{"id":{"repo_id":"cambridge","oai_identifier":"oai:www.repository.cam.ac.uk:1810/339647"},"canonical_url":"https://search.dev.ndltd.org/etd/cambridge/oai:www.repository.cam.ac.uk:1810/339647","repository":{"repo_id":"cambridge","name":"Cambridge University","base_url":"https://api.repository.cam.ac.uk/server/oai/request"},"display":{"title":"Projective twists and the Hopf correspondence","abstract":"This dissertation is the fruit of a research project on a class of symplectic automorphisms called $projective$ $twists$. In the first part of the thesis (Chapters 3,4) we use Picard$-$Lefschetz theory to introduce a new local model for the planar projective twists $\\tau_{\\mathbb{A}\\mathbb{P}^2} \\in \\mathrm{Symp}_{ct}(T^*\\mathbb{A}\\mathbb{P}^2), \\ \\mathbb{A} \\in \\{ \\mathbb{R}, \\mathbb{C} \\}$. In each case, we construct an exact Lefschetz fibration $\\pi\\colon T^*\\mathbb{A}\\mathbb{P}^2\\to \\mathbb{C}$ with three singular fibres, and define a compactly supported symplectomorphism $\\varphi \\in \\mathrm{Symp}_{ct}(T^*\\mathbb{A}\\mathbb{P}^2)$ on the total space. Given two disjoint Lefschetz thimbles $\\Delta_{\\alpha},\\Delta_{\\beta} \\subset T^*\\mathbb{A}\\mathbb{P}^2$, we compute the Floer cohomology groups $\\operatorname{HF}(\\varphi^k(\\Delta_{\\alpha}), \\Delta_{\\beta};\\mathbb{Z}/2\\mathbb{Z})$ and verify (partially for $\\mathbb{C}\\mathbb{P}^2$) that $\\varphi$ is indeed isotopic to (a power of) the standard local projective twist. The constructions we present are governed by $generalised$ $lantern$ $relations$, which provide an isotopy between the global monodromy of a Lefschetz fibration and a fibred twist along an $S^1$-fibred coisotropic submanifold of the smooth fibre. We also use these relations to study two classes of monotone Lagrangian submanifolds of $(T^*\\mathbb{C}\\mathbb{P}^2, d\\lambda_{T^*\\mathbb{C}\\mathbb{P}^2})$. In the second part of the thesis, starting from Chapter 5, we investigate the properties of projective twists within the symplectic mapping class group of Liouville/Stein manifolds. We define the $Hopf$ $correspondence$, a Lagrangian correspondence (in the sense of Wehrheim$-$Woodward) aimed at assigning Lagrangian spheres $L_1, \\dots , L_m$ of a Liouville manifold $(Y, \\Omega)$ to given Lagrangian (real, complex) projective spaces $K_1, \\dots , K_m$ of a Liouville manifold $(W, \\omega)$. When this correspondence can be established, it intertwines the (real, complex) projective twists $\\tau_{K_i} \\in \\pi_0(\\mathrm{Symp}_{ct}(W))$ (and the induced autoequivalences of the compact Fukaya category $\\operatorname{\\mathcal{F}uk}(W)$) with the Dehn twists $\\tau_{L_i} \\in \\pi_{0}(\\mathrm{Symp}_{ct}(Y))$ (and the corresponding autoequivalences of $\\operatorname{\\mathcal{F}uk}(Y)$), for $i=1, \\dots m$. Using the Hopf correspondence, we obtain a free generation result for projective twists in a clean plumbing of projective spaces and a result about products of positive powers of real projective twists in Liouville manifolds. The same techniques are also used to show that in infinitely many dimensions $n$, the Hamiltonian class of the local projective twist in $\\mathrm{Symp}_{ct}(T^*\\mathbb{C}\\mathbb{P}^n)$ does depend on a choice of framing, i.e a choice of smooth parametrisation of the Lagrangian projective space used to define the twist. Another application of the Hopf correspondence delivers smooth homotopy complex projective spaces $K\\simeq \\mathbb{C}\\mathbb{P}^n$, that do not admit Lagrangian embeddings into $(T^*\\mathbb{C}\\mathbb{P}^n, d\\lambda_{T^*\\mathbb{C}\\mathbb{P}^n})$, for $n=4,7$.","abstract_html":"This dissertation is the fruit of a research project on a class of symplectic automorphisms called $projective$ $twists$. In the first part of the thesis (Chapters 3,4) we use Picard$-$Lefschetz theory to introduce a new local model for the planar projective twists \\tau\\mathbb{A}\\mathbb{P}2 \\in Sympct(T*\\mathbb{A}\\mathbb{P}2), \\mathbb{A} \\in \\{ \\mathbb{R}, \\mathbb{C} \\}. In each case, we construct an exact Lefschetz fibration π\\colon T*\\mathbb{A}\\mathbb{P}2\\to \\mathbb{C} with three singular fibres, and define a compactly supported symplectomorphism \\varphi \\in Sympct(T*\\mathbb{A}\\mathbb{P}2) on the total space. Given two disjoint Lefschetz thimbles \\Deltaα,\\Deltaβ \\subset T*\\mathbb{A}\\mathbb{P}2, we compute the Floer cohomology groups \\operatorname{HF}(\\varphik(\\Deltaα), \\Deltaβ;\\mathbb{Z}/2\\mathbb{Z}) and verify (partially for \\mathbb{C}\\mathbb{P}2) that $\\varphi$ is indeed isotopic to (a power of) the standard local projective twist. The constructions we present are governed by $generalised$ $lantern$ $relations$, which provide an isotopy between the global monodromy of a Lefschetz fibration and a fibred twist along an S1-fibred coisotropic submanifold of the smooth fibre. We also use these relations to study two classes of monotone Lagrangian submanifolds of (T*\\mathbb{C}\\mathbb{P}2, d\\lambdaT*\\mathbb{C}\\mathbb{P}2). In the second part of the thesis, starting from Chapter 5, we investigate the properties of projective twists within the symplectic mapping class group of Liouville/Stein manifolds. We define the $Hopf$ $correspondence$, a Lagrangian correspondence (in the sense of Wehrheim$-$Woodward) aimed at assigning Lagrangian spheres L1, \\dots , Lm of a Liouville manifold $(Y, \\Omega)$ to given Lagrangian (real, complex) projective spaces K1, \\dots , Km of a Liouville manifold (W, ω). When this correspondence can be established, it intertwines the (real, complex) projective twists \\tauKi \\in π0(Sympct(W)) (and the induced autoequivalences of the compact Fukaya category $\\operatorname{\\mathcal{F}uk}(W)$) with the Dehn twists \\tauLi \\in π0(Sympct(Y)) (and the corresponding autoequivalences of $\\operatorname{\\mathcal{F}uk}(Y)$), for $i=1, \\dots m$. Using the Hopf correspondence, we obtain a free generation result for projective twists in a clean plumbing of projective spaces and a result about products of positive powers of real projective twists in Liouville manifolds. The same techniques are also used to show that in infinitely many dimensions $n$, the Hamiltonian class of the local projective twist in Sympct(T*\\mathbb{C}\\mathbb{P}n) does depend on a choice of framing, i.e a choice of smooth parametrisation of the Lagrangian projective space used to define the twist. Another application of the Hopf correspondence delivers smooth homotopy complex projective spaces K\\simeq \\mathbb{C}\\mathbb{P}n, that do not admit Lagrangian embeddings into (T*\\mathbb{C}\\mathbb{P}n, d\\lambdaT*\\mathbb{C}\\mathbb{P}n), for $n=4,7$.","abstract_has_math":true,"creators":["Torricelli, Brunella Charlotte"],"institution":"University of Cambridge","degree_name":"Doctor of Philosophy (PhD)","degree_level":"Doctoral","degree_discipline":null,"degree_department":null,"school":null,"contributors":[],"advisors":["Smith, Ivan"],"committee_chairs":[],"committee_members":[],"year":2022,"date_issued":"2022-01-26","date_published":"2022-01-26","updated_at":"2026-07-22T22:24:03Z","subjects":["symplectic topology","Dehn twist","Floer cohomology"],"languages":["eng"],"rights":[],"rights_urls":["https://www.rioxx.net/licenses/all-rights-reserved/"],"identifier_entries":[]},"links":{"outbound_url":"https://doi.org/10.17863/CAM.87065","outbound_label":"DOI","outbound_source":"dc:identifier.doi"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor.advisor","label":"Advisor","values":["Smith, Ivan"]},{"key":"dc:contributor.sponsor","label":"Sponsor","values":["EPSRC studentship"]},{"key":"dc:creator","label":"Author","values":["Torricelli, Brunella Charlotte"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date.issued","label":"Date","values":["2022-01-26"]},{"key":"dc:publisher.institution","label":"Dc Publisher Institution","values":["University of Cambridge"]},{"key":"dc:relation.isreferencedby.uri","label":"Dc Relation Isreferencedby URI","values":["https://www.repository.cam.ac.uk/handle/1810/339647"]},{"key":"dc:type","label":"Dc Type","values":["Thesis"]},{"key":"dc:type.qualificationlevel","label":"Dc Type Qualificationlevel","values":["Doctoral"]},{"key":"dc:type.qualificationname","label":"Dc Type Qualificationname","values":["Doctor of Philosophy (PhD)"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["symplectic topology","Dehn twist","Floer cohomology"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language","label":"Dc Language","values":["eng"]},{"key":"dc:rights","label":"Dc Rights","values":["https://www.rioxx.net/licenses/all-rights-reserved/"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier.doi","label":"DOI","values":["10.17863/CAM.87065"]},{"key":"dc:identifier.uri","label":"Identifier URI","values":["https://apollo8-f-pro.lib.cam.ac.uk/bitstreams/cfdc8ad3-578e-44b6-adc2-8905cab6a51a/download"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["This dissertation is the fruit of a research project on a class of symplectic automorphisms called $projective$ $twists$. In the first part of the thesis (Chapters 3,4) we use Picard$-$Lefschetz theory to introduce a new local model for the planar projective twists $\\tau_{\\mathbb{A}\\mathbb{P}^2} \\in \\mathrm{Symp}_{ct}(T^*\\mathbb{A}\\mathbb{P}^2), \\ \\mathbb{A} \\in \\{ \\mathbb{R}, \\mathbb{C} \\}$. In each case, we construct an exact Lefschetz fibration $\\pi\\colon T^*\\mathbb{A}\\mathbb{P}^2\\to \\mathbb{C}$ with three singular fibres, and define a compactly supported symplectomorphism $\\varphi \\in \\mathrm{Symp}_{ct}(T^*\\mathbb{A}\\mathbb{P}^2)$ on the total space. Given two disjoint Lefschetz thimbles $\\Delta_{\\alpha},\\Delta_{\\beta} \\subset T^*\\mathbb{A}\\mathbb{P}^2$, we compute the Floer cohomology groups $\\operatorname{HF}(\\varphi^k(\\Delta_{\\alpha}), \\Delta_{\\beta};\\mathbb{Z}/2\\mathbb{Z})$ and verify (partially for $\\mathbb{C}\\mathbb{P}^2$) that $\\varphi$ is indeed isotopic to (a power of) the standard local projective twist. The constructions we present are governed by $generalised$ $lantern$ $relations$, which provide an isotopy between the global monodromy of a Lefschetz fibration and a fibred twist along an $S^1$-fibred coisotropic submanifold of the smooth fibre. We also use these relations to study two classes of monotone Lagrangian submanifolds of $(T^*\\mathbb{C}\\mathbb{P}^2, d\\lambda_{T^*\\mathbb{C}\\mathbb{P}^2})$. In the second part of the thesis, starting from Chapter 5, we investigate the properties of projective twists within the symplectic mapping class group of Liouville/Stein manifolds. We define the $Hopf$ $correspondence$, a Lagrangian correspondence (in the sense of Wehrheim$-$Woodward) aimed at assigning Lagrangian spheres $L_1, \\dots , L_m$ of a Liouville manifold $(Y, \\Omega)$ to given Lagrangian (real, complex) projective spaces $K_1, \\dots , K_m$ of a Liouville manifold $(W, \\omega)$. When this correspondence can be established, it intertwines the (real, complex) projective twists $\\tau_{K_i} \\in \\pi_0(\\mathrm{Symp}_{ct}(W))$ (and the induced autoequivalences of the compact Fukaya category $\\operatorname{\\mathcal{F}uk}(W)$) with the Dehn twists $\\tau_{L_i} \\in \\pi_{0}(\\mathrm{Symp}_{ct}(Y))$ (and the corresponding autoequivalences of $\\operatorname{\\mathcal{F}uk}(Y)$), for $i=1, \\dots m$. Using the Hopf correspondence, we obtain a free generation result for projective twists in a clean plumbing of projective spaces and a result about products of positive powers of real projective twists in Liouville manifolds. The same techniques are also used to show that in infinitely many dimensions $n$, the Hamiltonian class of the local projective twist in $\\mathrm{Symp}_{ct}(T^*\\mathbb{C}\\mathbb{P}^n)$ does depend on a choice of framing, i.e a choice of smooth parametrisation of the Lagrangian projective space used to define the twist. Another application of the Hopf correspondence delivers smooth homotopy complex projective spaces $K\\simeq \\mathbb{C}\\mathbb{P}^n$, that do not admit Lagrangian embeddings into $(T^*\\mathbb{C}\\mathbb{P}^n, d\\lambda_{T^*\\mathbb{C}\\mathbb{P}^n})$, for $n=4,7$."]},{"key":"dc:format.checksum.md5","label":"Dc Format Checksum Md5","values":["6e768712c350d9c228a48a017b7c8863"]},{"key":"dc:title","label":"Title","values":["Projective twists and the Hopf correspondence"]}]}],"canonical_facts":{"dc:contributor.advisor":["Smith, Ivan"],"dc:contributor.sponsor":["EPSRC studentship"],"dc:creator":["Torricelli, Brunella Charlotte"],"dc:date.issued":["2022-01-26"],"dc:description.abstract":["This dissertation is the fruit of a research project on a class of symplectic automorphisms called $projective$ $twists$. In the first part of the thesis (Chapters 3,4) we use Picard$-$Lefschetz theory to introduce a new local model for the planar projective twists $\\tau_{\\mathbb{A}\\mathbb{P}^2} \\in \\mathrm{Symp}_{ct}(T^*\\mathbb{A}\\mathbb{P}^2), \\ \\mathbb{A} \\in \\{ \\mathbb{R}, \\mathbb{C} \\}$. In each case, we construct an exact Lefschetz fibration $\\pi\\colon T^*\\mathbb{A}\\mathbb{P}^2\\to \\mathbb{C}$ with three singular fibres, and define a compactly supported symplectomorphism $\\varphi \\in \\mathrm{Symp}_{ct}(T^*\\mathbb{A}\\mathbb{P}^2)$ on the total space. Given two disjoint Lefschetz thimbles $\\Delta_{\\alpha},\\Delta_{\\beta} \\subset T^*\\mathbb{A}\\mathbb{P}^2$, we compute the Floer cohomology groups $\\operatorname{HF}(\\varphi^k(\\Delta_{\\alpha}), \\Delta_{\\beta};\\mathbb{Z}/2\\mathbb{Z})$ and verify (partially for $\\mathbb{C}\\mathbb{P}^2$) that $\\varphi$ is indeed isotopic to (a power of) the standard local projective twist. The constructions we present are governed by $generalised$ $lantern$ $relations$, which provide an isotopy between the global monodromy of a Lefschetz fibration and a fibred twist along an $S^1$-fibred coisotropic submanifold of the smooth fibre. We also use these relations to study two classes of monotone Lagrangian submanifolds of $(T^*\\mathbb{C}\\mathbb{P}^2, d\\lambda_{T^*\\mathbb{C}\\mathbb{P}^2})$. In the second part of the thesis, starting from Chapter 5, we investigate the properties of projective twists within the symplectic mapping class group of Liouville/Stein manifolds. We define the $Hopf$ $correspondence$, a Lagrangian correspondence (in the sense of Wehrheim$-$Woodward) aimed at assigning Lagrangian spheres $L_1, \\dots , L_m$ of a Liouville manifold $(Y, \\Omega)$ to given Lagrangian (real, complex) projective spaces $K_1, \\dots , K_m$ of a Liouville manifold $(W, \\omega)$. When this correspondence can be established, it intertwines the (real, complex) projective twists $\\tau_{K_i} \\in \\pi_0(\\mathrm{Symp}_{ct}(W))$ (and the induced autoequivalences of the compact Fukaya category $\\operatorname{\\mathcal{F}uk}(W)$) with the Dehn twists $\\tau_{L_i} \\in \\pi_{0}(\\mathrm{Symp}_{ct}(Y))$ (and the corresponding autoequivalences of $\\operatorname{\\mathcal{F}uk}(Y)$), for $i=1, \\dots m$. Using the Hopf correspondence, we obtain a free generation result for projective twists in a clean plumbing of projective spaces and a result about products of positive powers of real projective twists in Liouville manifolds. The same techniques are also used to show that in infinitely many dimensions $n$, the Hamiltonian class of the local projective twist in $\\mathrm{Symp}_{ct}(T^*\\mathbb{C}\\mathbb{P}^n)$ does depend on a choice of framing, i.e a choice of smooth parametrisation of the Lagrangian projective space used to define the twist. Another application of the Hopf correspondence delivers smooth homotopy complex projective spaces $K\\simeq \\mathbb{C}\\mathbb{P}^n$, that do not admit Lagrangian embeddings into $(T^*\\mathbb{C}\\mathbb{P}^n, d\\lambda_{T^*\\mathbb{C}\\mathbb{P}^n})$, for $n=4,7$."],"dc:format.checksum.md5":["6e768712c350d9c228a48a017b7c8863"],"dc:identifier.doi":["10.17863/CAM.87065"],"dc:identifier.uri":["https://apollo8-f-pro.lib.cam.ac.uk/bitstreams/cfdc8ad3-578e-44b6-adc2-8905cab6a51a/download"],"dc:language":["eng"],"dc:publisher.institution":["University of Cambridge"],"dc:relation.isreferencedby.uri":["https://www.repository.cam.ac.uk/handle/1810/339647"],"dc:rights":["https://www.rioxx.net/licenses/all-rights-reserved/"],"dc:subject":["symplectic topology","Dehn twist","Floer cohomology"],"dc:title":["Projective twists and the Hopf correspondence"],"dc:type":["Thesis"],"dc:type.qualificationlevel":["Doctoral"],"dc:type.qualificationname":["Doctor of Philosophy (PhD)"]},"updated_at":"2026-07-22T22:24:03Z"}