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)
1. A spectral sequence is really a 3d grid, a #lattice of rank 3. It's points are labeled with integer coordinates p,q,r and to each point we associate a group E:
E_pq^r
By convention the coordinate p goes to the right, q up, and r picks a sheet. We'll be working with one sheet at a time.
To move around we'll need a 'boundary operator' or 'differential' d_r, which is just a function with a peculiar property:
d_r(d_r(E)) = 0
And 'bidegree' (-r,r-1).
This is chinese for you? Read on...
2. HOMOLOGY: If you haven't, read this wonderful introduction from our friends at wikipedia:
https://en.wikipedia.org/wiki/Homology_%28mathematics%29
Modern homology is a highly generalized subject. And we'll need some of that generality for spectral sequences to make sense. But to get started it's instructive to stick to classical #homology, on #manifolds.
For these, it's enough to give Betti numbers and torsion, if any. In general context you'd look at chain complexes and work with groups.
3. I'd like to look at de Serre's spectral sequence. It's about manifolds that have been decomposed into fibrations and
I hope it will nurture us with geometric intuition as we go.
A #fibration decomposes a space X into fibers F which intersect the base space B once:
F -> X -> B
Serre spectral sequence: the singular (co)homology of the total space X of a (Serre) fibration in terms of the (co)homology of the base space B and the fiber F.
4. A chain complex C(X) is a notation to encode information about a space X. It comes as a list of abelian-, which are very simple, groups
... E_2 -> E_1 -> E_0 -> 0
connected by maps (differentials, boundary operators), one for every arrow:
d_1: E_1 -> E_0
such that applying two consecutive ones to any element yields an identity element:
d_1(d_2(e)) = id_E_0
For example: ... (read on)
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