A+ CATEGORY SCIENTIFIC UNIT

On the structure of closed 3-manifolds

Volume 177 / 2003

Jan Mycielski Fundamenta Mathematicae 177 (2003), 193-208 MSC: Primary 57N10; Secondary 57M15. DOI: 10.4064/fm177-3-1

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.

Authors

  • Jan MycielskiDepartment of Mathematics
    University of Colorado
    Boulder, CO 80309-0395, U.S.A.
    e-mail

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