Here is the updated version.
Many things left to be done and to explain properly, like:
- the word tangent bundle and more flexible notions of smoothness, described a bit hermetically and by means of examples here, section 6,
- the relation between curvature and how to perform the Reidemeister 3 move, the story starts here in section 6 (coincidence)
- why the Baker-Campbell-Hausdorff formula can be deduced from self-similarity arguments (showing in particular that there is another interpretation of the Lie bracket than the usual one which says that the bracket is related to the commutator). This will be posted on arxiv soon. UPDATE: see this post by Tao on the (commutative, say) BCH formula. It is commutative because of his “radial homogeneity” axiom in Theorem 1, which is equivalent with the “barycentric condition” (Af3) in Theorem 2.2 from “Infinitesimal affine geometry…” article.
- a gallery of emergent algebras, in particular you shall see what “spirals” are (a kind of generalization of rings, amazingly connecting by way of an example with another field of interests of mine, convex analysis).