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 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 –sphere to give a topological condition for the cobordism to admit the above “pairing” phenomenon. By using Donaldson theory, we show that if has a certain minimality condition concerning the Chern–Simons invariants of the boundary components, then any must have a counterpart 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 –orbifold with two isolated singular points whose neighborhoods are homeomorphic to the cones over lens spaces and to admit a finite uniformization.
Citation
Yoshihiro Fukumoto. "On negative-definite cobordisms among lens spaces of type $(m,1)$ and uniformization of four-orbifolds." Algebr. Geom. Topol. 19 (4) 1837 - 1880, 2019. https://doi.org/10.2140/agt.2019.19.1837
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