# The Groove of Math

## 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.

## 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

## Filling the Details #3: Homotopy Fibrations and Long Exact sequence in Cohomology

Currently I’m reading the paper of Prof. Peter Teichner On the Signature of four-manifolds with universal covering spin, and I was stuck on the following passage:

The homotopy fibration $\tilde{M} \to M \to K(\pi, 1)$ induces an exact sequence in cohomology

$0 \to H^2(\pi ; Z/2) \to H^2(M;Z/2) \to H^2( \tilde{M} ; Z/2)$

I managed to find a solution to it now, and I thought it could be a good idea to write it here:

## Filling the Details #2: Representing 2-homology classes of a 4-manifold.

For the second article under the FtD tag, I’ve chosen a nice proposition which bothered me some times ago. It’s intuitively easy to believe but I found a scarce amount of details spelled out in the proofs one can find on the classical references. So here we have the following result:

Proposition: Every $2$-homology class of a smooth 4-manifold is generated by a surface

## Filling the Details #1: “Generalised Cohomology of the projective space via Atiyah-Hirzebruch Spectral Sequence”

So the aim of this article will be to fill out the details of the computation of $\mathbb{E}^*(\mathbb{C}P^n)$ one can find in several books like “Adams’ Stable Homotopy and Generalised Homology” Lemma 2.52.5 page 39 or “Kochman’s Bordism, Stable Homotopy, and Adams Spectral Sequence Proposition 4.3.2 page 132. I feel the amount of details one usually found in these references are quite scarce. It could be a problem, since the topic is rather technical instead. Here is my attempt in filling the details.