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In this paper, we introduce a category of graded commutative rings withcertain algebraic morphisms, to investigate the cobordism category of plumbed3-manifolds. In particular, we define a non-associative distributive algebrathat gives necessary conditions for an abstract morphism between the homologiesof two plumbed 3-manifolds to be realized geometrically by a cobordism. Here wealso consider the homology cobordism monoid, and give a necessary conditionusing w-invariants for the homology 3-spheres to belong to the inertia groupassociated to some homology 3-spheres.

Cobordism category of plumbed 3-manifolds and intersection product structures