#### Twigg

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The real projective plane ##\mathbb{RP}^2## comes to mind. It has no embeddings in ##\mathbb{R}^3## but it's most easily visualized using surfaces in ##\mathbb{R}^3## as visual aids, like the crosscap. You can even visualize it with a 2D figure if you count its fundamental rectangle. Those methods are much more useful than trying to visualize it as an embedding in ##\mathbb{R}^4##. The formal definition doesn't mention embeddings either, it's just a quotient space of the sphere with a smooth structure. The reason embeddings are useful and are used to characterize manifolds a lot is because they're a common, convenient starting point for constructing manifolds.Also, could you please give me an example of a manifold where not embedding it simplifies things? I am trying to remove this thought that I have that a manifold must be embedded in Euclidean space in order to be useful/interesting to us. I know that my point of view is wrong (based on more knowledgeable people telling me so), but I need to be more convinced.

The first volume of Spivak's Differential Geometry is another great resource. It dedicates a few solid chapters to differential topology, dedicating a chapter to each layer of structure in the manifold cake. Spivak was Milnor's student and those first few chapters are a good introduction to Milnor's work on exotic spheres, which helps puts a lot that's been said in this thread in perspective. Strongly recommend to JonnyG and anyone interested.The study of smooth manifolds without geometry is called Differential Topology. I do not know Lee's book but a short and wonderful introduction is Milnor's Topology from the Differentiable Viewpoint. The last chapter discusses the classification of smooth mappings of manifolds into spheres.