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

Yoshihiro Fukumoto

Algebraic & Geometric Topology 19 (2019) 1837–1880

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 1m 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.

Donaldson theory, orbifolds, homology cobordism, fundamental group
Mathematical Subject Classification 2010
Primary: 57R18, 57R57
Secondary: 57M05, 57R90
Received: 11 December 2017
Revised: 23 July 2018
Accepted: 2 December 2018
Published: 16 August 2019
Yoshihiro Fukumoto
Department of Mathematical Sciences
Faculty of Science and Engineering
Ritsumeikan University