>>14080041Some terminology one may see while playing with these are stuff like surviving and killing/getting killed. Say we have generated by , and , respectively. Assuming , we see the following: so it gets killed, while kills . is different and is neither hit by anything nor does it hit anything itself, so is in the kernel but not in the image and survives to the next page. Essentially, one wants to kill as much as possible and only the survivors matter.
A typical example would be the Serre SS related to a fibration . If that was a cofibration, it would give a long exact (co)homology sequence, but fibrations aren't that compatible with (co)homology, so we need some other machinery. Serre gives us a way to compute the homology of the total space from those of the base and the fibre: in the 2nd page thing converging to an interesting thing form it looks like . This can also be used for reverse-engineering stuff. Take the path-loop fibration of a space . Now the path space is contractible, so for , and so it follows that the homologies of the space and its loop space must kill one another. No mercy, no survivors!! Given any higher sphere , this gives us non-trivial homology for (and only for) degrees which are multiples of .
PS. Why did I link Get Out instead of Spectral Ecstasy? Silly me but w/e...
