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

Fukumoto, Yoshihiro

doi:10.2140/agt.2019.19.1837

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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

DOI: 10.2140/agt.2019.19.1837

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.

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Keywords

Donaldson theory, orbifolds, homology cobordism, fundamental group

Mathematical Subject Classification 2010

Primary: 57R18, 57R57

Secondary: 57M05, 57R90

References

Publication

Received: 11 December 2017

Revised: 23 July 2018

Accepted: 2 December 2018

Published: 16 August 2019

Authors

Yoshihiro Fukumoto
Department of Mathematical Sciences Faculty of Science and Engineering Ritsumeikan University Noji-Higashi Kusatsu Japan

Volume 19, issue 4 (2019)