# Learning Differential Topology

So, I'm currently at the point of learning about derivatives and integrals in Elementary Real Analysis, I'm also learning Point set topology and I've already gone over the main definition of topology, basis, subspace topology, quotient topology, product topology up to continuous functions thus far. I also understand elementary linear algebra and know enough Abstract Algebra.

My question is whether or not I can start reading Differential Topology by Guillemin and Pollack. It states a full year of analysis and semester of linear algebra but I'm not exactly sure if my current level understanding is sufficient to start studying differential topology.

If not should I start going over Hatcher's book on Algebraic Topology? I've read that knowing algebraic topology before differential topology is a good idea.

• I'd say Guillemin and Pollack is a good gentle start; very concrete. If you feel more ambitious John Lee's smooth manifolds book is also okay. I don't think alg top is a requisit for dif top; really one learns one of them picks up on the other as one goes. – chriseur Mar 24 '17 at 1:52
• Just a quick side question, I saw John Lee's Riemannian Manifolds and wondered if I really need to go through his smooth manifolds to understand basic Riemannian Manifolds? – Alexander King Mar 24 '17 at 2:08
• Not the whole book for sure. Basic chapters (first four or so I think?) + vector bundles would probably suffice, depending on how thoroughly one wants to understand it. – chriseur Mar 24 '17 at 2:13
• Thanks. I'm just trying to see what kind of prerequisites are really needed to study manifolds and even algebraic topology. – Alexander King Mar 24 '17 at 2:15

$$\textbf{linear algebra + multivariable calculus} \to \textbf{differential geometry} \to \textbf{differential topology}$$