The Standard Complex and the 3-dimensional Poincar\'e Conjecture

TL;DR

This paper introduces a new method for constructing and analyzing standard complexes in 3-manifolds, facilitating the calculation of invariants and embeddability, with applications to classical examples and a novel complex related to the Poincaré conjecture.

Contribution

It develops a systematic approach for constructing standard complexes and evaluating their embeddability, applying it to well-known 3-manifold spines and presenting a new complex with unique properties.

Findings

A method for calculating algebraic invariants of standard complexes.

Identification of embeddability criteria for complexes in 3-manifolds.

Construction of a complex with fundamental group Z2 containing a Klein bottle.

Abstract

We develop a method for constructing standard complexes which turns easy the calculation of their algebraic invariants and, as well, the precise evaluation of whether these complexes are embeddable or not in a 3-manifold. This method applies to all familiar spines of 3-manifolds and, in particular, to the Bing house with two rooms and the classical standard spine of the Poincar\'e sphere. Finally, we exhibit a compact, connected standard complex which is embeddable into an orientable 3-manifold, its fundamental group is $Z_{2}$ and it contains a Klein bottle. This standard complex is the spine of a reducible 3-manifold $M^3$, sum of a Seifert fiber space with a fake solid torus, whose universal covering space $W^3$ is a closed and simply connected 3-manifold that cannot be homeomorphic to $S^{3}$.