#### 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* : *M*^{n-1} → *M*^{n}
from closed (*n*-1)-manifolds *M*^{n-1}
into closed *n*-manifolds *M*^{n}
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

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