Types of Homology Theories| Types of Homology Theories |
In General > s.a. homology theory.
* Different coefficients:
Homology groups using \(\mathbb Q\) or \(\mathbb R\) instead of \(\mathbb Z\)
contain less information about the topological space; They cannot have any
torsion subgroup.
* Simplicial homology:
Defined for polyhedra, in a simple way; For a general space X, one
triangulates X with some polyhedron K, and shows that
\(H_q(K)\) does not depend on K.
* Singular simplicial homology:
A generalization of the simplicial one; Instead of triangulating the whole space,
one just looks at continuous (but possibly "singular", non-invertible)
maps from simplices into it; More formally, one constructs a functor to the
category of chain complexes, and from there another one to the Abelian groups.
* Singular cubical homology:
Same as singular simplicial homology, but with maps of cubes rather than simplices.
* Čech homology: Defined using simplices,
becomes independent of simplices through some limiting procedure.
@ Singular simplicial homology: Eilenberg AM(44)
[original paper, topologically-invariant manner].
@ K-homology: Douglas 80.
@ Morse homology:
Banyaga & Hurtubise 04.
@ Floer homology: Zois ht/05 [non-commutative];
> s.a. 4-manifolds; non-commutative field theory.
@ Knot homology: Gukov & Saberi in(14)-a1211-ln [and quantum curves];
Gorsky a1304 [colored homology of knots];
Nawata & Oblomkov a1510-proc.
Examples > s.a. causal
sets [stable homology]; graph invariants;
networks [persistent homology].
* Decomposition: In general,
we can write Hq(K)
= Gq ⊕ Tq,
where Gq is free and its rank gives the
number of (q+1)-dimensional holes in K, the q-th Betti number, while the
torsion subgrop Tq tells how K is twisted.
* For manifolds: The
0-th one is H0(M)
= \(\mathbb Z\) ⊕ \(\mathbb Z\) ⊕ ... ⊕ \(\mathbb Z\),
with as many \(\mathbb Z\)s as connected components in M; If M is
a manifold of dimension n, then Hq(M) = 0,
for all q > n (this is not true for the πq).
* For Rn:
Hm(\(\mathbb R\)n,
\(\mathbb R\)n \{0}) = δmn\(\mathbb Z\),
for m > 1.
* For spheres:
H1(S1)
= \(\mathbb Z\) (> s.a. Hurewicz Theorem).
Relative Homology
* Idea: We take a subpolyhedron
L of K and consider like the identity anything in K
belonging to L; Define the following chain complex,
Cp(K;
L):= Cp(K)/Cp(L)
, p > 0 , and
\(\bar\partial\)p: Cp(K; L)
→ Cp−1(K;
L) by \(\bar\partial\)p(cp +
Cp(L)):= ∂p cp + Cp−1(L)
, cp ∈ Cp(K)
.
* Motivation: The usefulness lies in the fact that Hp(K) → Hp(L) → Hp(K; L) is an exact sequence.
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