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October 25th, 2016, 08:38 PM   #1
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A "topology" preserving edges and vertices

Is there a topology, for a dimension D figure, that preserves its d<D-1 dimensional singularities even under deformation? Concerning 3-dimensional figures, those singularities would possibly be 1-dimensional edges possibly connected to zero-dimensional vertices.
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November 17th, 2016, 05:51 PM   #2
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What do you mean by singularities here? Does this mean the edges which generate trivial homology groups?
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November 18th, 2016, 03:55 AM   #3
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The best way I can describe a shape representative to my idea is a sphere transforming to a cube but barely differing from the former.

Would singularities like edges and vertices be geometrically definable, even if the shape were just differentially different from a sphere? (Singularities here are places nondifferentiable in a at least one direction.)

I now understand that for the shape being considered, under topology, genus zero would be conserved. If the consideration above is not one of topology, does it appear in math?
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