0. Okay, last post was fun, now let's get into SPECTRAL SEQUENCES, a device to compute #homology groups!
https://en.wikipedia.org/wiki/Spectral_sequence
I hope to unfold a little #tootorial' over the easter holidays, in realtime, pasting as I understand things better.
The uplink is a bit shaky here, and the usual disclaimers apply: about me being easily distracted or getting confused. After all, it's what I do until I finally get it!
#geometry -> #topology -> #homology (#spectralSequence)
5. EXAMPLE: You can cut up a torus in essentially two ways: Along a circle like the red longitudinal one, or along the blue latitudinal one. Both are examples for their class of boundaries.
Every boundary has an operator that sends the surface around it to the boundary. Like a projection sending a fiber to its base point!
So our torus has two 1-holes (circular paths that cannot be deformed into each other, nor shrunk to a point): it has ZxZ as 1d #homology group, and 2 as first Betti number.
Refurio Anachro