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f02f43881b4822d1981d16f6514a2dfa19b4fb4e | subsection | 42 | 153 | Floer homology from cycles of correspondences | Extend \mathbb {Y}(t) arbitrarily to a \tilde{Y}\in T_J (M_1,\dots , M_l), then replace \tilde{\mathbb {Y}} by \mathbb {Y}=\chi \tilde{\mathbb {Y}} where \chi is a bump function supported near (s,t). When the support of \chi is small enough, one will then have \int _{[0,1]\times }{\langle \eta , \mathbb {Y}\circ Du \ci... | {
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6a6ba58985d47a6115a5aefe4164f7efe2c2f536 | subsection | 43 | 153 | Floer homology from cycles of correspondences | Checking that this is a chain map is left to the reader.An object \mathsf {O} of {C} is, according to Definition REF , a finite collection (indexed by some set A) each of whose elements is a sequence of symplectic manifolds (M_1^\alpha ,\dots , M_{l^\alpha }^\alpha ) and a cycle of Lagrangian correspondences L_{i,i+1}^... | {
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385ecae21ba002bedb9bcb6e139a2c8d8c3f311a | subsection | 44 | 153 | Morphisms | A morphism in {C} is an isomorphism class of `cobordisms', where a cobordism is a matched collection of fibrations. The latter is defined to be a formidable assembly of data{F}=(S,\Gamma ; E,\pi ,\Omega ; Q; \lbrace \zeta _\alpha \rbrace _{\alpha \in \pi _0(\partial S)}),consisting of the following components:A quilted... | {
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592fc4306ea191e5519ef31e7bc1fc6f73b7803d | subsection | 45 | 153 | Morphisms | The minus sign is crucial (though to which factor it is attached is not).Component (iii):Definition 3.8 A Lagrangian matching condition Q \rightarrow \Gamma for E is a sub-fibre bundle Q\subset E_\Gamma , proper over \Gamma , such that (i) \Omega _\Gamma |_Q=0, and (ii) the fibres of Q have half the dimension of those ... | {
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6b048aba6f8fdaf932bdb6536b170aa6dd4c341c | subsection | 46 | 153 | Morphisms | The picture to have in mind is that one can cut the quilted surface (S,\Gamma ) (and the fibrations lying above it) along a collection of circles transverse to \Gamma ; sewing is just the inverse operation. For the sewing of fibrations, it is convenient to be able to assume that the LHFs are flat, hence (by Lemma REF )... | {
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5e608232195a92aa701d0ce3fa7b0494e5444389 | subsection | 47 | 153 | Morphisms | Form the
fibre productE_\Gamma : = (E|_{\Gamma ^+}) \times _{\Gamma } (E|_{\Gamma ^-}) .It becomes an LHF over \Gamma when we endow it with the two-form \Omega _\Gamma obtained as the restriction of the form (-\Omega ^+) \oplus \Omega ^- on (E|_{\Gamma ^+}) \times (E|_{\Gamma ^-}) to its subspace E_\Gamma . The minus s... | {
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3176c7bb40b41c57fdcd3fd16bfe3a137d476044 | subsection | 48 | 153 | Morphisms | A morphism in the category {C} is defined as an isomorphism class of cobordisms.The definition of `isomorphism' here is the obvious one, involving diffeomorphisms of the base surfaces and two-form-preserving bundle-maps; it is left to the reader to elaborate.Morphisms are composed by concatenation of cobordisms. The pi... | {
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af27d5e745d8390f0c20d7aac5723aa73b749615 | subsection | 49 | 153 | Cobordism-maps | We now turn to the construction of maps on Floer homology from matched collections of fibrations.We now turn to the construction of maps on Floer homology from matched collections of fibrations. | {
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0ec17e7417df9540a1aedb4b8eaa93d01b163522 | subsection | 50 | 153 | Cylindrical ends; almost complex structures | Let {F} be a matched collection of fibrations. We elongate the base surface S to a surface \widehat{S} with cylindrical ends. To do this, choose for each boundary component (\partial S)_\alpha a boundary collar e_\alpha \colon (-1,0]\times S^1 \hookrightarrow S such that e_\alpha ^{-1}(\Gamma ) is a union of line segme... | {
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fd1fbbaa9390fad234a96994f39eec88d8a4169e | subsection | 51 | 153 | Cylindrical ends; almost complex structures | We say J is adapted if it is partly compatible everywhere, fully compatible over the end, and translation-invariant (after shortening the end by increasing T).(d) When the cylindrical ends of \widehat{S} have seams \widehat{\Gamma } =(-\infty , -T) \times \lbrace z_1,\dots , z_l \rbrace \subset (-\infty , -T) \times S^... | {
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50648227ace845d87b13ab2aca21e0313a49b53b | subsection | 52 | 153 | Cylindrical ends; almost complex structures | The Lagrangian matching condition Q extends to a Lagrangian matching condition \widehat{Q} over \widehat{\Gamma } in a unique way; indeed, the requirement that it be isotropic implies that the extension must be obtained by symplectic parallel transport along the ends of \widehat{\Gamma }.Definition 3.10
(a) Given an L... | {
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830cccfd1896f66b9db1cbc78ea056ed92b2e035 | subsection | 53 | 153 | Holomorphic sections | The ends of the surface \widehat{S} are cylindrical, with seams \widehat{\Gamma }. Fix a complex structure j on \widehat{S} (inducing \hat{j} on \widehat{S}_\Gamma ) which is standard over the ends, as in Definition REF , and consider the space (\widehat{E}_\Gamma )=(\widehat{E}_\Gamma ,j) of j-adapted almost complex s... | {
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29bcd7cfe9d29ef524a604b20187bacd3c536b98 | subsection | 54 | 153 | Holomorphic sections | Given an adapted almost complex structure J, the moduli space of J-holomorphic sections is the subspace{Z}(\nu ;J) := \lbrace u\in {B}(\nu ): J\circ Du - Du\circ j = 0 ,\; \operatorname{\mathsf {A}}(u)<\infty \rbrace \subset {B}^p(\nu ) .Thus {Z}(\nu ;J) = \operatorname{\bar{\partial }}_J^{-1}(0) where \operatorname{\b... | {
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8619f5ebc7cc57f1938c60152ac46e9c42960053 | subsection | 55 | 153 | Holomorphic sections | Moreover, (U_\alpha ^{\prime }, U_\alpha ^{\prime } \cap \widehat{\Gamma }^{\prime }) is the image of an embedding of pairs whose domain is one of the following:
\begin{}
\item [(i)]
\left(D^2,\emptyset \right);
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53255ace4f1cccbd0320d4f290b4fb6b02546293 | subsection | 56 | 153 | Holomorphic sections | The operator D_{u,J} reduces to one of shape \frac{d}{ds} + L, where L=i \frac{d}{dt}\colon L^2_1([0,1/m], r; \Lambda _0,\Lambda _1)\rightarrow L^2([0,1/m],r) is formally self-adjoint and invertible. The operator \frac{d}{ds} + L\colon L^2_1(\times [0,1/m], r;\Lambda _0,\Lambda _1) \rightarrow L^2(\times [0,1/m], r) is... | {
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f5f0125a92f5c5871d6bc4633d3cccf9f3b86302 | subsection | 57 | 153 | Holomorphic sections | Fix a complex structure j on \widehat{S} (inducing \hat{j} on \widehat{S}_\Gamma ) which is standard over the ends, as in Definition REF , and consider the space (\widehat{E}_\Gamma )=(\widehat{E}_\Gamma ,j) of j-adapted almost complex structures in \widehat{E}_\Gamma . Over each end e, J is eventually translation inva... | {
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c2e7cd34a4b811c19cd2f82cd2fdbc912165a6ca | subsection | 58 | 153 | Holomorphic sections | Given an adapted almost complex structure J, the moduli space of J-holomorphic sections is the subspace{Z}(\nu ;J) := \lbrace u\in {B}(\nu ): J\circ Du - Du\circ j = 0 ,\; \operatorname{\mathsf {A}}(u)<\infty \rbrace \subset {B}^p(\nu ) .Thus {Z}(\nu ;J) = \operatorname{\bar{\partial }}_J^{-1}(0) where \operatorname{\b... | {
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0961708c6515a7787652b42725df8d821f395235 | subsection | 59 | 153 | Holomorphic sections | Moreover, (U_\alpha ^{\prime }, U_\alpha ^{\prime } \cap \widehat{\Gamma }^{\prime }) is the image of an embedding of pairs whose domain is one of the following:
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f399818dc81a19aba067b3f021c26e659f6ba64a | subsection | 60 | 153 | Holomorphic sections | The operator D_{u,J} reduces to one of shape \frac{d}{ds} + L, where L=i \frac{d}{dt}\colon L^2_1([0,1/m], r; \Lambda _0,\Lambda _1)\rightarrow L^2([0,1/m],r) is formally self-adjoint and invertible. The operator \frac{d}{ds} + L\colon L^2_1(\times [0,1/m], r;\Lambda _0,\Lambda _1) \rightarrow L^2(\times [0,1/m], r) is... | {
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ae2076c9a11c57e6faa518f5b5b787aafb0bc911 | subsection | 61 | 153 | Definition of the cobordism-maps | Let \mathsf {O}_- and \mathsf {O}_+ be objects in {C}, and {F} a matched collection of fibrations defining a cobordism between them. Let J_-^v and J^v_+ be regular complex structures defining the Floer complexes \operatorname{CF}_*(\mathsf {O}_\pm ;J^v_\pm ). Also pick J \in (\widehat{E}_\Gamma ; \lbrace J^v_\pm \rbrac... | {
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de59ecf4e2bfb955006a8d5f2636ddf9ac10e069 | subsection | 62 | 153 | Definition of the cobordism-maps | Given cycles of Hamiltonian chords \nu _- and \nu _+ for the respective objects, there is a locally constant action function \mathsf {A} on \widetilde{{Z}}(\nu _-,\nu _+;J) with sub-level sets \widetilde{{Z}}( \nu _-,\nu _+;J)_{\le c}:= \mathsf {A}^{-1}((-\infty ,c]). Assuming we are in the normalised monotone (or stro... | {
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98880f046ea51e1ec00c08a569f5e25fcfab3f3e | subsection | 63 | 153 | Quantum cap product | Assuming N monotone, the quantum cap productH^p(N;\Lambda _R) \otimes _{\Lambda _R} \operatorname{HF}_*(\mu )\rightarrow \operatorname{HF}_{*-p}(\mu ), \quad c\otimes x \mapsto c \frown x,could in principle be defined by means of cocycles for any ordinary cohomology theory in which one can make sense of transversality.... | {
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af8be362090239018a8a3b17e4a221b400776abf | subsection | 64 | 153 | Quantum cap product | By counting the points of fixed energies in the fibre products (REF ) one defines the matrix entries in a linear map\tau \frown \cdot \colon \operatorname{CF}_*(\mu )\rightarrow \operatorname{CF}_{*-p}(\mu ).The usual trajectory-breaking argument demonstrates that\partial (\tau \frown x)= (\partial \tau )\frown x + \ta... | {
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b26abfed381a62b585a5827d5b5c59bc67d061e2 | subsection | 65 | 153 | Quantum cap product | For \nu _\pm \in \operatorname{{H}}(\operatorname{T}(\mu )), one has the `continuation map' moduli spaces \widetilde{{Z}}(\nu _-,\nu _+;J), and evaluation maps\colon \widetilde{{Z}}(\nu _-,\nu _+;J) \rightarrow N,\quad u\mapsto u(0,[0]).Given any countable collection of smooth simplices \sigma ^k\colon \Delta ^{d(k)}\r... | {
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5571ebf0b720988b50224c34d0700942d8b299cc | subsection | 66 | 153 | Setting up the isomorphism | The maps \rho ^! and e(V)\frown \cdot in the Gysin sequence (REF ) fit into the field theory described in the previous section.
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c54151e3fa475795c5d65fe608c0809586e96a12 | subsection | 67 | 153 | Global angular cochains | The Euler class of an oriented sphere bundle vanishes when pulled back to the total space:\rho ^*e(V)=0.When V is an \operatornamewithlimits{\mathrm {SO}}(k+1)-bundle, this can be seen very simply: \rho ^*V has a tautological section. A more precise statement, valid for any smooth, R-oriented fibre bundle whose fibres ... | {
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f936cfa607466ec722a6f8bc71c7087575701a08 | subsection | 68 | 153 | Global angular cochains | Recall that the transgression \tau is the homomorphismd_{k+1}^{0,k}\colon E^{0,k}_{k+1}\rightarrow E^{k+1,0}_{k+1}in the Leray–Serre spectral sequence (E^{p,q}_r, d_r^{p,q} ) of a fibre bundle \pi \colon E\rightarrow B. One thinks of it as a map from a subgroup of H^k(F;R) to a quotient of H^{k+1}(B;R). The cochain int... | {
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338033b0a56593893ec38201ee2d9e6e7512c5e2 | subsection | 69 | 153 | Defining | The map C\rho is associated with a matched collection of fibrations over a quilted surface depicted schematically in Figure REF .
[Figure: The base of the matched pair of fibrations underlying C\rho .]The base surface S is a finite cylinder; its elongation \widehat{S} (see (REF )) an infinite cylinder. To be precise, w... | {
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fdf1a0cd65b97953f26344f20e65430161c374ec | subsection | 70 | 153 | Defining | \end{equation}
We denote this submanifold by Vp to emphasise that it lies in the product of the fibres over p S. Symplectic parallel transport along the arc , starting at p, carries Vp to a Lagrangian Vx(Px)-Qx for each x. Write
\begin{equation}
\widehat{\mathsf {V}} =\bigcup _{x\in \widehat{\Gamma }} {\widehat{V}_x}.
... | {
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663a0a1a434606ec54363bd4fe4c1359822e36fb | subsection | 71 | 153 | Defining | To be precise, we set \widehat{S}= 4i, andS=\lbrace [z]\in \widehat{S}: |\operatorname{Re}(z)| \le 2 \rbrace ,then define q\colon \widehat{S} as the quotient map. The arc \widehat{\Gamma }\subset S is the union of two line segments and a circular arcPedantically, we ought to smooth the joins between the line segments a... | {
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a9c3ff10f7589879db347acae4b77c36aa4cf3be | subsection | 72 | 153 | Defining | Write
\begin{equation}
\widehat{\mathsf {V}} =\bigcup _{x\in \widehat{\Gamma }} {\widehat{V}_x}.
\end{equation}
\begin{}
The map C\rho is the cobordism-map (\ref {cob map}) associated with the matched pair of fibrations (P,Q, \widehat{\mathsf {V}}). Thus it counts finite-action pairs (u,v), where u is a section of \pi ... | {
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1d64f30994b54a2a7b5bce1d7c87833ea0b91328 | subsection | 73 | 153 | The composite | The gluing (or sewing) theorem for fixed-point Floer homology expresses the composite C\rho \circ Ce in terms of a sewed fibration, as in Figure REF . Namely, Ce is defined using the LHF (R,\pi _R,\Omega _R) over \widehat{S} defined above; C\rho using a matched pair of LHFs over \widehat{S}. We can glue these together,... | {
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826a00f3a8377e16c6ead010962941151453639c | subsection | 74 | 153 | The nullhomotopy | The nullhomotopy H promised in (REF ) will be constructed as the concatenation of two homotopies, H_1 and H_2. The first homotopy, H_1, arises by moving the marked point from p^{\prime } to p, as in Figure REF .The matrix entries of this homotopy are defined as counts of isolated triples \lbrace (t,u,v)\rbrace , where ... | {
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902ee177751a3c0d5961082268a25a9cb37ad0c2 | subsection | 75 | 153 | The nullhomotopy | We have an evaluation map ev_1 \colon {Z}_k(x,y) \rightarrow \widehat{V}_1. We can choose the complex structures on P and Q so as to make ev_1 transverse to the codimension k cycle \rho ^! Z in \widehat{V}, by an argument explained by Seidel , for example.It follows that \mathsf {pr}_N \circ ev_1 is transverse to Z. Th... | {
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7870c0384bef87820a808f9e7abc4acfcda1918d | subsection | 76 | 153 | The nullhomotopy | The first homotopy, H_1, arises by moving the marked point from p^{\prime } to p, as in Figure REF .The matrix entries of this homotopy are defined as counts of isolated triples \lbrace (t,u,v)\rbrace , where (u,v) is a pair of pseudo-holomorphic sections as before, t \in [-1,0],
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e63c76bb8a14008d80f9e5ca6e288c407f2d3566 | subsection | 77 | 153 | The nullhomotopy | We can choose the complex structures on P and Q so as to make ev_1 transverse to the codimension k cycle \rho ^! Z in \widehat{V}, by an argument explained by Seidel , for example.It follows that \mathsf {pr}_N \circ ev_1 is transverse to Z. This projected evaluation map is the same as the evaluation map ev^v_1, where ... | {
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f73a2d918fbb89573dd449fd361d7e564ccd956c | subsection | 78 | 153 | Lagrangian intersections | The chain complexes defined in the previous section are well-defined up to homotopy-equivalence, and the maps between them are well-defined up to homotopy. The resulting homology modules are independent of choices up to isomorphism (in fact, canonical isomorphism) and the maps between them are canonical. However, the p... | {
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a2a3bbd2671dea1c9cc05c42a20461cec5ab4279 | subsection | 79 | 153 | Lagrangian intersections | Fix a Riemannian metric on V, and let W_\epsilon \subset W be the
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7d97768a779a0f214f9cc7fb8b1a88cd9b55cb99 | subsection | 80 | 153 | Lagrangian intersections | The Hamiltonian flow \phi _{\hat{m}} with respect to the canonical form -d\lambda _{can} is `vertical', and moves the zero
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73882d7261ad8a31732800b15093cb96e4e7bc79 | subsection | 81 | 153 | Lagrangian intersections | Now, \phi _{\chi H_m} preserves the fibre
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7e07a5807cbd101af4cbec41e1e0e38b66060ec5 | subsection | 82 | 153 | Lagrangian intersections | It is still true that
\phi _{K_{l,m}}-|\cdot |^2 preserves the fibre \rho ^{-1}(\rho (x)) when \rho (x)\in \operatorname{Fix}(\mu ). By the same argument as before, one sees that for any intersection point (x,\rho (x))\in \widehat{V}\cap (\phi _{K_{l,m}^{\prime } } \times \mu )\widehat{V} one has \rho (x)\in \operatorn... | {
"cite_spans": []
} | 0807.1863 | A symplectic Gysin sequence | [
"Timothy Perutz"
] | [
"math.SG",
"math.GT"
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9931d8d9e25823f90da5f75d02efbdf7dab8fe1c | subsection | 83 | 153 | Lagrangian intersections | To begin, \widehat{V} \cap (\mathrm {id}_{M_-} \times \mu )\widehat{V} is the set of
pairs (x,\rho x) such that x\in V and \rho x \in \operatorname{Fix}(\mu ):\mathsf {pr}_1 \left( \widehat{V} \cap (\mathrm {id}_{M_-} \times \mu )\widehat{V}\right)
= \rho ^{-1}( \operatorname{Fix}(\mu ) ).After a C^k-small Hamiltonian ... | {
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"raw": "H. Hofer and D. A. Salamon, Floer homology and Novikov rings, The Floer memorial volume, 483–524, Progr. Math., 133, Birkhäuser, Basel, 1995.",
"source_ref_id": "e5008dfc5ddb11cb991ed84eb515e1... | 0807.1863 | A symplectic Gysin sequence | [
"Timothy Perutz"
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a7dad0c60c262355d0cc21a4f46c9725a67d23ff | subsection | 84 | 153 | Lagrangian intersections | It is convenient to choose the connection form \alpha to be flat over a neighbourhood
U= \bigcup _{\bar{x}\in \operatorname{Fix}(\mu )}{U_{\bar{x}}} of the finite set \operatorname{Fix}(\mu ).
The Hamiltonian connection is then flat over U, hence (after shrinking U) symplectically trivial.
For each \bar{x}\in \operator... | {
"cite_spans": []
} | 0807.1863 | A symplectic Gysin sequence | [
"Timothy Perutz"
] | [
"math.SG",
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] | 2,008 | en | Mathematics | [
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77e79b8911bf6fd019d7d8621abe5c7733d343d5 | subsection | 85 | 153 | Lagrangian intersections | In particular, we may arrange that | \rho (y) - \rho \circ \phi _{\chi H_m}(y) | \le C/4 for all y\in W_\epsilon .Take a point (x,\rho (x)) = (\phi _{H_m} (y), \mu \rho (y)) \in \widehat{V} \cap (\phi _{\chi H_m} \times \mu )\widehat{V}. Then \rho (x)\in \operatorname{supp}(\eta _{\bar{x}}) for some \bar{x}\in \operato... | {
"cite_spans": []
} | 0807.1863 | A symplectic Gysin sequence | [
"Timothy Perutz"
] | [
"math.SG",
"math.GT"
] | 2,008 | en | Mathematics | [
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5428e74ffa2e897eb4ae64aa6babcd24781814b7 | subsection | 86 | 153 | Lagrangian intersections | The functions K_{l,m} are Morse; the critical points lie on V, and project to the critical points of l on N.Lemma 4.5 Lemma REF remains true when H_m is replaced by K^{\prime }_{l,m} or K_{l,m}.The proof is similar but a little more complicated.Given \epsilon >0, we can find a constant C^{\prime } so that if \mathrm {d... | {
"cite_spans": []
} | 0807.1863 | A symplectic Gysin sequence | [
"Timothy Perutz"
] | [
"math.SG",
"math.GT"
] | 2,008 | en | Mathematics | [
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2f24277fc249d86d7689f1db8ce8b0702fb9f0d4 | subsection | 87 | 153 | Lagrangian intersections | Then \rho (x)=\rho (y) and x\in \operatorname{crit}(m) as before.Proposition 4.6
When m is small in C^0 and has precisely two critical points, \rho induces a two-to-one map \beta \colon \widehat{V} \cap (\phi _{K_{l,m}} \times \mu )\widehat{V} \rightarrow \operatorname{Fix}(\mu ).Immediate from the previous lemma.Rema... | {
"cite_spans": []
} | 0807.1863 | A symplectic Gysin sequence | [
"Timothy Perutz"
] | [
"math.SG",
"math.GT"
] | 2,008 | en | Mathematics | [
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477ef238ec47aef8dd7aae0fa8dc8031e4352a4e | subsection | 88 | 153 | The definition of | Recall our standing hypothesis that F is has a Morse function with precisely two critical points. Thus, by an observation of Reeb (see ), F is homeomorphic to S^k.Conversely, by results due to Smale—the h-cobordism theorem and the non-existence of exotic 5-spheres —any F homeomorphic to S^k with k\ne 4 has such a Morse... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 163,
"openalex_id": "",
"raw": "J. Milnor, Morse theory, Annals of Mathematics Studies, No. 51. Princeton University Press, Princeton, N.J., 1963.",
"source_ref_id": "d90644ad91672a6b4df84133079bf36d2f4f79d9",
"start":... | 0807.1863 | A symplectic Gysin sequence | [
"Timothy Perutz"
] | [
"math.SG",
"math.GT"
] | 2,008 | en | Mathematics | [
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3cf41ed05b613de52f0db91f2bec1fa4b69d7ad9 | subsection | 89 | 153 | On | We continue to work on W \cong T^*_v V (REF ). Notice that the connection \alpha induces isomorphismsT^*W & \cong R^*(T^*V) \oplus R^* (T^*_v V) \\
& \cong R^*(T^*_v V)\oplus \left( R^* (T_v^* V) \oplus R^* \rho ^*(T^*N) \right).We shall consider almost complex structures I which have a block decompositionI = \left[ \b... | {
"cite_spans": []
} | 0807.1863 | A symplectic Gysin sequence | [
"Timothy Perutz"
] | [
"math.SG",
"math.GT"
] | 2,008 | en | Mathematics | [
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de6070cc253eef8421b7856febb2281d0a977387 | subsection | 90 | 153 | On | Notice that the connection \alpha induces isomorphismsT^*W & \cong R^*(T^*V) \oplus R^* (T^*_v V) \\
& \cong R^*(T^*_v V)\oplus \left( R^* (T_v^* V) \oplus R^* \rho ^*(T^*N) \right).We shall consider almost complex structures I which have a block decompositionI = \left[ \begin{array}{ccc}
0 & - \mathrm {id}& 0 \\
\math... | {
"cite_spans": []
} | 0807.1863 | A symplectic Gysin sequence | [
"Timothy Perutz"
] | [
"math.SG",
"math.GT"
] | 2,008 | en | Mathematics | [
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c9d8f1944cd934c959c08cd310032ec5b4a6431f | subsection | 91 | 153 | On the matched pair of fibrations | Recall that \pi _P\colon P\rightarrow B is a trivial M-bundle, whilst \pi _Q\colon Q\rightarrow C is an N-bundle. We endowed Q with the closed 2-form \Omega _Q which makes it a flat LHF with monodromy \mu around the puncture. We gave P=B\times M the closed 2-form \Omega _P = \omega _M + d( K dt) ,
where t is the vertic... | {
"cite_spans": []
} | 0807.1863 | A symplectic Gysin sequence | [
"Timothy Perutz"
] | [
"math.SG",
"math.GT"
] | 2,008 | en | Mathematics | [
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9364689364c0e16c77b7bb03934c6878f0e5c58d | subsection | 92 | 153 | On the matched pair of fibrations | We gave P=B\times M the closed 2-form \Omega _P = \omega _M + d( K dt) ,
where t is the vertical (or imaginary) coordinate on B and K\in C^\infty (M); in Lemma REF we observed that this LHF is also flat. Remember also that we built the Lagrangian matching condition \widehat{\mathsf {V}} using the symplectic parallel tr... | {
"cite_spans": []
} | 0807.1863 | A symplectic Gysin sequence | [
"Timothy Perutz"
] | [
"math.SG",
"math.GT"
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5547fdf3b995c6318457027838caf94d03635217 | subsection | 93 | 153 | Tweaking the almost complex structures | The definition of {J}(M,Q) suffers from a predictable defect: it is
so stringent that {J}(M,Q) might not contain any regular almost complex structures. To get around this, we will take some (J_M, J_Q)\in (M,Q) and tweak it slightly, without disturbing the features which are useful to us.Consider the region U_B = \lbrac... | {
"cite_spans": []
} | 0807.1863 | A symplectic Gysin sequence | [
"Timothy Perutz"
] | [
"math.SG",
"math.GT"
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2a5fbd99f0c7f976c92babab838d9c02cbead9b1 | subsection | 94 | 153 | Tweaking the almost complex structures | The quotient map qS=Z is injective on (UB) and maps it to C; let UCC denote q(UB). We shall write also for the induced map UBUC, which is a diffeomorphism, mapping UB to UC by the identity map. Notice that the closure UC wraps all the way around the cylinder: one has q(1+i )UCUB.
[Figure: Schematic of S, pictured as a... | {
"cite_spans": []
} | 0807.1863 | A symplectic Gysin sequence | [
"Timothy Perutz"
] | [
"math.SG",
"math.GT"
] | 2,008 | en | Mathematics | [
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61df9491b09f41e511772a8d6ccdc6da61347801 | subsection | 95 | 153 | Low-energy pseudo-holomorphic sections | We consider pairs (u,v) where u is a J_P-holomorphic section of \pi _P, v a J_Q-holomorphic section of \pi _Q, and (u(x),v(x))\in \widehat{V}_x when x lies in \widehat{\Gamma }. We first identify a subset of the J_P-holomorphic sections of \pi _P. The latter is, by construction, a trivial fibration, and a J_P-holomorph... | {
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f62bfc5d2954333e788bc1607d0d0fc145f40a7a | subsection | 96 | 153 | Low-energy pseudo-holomorphic sections | But from Lemma REF , when J_M\in (M;V), \nabla K is everywhere tangent to V.We can precisely identify the moduli space ^{\mathrm {grad}}(J_M,J_Q) of pairs (u,v)\in {Z}(J_M,J_Q), where u is a gradient section and v is horizontal: there is a canonical identification^{\mathrm {grad}}(J_M,J_Q) \cong \bigcup _{x\in \operato... | {
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b0334fce70469bc76516ce17593062e00d4605a8 | subsection | 97 | 153 | Low-energy pseudo-holomorphic sections | Working in P\rightarrow B, with the symplectic form \Omega _P+ \pi _P^*(ds\wedge dt), and considering the natural linearised symplectic connection in \nabla ^u in u^*T^{\mathrm {v}}P, we haveD_{u}^*D_{u} X &= \frac{1}{2} \nabla ^{u*} \nabla ^{u} X + \frac{J}{2} F_\nabla (\partial _s,\partial _t) X , && X\in C^\infty _c... | {
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76e5caaad15c5915941daec1324391576fdc6f50 | subsection | 98 | 153 | Low-energy pseudo-holomorphic sections | But \ker \nabla _v^*=0 because K is Morse–Smale, and \ker \nabla _u^*=0 because \mu is non-degenerate.Lemma 4.10
The number \# ^{\mathrm {grad}}(\bar{x}, x)_0 of isolated gradient-type pairs asymptotic to \bar{x}\in \operatorname{Fix}(\mu ) and to x\in \widehat{V}\cap \widehat{V}^{\prime } is equal to 1 if \rho (x)=\b... | {
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40103f1b27fe653f7e978e2ca0ee79df5759b4aa | subsection | 99 | 153 | Low-energy pseudo-holomorphic sections | Notice also that the set U^{\prime }:=\overline{U_B\cup U_C}\subset \widehat{S} has the property that any pair of horizontal sections (u,v) over U^{\prime } (or rather, over its image in \widehat{S}_\Gamma ), satisfying the matching condition \widehat{\mathsf {V}}, extends to an element of ^{\mathrm {grad}}(J_M,J_Q). B... | {
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2faa8c0bf95bffc42ac5194ccd4b1fbee891d9c6 | subsection | 100 | 153 | Low-energy pseudo-holomorphic sections | It is better to replace u_n by \tilde{u}_n, so that Floer's equation (REF ) holds, because (\tilde{u}_n,v_n) is subject to a Lagrangian matching condition which is independent of n. Moreover, and satisfies a version of Floer's equation (REF ) whose coefficients converge in C^\infty as n\rightarrow \infty . Gromov–Floer... | {
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6a2f115af8464347b07cc02f2b1c9970e0f17be9 | subsection | 101 | 153 | Low-energy pseudo-holomorphic sections | The deformation operator {D}=D_{(0,w)}\Phi is given by{D} (t, \dot{w}) = D_0 \dot{w} + \frac{t}{2} ( \beta \circ J - J\circ \beta ) \circ dw \circ j ,where \beta = (d/dt)(\alpha _t)|_{t=0}. But \beta (T^{\mathrm {h}}E_\Gamma )=0, and so if w=(u_\infty ,v_\infty ) is our pair of horizontal sections then ( \beta \circ J ... | {
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29327e882a49486554e5df98f3ae588f7a72472e | subsection | 102 | 153 | Low-energy pseudo-holomorphic sections | This contradicts regularity of (u_n,v_n).Thus A is an isomorphism, and there exist coordinates in which \kappa is given by the map\times ^k \rightarrow ^k, \quad (\lambda , x)\mapsto \lambda x,and \mathsf {pr}_2 \colon \times ^k \rightarrow corresponds to the projection {Z}^{\mathrm {par}}\rightarrow . The strand \time... | {
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ebe0c1e943cd15194083372475fd08ff10e21314 | subsection | 103 | 153 | Low-energy pseudo-holomorphic sections | But from Lemma REF , when J_M\in (M;V), \nabla K is everywhere tangent to V.We can precisely identify the moduli space ^{\mathrm {grad}}(J_M,J_Q) of pairs (u,v)\in {Z}(J_M,J_Q), where u is a gradient section and v is horizontal: there is a canonical identification^{\mathrm {grad}}(J_M,J_Q) \cong \bigcup _{x\in \operato... | {
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11aae31eca2d8b850417d06bf6d07c0b822502d5 | subsection | 104 | 153 | Low-energy pseudo-holomorphic sections | Working in P\rightarrow B, with the symplectic form \Omega _P+ \pi _P^*(ds\wedge dt), and considering the natural linearised symplectic connection in \nabla ^u in u^*T^{\mathrm {v}}P, we haveD_{u}^*D_{u} X &= \frac{1}{2} \nabla ^{u*} \nabla ^{u} X + \frac{J}{2} F_\nabla (\partial _s,\partial _t) X , && X\in C^\infty _c... | {
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d2cd6bd137f5eb61fef61845e25911c7c5b1b1c0 | subsection | 105 | 153 | Low-energy pseudo-holomorphic sections | But \ker \nabla _v^*=0 because K is Morse–Smale, and \ker \nabla _u^*=0 because \mu is non-degenerate.Lemma 4.10
The number \# ^{\mathrm {grad}}(\bar{x}, x)_0 of isolated gradient-type pairs asymptotic to \bar{x}\in \operatorname{Fix}(\mu ) and to x\in \widehat{V}\cap \widehat{V}^{\prime } is equal to 1 if \rho (x)=\b... | {
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7de8692cdb64ed5b9b6d7a59ca0e30ce1075fb87 | subsection | 106 | 153 | Low-energy pseudo-holomorphic sections | Notice also that the set U^{\prime }:=\overline{U_B\cup U_C}\subset \widehat{S} has the property that any pair of horizontal sections (u,v) over U^{\prime } (or rather, over its image in \widehat{S}_\Gamma ), satisfying the matching condition \widehat{\mathsf {V}}, extends to an element of ^{\mathrm {grad}}(J_M,J_Q). B... | {
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0686a2f2146c4158e4db92bce76418c5ad3f118a | subsection | 107 | 153 | Low-energy pseudo-holomorphic sections | It is better to replace u_n by \tilde{u}_n, so that Floer's equation (REF ) holds, because (\tilde{u}_n,v_n) is subject to a Lagrangian matching condition which is independent of n. Moreover, and satisfies a version of Floer's equation (REF ) whose coefficients converge in C^\infty as n\rightarrow \infty . Gromov–Floer... | {
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eb45ba538c3504d4980dd5e2243861c919142d78 | subsection | 108 | 153 | Low-energy pseudo-holomorphic sections | The deformation operator {D}=D_{(0,w)}\Phi is given by{D} (t, \dot{w}) = D_0 \dot{w} + \frac{t}{2} ( \beta \circ J - J\circ \beta ) \circ dw \circ j ,where \beta = (d/dt)(\alpha _t)|_{t=0}. But \beta (T^{\mathrm {h}}E_\Gamma )=0, and so if w=(u_\infty ,v_\infty ) is our pair of horizontal sections then ( \beta \circ J ... | {
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fe2e41653909fd40e841bf5d91f71fe8b5258aa8 | subsection | 109 | 153 | Low-energy pseudo-holomorphic sections | This contradicts regularity of (u_n,v_n).Thus A is an isomorphism, and there exist coordinates in which \kappa is given by the map\times ^k \rightarrow ^k, \quad (\lambda , x)\mapsto \lambda x,and \mathsf {pr}_2 \colon \times ^k \rightarrow corresponds to the projection {Z}^{\mathrm {par}}\rightarrow . The strand \time... | {
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0989fefebd236627495f09b46cf48756895a726c | subsection | 110 | 153 | Body | The algebraic mechanism we use to prove Theorem REF is closely related to that used by Seidel to establish the exactness of the sequence describing the effect of Dehn twists on Floer homology. However, because the the symplectic action functional is not exact and the action spectrum not necessarily discrete, two new in... | {
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74f7bde3b8feae47dcf25b64980cfb06caea63dc | subsection | 111 | 153 | Body | Completion is functorial: a homomorphism f of degree [0,\infty ) extends `by continuity' to a homomorphism \hat{f} between completions. We can still speak of a map F \colon \hat{V} \rightarrow \hat{V^{\prime }} having order I: we mean that, for all r, we haveF(\hat{V}_r) \subset \big ( \bigoplus _{s\in I}{V^{\prime }_{... | {
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ad63d3603dfd111eacae49b1eba485b61911c407 | subsection | 112 | 153 | Body | \delta =\delta _{\mathrm {low}}+\delta _{\mathrm {high}}, where \delta _{\mathrm {low}} is equal to \widehat{\underline{d}}, the differential induced by d, and \delta _{\mathrm {high}} is a homomorphism of order [2\epsilon ,\infty ).
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57368d59008a148640ea9f8af5f964034d955f81 | subsection | 113 | 153 | Body | Let us give property (ii) a name: given R-modules A and B and a \Lambda _R-linear map f\colon A\otimes \Lambda _R\rightarrow A\otimes \Lambda _R, let us say f is positive if f(a)=\sum {b_i t^{r_i}} for elements b_i\in B and r_i\ge 0.The following lemma is a generalisation of a rotated version of .Lemma 5.4 (Double mapp... | {
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9c30b1864a6339fe92a200145c3c7728c7331ea1 | subsection | 114 | 153 | Body | One can write c=c_{\mathrm {low}}+c_{\mathrm {high}} and h=h_{\mathrm {low}}+h_{\mathrm {high}}, where the linear maps c_{\mathrm {low}} and h_{\mathrm {low}} have order [0,\epsilon ) while c_{\mathrm {high}} and h_{\mathrm {high}} have order [ 2\epsilon ,\infty ).Further assumethe maps c_{\mathrm {low}} and h_{\mathrm... | {
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0ff7f2f8404e6cd1dcce98397f2f4f08f628bf4b | subsection | 115 | 153 | Body | We use ord 2 to decompose the differential as \delta = \delta _{\mathrm {low}} + \delta _{\mathrm {high}}, where\delta _{\mathrm {low}}= \left[ \begin{array}{ccc}
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90f9d52a4487135a2bf32f1dce6fbf6ea9e8bdeb | subsection | 116 | 153 | Body | A homomorphism f\colon V\rightarrow V^{\prime } between -graded modules has order I if f(V_r)\subset \bigoplus _{s\in I}{V^{\prime }_{r+s}} for all r.The following lemma is Seidel's variation on a well-known principle.Lemma 5.1 Suppose that (D,\delta ) is a finitely-supported -graded R-module with gap [\epsilon ,2\epsi... | {
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b22e02c72500b58720fb05c2379c3096366fb4fd | subsection | 117 | 153 | Body | We can still speak of a map F \colon \hat{V} \rightarrow \hat{V^{\prime }} having order I: we mean that, for all r, we haveF(\hat{V}_r) \subset \big ( \bigoplus _{s\in I}{V^{\prime }_{r+s} }\big )\, \hat{} ,where \hat{V}_r\subset \hat{V} is the image of V_r under V\rightarrow \hat{V}, and the hat on the right-hand side... | {
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feaa42870b13fe273911d7af8c9757986a09db15 | subsection | 118 | 153 | Body | \delta =\delta _{\mathrm {low}}+\delta _{\mathrm {high}}, where \delta _{\mathrm {low}} is equal to \widehat{\underline{d}}, the differential induced by d, and \delta _{\mathrm {high}} is a homomorphism of order [2\epsilon ,\infty ).
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21b769ff91b7417c7a601852bd1ce843a0217a09 | subsection | 119 | 153 | Body | Let us give property (ii) a name: given R-modules A and B and a \Lambda _R-linear map f\colon A\otimes \Lambda _R\rightarrow A\otimes \Lambda _R, let us say f is positive if f(a)=\sum {b_i t^{r_i}} for elements b_i\in B and r_i\ge 0.The following lemma is a generalisation of a rotated version of .Lemma 5.4 (Double mapp... | {
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39070ada23b9c9157ec162530c3629e284141af6 | subsection | 120 | 153 | Body | One can write c=c_{\mathrm {low}}+c_{\mathrm {high}} and h=h_{\mathrm {low}}+h_{\mathrm {high}}, where the linear maps c_{\mathrm {low}} and h_{\mathrm {low}} have order [0,\epsilon ) while c_{\mathrm {high}} and h_{\mathrm {high}} have order [ 2\epsilon ,\infty ).Further assumethe maps c_{\mathrm {low}} and h_{\mathrm... | {
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aaa40f11a0d24cb4a250f4bc2ef357e151ed2584 | subsection | 121 | 153 | Body | We use ord 2 to decompose the differential as \delta = \delta _{\mathrm {low}} + \delta _{\mathrm {high}}, where\delta _{\mathrm {low}}= \left[ \begin{array}{ccc}
0 & 0 & 0 \\
0 & 0 & 0 \\
-h_{\mathrm {low}} & c_{\mathrm {low}} & 0 \end{array} \right] , \quad \delta _{\mathrm {high}} = \left[ \begin{array}{ccc}
d_{C^{\... | {
"cite_spans": []
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"Timothy Perutz"
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32d1becca514f99c5a1b131b897ffb2ba5d3ba5e | subsection | 122 | 153 | Completing the proof | All the ingredients in our proof of Theorem REF are now to close to hand.
Recall from the introduction that the Gysin sequence is derived from the exact sequence of a mapping cone: indeed, it is an immediate consequence of Theorem REF . The definition of \rho _* is algebraic. To prove Theorem REF we shall invoke the do... | {
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"Timothy Perutz"
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8990e0c4cd407b1daec34f14cab0805a2c1ecbf6 | subsection | 123 | 153 | Completing the proof | It follows that h and c are positive, so pos holds.This explains why we chose to concentrate A and A^{\prime } in degree 0 rather than grading them by representatives the action functional. Moreover, ord 2 holds because \epsilon <\epsilon _4. The crucial condition low 2 comes from Lemmas REF , REF and REF .This complet... | {
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} | 0807.1863 | A symplectic Gysin sequence | [
"Timothy Perutz"
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0d9c3657d97f12b24dea5d980ce1cd0817ac6b60 | subsection | 124 | 153 | Completing the proof | From Proposition REF , we have a canonical bijection between \widehat{V}\cap \widehat{V}^{\prime } and \operatorname{Fix}(\mu )\amalg \operatorname{Fix}(\mu ). We assign -degrees to elements of \widehat{V}\cap \widehat{V}^{\prime } so that the map (h_0,c_0) induced by this bijection preserves degree.
We choose \epsilo... | {
"cite_spans": []
} | 0807.1863 | A symplectic Gysin sequence | [
"Timothy Perutz"
] | [
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c6d1f7725489c31785b521b669fc2ccd84cede5a | subsection | 125 | 153 | Completing the proof | Then \operatorname{Fix}(\mu )=\operatorname{crit}(H), and the Floer complex \operatorname{CF}(\phi _H) is canonically identified with the Morse complex for H. Similarly, \operatorname{CF}(\widehat{V}, (\phi _K\times \phi _H)\widehat{V}) is identified with the Morse complex for H\times K restricted to V. We have establi... | {
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4d8b8bf5ad42e68365f928daf65df60d4a9aed03 | subsection | 126 | 153 | On the borderline | We now analyse the breakdown of the Gysin sequence in the borderline case where mon holds but mas just fails because m^{\min }_{\widehat{V}}=k+1. We saw at the outset (Example REF ) that Theorem REF can then also fail.Consider the moduli space \widetilde{{N}} of parametrized J-holomorphic discs \delta \colon (D,\partia... | {
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0e8f15c06db1c07fc3c25ad0aad94e44e0702028 | subsection | 127 | 153 | On the borderline | The conclusions of Theorem REF hold when one replaces the map e by e+ t^{(k+1)\lambda } \nu _Y\,\mathrm {id}, where \lambda is the monotonicity constant of \widehat{V}.The construction of the map e obviously goes through. Less obviously, so does that of \rho ^!: limits of sequences of sections of index 0 or 1 cannot bu... | {
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c8f9b2e2d41ee09ea12372e8d76a9f2cde07f172 | subsection | 128 | 153 | On the borderline | \circ e + \nu _Y t^{ (k+1) \lambda } \, \rho ^! (the Novikov weight \lambda (k+1) is the area of the minimal Maslov-index discs). That is, h is a nullhomotopy of \rho ^!+ t^{\lambda (k+1)} \nu _Y \mathrm {id}.The analysis of low-action sections goes through unchanged. We may assume that (k+1)\lambda \gg \epsilon , whic... | {
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e5972a8580ecd5cb297060966fe2d6be1199425d | subsection | 129 | 153 | On the borderline | The space
(_1^{-1}(\lbrace x\rbrace )\cap _{-1}^{-1}(Y)) / is a compact 0-manifold.Definition 6.1 Define
\nu _Y= \# ( (_1^{-1}(x)\cap _{-1}^{-1}(Y)) /) \in /2.Note that \nu _Y is independent of x and of (regular) J. For example, \nu _Y=0 if, for some J, there is a point x\in \widehat{V} which does not lie on a J-holom... | {
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"Timothy Perutz"
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9d7510b8cc6e5a42b196f002fabae9aac91014d6 | subsection | 130 | 153 | On the borderline | The moduli space now has ends corresponding to configurations consisting of:A pair (u,v) of pseudo-holomorphic sections of index 0; and
a bubble \delta \colon (D,\partial D) \rightarrow (M_-\times N,\widehat{V}) such that \delta (1)=v(p) and \delta (-1) \in \operatorname{im}Y.These configurations contribute the follow... | {
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c5ffde7c65216188e0ddc3b60d1b740865a411ec | subsection | 131 | 153 | Compatibility with quantum cap product | The quantum cohomology QH^*(N)=H^*(N;\Lambda _{/2}) acts on \operatorname{HF}_*(\mu ) by quantum cap product. This makes it a module over the algebra QH^*(N) with its quantum product. Since the map e is itself defined as the quantum cap product by the Euler class of \rho \colon V\rightarrow N, and since QH^*(N) is (sup... | {
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"Timothy Perutz"
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19c5cdebe15902c2c7cf406bdf2bf5e424256512 | subsection | 132 | 153 | Orientations | To define Floer homology over \Lambda _, one needs to specify a system of coherent orientations. This can be reduced to topology: to specify coherent orientations for the Lagrangian Floer homology \operatorname{HF}_*(L_0,L_1) of L_0,L_1\subset P , it suffices to give relative spin-structures on the two Lagrangians, tha... | {
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e2e5dccc5eaf276bcabcdd99e77ef74ddea35aa8 | subsection | 133 | 153 | Orientations | Applying this last principle to U=T V, we see that to give a spin-structure on \xi |_{ \widehat{V}} \oplus T\widehat{V} it suffices to give a spin-structure \sigma in N_{V/M}.Now, \sigma also induces a spin-structure (\mathrm {id}\times \mu )^* \sigma on the normal bundle to \widehat{V}^{\prime }= (\mathrm {id}\times \... | {
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a158b04b308dabbd97966530515c7f69222c3507 | subsection | 134 | 153 | Orientations | This can be reduced to topology: to specify coherent orientations for the Lagrangian Floer homology \operatorname{HF}_*(L_0,L_1) of L_0,L_1\subset P , it suffices to give relative spin-structures on the two Lagrangians, that is, to give a stable vector bundle \xi \rightarrow P, together with stable spin-structures on \... | {
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c18f2a7278ee77b84e39ff27a3b96e9a53eced63 | subsection | 135 | 153 | Orientations | Applying this last principle to U=T V, we see that to give a spin-structure on \xi |_{ \widehat{V}} \oplus T\widehat{V} it suffices to give a spin-structure \sigma in N_{V/M}.Now, \sigma also induces a spin-structure (\mathrm {id}\times \mu )^* \sigma on the normal bundle to \widehat{V}^{\prime }= (\mathrm {id}\times \... | {
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9375186bf135fa2569393d3045e59d31438937cd | subsection | 136 | 153 | Connected sums of three-manifolds | This final section explains the (conjectural) connection between the symplectic Gysin sequence and gauge theory on 3- and 4-manifolds.This final section explains the (conjectural) connection between the symplectic Gysin sequence and gauge theory on 3- and 4-manifolds. | {
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b17016e1415e572eb6a67ca8d49ed6c5df7b4766 | subsection | 137 | 153 | Connected sums and indefinite singularities | Let Y_1 and Y_2 be closed, oriented smooth 3-manifolds. When f_1\colon Y_1 \rightarrow S^1 and f_2 \colon Y_2\rightarrow S^1 are harmonic Morse functions, that is, circle-valued Morse functions with only indefinite critical points, the connected sum Y_1\, \# \,Y_2 inherits a harmonic Morse function with (\# \operatorna... | {
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c7f164e17a79d2cbf515a447280e30daa564069f | subsection | 138 | 153 | Connected sums and indefinite singularities | Let H = \lbrace z: |z| \le 1, \operatorname{Im}(z) \ge 0 \rbrace \subset be a closed half-disc, and let m H[0,1] be the modulus function, z|z|. Let X0 = m* W; so X0 carries a natural map F0X0H.
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d5690b405274f945de0b32ee71f46a6a11c5b890 | subsection | 139 | 153 | Connected sums and indefinite singularities | There is an elementary cobordism W from \Sigma _1 \amalg \Sigma _2 to \Sigma _1\#\Sigma _2, carrying a Morse function with a single critical point c_2, of index 2. Likewise, there is an elementary cobordism W^{\prime } from \Sigma _1\#\Sigma _2
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7767e0d757855cdf6c41ed101bd42bd21e41e47a | subsection | 140 | 153 | Connected sums and indefinite singularities | Trivialise F_1 over H (extending the existing trivialisation over the straight edge). Now define X by excising F_1^{-1} (H) from (Y_1 \amalg Y_2) \times [0,1] and gluing in X_0 in its place, in a way which should be clear. ThusX = X_0 \cup \left( (Y_1 \amalg Y_2) \times [0,1] \setminus F^{-1}(H) \right).The map F\colon... | {
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e51673e9d0e92c76a06df6510d6dbee289983699 | subsection | 141 | 153 | Symmetric products | This section describes a class of examples of the symplectic Gysin sequence.
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