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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
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