Terminology for topologies that can't be embedded in a plane (without severing the topology)?

Question

In some code I've written, I justify the existence of a certain structure as follows:

X is needed because we support topologies that cannot result in a planar embedding, e.g. cube, sphere, torus, cylinder, Möbius strip.

Another alternative is

X is needed because we support geometries whose topologies are not equivalent / homeomorphic to a finite plane.

Any more accurate ways to put this? It would be nice to have a single go-to word for this concept, e.g.

X is needed because of the existence of blartigons.

Answer 1

Perhaps

We support topologies that are not homeomorphic to subsets of the plane.

You can then list a few examples for your readers.

Answer 2

X is needed because of the existence of closed orientable 2-manifolds, which have no planar embeddings homeomorphic to their original topology.