In the previous post I mentioned the two categories of manifolds with which we’re going to be mostly concerned: smooth manifolds, equipped with an atlas whose transition maps are \(C^\infty\), and topological manifolds (locally Euclidean spaces). Smooth manifolds are, of course, the natural outgrowth of 19th and 20th century mathematicians’ concern with differential equations and their solutions. Topological manifolds still give rise to strong opinions: some think of nonsmooth constructions as monstrosities (like Hermite in the 19th century: “I turn away in horror from this lamentable plague of functions without derivatives”) while others regard the topological definition as the “natural” one and smooth structure as an artificial convention to be jettisoned as soon as possible (as in Siebenmann’s celebration of the “naked homeomorphism”, 1970 – “One can expect that mathematicians will consequently come to use freely the notions of homeomorphism and topological manifold untroubled by the frustrating difficulties that worried their early history”).
To further enrich this sectarian dispute, there are other categories of manifolds (other religions, one might almost say), among which the category of piecewise linear or PL manifolds is the most well-known. A PL manifold can be defined (analogously to the smooth case) as a topological manifold with an atlas whose transition functions are PL homeomorphisms, or equivalently as a polyhedron such that each point has a neighborhood PL homeomorphic to Euclidean space. True believers from both the smooth and topological worlds will mock the clunky, angular nature of these combinatorial objects. On the other hand, if you want actually to represent a manifold (or anything else) on a computer you will eventually end up with something combinatorial, so why not embrace it? Several views on these basic questions of PL theology can be found in this MathOverflow post.
In keeping with its amateurish nature, the book will concentrate on the smooth and topological cases (an amateur is someone coming from outside, and as far as I can see, by the time you have started seriously studying PL topology you are no longer an outsider). The PL case is “in between” smooth and topological: this is because of a theorem of Whitehead, which says that every smooth manifold admits an (essentially unique) PL manifold structure. A recent exposition of this and some other basic results of PL topology is found in the lecture notes of Lurie.
One final remark: there are no (high-dimensional) exotic spheres in the PL or TOP categories. As we will see, the smooth exotic spheres \(\Sigma^7\) constructed by Milnor (say in dimension 7) arise by gluing two 7-disks along a diffeomorphism of their boundary 6-spheres. If this diffeomorphism extended over one of the 7-disks, the resulting \(\Sigma^7\) would clearly be diffeomorphic to the standard \(S^7\). But in the topological or PL categories, it is trivial to make such an extension: we just “form the cone”, in other words, work in polar coordinates, map the radial coordinate to itself, and let the action on the boundary sphere determine the angular coordinate. It is a surprising fact, then, that the techniques of surgery theory, developed to study exotic spheres in the smooth category, provide nontrivial information also in the PL and topological categories, where no such examples exist.