How can I tell if a mesh is a plane from a topological point of view?

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I need to distinguish between meshes that can be topologically transformed to a plane (a triangulated square, for example) and those that can not (a cube or a spherical mesh), and among those toruses.

How would I do that - if the mesh is represented with nodes, edges and the resulting triangles?

Sorry if this is imprecise, newbie, and non-mathematician here.