Čech cohomology

These are unfinished notes regarding the definitions necessary to understand the Čech cohomology of a given presheaf. It itroduces presheaves and sheaves, directed limts, stalks and sheafifications, stalk spaces and exact sequences of presheaves. Note that in the literature, one can sometimes encounter Čech cohomology to be defined using only the alternating cochains, but the resulting homological groups are isomorphic. This is not often discussed in detail in the literature and we address this here.

Štefan – Sussmann integrability

The integrability of singular distributions has an interesting history. It was treated independently by an Argentinian mathemacian Héctor J. Sussmann and a Slovak mathematician Peter Štefan. Both approaches are completely different and both are brilliant. In fact, Štefan's paper is a lot more general and offers interesting insights into the geometry of leaves of singular foliations.

In these little notes, I go through both papers and discuss some mistakes in the original paper of Sussmann. In fact, there is a paper by Balan finding a counterexample. However, it finds a mistake where there is none, and misses the actual one. Interestingly, this error was not done by Sussmann himself, but it appears already in the cited paper by Lobry. Oddly enough, this oversight is spotted ten years by Štefan in another paper. We offer a modification closing the gap in the Lobry's paper.

Cartan – Dieudonné theorem

This just a nice proof to the famous theorem stating that every orthogonal automorphism of \(n\)-dimensional quadratic vector space can be written as a composition of at most \(n\) reflections. I did probably stole this proof from this amazing book. Note that it is quite general since it does not use the Euclidean signature.