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

Abstract

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.