| Summary: | <p>The <em>spin<sup>c</sup></em>-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 <em>M'</em> of '1-instantons', which is the subspace of the moduli space <em>M</em> of anti-self-dual connections parametrizing coupled (<em>spin<sup>c</sup></em>) Dirac operators with non-trivial kernel.</p> <p>Our main goal is to study the relationship between these <em>spin<sup>c</sup></em>-invariants and the Donaldson polynomial invariants. The 'jumping subset' <em>M'</em> defined a cohomology class <em>P</em> of <em>M</em> which is given by the generalised Porteous formula. When the index <em>l</em> of the coupled Dirac operator is 1, the two smooth invariants are the same by definition. When <em>l</em> = 0 (or when <em>M</em> is compact), the <em>spin<sup>c</sup></em>-invariants are expressable as a Donaldson polynomial evaluating the 'Porteous class' <em>P</em>. Our main results concern the first two non-trivial cases <em>l</em> = -1 and -2, when the generalised Porteous formula can not be applied directly. Using cut-and-paste arguments to the moduli space <em>M</em>, we show that for the former case the <em>spin<sup>c</sup></em>-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 <em>S</em><sup>4</sup>, which is obtained by computing an example (the <em>K</em>3 surface). The case when <em>l</em> = -2 and <em>dimM</em> = 8 is a parametrized version of the <em>l</em> = -1 situation and the correction term, which involves the same 'linking invariant', is obtained from a suitable obstruction theory.</p>
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