On spinc[sic]-invariants of four-manifolds

Thesis



The spin^c-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 (spin^c) Dirac operators with non-trivial kernel.

Our main goal is to study the relationship between these spin^c-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 spin^c-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 spin^c-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.

Authors: Leung, W; Leung, Wai-Man Raymond

• Publication date: 1995
• Type of award: DPhil
• Level of award: Doctoral
• Awarding institution: University of Oxford
• Language: English
• Subjects: Four-manifolds (Topology); Topological spaces; Manifolds (Mathematics)