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

Kazufumi Eto; Shosaku Matsuzaki; Makoto Ozawa

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arXiv · 1503.02170

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

Kazufumi Eto·Shosaku Matsuzaki·Makoto Ozawa

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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.

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Kazufumi Eto, Shosaku Matsuzaki, Makoto Ozawa. 2015-03-07. An obstruction to embedding $2$-dimensional complexes into the $3$-sphere. https://arxiv.org/abs/1503.02170

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