{"id":{"repo_id":"adelaide","oai_identifier":"oai:digital.library.adelaide.edu.au:2440/118136"},"canonical_url":"https://search.dev.ndltd.org/etd/adelaide/oai:digital.library.adelaide.edu.au:2440/118136","repository":{"repo_id":"adelaide","name":"University of Adelaide","base_url":"https://digital.library.adelaide.edu.au/server/oai/request"},"display":{"title":"Positive scalar curvature and Callias-type index theorems for proper actions","abstract":"This thesis by publication is a study of the equivariant index theory of Dirac operators and Callias-type operators in two distinct settings, namely on cocompact and non-cocompact manifolds with a Lie group action. The first two chapters are a short resumé of Dirac operators and index theory and form a common introduction to the papers in the appendices. Appendix A is joint work with my supervisors, Elder Professor Mathai Varghese and Dr. Hang Wang. For G an almost-connected Lie group acting properly and cocompactly on a manifold M, we study G-index theory of G- invariant Dirac operators. By establishing Poincaré duality for equivariant K-theory and K-homology, we are able to extend the scope of our results to include all elements of equivariant analytic K-homology, which we also show is isomorphic to equivariant geometric K-homology. Our results are applied to prove: a rigidity result for almost-complex manifolds, generalising a vanishing theorem of Hattori; an analogue of Petrie's conjecture; and Lichnerowicz-type obstructions to G-invariant Riemannian metrics on M. Appendix B studies the much more general situation when the quotient M=G is non-compact and G is an arbitrary Lie group. I define G-Callias- type operators and show that they are C*(G)-Fredholm by adapting analysis of Kasparov to new Hilbert C*(G)-module analogues of Sobolev spaces. Questions of adjointability, regularity and essential self-adjointness are addressed in detail. The estimates on G-Callias-type operators are based on the work of Bunke [8] in the non-equivariant context. We construct explicit admissible endomorphisms for G-Callias-type operators from the K-theory of the Higson G-corona of M, a highly non-trivial group. The index theory developed here is applied to prove a general obstruction theorem for G- invariant metrics of positive scalar curvature in the non-cocompact setting.","abstract_html":"This thesis by publication is a study of the equivariant index theory of Dirac operators and Callias-type operators in two distinct settings, namely on cocompact and non-cocompact manifolds with a Lie group action. The first two chapters are a short resumé of Dirac operators and index theory and form a common introduction to the papers in the appendices. Appendix A is joint work with my supervisors, Elder Professor Mathai Varghese and Dr. Hang Wang. For G an almost-connected Lie group acting properly and cocompactly on a manifold M, we study G-index theory of G- invariant Dirac operators. By establishing Poincaré duality for equivariant K-theory and K-homology, we are able to extend the scope of our results to include all elements of equivariant analytic K-homology, which we also show is isomorphic to equivariant geometric K-homology. Our results are applied to prove: a rigidity result for almost-complex manifolds, generalising a vanishing theorem of Hattori; an analogue of Petrie's conjecture; and Lichnerowicz-type obstructions to G-invariant Riemannian metrics on M. Appendix B studies the much more general situation when the quotient M=G is non-compact and G is an arbitrary Lie group. I define G-Callias- type operators and show that they are C*(G)-Fredholm by adapting analysis of Kasparov to new Hilbert C*(G)-module analogues of Sobolev spaces. Questions of adjointability, regularity and essential self-adjointness are addressed in detail. The estimates on G-Callias-type operators are based on the work of Bunke [8] in the non-equivariant context. We construct explicit admissible endomorphisms for G-Callias-type operators from the K-theory of the Higson G-corona of M, a highly non-trivial group. The index theory developed here is applied to prove a general obstruction theorem for G- invariant metrics of positive scalar curvature in the non-cocompact setting.","abstract_has_math":false,"creators":["Guo, Hao"],"institution":null,"degree_name":null,"degree_level":null,"degree_discipline":null,"degree_department":null,"school":null,"contributors":[],"advisors":["Varghese, Mathai","Wang, Hang"],"committee_chairs":[],"committee_members":[],"year":2018,"date_issued":"2018","date_published":"2018","updated_at":"2026-07-24T00:51:02Z","subjects":["Positive scalar curvature","Callias-type operators","index theory","equivariant index"],"languages":["en"],"rights":[],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"http://hdl.handle.net/2440/118136","outbound_label":"Handle","outbound_source":"dc:identifier.uri"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor.advisor","label":"Advisor","values":["Varghese, Mathai","Wang, Hang"]},{"key":"dc:creator","label":"Author","values":["Guo, Hao"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date.issued","label":"Date","values":["2018"]},{"key":"dc:type","label":"Dc Type","values":["Thesis"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Positive scalar curvature","Callias-type operators","index theory","equivariant index"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language.iso","label":"Language (ISO)","values":["en"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier.uri","label":"Identifier URI","values":["http://hdl.handle.net/2440/118136"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["This thesis by publication is a study of the equivariant index theory of Dirac operators and Callias-type operators in two distinct settings, namely on cocompact and non-cocompact manifolds with a Lie group action. The first two chapters are a short resumé of Dirac operators and index theory and form a common introduction to the papers in the appendices. Appendix A is joint work with my supervisors, Elder Professor Mathai Varghese and Dr. Hang Wang. For G an almost-connected Lie group acting properly and cocompactly on a manifold M, we study G-index theory of G- invariant Dirac operators. By establishing Poincaré duality for equivariant K-theory and K-homology, we are able to extend the scope of our results to include all elements of equivariant analytic K-homology, which we also show is isomorphic to equivariant geometric K-homology. Our results are applied to prove: a rigidity result for almost-complex manifolds, generalising a vanishing theorem of Hattori; an analogue of Petrie's conjecture; and Lichnerowicz-type obstructions to G-invariant Riemannian metrics on M. Appendix B studies the much more general situation when the quotient M=G is non-compact and G is an arbitrary Lie group. I define G-Callias- type operators and show that they are C*(G)-Fredholm by adapting analysis of Kasparov to new Hilbert C*(G)-module analogues of Sobolev spaces. Questions of adjointability, regularity and essential self-adjointness are addressed in detail. The estimates on G-Callias-type operators are based on the work of Bunke [8] in the non-equivariant context. We construct explicit admissible endomorphisms for G-Callias-type operators from the K-theory of the Higson G-corona of M, a highly non-trivial group. The index theory developed here is applied to prove a general obstruction theorem for G- invariant metrics of positive scalar curvature in the non-cocompact setting."]},{"key":"dc:title","label":"Title","values":["Positive scalar curvature and Callias-type index theorems for proper actions"]}]}],"canonical_facts":{"dc:contributor.advisor":["Varghese, Mathai","Wang, Hang"],"dc:creator":["Guo, Hao"],"dc:date.issued":["2018"],"dc:description.abstract":["This thesis by publication is a study of the equivariant index theory of Dirac operators and Callias-type operators in two distinct settings, namely on cocompact and non-cocompact manifolds with a Lie group action. The first two chapters are a short resumé of Dirac operators and index theory and form a common introduction to the papers in the appendices. Appendix A is joint work with my supervisors, Elder Professor Mathai Varghese and Dr. Hang Wang. For G an almost-connected Lie group acting properly and cocompactly on a manifold M, we study G-index theory of G- invariant Dirac operators. By establishing Poincaré duality for equivariant K-theory and K-homology, we are able to extend the scope of our results to include all elements of equivariant analytic K-homology, which we also show is isomorphic to equivariant geometric K-homology. Our results are applied to prove: a rigidity result for almost-complex manifolds, generalising a vanishing theorem of Hattori; an analogue of Petrie's conjecture; and Lichnerowicz-type obstructions to G-invariant Riemannian metrics on M. Appendix B studies the much more general situation when the quotient M=G is non-compact and G is an arbitrary Lie group. I define G-Callias- type operators and show that they are C*(G)-Fredholm by adapting analysis of Kasparov to new Hilbert C*(G)-module analogues of Sobolev spaces. Questions of adjointability, regularity and essential self-adjointness are addressed in detail. The estimates on G-Callias-type operators are based on the work of Bunke [8] in the non-equivariant context. We construct explicit admissible endomorphisms for G-Callias-type operators from the K-theory of the Higson G-corona of M, a highly non-trivial group. The index theory developed here is applied to prove a general obstruction theorem for G- invariant metrics of positive scalar curvature in the non-cocompact setting."],"dc:identifier.uri":["http://hdl.handle.net/2440/118136"],"dc:language.iso":["en"],"dc:subject":["Positive scalar curvature","Callias-type operators","index theory","equivariant index"],"dc:title":["Positive scalar curvature and Callias-type index theorems for proper actions"],"dc:type":["Thesis"]},"updated_at":"2026-07-24T00:51:02Z"}