Should I read about Manifolds or Algebraic Topology?

Question

I really enjoy doing maths and it fills quite a lot of my spare time. I'm starting my first year in the university on october and I probably won't have that much time for independent reading once there.

I have been reading maths (on and off) for years now and have the necessary background in topology, analysis (although i would say my grip on analysis ins't the best - i.e. I wouldn't be able to really enjoy a serious book on functional analysis) and algebra to read in one of two categories that interest me:

(1) Differential Topology\Manifolds - (co-)tangent space, fibre bundles, vector fields... etc.

(2) Algebraic Topology - (co-)homology, homotopy, covering spaces... etc.

My question is what of the two would I be more likely to encounter in my future studies?

In other words: what will be more valuable for me to study independently while i still have the time?

I already have books I like but am unsure about what topic to pick.

The books are: Manifolds and Differential Geometry, Elements of Algebraic Topology.

Answer 1

Read about both simultaneously in the same book: Differential forms in Algebraic Topology !
(Bott was one of the best twentieth century geometers, and it shows in this extraordinary book)

Answer 2

I can suggest two nice books covering basics of the above subjects. Hopefully they give you some perspective.

1. Topology and Geometry, by Breadon is book primarily on algebraic topology but it treats the subject using differential manifolds, so you probably enjoy reading its 2-3 chapters. Here is the link.

2. Foundations of Differentiable Manifolds and Lie Groups, by Warner. It is a book on differential manifolds but it also discusses several cohomology theories on manifolds, and so has a algebraic topology flavor too. Here is the link.

Answer 3

The two are fairly closely linked, and it wouldn't be a bad idea to study them concurrently. That said, I think if you have to pick one, go with algebraic topology first.

The first reason is historical. Before the modern theory of differential topology was developed, algebraic (and combinatorial) methods were employed. It is usually much easier to calculate the cohomology of a manifold using the smooth theory than it is using the algebraic theory.

Secondly, there are methods in the modern theory of manifolds that will not make much sense if you have studied the basics of algebraic topology. de Rham cohomology might not make much sense if you don't know what cohomology is.

That isn't to say one couldn't study differential topology before algebraic topology, though. It shouldn't be too hard to pick up de Rham cohomology without having worked your way through an algebraic topology textbook, and there are many parts of differential geometry that don't require any real topological machinery, like curvature and volume and differential forms (diffy forms are very useful in topology, but you don't need to understand these particular applications to make use of them in the geometry).

So, all in all, I'd say you are better off working through them both together. If you already have the books, you might as well!

Answer 4

I wholeheartedly agree with Georges Elencwajg: It is very appropriate for your first introduction to cohomology to be de Rham cohomology. You will learn a lot more multivariable calculus, many tools from differential geometry, and you will meet a lot of the basics of algebraic topology in an easily motivated form.

From what I gather about your background, however, I think that Bott and Tu may be a little advanced. You might instead check out Fulton's book "Algebraic Topology: an introduction".