On negative-definite cobordisms among lens spaces of type (m, 1) and uniformization of four-orbifolds

摘要

Connected sums of lens spaces which smoothly bound a rational homology ball are classified by P Lisca. In the classification, there is a phenomenon that a connected sum of a pair of lens spaces L(a, b) # L(a,-b) appears in one of the typical cases of rational homology cobordisms. We consider smooth negative-definite cobordisms among several disjoint union of lens spaces and a rational homology 3-sphere to give a topological condition for the cobordism to admit the above “pairing" phenomenon. By using Donaldson theory, we show that if 1=m has a certain minimality condition concerning the Chern-Simons invariants of the boundary components, then any L(m, 1) must have a counterpart L(m,-1) in negative-definite cobordisms with a certain condition only on homology. In addition, we show an existence of a reducible flat connection through which the pair is related over the cobordism. As an application, we give a sufficient condition for a closed smooth negative-definite 4-orbifold with two isolated singular points whose neighborhoods are homeomorphic to the cones over lens spaces L(m, 1) and L(m,-1) to admit a finite uniformization.