An obstruction to embedding $2$-dimensional complexes into the $3$-sphere

TL;DR

The paper introduces an algebraic obstruction based on homology and linear systems to determine when certain 2-dimensional complexes cannot be embedded into the 3-sphere.

Contribution

It provides a novel algebraic criterion using linear equations derived from the dual graph to obstruct embeddings of 2-complexes into the 3-sphere.

Findings

Identifies complexes that cannot embed into the 3-sphere due to non-solvability of associated linear systems.

Develops a method to construct dual graphs and analyze their homology.

Establishes a new obstruction criterion for 2-complex embeddings.

Abstract

We consider an embedding of a $2$-dimensional CW complex into the $3$-sphere, and construct it's dual graph. Then we obtain a homogeneous system of linear equations from the $2$-dimensional CW complex in the first homology group of the complement of the dual graph. By checking that the homogeneous system of linear equations does not have an integral solution, we show that some $2$-dimensional CW complexes cannot be embedded into the 3-sphere.