# 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 #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