Now I’ll begin to post material from the book proper. Here is the first section of Chapter I. I’m going to try to post stuff in bite-size pieces like this, rather than a whole chapter at a time. Of course, there will probably be an ongoing process of revision as well.
In this installment, we introduce the main objects that surgery theory is about – manifolds, especially (the reason for this will become apparent later) manifolds of high dimension, which typically means dimension at least five. Even as regards the definitions, though, there are important distinctions to be made. The most natural notion of a manifold is simply “a (nice) topological space locally homeomorphic to Euclidean space” – that is, a topological manifold. Historically, however, manifolds arose in connection with problems in differential geometry and analysis and thus possessed not just a topological but a smooth structure (i.e., it makes sense to talk about infinitely differentiable functions on such a manifold, not just about continuous ones.) One might naturally suppose that a topological manifold can always be “smoothed out”, just as a continuous function can always be approximated by a differentiable one. However, this is not the case. In fact, our story really gets its start in the middle fifties, when Milnor discovers that \( S^7 \) admits several distinct differentiable structures.
I am very interested to hear how accessible readers find this chapter. Could you follow the presentation/do the exercises? I should mention that I learned some of the ideas in Chapter 1 from some lecture notes of Tom Farrell, “Introduction to High-Dimensional Manifold Topology”, 2001.