Geodesic
Virtual $3$-manifolds
Sibirskie èlektronnye matematičeskie izvestiâ, Tome 6 (2009), pp. 518-521.

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We generalize the class if all compact $3$-manifolds to a class of new objects called virtual $3$-manifolds. Each virtual $3$-manifold determines a $3$-manifold with singularities of the type $\mathrm{Con}(RP^2)$ and may be presented by a triangulation as well as by a special spine. Many properties and invariants of $3$-manifolds can be extended to the virtual ones. We restrict ourselves to mentioning Turaev–Viro invariants and two-sheeted branched coverings of virtual $3$-manifolds.
Keywords: $3$-manifold, special spine, virtual $3$-manifold. 
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     author = {S. V. Matveev},
     title = {Virtual $3$-manifolds},
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S. V. Matveev. Virtual $3$-manifolds. Sibirskie èlektronnye matematičeskie izvestiâ, Tome 6 (2009), pp. 518-521. http://geodesic.mathdoc.fr/item/SEMR_2009_6_a29/

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