Geodesic
Smooth 4-Dimensional Thickening of Singular 2-Dimensional Complex in the non Compact Case
Bollettino della Unione matematica italiana, Série 9, Tome 2 (2009) no. 2, pp. 467-482.

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The present paper extends the theory of 4-dimensional thickening [3], [4], in the NON-PROPER case. Concrete uses of this extensions are to be founded in [2]. 
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Poénaru, Valentin; Tanasi, Corrado. Smooth 4-Dimensional Thickening of Singular 2-Dimensional Complex in the non Compact Case. Bollettino della Unione matematica italiana, Série 9, Tome 2 (2009) no. 2, pp. 467-482. http://geodesic.mathdoc.fr/item/BUMI_2009_9_2_2_a11/

[1] D. Gabai, Valentin Poénaru's Program for the Poincaré Conjecture, in volume Geometry, Topology and Physics for Raoul Bott, S.T. Yau editor, International Press (1994), 139-166 | MR

[2] V. Poénaru, Discrete Symmetry with Compact Fundamental Domain and Geometric Simple Connectivity - A provisional Outline of Work in Progress - Preprint Univ. Paris-Sud 2007-16.

[3] V. Poénaru, Infinite processes and the 3-dimensional Poincaré Conjecture I: The collapsible pseudo-spine representation theorem,Topology, vol. 31, n. 3 (1992), 625-656. | DOI | MR

[4] V. Poénaru, Infinite processes and the 3-dimensional Poincaré Conjecture II: The Honeycom representation theorem (second revised version), Prépublications d'Orsay 93-14 (1994).

[5] V. Poénaru, Geometric simple connectivity in smooth four dimensional Topology, Part A (this is an expanded version, with full details of the previous item) Préprint IHES/M/01/45 (2001).

[6] V. Poénaru - C. Tanasi, Equivariant Almost-Arborescent representations of open simply-connected 3-manifolds; A finiteness result; Memoirs of the AMS n. 800 May 2004. vol. 169. | DOI | MR | Zbl

[7] V. Poénaru - C. Tanasi, Some Remarks on Geometric Simple connectivity in Dimension Four. Part A, Rend. Sem. Mat. e Pol. Torino, vol. 65 no. 3 (2007). | MR | Zbl