Cohomology
Theory| Cohomology
Theory |
In General
* History: Invented by H Whitney.
* Idea: A way to construct
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 CP3,
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
...
Related Concepts
* 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 
cc',
:=
(–1) c,
front m-face of c',
back n-face of
;
It satisfies 
(cc')
= (c) c'
+ (–1) c (c');
Examples: For forms, [
] [
]:=
[ 
].
* Ring structure: The space H*(X; 
):=
p>0 Hp(X; ),
is a ring, with the cup product.
* Kronecker index: Given
a cohomology class v 
Hn(M; Z/2)
for a manifold M, its Kronecker index is v[M]:= v, 
Z/2,
where M is
the fundamental homology class of M.
* Poincaré duality: If M is a compact, oriented n-manifold,
then Hi(M) is isomorphic
to Hn–i(M) under a → a 
M,
where M is
the fundamental
homology class of M.
> Other related concepts:
see Cap Product; yang-mills
theories [operator
complexes].
Examples > s.a. BRST; lie
algebra; quantum
group; tilings; types
of cohomology.
* Connected, simply
connected M: H1(M; R)
= 0; Otherwise, the dimension of H1 is the
number of holes in M.
* Compact,
connected, orientable, n-dimensional M: Hn(M;
R) = R.
* Compact,
connected, non-orientable, n-dimensional M: Hn(M;
R) = 0.
* Non-compact,
connected, n-dimensional M: Hn(M; R)
= 0.
* Spheres: H0(Sn;
R) = R; Hp(Sn; R)
= 0 for 1 
p < n or p > n,
Hn(Sn; R)
= R; H0(Rn;
R) = R.
* Projective spaces: Hp(RPn; Z/2)
= 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; topology
in physics.
@ General: De Rham 60; in Nash & Sen 83.
@ For groups: Weiss 69; K S Brown 82.
@ Non-Abelian: Eilenberg & MacLane AM(47); Andersson 86.
@ Quantum: Ruan & Tian JDG(95).
In Physics > s.a. BRST cohomology; classical
particles; crystals; formulations
of general relativity;
lagrangian dynamics.
@ Applications: Azcárraga & Izquierdo 95; Gross JMP(96);
Forrest
et al mp/00-in
[quasicrystals].
Main page – Abbreviations – Journals – Comments – Other
sites – Acknowledgements
Send feedback and suggestions to bombelli at olemiss.edu – Modified
22 jun 2008