# 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 sincere as possible, especially when dealing with math! I hope to be able to provide the complete classification asap!

## Random Exercise #4: The nonexistent 5-Manifold in Bredon’s book

Dear all, sorry for not having posted anything so far but my master thesis (stable classification of certain families of four manifolds) and PhD interviews have been keeping me very busy 🙂

This is a quick exercise which I was asked to solve by a friend of mine and since I found it funny I decided to spell some more details about it in order to show the importance of the Bockstein exact sequence. This could be a typical exam exercise in “Topology 2” here in Bonn, so dear fellows students, read and comment if anything is unclear!

The following is exercise 7, page 366 from Bredon’s Topology and Geometry:

Show that there can be no $5$-manifold $M$ with