I apologize if this question is badly worded or obvious, but I have no formal topology background. I have put some effort into trying to find something, but nothing turned up, perhaps due to my lack of vocabulary knowledge so I turned here.
First, I think I wanted to ask something I was thinking about simple connectedness. Would it be equivalent to say that removing all of the points from any given closed path cuts the manifold, that is, renders it disconnected? For example, given a 2-sphere, if you remove a path from it you get two disconnected parts; but if you have a torus, you can chose a circle which passes through the hole (like cutting a doughnut in half with a knife), and the torus remains connected even after removing this path.
Second, I was wondering about a discrete analogue of this idea. I am interested in examining a hypergraph and being able to note if it resembles an n-sphere topologically, in some manner of speaking. This would be useful for being able to tell if a combinatorially defined object is in fact convex polytope.
From what I can tell, it seems as if in 3 dimensions, if we examine every possible closed path of edges, cutting along this path in the polytope must disconnect the polytope. This would be obvious when considering their realizations (hinging upon the response to the first question), but if we want to consider the combinatorial structure of the polytope (i.e. the face lattice), it becomes less trivial.
Below I attached an image of an object which is not a convex polytope. Imagine it as a picture of a combinatorial structure rather than a 3-dimensional object.
If we select a closed path like {D,H,G,C}, we can see that removing this path isolates the facet containing all of these points. This would be analogous to a choice of path which cuts the object into 2 parts in Euclidean space.
However, if we select another closed path like {D,H,P,L} or {A,B,C,D}, we can see that removing it does not isolate anything. This would be analogous to a choice of path which does not cut the object into 2 parts in Euclidean space.
I wonder if this holds for every combinatorial description of polyhedra, and if it can be generalized to any polytope, perhaps not only by considering edge paths (which are actually polygons themselves), but also sets of 2-faces which form polyhedra, and so on.
Hopefully I at least talked some sense. I would appreciate any direction on something to look at or read, as well as a response to my question (if you can figure out what I'm trying to ask!).
