What should I study to understand Milnor's exotic 7-spheres?

I am very curious about the Milnor's exotic 7-spheres and I want to understand the details of the construction.

I have some basic idea of topological manifold (learned the book GTM202, Introduction to topological manifolds). What should I learn next to achieve my goal?

Solution 1:

My suggestion is to learn a bit more algebraic topology, namely, homotopy groups. Then vector bundles, classifying spaces and characteristic classes (Pontryagin classes; signature). You will also need basic understanding of the Morse theory. If you like geometric topology, you should learn these anyhow.

Now, you are ready. Peter May suprvised many excellent REU projects of University of Chicago undergrad students, one of which is a detailed discussion of Milnor's construction of exotic spheres:

Rachel Mcenroe, Milnor's Construction of Exotic Spheres

which is more detailed than Milnor's original paper. She even provides some basic definitions of Morse theory and characteristic classes.