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. 
@article{SEMR_2009_6_a29,
     author = {S. V. Matveev},
     title = {Virtual $3$-manifolds},
     journal = {Sibirskie \`elektronnye matemati\v{c}eskie izvesti\^a},
     pages = {518--521},
     publisher = {mathdoc},
     volume = {6},
     year = {2009},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SEMR_2009_6_a29/}
}
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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/