The Groove of Math

Random Exercise #5: The homotopy fibre of CP^1 ↪ CP^∞ and a Serre Spectral Sequence

[It is really annoying that the title cannot contain any tex code:( ]

Hi all, sorry for not having written much in the blog lately, i was kind of busy. First of all, I’m happy to announce that next I’ll be a graduate student at UT in Austin. I’m so happy and I’m looking forward to start working there. Ok, back to business, on today’s random exercise I will deal with a problem I encountered today. Since I found it funny and interesting I will write it down here. We are asked to study the homotopy fibre of the map

$\mathbb{C}P^1 \hookrightarrow \mathbb{C}P^{\infty}$.

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

Random Exercise #3: K(G,1) is a model for the classifying space BG via covering theory (NOT l.e.s. of a fibration)

I decided to write down this nice exercise since it is the kind of exercise which is readily solved with the right tools (i.e. l.e.s. of a fibration), but can become non-trivial if one doesn’t know them.

In particular, I want to prove that the Eilenberg MacLance space $K(G,1)$, for a discrete group $G$ is a model for the classifying space $BG$.

Filling the Details #5: Two words about the Atiyah-Hirzebruch Spectral Sequence

It’s finally time we shed some light on the technicalities behind this powerful tool. It took me some time digesting all these materials and I have to thank all the people who helped me understanding these passages. We will identify the differentials of the AHSS for spin-bordism (since it’s the one I need for my thesis), but the method we are going to use can be generalised to any AHSS, provided we have enough informations about the cohomology group of the spectra involved.

Notes of a Seminar in Group Cohomology

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 $SL_n(Z)$.

There might be some errors here and there, please read with cautions and let me know if you find something unclear!

pedrotti-riccardo-cohomology-of-groups

Filling the Details #4: Maps of Eilenberg MacLane Spectra induces stable cohomology operations

First of all, sorry for not having written lately. I’ve just done the GRE and GRE math examinations, together with the TOEFL. I really hope the result will be good, since I’m going to apply in the USA!

Ok, back to Math 🙂  As the title suggests, I want to write something about this known result, since I needed it for understanding why the first non-vanishing differential in the Atiyah-Hirzebruch Spectral Sequence is a stable operation. What I’m going to show is not the nicest way to prove it, but still I think it’s worth to take a look at a more “concrete” proof