {
"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 | [
-0.06971944868564606,
0.037057533860206604,
-0.013087659142911434,
-0.02351962961256504,
0.006639220751821995,
-0.003998795058578253,
-0.03464604914188385,
-0.004185761325061321,
0.0190324317663908,
-0.01843719184398651,
-0.000654859934002161,
0.015781503170728683,
0.0034607890993356705,
0... |
db9097128c1e9be38d92bcb7a2bbaaedc49ae244 | subsection | 244 | 279 | The coefficients in the differential | To prove that
\operatorname{GW}_{L_1+L_2,1,(2)}^{\mathbb {CP}^1 \times \mathbb {CP}^1, \Delta }\left(pt; pt \right) = 1
we show that, up to domain automorphism, there is a unique holomorphic map U: \mathbb {CP}^1 \rightarrow \mathbb {CP}^1 \times \mathbb {CP}^1, such that U_*[\mathbb {CP}^1] = L_1+L_2 \in H_2(\mathbb... | {
"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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cba24113f8a0fc85c34b00b4f6ecff21467fcd49 | subsection | 245 | 279 | The group | We can now compute the symplectic homology chain complex for T^*S^2. Using the functions f_\Sigma and f_W introduced above, we haveSC_*(T^*S^2) = \mathbb {Z}\left\langle e,c\right\rangle \oplus \bigoplus _{k>0} \mathbb {Z}\left\langle {m}_k, \widehat{m}_k, {M}_k, \widehat{M}_k \right\rangleWe can use Theorem REF , Lemm... | {
"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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029154022571e18cfb87f3dbbb287988217c01d4 | subsection | 246 | 279 | The group | Given k\ge 2,\partial {m}_{k+1} &= 2 \, n_L(M,m) \, \widehat{M}_{k-1} \,+ \,\big ( n_{L_1} \, + \, n_{L_2} \big )\, \widehat{m}_k = \\
&= 2 \times 1 \, \widehat{M}_{k-1} \, + \, \big (1 + 1 \big )\, \widehat{m}_k = 2 \, \widehat{M}_{k-1} + 2 \, \widehat{m}_kand\partial {M}_{k} &= \big ( n_{L_1} \, + \, n_{L_2} \big )\,... | {
"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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a1c6a5dc100705f74d539f1642f165787a1bb891 | subsection | 247 | 279 | The group | We can conclude the following.Proposition 11.1SH_*(T^*S^2 ; \mathbb {Z}) = \mathbb {Z}\left\langle c, {m}_{1}, e, {M}_{k} - {m}_{k+1}, \widehat{M}_{k} \right\rangle \oplus \mathbb {Z}/ 2 \left\langle e + \widehat{m}_{1}, \widehat{M}_{k} + \widehat{m}_{k+1} \right\ranglewhere we take all k \ge 1.We can compare these res... | {
"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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452231d946645c7704d16f3a07b62cacfa568e02 | subsection | 248 | 279 | The group | Phys.,
volume=24,
publisher=Eur. Math. Soc., Zürich,
pages=271485,
review=3444367,AlbersFrauenfelderNegativeLineBundlesarticle
author=Albers, Peter,
author=Frauenfelder, Urs,
title=Floer homology for negative line bundles and Reeb chords in
prequantization spaces,
date=2009,
ISSN=1930-5311,
journal=J. Mod. Dyn.,
volume... | {
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} | 10.1112/topo.12105 | 1804.08014 | Symplectic Homology of complements of smooth divisors | [
"Luís Diogo",
"Samuel T. Lisi"
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"math.SG"
] | 2,018 | en | Mathematics | [
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2eb40f203ce5843e1c6ef0cea4e076962b06c989 | subsection | 249 | 279 | The group | Topol.,
volume=7,
pages=799888,
url=https://doi.org/10.2140/gt.2003.7.799,
review=2026549,BiranBarriersarticle
author=Biran, Paul,
title=Lagrangian barriers and symplectic embeddings,
date=2001,
ISSN=1016-443X,
journal=Geom. Funct. Anal.,
volume=11,
number=3,
pages=407464,
url=http://dx.doi.org/10.1007/PL00001678,
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"math.SG"
] | 2,018 | en | Mathematics | [
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... |
caf57b3078ddf052582ae7f7a0f2e274a23ae60c | subsection | 250 | 279 | The group | S.,
author=Yan, Jun,
title=The loop homology algebra of spheres and projective spaces,
date=2004,
booktitle=Categorical decomposition techniques in algebraic topology
(Isle of Skye, 2001),
series=Progr. Math.,
volume=215,
publisher=Birkhäuser, Basel,
pages=7792,
review=2039760,DiogoLisiSplitunpublished
author=Diogo, Lu... | {
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"Luís Diogo",
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"math.SG"
] | 2,018 | en | Mathematics | [
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3cb46550851653bcaae7e6c68c2e492d956b1e8e | subsection | 251 | 279 | The group | Math.,
volume=41,
number=6,
pages=775813,
url=http://dx.doi.org.umiss.idm.oclc.org/10.1002/cpa.3160410603,
review=948771,GanatraThesisbook
author=Ganatra, Sheel,
title=Symplectic Cohomology and Duality for the Wrapped Fukaya
Category,
publisher=ProQuest LLC, Ann Arbor, MI,
date=2012,
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"Luís Diogo",
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] | 2,018 | en | Mathematics | [
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fda97215b3ee1aed303b90bd94e5950bd9f1977e | subsection | 252 | 279 | The group | Math.,
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"Luís Diogo",
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"math.SG"
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4c903e95853833b19ba1789be9bafffdd35c88ab | subsection | 253 | 279 | The group | Math.,
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number=8,
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"math.SG"
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9f5ece6d29b10a467b45a8bbc1848a810227186e | subsection | 254 | 279 | The group | Topol.,
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number=4,
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url=http://dx.doi.org.umiss.idm.oclc.org/10.2140/gt.2009.13.1877,
review=2497314,McLeanMBSequencemisc
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sequences),
date=2016,
url=http://www.math.stonybrook.edu/ markmclean/talks/spec... | {
"cite_spans": []
} | 10.1112/topo.12105 | 1804.08014 | Symplectic Homology of complements of smooth divisors | [
"Luís Diogo",
"Samuel T. Lisi"
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"math.SG"
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9b86b3547d8503e9e92ebfdb16ca7c989cc4ae82 | subsection | 255 | 279 | The group | Symplectic Geom.,
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number=1,
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url=http://projecteuclid.org.umiss.idm.oclc.org/euclid.jsg/1362146735,
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"math.SG"
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6c50a155f5457077447849d16afc5232d1daaf8c | subsection | 256 | 279 | The group | Ann.,
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number=1-2,
pages=367390,
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"Luís Diogo",
"Samuel T. Lisi"
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"math.SG"
] | 2,018 | en | Mathematics | [
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e40d956289a902bb790ec624c6b023d9e29ef465 | subsection | 257 | 279 | The group | Given k\ge 2,\partial {m}_{k+1} &= 2 \, n_L(M,m) \, \widehat{M}_{k-1} \,+ \,\big ( n_{L_1} \, + \, n_{L_2} \big )\, \widehat{m}_k = \\
&= 2 \times 1 \, \widehat{M}_{k-1} \, + \, \big (1 + 1 \big )\, \widehat{m}_k = 2 \, \widehat{M}_{k-1} + 2 \, \widehat{m}_kand\partial {M}_{k} &= \big ( n_{L_1} \, + \, n_{L_2} \big )\,... | {
"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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7605fd96f3c8fd4c150ce9edd50f61e0b130b059 | subsection | 258 | 279 | The group | We can conclude the following.Proposition 11.1SH_*(T^*S^2 ; \mathbb {Z}) = \mathbb {Z}\left\langle c, {m}_{1}, e, {M}_{k} - {m}_{k+1}, \widehat{M}_{k} \right\rangle \oplus \mathbb {Z}/ 2 \left\langle e + \widehat{m}_{1}, \widehat{M}_{k} + \widehat{m}_{k+1} \right\ranglewhere we take all k \ge 1.We can compare these res... | {
"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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de2a8a5d753a023d4e40caf1d92ed6077537ddc0 | subsection | 259 | 279 | The group | Phys.,
volume=24,
publisher=Eur. Math. Soc., Zürich,
pages=271485,
review=3444367,AlbersFrauenfelderNegativeLineBundlesarticle
author=Albers, Peter,
author=Frauenfelder, Urs,
title=Floer homology for negative line bundles and Reeb chords in
prequantization spaces,
date=2009,
ISSN=1930-5311,
journal=J. Mod. Dyn.,
volume... | {
"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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18e03ed34f6948804572a2d043412dd4f07e745e | subsection | 260 | 279 | The group | Topol.,
volume=7,
pages=799888,
url=https://doi.org/10.2140/gt.2003.7.799,
review=2026549,BiranBarriersarticle
author=Biran, Paul,
title=Lagrangian barriers and symplectic embeddings,
date=2001,
ISSN=1016-443X,
journal=Geom. Funct. Anal.,
volume=11,
number=3,
pages=407464,
url=http://dx.doi.org/10.1007/PL00001678,
revi... | {
"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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... |
cd698903096e078a50b256b476be63ee81032a17 | subsection | 261 | 279 | The group | S.,
author=Yan, Jun,
title=The loop homology algebra of spheres and projective spaces,
date=2004,
booktitle=Categorical decomposition techniques in algebraic topology
(Isle of Skye, 2001),
series=Progr. Math.,
volume=215,
publisher=Birkhäuser, Basel,
pages=7792,
review=2039760,DiogoLisiSplitunpublished
author=Diogo, Lu... | {
"cite_spans": []
} | 10.1112/topo.12105 | 1804.08014 | Symplectic Homology of complements of smooth divisors | [
"Luís Diogo",
"Samuel T. Lisi"
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"math.SG"
] | 2,018 | en | Mathematics | [
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525a52f517136dccf2a9e03a495abc5cf57aa708 | subsection | 262 | 279 | The group | Math.,
volume=41,
number=6,
pages=775813,
url=http://dx.doi.org.umiss.idm.oclc.org/10.1002/cpa.3160410603,
review=948771,GanatraThesisbook
author=Ganatra, Sheel,
title=Symplectic Cohomology and Duality for the Wrapped Fukaya
Category,
publisher=ProQuest LLC, Ann Arbor, MI,
date=2012,
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"cite_spans": []
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03b24665d6b58b796b099dfb55aeacd5a6b87a75 | subsection | 263 | 279 | The group | Math.,
volume=82,
number=2,
pages=307347,
url=https://doi.org/10.1007/BF01388806,
review=809718,HuRuanarticle
author=Hu, JianXun,
author=Ruan, YongBin,
title=Positive divisors in symplectic geometry,
date=2013,
ISSN=1674-7283,
journal=Sci. China Math.,
volume=56,
number=6,
pages=11291144,
url=http://dx.doi.org/10.1007/... | {
"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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