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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.
Keywords: Donaldson theory, orbifolds, homology cobordism, fundamental group
Fukumoto, Yoshihiro 1
@article{10_2140_agt_2019_19_1837,
author = {Fukumoto, Yoshihiro},
title = {On negative-definite cobordisms among lens spaces of type (m,1) and uniformization of four-orbifolds},
journal = {Algebraic and Geometric Topology},
pages = {1837--1880},
publisher = {mathdoc},
volume = {19},
number = {4},
year = {2019},
doi = {10.2140/agt.2019.19.1837},
url = {http://geodesic.mathdoc.fr/articles/10.2140/agt.2019.19.1837/}
}
TY - JOUR AU - Fukumoto, Yoshihiro TI - On negative-definite cobordisms among lens spaces of type (m,1) and uniformization of four-orbifolds JO - Algebraic and Geometric Topology PY - 2019 SP - 1837 EP - 1880 VL - 19 IS - 4 PB - mathdoc UR - http://geodesic.mathdoc.fr/articles/10.2140/agt.2019.19.1837/ DO - 10.2140/agt.2019.19.1837 ID - 10_2140_agt_2019_19_1837 ER -
%0 Journal Article %A Fukumoto, Yoshihiro %T On negative-definite cobordisms among lens spaces of type (m,1) and uniformization of four-orbifolds %J Algebraic and Geometric Topology %D 2019 %P 1837-1880 %V 19 %N 4 %I mathdoc %U http://geodesic.mathdoc.fr/articles/10.2140/agt.2019.19.1837/ %R 10.2140/agt.2019.19.1837 %F 10_2140_agt_2019_19_1837
Fukumoto, Yoshihiro. On negative-definite cobordisms among lens spaces of type (m,1) and uniformization of four-orbifolds. Algebraic and Geometric Topology, Tome 19 (2019) no. 4, pp. 1837-1880. doi: 10.2140/agt.2019.19.1837
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