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Cobordism category of plumbed 3-manifolds and intersection product structures

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Yoshihiro Fukumoto

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