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3-manifold spines and bijoins.

Luigi Grasselli (1990)

Revista Matemática de la Universidad Complutense de Madrid

We describe a combinatorial algorithm for constructing all orientable 3-manifolds with a given standard bidimensional spine by making use of the idea of bijoin (Bandieri and Gagliardi (1982), Graselli (1985)) over a suitable pseudosimplicial triangulation of the spine.

A non-𝒵-compactifiable polyhedron whose product with the Hilbert cube is 𝒵-compactifiable

C. R. Guilbault (2001)

Fundamenta Mathematicae

We construct a locally compact 2-dimensional polyhedron X which does not admit a 𝒵-compactification, but which becomes 𝒵-compactifiable upon crossing with the Hilbert cube. This answers a long-standing question posed by Chapman and Siebenmann in 1976 and repeated in the 1976, 1979 and 1990 versions of Open Problems in Infinite-Dimensional Topology. Our solution corrects an error in the 1990 problem list.

Cell-like resolutions of polyhedra by special ones

Dušan Repovš, Arkady Skopenkov (2000)

Colloquium Mathematicae

Suppose that P is a finite 2-polyhedron. We prove that there exists a PL surjective map f:Q → P from a fake surface Q with preimages of f either points or arcs or 2-disks. This yields a reduction of the Whitehead asphericity conjecture (which asserts that every subpolyhedron of an aspherical 2-polyhedron is also aspherical) to the case of fake surfaces. Moreover, if the set of points of P having a neighbourhood homeomorphic to the 2-disk is a disjoint union of open 2-disks, and every point of P...

Combinatorial mapping-torus, branched surfaces and free group automorphisms

François Gautero (2007)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

We give a characterization of the geometric automorphisms in a certain class of (not necessarily irreducible) free group automorphisms. When the automorphism is geometric, then it is induced by a pseudo-Anosov homeomorphism without interior singularities. An outer free group automorphism is given by a 1 -cocycle of a 2 -complex (a standard dynamical branched surface, see [7] and [9]) the fundamental group of which is the mapping-torus group of the automorphism. A combinatorial construction elucidates...

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