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 $L\left(a,b\right)\#L\left(a,-b\right)$ 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 $L\left(m,1\right)$ must have a counterpart $L\left(m,-1\right)$ 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 $L\left(m,1\right)$ and $L\left(m,-1\right)$ to admit a finite uniformization.