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.
Keywords
Donaldson theory, orbifolds, homology cobordism,
fundamental group
