Graph homeomorphic to Sphere

In summary: Clément Deymier is asking for a mathematical strategy to prove that a graph represents a manifold. They mention using path based properties or homotopy groups, but they are not familiar with these methods and are seeking assistance. They also mention having difficulty with higher dimensions. The respondent suggests using the Simplex, a generalization of the concept of a triangle, to check if the graph represents a manifold. However, the stitching method used must be done in a specific way to ensure an orientable manifold. The person asking the question clarifies that they want to check if the graph represents a manifold generated by a complex software.
  • #1
kroni
80
10
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
 
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  • #2
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.
 
  • #3
I totally agree with that but i get a graph from a extremely complex software and i want to check if this graph represent a manifold.
 
  • #4
kroni said:
i want to check if this graph represent a manifold
What does that mean?
 

1. What does it mean for a graph to be homeomorphic to a sphere?

Two objects are considered homeomorphic if they can be transformed into one another without tearing or gluing, similar to how a rubber band can be stretched and reshaped without breaking. Therefore, a graph that is homeomorphic to a sphere means that the graph can be continuously deformed into a sphere without any cuts or tears.

2. How can you determine if a graph is homeomorphic to a sphere?

To determine if a graph is homeomorphic to a sphere, you can use the Euler characteristic formula. The formula states that for any connected graph, the number of vertices minus the number of edges plus the number of faces is always equal to 2. If a graph satisfies this formula, it is homeomorphic to a sphere.

3. Can a graph be homeomorphic to a sphere if it has holes or handles?

Yes, a graph can still be homeomorphic to a sphere even if it has holes or handles. As long as the graph can be transformed into a sphere without any cuts or tears, it is considered homeomorphic. This is similar to how a doughnut can be reshaped into a sphere without breaking it.

4. Are there any real-world applications of studying graphs that are homeomorphic to a sphere?

Yes, there are many real-world applications of studying graphs that are homeomorphic to a sphere. For example, in computer graphics and animation, understanding the topological properties of graphs can help create more realistic and efficient 3D models. In biology, studying the topological properties of protein structures can provide insights into their functions and interactions.

5. What are some other common examples of objects that are homeomorphic to a sphere?

Besides graphs, other common examples of objects that are homeomorphic to a sphere include a balloon, a soap bubble, a globe, and a rubber ball. These objects can all be continuously reshaped into a sphere without any cuts or tears.

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