How to compute Characteristic Classes from scratch and get which structure it capture

Question

For our Differential Geometry II course we are strictly follow Differenetial Geometry, Loring W. Tu. Currently, I am reading chapter $5$, Vector bundles and Characteristic Classes. I feel the author focus only the theory part. Like I didn't find example where shown that how to compute those Classes. Besides, The book doesn't introduce Cohomology in details. I googled a lot and read that: Characteristic classes provide algebraic invariants of geometric information, certain structures on manifold (orientation,spin, etc), invariant under vector bundle isomorphism. For this I try to understand that with trivial bundle $E_1$ and Mobius bundle $E_2$ over the circle. As, $E_1$ and $E_2$ are not isomorphic (trivial,orientable and non-trivial, non-orientable resp.) then definitely that should be reflect on the characteristic classes. But I didn't know how to start. Like do I need to follow the path, $$\text{construct metric}\rightarrow\text{find connection}\rightarrow\text{curvature matrix }\Omega\rightarrow\text{apply formula}$$ And there are different chacteristic classes: euler, pontrjagin, chern and Stiefel–Whitney class. I guess each of them capture distinct structure but didn't get the complete picture yet. So, it will be great if anyone help me to see those capturing thing as well as right computation path for my example.

Answer 1

First of all, let's note that L. Tu provides his introduction to the topic of cohomology in his book "Introduction to Manifolds". It might be considered a prerequisite to the sequel, i.e. the book you are using.

Secondly, I would say that your assessment of what characteristic classes are is more or less accurate. Characteristic classes are a way of capturing the "twistedness" of a vector bundle. They come in several flavours, as you've observed:

1. Chern classes. These are invariants of complex vector bundles only.
2. Pontrjagin classes. These are invariants of real vector bundles. Of course, every complex vector bundle is also a real vector bundle, and the Pontrjagin and Chern classes of a complex vector bundle are related - the Pontrjagin classes are completely determined by the Chern classes (an exercise in the naturality of characteristic classes).
3. The Euler class. Unlike the previous two, the Euler class doesn't exist in multiple cohomology degrees, it only exists in $H^r(M,\mathbb{R})$, where $r=\text{rank}(E)$.
4. Stiefel-Whitney classes. Although these classes are relevant for differential geometric constructions like spin structures, these characteristic classes aren't obtained by differential geometric means; they are elements of $H^i(M,\mathbb{Z}/2\mathbb{Z})$. This is the domain of algebraic topology, so I will not address this further as it is outside the scope of Tu's book.

Now that you have those three classes (Chern, Pontrjagin, Euler), you would like to know which one to choose. Well, the answer is simple for the example that you'd like to consider. Indeed, Chern classes only exist for complex vector bundles. Pontrjagin classes only exist in degrees $4k$, and clearly, $H^{4k}(S^1)=0$. So one option remains: the Euler class. How would you construct it? I refer you to my answer here: How restricted is the curvature of a manifold by choice of topology keeping the underlying set fixed?

Whilst I didn't work out the details, I think the steps are clear:

1. Choose a metric.
2. Obtain a local orthonormal frame
3. Represent the curvature $2$-form w.r.t. this frame
4. Write down the Euler form in these coordinates
5. Repeat for an open cover of the entire manifold
6. Glue together

In the case of the Mobius bundle, you can do this explicitly, and I would recommend doing the same for $S^2$ and its tangent bundle afterwards.

Then how would you prove whether or not the resulting differential form is trivial in cohomology? Well, you integrate. If you get a non-zero integral, you have your answer.

Note: For Pontrjagin and Chern classes, you need not choose a metric. You need only choose a connection. The Euler class is unique in this regard, when computed through differential geometric means, because it requires you to choose an orthonormal frame.