Faithful interpretations of Matveev’s virtual manifolds

Abstract

<p> 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 <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="3"> <mml:semantics> <mml:mn>3</mml:mn> <mml:annotation encoding="application/x-tex">3</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -manifolds with  <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper R upper P squared"> <mml:semantics> <mml:mrow> <mml:mi>R</mml:mi> <mml:msup> <mml:mi>P</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">RP^2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -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 <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="3"> <mml:semantics> <mml:mn>3</mml:mn> <mml:annotation encoding="application/x-tex">3</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -manifolds with <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper R upper P squared"> <mml:semantics> <mml:mrow> <mml:mi>R</mml:mi> <mml:msup> <mml:mi>P</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">RP^2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -singularities equipped with collections of arcs of a special type is injective. </p>