# Topology of manifolds and transition functions

let me start by describing some examples which may well demonstrate the motivation:

A manifold is obtained by glueing together Euclidean spaces, and there is a transition function on the overlap of two charts.

It is conceivable that the properties of manifolds may be determined by the the transitions functions.

1.No restriction on transition function. The manifolds are TOP.

2.Using affine homeomorphisms as transition function. The manifolds are called affine manifolds. The Chern's conjecture (still open) says that Euler Characteristic of Affine manifolds should be vanishing.

3.Foliation.

4.Using diffeomorphism as transition function.The manifolds are Diff. Donaldson's theorem says the positive definite intersection form of a simply connected SMOOTH four-manifolds is Identity, which is in sharp contrast to Freedman's result for TOP.

My questions are:

1. Is there a place where i could find discussion of above results(or at least of the same style) in a local to global way? (restriction on transition functions is local, but the topological properties are global) There is probably a classifying space for each restriction and hence a characteristic class theory. How to see the local-to-global process clearly?

2.For the 4th example, Are there some other TOP properties enjoyed by SMOOTH manifolds? There is Rokhlin's theorem on signature, are there more?

• I think you are looking at a zoo of different geometric notions, which don't have enough in common to have a single discussion go in any depth into all of them. The global theory of affine structures has almost nothing in common with the global theory of foliations, as far as I know, and these are nothing like the theory of topological 4-manifolds or smooth 4-manifolds. But I would like to be wrong. Feb 4, 2017 at 10:39
• Chapter 3 of press.princeton.edu/titles/6086.html discusses various structures on manifolds in the spirit of your question. This is "Three-Dimensional Geometry and Topology" by William P. Thurston, Edited by Silvio Levy. (The text is different from perhaps more famous Thurston's notes which are freely available from MSRI). Feb 4, 2017 at 13:32