\chapter{Homology}
Homology is a key algebraic tool: in various settings, it is a $\bZ$-graded
set of functors from some category to a category of modules.
These come in dual pairs: you get a homology theory and a cohomology theory.
Pragmatically, they let you apply linear algebra, and hence is quite powerful.
Structurally, it reveals hidden linear-algebraic structures.
EGs:
- topological spaces: singular (co)homology, and generalize (co)homology theories
- groups, especially Galois groups
- chain complexes
Simplest shadow: Euler number
Simplest EG: distinguishing S^2 and T^2:
compute for 'em both via simplicial homology
(yes, you can do that via \pi_1
Fine. Distinguish S^2 and S^3)
(Co)Homology comes from two places:
. topological spaces
. chain complexes
Used in class field theory (abelian Galois theory)
EG of Braurer group
(classifying division algebras)
\textrm{Br}(K) \cong H^2(\textrm{Gal} (K^s/K), {K^s}^*).
Topologically, a cohomology theory is a nice set of functors
(indeed, they are all represented by spectra, and that characterizes
them).
Algebraically, (co)homology is a functor on chain complexes.
[only need filtration, not grading, right?]
They come from:
. naturally occurring chain complexes (simplicial/singular chain
complex, de Rham complex)
. measure failure of functors to be exact (start with exact sequence,
apply functor, it's not exact anymore: take homology)
[-> leads to computation by resolutions]
also, can take homology of a module
(concretely, can compute via resolutions);
[group homology is homology of ZG-modules]
an interpretation is "measures failure of module to be free",
from the "free resolution" POV
[but generalized: use projective, not free]
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Stated this way,
homology is (just) a computation.
more deeply, it's deep algebraic structure
(just as you understand operators on vector spaces
by going to K[T]-modules,
you understand other stuff
by looking at fancier algebraic categories,
like chain complexes.
[EG: understand group by looking at chain complexes
of ZG-modules and homology thereof]
...though chain complexes feel less natural, more a tool.)
In algebra, homology can be very mysterious:
group cohomology is easily defined in this chain-complex way,
but its interpretation is subtle:
* H^2(G,A) as central extensions
* Galois cohomology
* Lie algebra cohomology
Galois cohomology measures failure of "Galois invariants"
(this is a natural group action)
...and is zero-dimensional bit of "etale cohomology"
That is, given a field extension, the base field is the elements invariant under
the Galois group -- but to what extent can you do this for structures derived from the field?
(ok, so points of a variety in the base field are the invariant points,
but what about other settings?)
(invariant -> left derived, cohomology; not quotient = homology)
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"twists" (groan): complications
ick notation: relative (X,A),
coefficients/sheaf: (X;F)
[there are also triples and triads: (X,A,B) and (X;A,B)
...but that's a different notation,
and mighty confusing at first.
Get straight:
- the *types* of objects and what they are
(X,A): space and subspace; (X;F) space and ring or module or sheaf
(X;A,B): space and 2 subspaces that add up to it etc.
- how it relates / connects to others (LES, universal coefficient, etc.)
]
- relative homology: H(X,A)
aka, excision; more elegant form of "patching"
gives LES (that's the key point! bdding map key geometry)
interpret as "trivial over subset" (subtle: EG of K-theory: trivializ*ed*)
- sheaf (co)homology: H(X;F)
EG of group cohomology of a module as sheaf cohomology
of that module, thought of as sheaf on K(\pi,1) space.
Topologically, singular cohomology as cohomology of constant sheaf.
[hmmm...sheaf cohomology represented by maps to K(\pi,*) with a non-trivial
sheaf over them?? We don't talk about sheaf cohomology much
in topology -- it's more an alg. geo. thing afaict]
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Where do spectral sequences come from?
. homology of composition of functors in terms of homology of
constituents
. homology of fiber bundle (Leray-Serre spectral sequence)
. many others