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 *H*_{q}(*K*)
= *G*_{q} ⊕ *T*_{q},
where *G*_{q} is free and its rank gives the
number of (*q*+1)-dimensional holes in *K*, the *q*-th Betti number, while the
torsion subgrop *T*_{q} tells how *K* is twisted.

* __For manifolds__: The
0-th one is *H*_{0}(*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 *H*_{q}(*M*) = 0,
for all *q* > *n* (this is not true for the π_{q}).

* __For R__

*

**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,

*C*_{p}(*K*;
*L*):= *C*_{p}(*K*)/*C*_{p}(*L*)
, *p *> 0 , and

\(\bar\partial\)_{p}: *C*_{p}(*K*;* L*)
→ *C*_{p–1}(*K*;
*L*) by \(\bar\partial\)_{p}(*c*_{p} +
*C*_{p}(*L*)):= ∂_{p} *c*_{p} + *C*_{p–1}(*L*)
, *c*_{p} ∈ *C*_{p}(*K*)
.

* __Motivation__: The usefulness
lies in the fact that *H*_{p}(*K*)
→ *H*_{p}(*L*) → *H*_{p}(*K*; *L*)
is an exact sequence.

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