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
{Ω^{p}},
the duality with homology is through integration on chains, d is the exterior
derivative; Thus cocycles Z^{p} are
closed forms, coboundaries B^{p} are
exact forms, and the cohomology groups are H^{p}(*X*; \(\mathbb R\)):=
Z^{p}(*X*) / B^{p}(*X*).

* __Consequence__: For an *n*-dimensional *X*,
only H^{p} for
0 ≤ *p* ≤ *n* can be non-trivial.

* __And homology__: H^{p} is the dual space of H^{p},
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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24 dec 2016