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.
Journal of Homotopy and Related Structures, Vol. 4(2009), No. 1, pp. 39-68