I'm currently up to connectedness and Compactness in Topology and I was wondering if I should start algebraic topology first or begin with differential topology. I'm looking at Hatcher's algebraic topology and using lee's topological manifolds as a supplement alongside it. Then there's Guillemins differential topology.
Which subject would be ideal to study first? Any advice would be appreciated tremendously.
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Which subject you study first, given your two choices of Algebraic Topology and Differential Topology, are probably more a matter of taste than anything else.
For Algebraic Topology, Hatcher is a good choice (though for some it may be a challenging first read). Just make sure you have gone through the necessary algebraic prerequisites.
For Differential Topology, Introduction to Topological Manifolds by Lee is again another good choice. You can supplement it with Differential Topology by Guillemin & Pollack, or with Lee's sequel, Introduction to Smooth Manifolds.
At a higher level, if you want a mix of both fields, you could take a look at Differential Forms in Algebraic Topology by Bott and Tu
I think that it may be best to finish studying in Algebraic Topology and then start Differential Topology.
My experience is this: To gain a deep understanding of differential topology and the power of its abstract nature you need to have a good and working experience with 1 and 2 manifolds, namely curves and surfaces (my favourite book on the area is probably Tapp's Differential Geometry of Curves and Surfaces) which can be easily visualized and provide a great deal of motivation.
On the other hand I think that Algebraic Topology doesn't really have such a prerequisite (other than some familiriaty with General Topology which I undestand you have). Also, at least in some sense, knowing Algebraic Topology will help you with Differential Topology but not the other way around (one very good book to do so is Massey's : An introduction to Algebraic Toplogy). Nevertheless, you may be willing to learn about the "inbetween" area of differential forms and De-Rham cohomology (and for that I would suggest for first reading Bachmann's "A geometric approach to Differential Forms" and then a somewhat sterile but very complete book of Madsen & Tornehave: From Calculus to Cohomology)
Nevertheless, if you want to try flexing your muscles in Differential Topology Guillemin & Pollack offer a very readble introduction. For my though , Spivak's Differential Geometry Vol 1 did the trick and made me reallize the intrinsic beauty of the subject (and it also has a chapter dedicated to Algebraic topology too!)