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What are the prerequisites to learning both or one of the items?

Consider that one will have done some of the "core" classes like Differential Geometry, Real Analysis, Abstract Algebra and POint-Set Topology.

What is necessary for Differential Topology and/or Geometric Topology?

Thank you

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The only thing that's necessary is a good course in point-set topology, for example one out of Munkres's book. –  Potato Oct 5 '12 at 5:33
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As is indicated by the subject names, having some background in general topology is usually a good idea. However, as it turns out, the topologies typically introduced in differential topology are very "nice" comparing to the study of general topological spaces, so a full course in general topology is not necessary.

My personal view is that one should at least have a solid background in Euclidean analysis, that is, some background in differentiation and integration between functions $\mathbb{R}^n \rightarrow \mathbb{R}^n$. A large part of differential topology is the study of smooth maps between manifolds (or, if you're a masochist, $C^k$ maps between manifolds), which are defined by behaving locally like in the Euclidean case. Therefore I think it is natural both from a theoretical and also from an intuition standpoint to have a good understanding of the Euclidean case first.

Some very light group theory is also worth knowing, as manifolds can be compared topologically by considering various algebraic invariants like the fundamental group.

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What about Euclidean Geometry? –  sidht Oct 6 '12 at 1:46
@jak, I don't think that Euclidean geometry is necessary. –  Christopher A. Wong Oct 6 '12 at 22:12
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