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We discuss gauge theory with a topological N=2 symmetry. This theory capturesthe de Rham complex and Riemannian geometry of some underlying moduli space$\cal M$ and the partition function equals the Euler number of $\cal M$. Weexplicitly deal with moduli spaces of instantons and of flat connections in twoand three dimensions.  To motivate our constructions we explain the relation between theMathai-Quillen formalism and supersymmetric quantum mechanics and introduce anew kind of supersymmetric quantum mechanics based on the Gauss-Codazziequations. We interpret the gauge theory actions from the Atiyah-Jeffrey pointof view and relate them to supersymmetric quantum mechanics on spaces ofconnections. As a consequence of these considerations we propose the Eulernumber of the moduli space of flat connections as a generalization to arbitrarythree-manifolds of the Casson invariant. We also comment on the possibility ofconstructing a topological version of the Penner matrix model.Comment: 50 p. (with a4.sty: 40 p.

N=2 topological gauge theory, the Euler characteristic of moduli spaces, and the Casson invariant