# The Groove of Math

## 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: