Algebraic & Geometric Topology
- Algebr. Geom. Topol.
- Volume 19, Number 4 (2019), 1837-1880.
On negative-definite cobordisms among lens spaces of type $(m,1)$ and uniformization of four-orbifolds
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.
Algebr. Geom. Topol., Volume 19, Number 4 (2019), 1837-1880.
Received: 11 December 2017
Revised: 23 July 2018
Accepted: 2 December 2018
First available in Project Euclid: 22 August 2019
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Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
Primary: 57R18: Topology and geometry of orbifolds 57R57: Applications of global analysis to structures on manifolds, Donaldson and Seiberg-Witten invariants [See also 58-XX]
Secondary: 57M05: Fundamental group, presentations, free differential calculus 57R90: Other types of cobordism [See also 55N22]
Fukumoto, Yoshihiro. On negative-definite cobordisms among lens spaces of type $(m,1)$ and uniformization of four-orbifolds. Algebr. Geom. Topol. 19 (2019), no. 4, 1837--1880. doi:10.2140/agt.2019.19.1837. https://projecteuclid.org/euclid.agt/1566439276