I have recently come across the concept of a genus, and I was wondering, is 3D space mathematically equivalent to a surface with an infinite number of handles? I ask this because I asked a question a few weeks ago on whether a graph can be 'non-planar' in 3D, or more clearly, whether a graph exists, that when drawn in 3D still has an edge crossing which cannot be avoided. Clearly no such graph exists, but, if the answer to my above question is yes, then I can prove so by saying:
Since every graph has a genus, let us call it g, this means that the graph can be drawn, without edge-crossings, on every surface with g or more handles. This would effectively mean that every graph can be drawn, without edge-crossings, on a surface with an infinite number of handles, i.e., 3D space. Hence no graph exists which cannot be drawn without edge-crossings in 3D space.
The more I look at this question, the less confident I become of being correct, but I'm extremely curious.
Thanks :)
One argument you can make along those lines, if you really wanted to, is that for any $n$, you can embed an $n$-holed torus in $\mathbb R^3$.
So if you want to draw a graph in $\mathbb R^3$, you can draw it on an $n$-holed torus (for sufficiently large $n$), embed the torus in $\mathbb R^3$, and get a drawing of the graph in $\mathbb R^3$ as a result.
What's missing is the converse: there are objects you can embed in $\mathbb R^3$ but not on any surface. But none of those objects are graphs.