On the structure of closed 3-manifolds
Volume 177 / 2003
Abstract
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.
