On the structure of closed 3-manifolds

Jan Mycielski¹

¹ Department of Mathematics University of Colorado Boulder, CO 80309-0395, U.S.A.

Fundamenta Mathematicae, Tome 177 (2003) no. 3, pp. 193-208.

Résumé

We will show that for every irreducible closed 3-manifold $M$, other than the real projective space $P^3$, there exists a piecewise linear map $f : S \rightarrow M$ where $S$ is a non-orientable closed 2-manifold of Euler characteristic $\chi \equiv 2$ (mod 3) such that $|f^{-1} (x)| \leq 2$ for all $x\in M$, the closure of the set $\{ x \in M : |f^{-1} (x)| = 2\} $ is a cubic graph $G$ such that $S-f^{-1} (G)$ consists of ${1\over 3} (2-\chi ) + 2$ simply connected regions, $M-f(S)$ consists of two disjoint open 3-cells such that $f(S)$ is the boundary of each of them, and $f$ has some additional interesting properties. The pair $(S, f^{-1} (G))$ fully determines $M$, and the minimal value of ${1\over 3} (2-\chi )$ is a natural topological invariant of $M$. Given $S$ there are only finitely many $M$'s for which there exists a map $f : S\rightarrow M$ with all those properties. Several open problems concerning the relationship between $G$ and $M$ are raised.

Détail

DOI : 10.4064/fm177-3-1

Keywords: every irreducible closed manifold other real projective space there exists piecewise linear map rightarrow where non orientable closed manifold euler characteristic chi equiv mod leq closure set cubic graph s f consists chi simply connected regions m f consists disjoint cells boundary each has additional interesting properties pair fully determines minimal value chi natural topological invariant given there only finitely many which there exists map rightarrow those properties several problems concerning relationship between raised