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Title: On spin c-invariants of four-manifolds
Author: Leung, Wai-Man Raymond
ISNI:       0000 0001 3608 550X
Awarding Body: University of Oxford
Current Institution: University of Oxford
Date of Award: 1995
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The spinc-invariants for a compact smooth simply-connected oriented four-manifold, as defined by Pidstrigach and Tyurin, are studied in this thesis. Unlike the Donaldson polynomial invariants, they are defined by cutting down the moduli space M' of '1-instantons', which is the subspace of the moduli space M of anti-self-dual connections parametrizing coupled (spinc) Dirac operators with non-trivial kernel. Our main goal is to study the relationship between these spinc-invariants and the Donaldson polynomial invariants. The 'jumping subset' M' defined a cohomology class P of M which is given by the generalised Porteous formula. When the index l of the coupled Dirac operator is 1, the two smooth invariants are the same by definition. When l = 0 (or when M is compact), the spinc-invariants are expressable as a Donaldson polynomial evaluating the 'Porteous class' P. Our main results concern the first two non-trivial cases l = -1 and -2, when the generalised Porteous formula can not be applied directly. Using cut-and-paste arguments to the moduli space M, we show that for the former case the spinc-invariants and the contracted Donaldson invariants differ by a correction term. It is the number of points in the immediate lower stratum of the Uhlenbeck compactification times a universal 'linking invariant' on S4, which is obtained by computing an example (the K3 surface). The case when l = -2 and dimM = 8 is a parametrized version of the l = -1 situation and the correction term, which involves the same 'linking invariant', is obtained from a suitable obstruction theory.
Supervisor: Donaldson, S. K. Sponsor: Not available
Qualification Name: Thesis (Ph.D.) Qualification Level: Doctoral
EThOS ID:  DOI: Not available
Keywords: Manifolds (Mathematics) ; Topological spaces ; Four-manifolds (Topology)