On the structure of closed 3-manifolds

• Autorzy: Jan Mycielski
• Języki publikacji: EN

Abstrakty

EN

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.

Kategorie tematyczne

• 57N10: Topology of general 3 -manifolds
• 57M15: Relations with graph theory

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Fundamenta Mathematicae
• Rocznik: 2003
• Tom: 177
• Numer: 3
• Strony: 193-208

Daty

wydano

2003

Twórcy

autor

Jan Mycielski

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

Identyfikatory

DOI

10.4064/fm177-3-1