### 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-1Mn 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.

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DOI: 10.3336/gm.44.1.16

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