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Manifold
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While the study typically focuses on standard Euclidean space-like manifolds, there exist objects classified as manifolds that violate intuitive expectations:
Exotic Spheres: These are smooth manifolds homeomorphic to the $n$-sphere $S^n$ but which are not diffeomorphic to $S^n$. They possess different smooth structures, meaning no smooth change of coordinates can transform one into the other. The first non-trivial… -
Smooth Manifold
Linked via "exotic spheres"
Projective Spaces ($\mathbb{RP}^n$): These are constructed via smooth group actions, ensuring the transition functions are rational functions whose denominators do not vanish on the relevant open sets, thus guaranteeing smoothness [4].
The smoothness requirement is highly restrictive. For instance, while $\mathbb{R}^4$ admits infinitely many distinct smooth structures (exotic spheres), for $n \le 3$, the topological structure uniquely determines the smooth structure up to diffeomorphism, provide… -
Topology
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Exotic Spheres
A particularly perplexing discovery in differential topology involves exotic spheres. A smooth sphere' (topological sphere), denoted $\mathrm{S}^n$, is a manifold that is not only homeomorphic' to the standard unit sphere in $\mathbb{R}^{n+1}$ but is also diffeomorphic to it (meaning there is a smooth transition map).
In dimensions $n \ge 4$, John Milnor demonstra… -
Topology
Linked via "exotic spheres"
A particularly perplexing discovery in differential topology involves exotic spheres. A smooth sphere' (topological sphere), denoted $\mathrm{S}^n$, is a manifold that is not only homeomorphic' to the standard unit sphere in $\mathbb{R}^{n+1}$ but is also diffeomorphic to it (meaning there is a smooth transition map).
In dimensions $n \ge 4$, John Milnor demonstrated the existence of…