Hi all! Since I was asked to give a talk about my Master’s Thesis, I decided to write some brief notes in which I explained the main definitions and the tools used and which results I obtained so far. Clearly I skipped lots of technical results and computations which I clearly find interesting but are not suited for this kind of seminar.

Let me know if something is unclear, I will do my best to explain it!

Riccardo Pedrotti – Stable classification of four manifolds and signature related questions

**[Update #1 (4 April 2017)]** I **may **have completed the stable diffeomorphism classification for smooth oriented closed four manifolds with fundamental group the dihedral group. The two missing -types were and in the case where the in the Dihedral group was a multiple of . I have to triple-check what I’ve done so far, but the idea was to exploit the cofibre sequence given by James:

For vector bundles and over a space , with is the projection of the sphere bundle of . With such cofibre sequence and some algebraic observations I was able to compute the stable page of the relevant AHSS/JSS and therefore providing an answer to the missing cases. I am so excited! I will try to address now the *hard *part, provide some unstable result, before June.

I will update the pdf above as soon as I have some time, now I’m quite busy with the deadlines of the PhD programs I was admitted/ I’m on the waitlist

**[Update #2 (13 May 2017)] **I uploaded an updated version of the notes, with some minor corrections and updates on my results. A correction which is worth noticing is that it seems (according to my computations at least) that the secondary invariant is not a relevant stable diffeomorphism in our cases, since it is always possible to change the normal -type of an almost spin manifold in such a way that the sec-invariant is always trivial.

**[Update #3] **Sadly, I found an error in my computations. I’m trying to fix it, but for sure at least a case in the stable diffeomorphism classification has to be checked again! I’m really sorry for this but I try to be as much precise and sincere as possible, especially when dealing with math! I hope to be able to provide the complete classification asap!