Cohomology Theory |

**In General**

* __Idea__: A framework,
invented by H Whitney, for constructing algebraic quantities that
provide a partial classification of topological spaces, like homology
and homotopy, in which the structures are dual to homology classes; The
way the duality is defined may differ, giving rise to different cohomology theories.

* __Advantages__: It is a more
powerful and easier to use tool than homology theory, and its nice extra
algebraic structure permits in some cases to tell that two topological
spaces are not homeomorphic even if they have the same cohomology groups,
from the different ring structures [e.g., S^{2} ×
S^{4} and \(\mathbb C\)P^{3},
below], and it uses a local operator (d), instead of a global one (∂).

$ __Def__: A cohomology theory
(H*, d) consists of (a) A contravariant functor H from differentiable
manifolds and smooth maps to ... ; (b) A transformation d ...

> __Online resources__:
see Wikipedia page.

**Related Concepts** > s.a. Schubert Calculus.

* __Cup product__: A map ∪:
H^{p}(*M*; *X*)
× H^{q}(*M*; *X*)
→ H^{p+q}(*M*;
*X*), or ∪: H*(*M*; *X*) × H*(*M*;
*X*) → H*(*M*; *X*), defined by [*c*] ∪
[*c*']:= [*cc*'], where \(\langle\)*cc*', *σ*\(\rangle\):=
(−1) \(\langle\)*c*, front *m*-face of *σ*\(\rangle\)
\(\langle\)*c*', back *n*-face of *σ*\(\rangle\);
It satisfies δ(*cc*') = (δ*c*) *c*' + (−1)
*c* (δ*c*'); __Examples__: For forms, [*ω*]
∪ [*η*]:= [*ω* ∧ *η*].

* __Ring structure__: The space
H*(*X*; Λ):= ⊕_{p>0}
H^{p}(*X*; Λ) is a ring,
with the cup product.

* __Kronecker index__: Given a
cohomology class ν ∈ H^{n}(*M*;
\(\mathbb Z\)/2) for a manifold *M*, its Kronecker index is
\(\nu[M]:= \langle\nu\), \(\nu\rangle \in {\mathbb Z}/2\),
where *μ*_{M}
is the fundamental homology class of *M*.

* __Poincaré duality__:
If *M* is a compact, oriented *n*-manifold, then
H^{i}(*M*) is isomorphic
to H_{n−i}(*M*)
under *a* → *a* ∩ *μ*_{M},
where *μ*_{M} is the fundamental
homology class of *M*.

> __Other related concepts__:
see Cap Product; yang-mills
theories [operator complexes].

**Examples**
> s.a. BRST transformations; lie algebras;
quantum groups; tilings;
types of cohomology [including generalizations].

* __Connected, simply connected
M__: In this case, H

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**References**
> s.a. algebraic topology; crystals.

@ __General__: De Rham 60;
in Nash & Sen 83.

@ __For groups__: Weiss 69;
Brown 82;
Totaro 14;
> s.a. group theory.

@ __Non-Abelian__: Eilenberg & MacLane AM(47);
Andersson 86.

@ __Quantum__: Ruan & Tian JDG(95).

**In Physics**
> s.a. BRST cohomology; topology in physics.

@ __Field theory on curved spacetime__: Khavkine a1404 [De Rham cohomology with causally restricted supports].

@ __Other applications__: Azcárraga & Izquierdo 95;
Gross JMP(96);
Forrest et al mp/00-proc [quasicrystals];
Alexandradinata et al PRX(16) [topological insulators].

> __Related topics__:
see classical particles; crystals;
formulations of general relativity; lagrangian dynamics;
renormalization theory.

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send feedback and suggestions to bombelli at olemiss.edu – modified 18 apr 2019