Homology Theory |

**In General** > s.a. Betti
Numbers; Cap Product;
cohomology; Hurewicz
Theorem [relation to homotopy].

* __History__: Homology groups were introduced by Betti.

* __Idea__: A method
for spotting holes in a topological space that is different from homotopy;
We take some class of subsets of the topological space, without boundary, check
which ones are not boundaries of higher-dimensional subsets, and classify them;
The groups are often easier to compute than fundamental groups in checking homeomorphisms
of topological spaces, but contain less information about the underlying topological
space (for example, the homology version of the Poincaré conjecture would be false).

$ __Def__: In an abstract
sense, a homology theory (*H*, ∂) consists of
[@ Spanier 66] (a) A covariant functor
*H* from topological pairs and maps to graded Abelian groups and homomorphisms of
degree 0; (b) A natural transformation ∂ (boundary operator) of degree −1 from
the functor *H* on (*X*, *A*) to the functor *H* on
(*A*, 0), satisfying the following axioms:

*- Homotopy*: If *f*, *g*:
(*X*, *A*) → (*Y*, *B*) are homotopic, then
*H*(*f*) = *H*(*g*), the relevant arrows in the domain
category are homotopy classes of (continuous) maps;

*- Exactness*: For any
pair (*X*, *A*) with inclusion maps
*i*: (*A*, Ø) ⊂ (*X*, Ø)
and *j*: (*X*, Ø) ⊂ (*X*, *A*)
there is an exact sequence

... H_{q}(*A*)
→_{H(i)} H_{q}(*X*)
→_{H(j)} H_{q}(*X*,*A*)
→_{∂} H_{q−1}(*A*)
→ ... ;

- *Excision axiom*:
For any pair (*X*, *A*), if *U* is an open set
in *X* such that *U*-bar ⊂ int *A*, then the
excision map *j*: (*X*−*U*, *A*−*U*)
→ (*X*, *A*) induces an isomorphism H(*j*): H(*X*−*U*,
*A*−*U*) →_{≅} H(*X*, *A*) on the homology groups;

- *Dimension axiom*: On the full subcategory of
1-point spaces, there is a natural equivalence of *H* with the constant functor, i.e.,
if *P* is a 1-point space, then *H*_{q}(*P*)
= 0 for *q* ≠ 0, \(\mathbb Z\) for *q* = 0.

The first three
give the structure of a homology theory, the fourth one characterizes
it in terms of geometrical objects.

* __Construction of homology groups__:
Take a topological space, construct a chain sequence, define from these the cycles
*Z*_{q}(*X*) and the
boundaries *B*_{q}(*X*),
and finally *H*_{q}(*X*):=
*Z*_{q}(*X*)/*B*_{q}(*X*).

* __Generalizations__: Actually,
there are appropriate homology groups for objects other than topological spaces,
e.g., groups or associative algebras, and *homological algebra* is studied in its own right.

> __Online resources__:
see Wikipedia page.

**Related Concepts** > s.a. Boundary;
exact sequence; Fundamental
Homology Class; Homological Algebra;
types of homology.

* __Künneth formula / theorem__:
A formula giving the *k*-th homology (cohomology) group of a product space as

H_{k}(*X* × *Y*;
\(\mathbb Q\)) = ⊕_{p+q=k}
H_{p}(*X*; \(\mathbb Q\))
⊗ H_{q}(*Y*; \(\mathbb Q\)) ,

H_{k}(*X* × *Y*; \(\mathbb Z\)/2)
= H_{k}(*X*; \(\mathbb Z\)/2)
⊗ H_{k}(*Y*; \(\mathbb Z\)/2) .

It holds for *X* and *Y* CW-complexes such that H^{k}(*X*)
is torsion-free, and *Y* has only finitely many cells in each dimension;
For integer coefficients (torsion subgroup present), @ see Massey 80.

@ __Künneth formula__: in Spanier 66, p247.

**References** > s.a. algebraic topology.

@ __General__: Cartan & Eilenberg 56;
Hilton & Wylie 62;
MacLane 63;
Hu 66;
Massey 78;
Massey 80;
Vick 94.

@ __Introductions__:
Nadathur ln(07);
Lerner-Brecher & Yamakawa ln(19).

@ __And mathematical physics__:
Krasil'schik & Verbovetsky m.DG/98-ln;
Benini et LMP(15)-a1503 [chain complexes of field configurations and observables for Abelian gauge theory];
Benini et al LMP(19)-a1805
[general framework for chain-complex-valued algebraic quantum field theories];
> s.a. electromagnetism in non-trivial backgrounds [on a chain complex].

@ __Variations, generalizations__: Zomorodian& Carlsson CG(08) [localized].

@ __Related topics__: Chen & Freedman CG(09) [computational].

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