Glasnik Matematicki, Vol. 44, No.1 (2009), 255-258.

CONSTRUCTING NEAR-EMBEDDINGS OF CODIMENSION ONE MANIFOLDS WITH COUNTABLE DENSE SINGULAR SETS

D. Repovš, W. Rosicki, A. Zastrow and M. Željko

Institute of Mathematics, Physics and Mechanics and Faculty of Mathematics and Physics, University of Ljubljana, Jadranska 19, Ljubljana 1001, Slovenia
e-mail: dusan.repovs@guest.arnes.si
e-mail: matjaz.zeljko@fmf.uni-lj.si

Institute of Mathematics, Gdansk University, ul. Wita Stwosza 57, 80-952 Gdansk, Poland
e-mail: wrosicki@math.univ.gda.pl
e-mail: zastrow@math.univ.gda.pl

Abstract.   We present, for all n ≥ 3, very simple examples of continuous maps f : Mn-1 → Mn from closed (n-1)-manifolds Mn-1 into closed n-manifolds Mn such that even though the singular set S(f) of f is countable and dense, the map f can nevertheless be approximated by an embedding, i.e. f is a near-embedding. In dimension 3 one can get even a piecewise-linear approximation by an embedding.

2000 Mathematics Subject Classification.   57Q55, 57N35, 54B15, 57N60.

Key words and phrases.   Near-embedding, singular set, Bing conjecture, recognition problem, space filling map, cellular decomposition, shrinkability.

Full text (PDF) (free access)

DOI: 10.3336/gm.44.1.16

References:

  1. E. H. Anderson, Approximations of certain continuous functions of S2 into E3, Proc. Amer. Math. Soc. 18 (1967), 889-891.
    MathSciNet     CrossRef

  2. R. H. Bing, Approximating surfaces with polyhedral ones, Ann. of Math. (2) 65 (1957), 456-483.
    MathSciNet     CrossRef

  3. M. V. Brahm, The Repovs Conjecture, Doctoral Dissertation, The University of Texas, Austin, 1989.

  4. M. V. Brahm, Approximating maps of 2-manifolds with zero-dimensional nondegeneracy sets, Topology Appl. {\bf 45} (1992), 25-38.
    MathSciNet     CrossRef

  5. M. V. Brahm, A space filling map from I2 to I3 with a zero-dimensional singular set, Topology Appl. 57 (1994), 41-46.
    MathSciNet     CrossRef

  6. R. J. Daverman, Decomposition of Manifolds Academic Press, Inc., Orlando, 1986.
    MathSciNet

  7. R. J. Daverman and D. Repovs, A new 3-dimensional shrinking criterion, Trans. Amer. Math. Soc. 315 (1989), 219-230.
    MathSciNet     CrossRef

  8. D. Repovs, The recognition problem for topological manifolds: A survey, Kodai Math. J. 17 (1994), 538-548.
    MathSciNet     CrossRef

  9. G. T. Whyburn, Analytic Topology, American Mathematical Society, Providence, 1963.
    MathSciNet

Glasnik Matematicki Home Page