  In this paper, we introduce a category of graded commutative rings with
certain algebraic morphisms, to investigate the cobordism category of plumbed
3-manifolds. In particular, we define a non-associative distributive algebra
that gives necessary conditions for an abstract morphism between the homologies
of two plumbed 3-manifolds to be realized geometrically by a cobordism. Here we
also consider the homology cobordism monoid, and give a necessary condition
using w-invariants for the homology 3-spheres to belong to the inertia group
associated to some homology 3-spheres.
