# The Groove of Math

## About my Master’s Thesis: Notes for a brief seminar [Update #1]

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)]may have completed the stable diffeomorphism classification for smooth oriented closed four manifolds with fundamental group the dihedral group. The two missing $w$-types were $w=x^2$ and $w=y^2$ in the case where the $n$ in the Dihedral group $D_{2n}$ was a multiple of $4$. I have to triple-check what I’ve done so far, but the idea was to exploit the cofibre sequence given by James:

$Tq^*E \to TE \to \Sigma^{\dim F}T(E\oplus F)$

For vector bundles $E$ and $F$ over a space $X$, with $q\colon S(F)\to X$ is the projection of the sphere bundle of $F$. 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 $1$-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 honest with you, (especially when dealing with math!) I hope to be able to provide the complete classification asap!

I decided to share the notes I prepared for a Seminar I gave about Group Cohomology. To be more precise, I spoke about Finiteness Conditions and I studied a particular subgroup  of $SL_n(Z)$.