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I am looking for software, websites and/or mobile apps about visualizating topological concepts using 3d graphics such as homeomorphisms between spaces, different kinds of decompositions (like handle decompositions), fundamental polygons, operations between spaces (like the connected sum), deformation retracts and so on.

Do such software exist? If yes, can you suggest some?

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The (now defunct) Geometry Center developed software for visualising 3-manifolds. You can find some of it on Jeff Weeks' website here and also SnapPy here.

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