# Graph homeomorphic to Sphere

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## Main Question or Discussion Point

Hello,

I want to prove that a graph represent a manifold, for this i take the opposites edges of a vertex (edge connected between vertex connected to the current vertex) and this subgraph need to be homeomorphic for example to the 1-sphere if i want a 2 manifold. This criterion ensure that my graph represent a manifold.

In 2 dimension its easy (opposite edge homeomorphic to S1) but i have difficulty with higher dimension. Do you know a strategy by using a mathematical approach to prove that ? i think using path based property or homotopy group ? but i am more a physician.

Thanks

Clément Deymier

andrewkirk
Homework Helper
Gold Member
Graphs are the wrong tool to use, as they do not have variable dimension.

A tool that might do what you want is the Simplex, which is the generalisation of the notion of a triangle to higher and lower dimensions. It is described here.

A solid triangle is a 2-simplex (two-dimensional simplex). A 1-simplex is a line segment. A 3-simplex is a solid tetrahedron.

Just as the ring $S^1$ is homeomorphic to what we get by joining two 1-simplices at both ends, the hollow sphere $S^2$ is homeomorphic to what we get by joining ('sewing' or 'pasting') two 2-simplices (solid triangles) along their edges.

For two dimensions and higher, we need to sew the two simplices together in the most natural way, ie by laying them on top of one another and sewing the adjacent edges together. If we sew them with a different orientation we can get weird non-orientable manifolds.

I totally agree with that but i get a graph from a extremly complex software and i want to check if this graph represent a manifold.

andrewkirk