Please use this identifier to cite or link to this item: http://hdl.handle.net/2440/119920
Type: Thesis
Title: Geometric K-homology and the Atiyah-Singer index theorem
Author: Mills, Samuel
Issue Date: 2019
School/Discipline: School of Mathematical Sciences
Abstract: This thesis presents a proof of the Atiyah{Singer index theorem for twisted Spinc- Dirac operators using (geometric) K-homology. The case of twisted Spinc-Dirac operators is the most important case to resolve, and will proceed as a corollary of the computation that the K-homology of a point is Z. We introduce the topological index of a pair (M;E), indt(M;E) = (ch(E) [ Td(M))[M] and the analytic index inda(M;E) = dim(kerDE)⁺- dim(kerDE)- and show that they agree for a \test computation" on a pair of index 1. The main result is that both inda and indt are well-defined on classes [(M;E)] ∈ K0(·) and that there exists a representative on each class for which the analytic and topological indices agree, proving the index theorem for twisted Spinc-Dirac operators. We also present a description of an analogue the Atiyah{Singer index theorem when a compact Lie group action is introduced to (M;E) and an overview of the steps required prove this result.
Advisor: Hochs, Peter
Dissertation Note: Thesis (MPhil.) -- University of Adelaide, School of Mathematical Sciences, 2019
Keywords: K-theory
index theory
K-homology
Provenance: This electronic version is made publicly available by the University of Adelaide in accordance with its open access policy for student theses. Copyright in this thesis remains with the author. This thesis may incorporate third party material which has been used by the author pursuant to Fair Dealing exceptions. If you are the owner of any included third party copyright material you wish to be removed from this electronic version, please complete the take down form located at: http://www.adelaide.edu.au/legals
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