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

... Hq(A) →H(i) Hq(X) →H(j) Hq(X,A) → Hq−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: (XU, AU) → (X, A) induces an isomorphism H(j): H(XU, AU) → 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 Hq(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 Zq(X) and the boundaries Bq(X), and finally Hq(X):= Zq(X)/Bq(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

Hk(X × Y; \(\mathbb Q\)) = ⊕p+q=k Hp(X; \(\mathbb Q\)) ⊗ Hq(Y; \(\mathbb Q\)) ,

Hk(X × Y; \(\mathbb Z\)/2) = Hk(X; \(\mathbb Z\)/2) ⊗ Hk(Y; \(\mathbb Z\)/2) .

It holds for X and Y CW-complexes such that Hk(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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