Types of Cohomology Theory| 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 H\(^p(X; {\mathbb R})\):=
Z\(^p(X)\)/B\(^p(X)\).
* Consequence: For an
n-dimensional X, only Hp
for 0 ≤ p ≤ n 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;
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
– abbreviations
– journals – comments
– other sites – acknowledgements
send feedback and suggestions to bombelli at olemiss.edu – modified 25 may 2019