Types of Cohomology Theory  

De Rham Cohomology > s.a. Betti Numbers; cohomology [and physics]; de Rham Theorem.
$ Def: A cohomology theory based on p-forms ω, and therefore only available for differentiable manifolds; Cochains are p-forms {Ωp}, the duality with homology is through integration on chains, d is the exterior derivative; Thus cocycles Zp are closed forms, coboundaries Bp are exact forms, and the cohomology groups are Hp(X; \(\mathbb R\)):= Zp(X) / Bp(X).
* Consequence: For an n-dimensional X, only Hp for 0 ≤ pn can be non-trivial.
* And homology: Hp is the dual space of Hp, with ([ω],[C]):= ∫C ω.
* Ring structure: The cup product is wedge product of forms.
@ References: Wilson math/05 [algebraic structures on simplicial cochains]; Ivancevic & Ivancevic a0807-ln; Catenacci et al JGP(12)-a1003 [integral forms].

Čech Cohomology > s.a. Čech Complex.
* Idea: A cohomology theory based on the intersection properties of open covers of a topological space.
@ References: Álvarez CMP(85); Mallios & Raptis IJTP(02) [finitary]; Catenacci et al JGP(12)-a1003 [integral forms].
> Online resources: see Wikipedia page.

Equivariant Cohomology
* Applications: Kinematical understanding of topological gauge theories of cohomological type.
@ References: Stora ht/96, ht/96.

Étale Cohomology > s.a. math conjectures [Adams, Weil].
* Idea: A very useful unification of arithmetic and topology.
* History: Conceived by Grothendieck, and realized by Artin, Deligne, Grothendieck and Verdier in 1963.
@ References: Milne 79; Fu 15.
> Online resources: see Wikipedia page.

Floer Cohomology
@ Equivalence with quantum cohomology: Sadov CMP(95).

Sheaf Cohomology > s.a. locality in quantum theory.
@ References: Warner 71; Griffiths & Harris 78; Strooker 78; Wells 80.

Other Types > s.a. cohomology / K-Theory.
@ Lichnerowicz-Poisson cohomology: de León et al JPA(97).
@ Cyclic cohomology: Herscovich & Solotar JRAM-a0906 [and Yang-Mills algebras]; Khalkhali a1008-proc [A Connes' contributions].
@ Hochschild cohomology: Zharinov TMP(05) [of algebra of smooth functions on torus]; Kreimer AP(06)ht/05 [in quantum field theory]; > s.a. algebraic quantum field theory; deformation quantization.
@ Other types: Frégier LMP(04) [related to deformations of Lie algebra morphisms]; Papadopoulos JGP(06) [spin cohomology]; Blumenhagen et al JMP(10)-a1003 [line-bundle valued cohomology]; De Sole & Kac JJM(13)-a1106 [variational Poisson cohomolgy]; Becker et al RVMP(16)-a1406 [differential cohomology, and locally covariant quantum field theory].
> Related topics: see N-Complexes [generalized cohomology]; Figueroa-O'Farrill's lecture notes on BRST cohomology.

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