Cell-like resolutions of polyhedra by special ones

Dušan Repovš; Arkady Skopenkov

Colloquium Mathematicae (2000)

  • Volume: 86, Issue: 2, page 231-237
  • ISSN: 0010-1354

Abstract

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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 has an arbitrarily small 2-dimensional neighbourhood, then we may additionally conclude that Q is a special 2-polyhedron.

How to cite

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Repovš, Dušan, and Skopenkov, Arkady. "Cell-like resolutions of polyhedra by special ones." Colloquium Mathematicae 86.2 (2000): 231-237. <http://eudml.org/doc/210852>.

@article{Repovš2000,
abstract = {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 has an arbitrarily small 2-dimensional neighbourhood, then we may additionally conclude that Q is a special 2-polyhedron.},
author = {Repovš, Dušan, Skopenkov, Arkady},
journal = {Colloquium Mathematicae},
keywords = {banana and pineapple trick; cell-like resolution; Whitehead conjecture; special polyhedron; fake surface; resolutions of polyhedra; Whitehead's asphericity question; special 2-polyhedron; Whitehead's asphericity conjecture},
language = {eng},
number = {2},
pages = {231-237},
title = {Cell-like resolutions of polyhedra by special ones},
url = {http://eudml.org/doc/210852},
volume = {86},
year = {2000},
}

TY - JOUR
AU - Repovš, Dušan
AU - Skopenkov, Arkady
TI - Cell-like resolutions of polyhedra by special ones
JO - Colloquium Mathematicae
PY - 2000
VL - 86
IS - 2
SP - 231
EP - 237
AB - 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 has an arbitrarily small 2-dimensional neighbourhood, then we may additionally conclude that Q is a special 2-polyhedron.
LA - eng
KW - banana and pineapple trick; cell-like resolution; Whitehead conjecture; special polyhedron; fake surface; resolutions of polyhedra; Whitehead's asphericity question; special 2-polyhedron; Whitehead's asphericity conjecture
UR - http://eudml.org/doc/210852
ER -

References

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  1. [Da86] R. J. Daverman, Decomposition of Manifolds, Academic Press, New York, 1986. 
  2. [HMS93] C. Hog-Angeloni, W. Metzler and A. J. Sieradski (eds.), Two-Dimensional Homotopy and Combinatorial Group Theory, London Math. Soc. Lecture Note Ser. 197, Cambridge Univ. Press, Cambridge, 1993. Zbl0788.00031
  3. [La77] R. C. Lacher, Cell-like mappings and their generalizations, Bull. Amer. Math. Soc. 83 (1977), 336-552. Zbl0364.54009
  4. [Ma73] S. V. Matveev, Special skeletons of PL manifolds, Mat. Sb. 92 (1973), 287-293 (in Russian). 
  5. [MR88] W. J. R. Mitchell and D. Repovš, The topology of cell-like mappings, Rend. Sem. Fac. Sci. Univ. Cagliari Suppl. 58 (1988), 265-300. 
  6. [RS72] C. P. Rourke and B. J. Sanderson, Introduction to Piecewise-Linear Topology, Ergeb. Math. Grenzgeb. 69, Springer, Berlin, 1972. Zbl0254.57010
  7. [Sa] K. Salikhov, Non-existence of special resolutions of 2-polyhedra by special ones, preprint. 

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