Cobordism category of plumbed 3-manifolds and intersection product structures

TL;DR

This paper develops an algebraic framework to study cobordisms of plumbed 3-manifolds, introducing new algebraic structures and conditions that relate homology morphisms to geometric cobordisms.

Contribution

It introduces a category of graded rings and a non-associative algebra to analyze cobordisms, providing algebraic criteria for geometric realizability and homology cobordism properties.

Findings

Defined a category of graded commutative rings for cobordism analysis

Established algebraic conditions for homology morphisms to be geometrically realizable

Provided necessary conditions using w-invariants for homology 3-spheres in the inertia group

Abstract

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.