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The $$L^ 2$$-moduli space and a vanishing theorem for Donaldson polynomial invariants. (English) Zbl 0830.58005
Monographs in Geometry and Topology. 2. Cambridge, MA: International Press. 222 p. (1994).
This monograph is a continuation of [C. H. Taubes, ‘$$L^2$$-moduli spaces on 4-manifolds with cylindrical ends’, Monographs in Geometry and Topology. 1 (1993)] reviewed above. It is divided into three parts. The first deals with $$L^2$$-moduli spaces on a manifold with cylindrical ends providing a local description of them when the energy is fixed. The second introduces a local thicking of the $$L^2$$-moduli spaces and proves various local structure theorems for it. The third discusses the example of a four-manifold with an end modeled on a circle bundle over a Riemann surface. As an illustration of the proposed techniques a vanishing theorem of the Donaldson invariants which contain an embedded nontrivial circle bundle over a torus is proved.

##### MSC:
 58D27 Moduli problems for differential geometric structures 58-02 Research exposition (monographs, survey articles) pertaining to global analysis