# Question about transversality of manifolds

I am a PhD student. I work on transversal submanifolds. I want to know if someone has an example of application of transversality in physics or engineering or in any other area. I hear that there is an application in the construction of plans but I didn't find it.

• The reason transversality was invented was to calculate cobordism groups through the Pontrjagin-Thom construction. In physics, these are used to understand topological phases of matter. – Arun Debray Jul 4 '17 at 3:03
• Thank you for your answer, have you any document or title of books speak about the physical applications. Thank you in advance. – Mohammed Mohammed Oct 28 '17 at 22:41
• This stuff is new enough that there aren't any books on it yet, but here are a few papers: arxiv.org/abs/1403.1467, arxiv.org/abs/1406.7329, arxiv.org/abs/1604.06527. They don't explicitly discuss transversality, but it's an important ingredient in the construction of Thom spectra that calculate cobordism groups: the Pontrjagin-Thom construction is the thing that makes it work, and can be read about in many places, e.g. here. – Arun Debray Oct 29 '17 at 15:40
• Thank you for your answer, I took a look to this documents, my idea is to get a concret example, I asked the same question to my professor and he said that there is an axample in physics, exactly in the construction of planes, they use the notion of transversality but till now I didn't find thsi example. – Mohammed Mohammed Oct 29 '17 at 20:19

The key point of transversality is that it is stable. A transverse intersection persists under perturbations. Moreover: any intersection can be perturbed to be transverse. Moreover: almost any perturbation will do this!

Measurements are never precise: They are always perturbations of the real state. When I say "real state" I mean this in the classical sense, I am not talking about quantum mechanics here.

That means that our measurements will always measure transverse states. You cannot measure any non-transverse situation! Transversality is the only thing we observe!