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A smooth 4-manifold is a 4-manifold with a smooth structure. In dimension four, in marked contrast with lower dimensions, topological and smooth manifolds are quite different.
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Theoretical physics: Why not just $\mathbb{R}^4$?
You and I are having a conversation:
"Okay," I say, "I think I get it. The gauge groups we know and love arise naturally as symmetries of state spaces of particles."
"Something like that."
"...And the …