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 for all x ∈ M, the closure of the set is a cubic graph G such that 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 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.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-fm177-3-1,
author = {Jan Mycielski},
title = {On the structure of closed 3-manifolds},
journal = {Fundamenta Mathematicae},
volume = {177},
year = {2003},
pages = {193-208},
zbl = {1030.57030},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm177-3-1}
}
Jan Mycielski. On the structure of closed 3-manifolds. Fundamenta Mathematicae, Tome 177 (2003) pp. 193-208. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm177-3-1/