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;
Wedhorn 16.

> __Online resources__:
see Wikipedia page.

**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.

main page
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– other sites – acknowledgements

send feedback and suggestions to bombelli at olemiss.edu – modified 25 may 2019