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.