Complexity of three-dimensional manifolds: problems and results (Q1876458)

The complexity of a three-dimensional manifold is the minimal number of vertices belonging to its almost simple spine. This number rather well reflects the complexity of a manifold in some informal sense. The article is a small overview of the current results on estimation and calculation of complexity. This is an English translation of the author's article published in the book [\textit{Yu. G. Reshetnyak} (ed.) et al., Proceedings of the conference `Geometry and applications' dedicated to the 70th anniversary of Prof. Victor Toponogov. Novosibirsk: Izdatel'stvo Instituta Matematiki (2001; Zbl 1004.57016)].