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., S2 × S4 and \(\mathbb C\)P3, 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 ∪: Hp(M; X) × Hq(M; X) → Hp+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') = (δc) c' + (−1) cc'); Examples: For forms, [ω] ∪ [η]:= [ωη].
* Ring structure: The space H*(X; Λ):= ⊕p>0 Hp(X; Λ) is a ring, with the cup product.
* Kronecker index: Given a cohomology class ν ∈ Hn(M; \(\mathbb Z\)/2) for a manifold M, its Kronecker index is \(\nu[M]:= \langle\nu\), \(\nu\rangle \in {\mathbb Z}/2\), where μM is the fundamental homology class of M.
* Poincaré duality: If M is a compact, oriented n-manifold, then Hi(M) is isomorphic to Hni(M) under aaμM, where μM 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, H1(M; \(\mathbb R\)) = 0; Otherwise, the dimension of H1 is the number of holes in M.
* Compact, connected, orientable, n-dimensional M: Hn(M; \(\mathbb R\)) = \(\mathbb R\).
* Compact, connected, non-orientable, n-dimensional M: Hn(M; \(\mathbb R\)) = 0.
* Non-compact, connected, n-dimensional M: Hn(M; \(\mathbb R\)) = 0.
* Spheres: H0(Sn; \(\mathbb R\)) = \(\mathbb R\); Hp(Sn; \(\mathbb R\)) = 0 for 1 ≤ p < n or p > n, Hn(Sn; \(\mathbb R\)) = \(\mathbb R\); H0(\(\mathbb R\)n; \(\mathbb R\)) = \(\mathbb R\).
* Projective spaces: Hp(\(\mathbb R\)Pn; \(\mathbb Z\)/2) = \(\mathbb Z\)/2 for 0 < p < n and, if the generator of H is a, that of Hp is ap.

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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