{"id":{"repo_id":"mit","oai_identifier":"oai:dspace.mit.edu:1721.1/157113"},"canonical_url":"https://search.dev.ndltd.org/etd/mit/oai:dspace.mit.edu:1721.1/157113","repository":{"repo_id":"mit","name":"MIT","base_url":"https://dspace.mit.edu/oai/request"},"display":{"title":"Surgery Exact Triangles in Instanton Theory","abstract":"The introduction of instanton Floer theory and Donaldson polynomial invariants in the 1980s revolutionised the study of low dimensional topology. Since then, many Floer theories have been introduced with different structural properties and qualitative features. One of these Floer theories, Heegaard Floer theory, is popular due to its computational ease and rich algebraic structure. One of the computational tools absent in other Floer theories is the integer surgery formula that computes Heegaard Floer homology of 3-manifolds obtained by surgery along knot(s) in them. This thesis establishes a new surgery formula in instanton Floer theory. The algebraic language to express this formula is that of the derived category of chain complexes. The first part of the thesis describes this surgery formula whose statement and proof are inspired by the Atiyah-Floer conjectures. The second part then contrasts with the Heegaard Floer analogue by showing that instanton and Heegaard Floer theory cannot agree over integers.","abstract_html":"The introduction of instanton Floer theory and Donaldson polynomial invariants in the 1980s revolutionised the study of low dimensional topology. Since then, many Floer theories have been introduced with different structural properties and qualitative features. One of these Floer theories, Heegaard Floer theory, is popular due to its computational ease and rich algebraic structure. One of the computational tools absent in other Floer theories is the integer surgery formula that computes Heegaard Floer homology of 3-manifolds obtained by surgery along knot(s) in them. This thesis establishes a new surgery formula in instanton Floer theory. The algebraic language to express this formula is that of the derived category of chain complexes. The first part of the thesis describes this surgery formula whose statement and proof are inspired by the Atiyah-Floer conjectures. The second part then contrasts with the Heegaard Floer analogue by showing that instanton and Heegaard Floer theory cannot agree over integers.","abstract_has_math":false,"creators":["Bhat, Deeparaj"],"institution":"Massachusetts Institute of Technology","degree_name":"Doctoral","degree_level":null,"degree_discipline":null,"degree_department":"Massachusetts Institute of Technology. Department of Mathematics","school":null,"contributors":[],"advisors":["Mrowka, Tomasz"],"committee_chairs":[],"committee_members":[],"year":2024,"date_issued":"2024-09","date_published":"2024-09","updated_at":"2026-07-22T22:22:26Z","subjects":[],"languages":[],"rights":["In Copyright - Educational Use Permitted","Copyright retained by author(s)"],"rights_urls":["https://rightsstatements.org/page/InC-EDU/1.0/"],"identifier_entries":[]},"links":{"outbound_url":"https://hdl.handle.net/1721.1/157113","outbound_label":"Handle","outbound_source":"dc:identifier.uri"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor.advisor","label":"Advisor","values":["Mrowka, Tomasz"]},{"key":"dc:contributor.department","label":"Department","values":["Massachusetts Institute of Technology. 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Since then, many Floer theories have been introduced with different structural properties and qualitative features. One of these Floer theories, Heegaard Floer theory, is popular due to its computational ease and rich algebraic structure. One of the computational tools absent in other Floer theories is the integer surgery formula that computes Heegaard Floer homology of 3-manifolds obtained by surgery along knot(s) in them. This thesis establishes a new surgery formula in instanton Floer theory. The algebraic language to express this formula is that of the derived category of chain complexes. The first part of the thesis describes this surgery formula whose statement and proof are inspired by the Atiyah-Floer conjectures. The second part then contrasts with the Heegaard Floer analogue by showing that instanton and Heegaard Floer theory cannot agree over integers."]},{"key":"dc:description.degree","label":"Dc Description Degree","values":["Ph.D."]},{"key":"dc:title","label":"Title","values":["Surgery Exact Triangles in Instanton Theory"]}]}],"canonical_facts":{"dc:contributor.advisor":["Mrowka, Tomasz"],"dc:contributor.department":["Massachusetts Institute of Technology. Department of Mathematics"],"dc:creator":["Bhat, Deeparaj"],"dc:date.accessioned":["2024-10-02T17:31:20Z"],"dc:date.available":["2024-10-02T17:31:20Z"],"dc:date.issued":["2024-09"],"dc:description.abstract":["The introduction of instanton Floer theory and Donaldson polynomial invariants in the 1980s revolutionised the study of low dimensional topology. Since then, many Floer theories have been introduced with different structural properties and qualitative features. One of these Floer theories, Heegaard Floer theory, is popular due to its computational ease and rich algebraic structure. One of the computational tools absent in other Floer theories is the integer surgery formula that computes Heegaard Floer homology of 3-manifolds obtained by surgery along knot(s) in them. This thesis establishes a new surgery formula in instanton Floer theory. The algebraic language to express this formula is that of the derived category of chain complexes. The first part of the thesis describes this surgery formula whose statement and proof are inspired by the Atiyah-Floer conjectures. The second part then contrasts with the Heegaard Floer analogue by showing that instanton and Heegaard Floer theory cannot agree over integers."],"dc:description.degree":["Ph.D."],"dc:identifier.uri":["https://hdl.handle.net/1721.1/157113"],"dc:publisher":["Massachusetts Institute of Technology"],"dc:rights":["In Copyright - Educational Use Permitted","Copyright retained by author(s)"],"dc:rights.uri":["https://rightsstatements.org/page/InC-EDU/1.0/"],"dc:title":["Surgery Exact Triangles in Instanton Theory"],"dc:type":["Thesis"],"thesis:degree_name":["Doctoral","Doctor of Philosophy"]},"updated_at":"2026-07-22T22:22:26Z"}