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On the structure of closed 3-manifolds

Jan Mycielski (2003)

Fundamenta Mathematicae

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....

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