Abstract
A Dehn sphere Σ [C. D. Papakyriakopoulos, On Dehn's Lemma and the asphericity of knots, Ann. Math. (2) 66 (1957) 1–26] in a closed 3-manifold M is a sphere immersed in M with only double curve and triple point singularities. The Dehn sphere Σ ⊂ M lifts to M × I, where I is an interval, if there exists an embedded sphere in M × I that projects onto Σ. Every closed 3-manifold has a filling Dehn sphere [J. M. Montesinos-Amilibia, Representing 3-Manifold by Dehn Spheres, Contribuciones Matemáticas: Homenaje a Joaquín Arregui Fernández (Editorial Complutense, 2000), pp. 239–247], i.e. a Dehn sphere that defines a cell decomposition of M. In [R. Vigara, Representación de 3-variedades por esferas de Dehn rellenantes, Ph.D. Thesis, UNED, Madrid (2006)], it is shown that every closed 3-manifold M has a filling Dehn sphere that lifts to M × I. In this paper it is proved that every closed 3-manifold has a filling Dehn sphere that does not lift to M × I. These results solve a question of Roger Fenn.
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