On the structure of closed 3-manifolds

Jan Mycielski. "On the structure of closed 3-manifolds." Fundamenta Mathematicae 177.3 (2003): 193-208. <http://eudml.org/doc/282725>.

@article{JanMycielski2003,
abstract = {We will show that for every irreducible closed 3-manifold M, other than the real projective space P³, there exists a piecewise linear map f: S → M where S is a non-orientable closed 2-manifold of Euler characteristic χ ≡ 2 (mod 3) such that $|f^\{-1\}(x)| ≤ 2$ for all x ∈ M, the closure of the set $\{x ∈ M : |f^\{-1\}(x)| = 2\}$ is a cubic graph G such that $S - f^\{-1\}(G)$ consists of 1/3(2-χ) + 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 1/3(2-χ) is a natural topological invariant of M. Given S there are only finitely many M’s for which there exists a map f: S → M with all those properties. Several open problems concerning the relationship between G and M are raised.},
author = {Jan Mycielski},
journal = {Fundamenta Mathematicae},
keywords = {3-manifold; construction; graphs; nonorientable surface; cubic graph},
language = {eng},
number = {3},
pages = {193-208},
title = {On the structure of closed 3-manifolds},
url = {http://eudml.org/doc/282725},
volume = {177},
year = {2003},
}

TY - JOUR
AU - Jan Mycielski
TI - On the structure of closed 3-manifolds
JO - Fundamenta Mathematicae
PY - 2003
VL - 177
IS - 3
SP - 193
EP - 208
AB - We will show that for every irreducible closed 3-manifold M, other than the real projective space P³, there exists a piecewise linear map f: S → M where S is a non-orientable closed 2-manifold of Euler characteristic χ ≡ 2 (mod 3) such that $|f^{-1}(x)| ≤ 2$ for all x ∈ M, the closure of the set ${x ∈ M : |f^{-1}(x)| = 2}$ is a cubic graph G such that $S - f^{-1}(G)$ consists of 1/3(2-χ) + 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 1/3(2-χ) is a natural topological invariant of M. Given S there are only finitely many M’s for which there exists a map f: S → M with all those properties. Several open problems concerning the relationship between G and M are raised.
LA - eng
KW - 3-manifold; construction; graphs; nonorientable surface; cubic graph
UR - http://eudml.org/doc/282725
ER -