Research demonstrates the injectiveness of refined maps in virtual and 3-manifolds, suggesting new interpretations.
S. V. Matveev introduced the concept of virtual manifolds and posed the question of whether the natural map from the set of virtual manifolds to the set of 3 3 -manifolds with R P 2 RP^2 -singularities is injective. Here it is proved that this map is not injective, but the refined map from the set of virtual manifolds to the set of 3 3 -manifolds with R P 2 RP^2 -singularities equipped with collections of arcs of a special type is injective.
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Malyutin et al. (2026) studied this question.
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