Homology cobordism among lens spaces are completely classified by P. Lisca in 2007.
In his classification, there can be seen a phenomenon that a pair of lens spaces
L(m,1) and L(m,-1) appears in a special case of homology cobordisms.
Then our problem is that what homological structure of cobordisms can be seen
if we consider "negative-definite" cobordism from several disjoint union of lens spaces
to a rational homology 3-sphere?
In a joint work with Furuta, we proved that if m satisfies a condition related to
Chern-Simons invariants then any lens space L(m,1) must have a partner L(m,-1)
in such negative-definite cobordisms, and moreover these are related through a
U(1) flat connection on the cobordism.
In this talk, we introduce these arguments and their extension to the case of
general lens spaces L(a,b). If it is possible, we want to mention about a generalization to
the case of spherical space forms.