under construction
group cohomology, nonabelian group cohomology, Lie group cohomology
cohomology with constant coefficients / with a local system of coefficients
differential cohomology
symmetric monoidal (∞,1)-category of spectra
synthetic differential geometry
Introductions
from point-set topology to differentiable manifolds
geometry of physics: coordinate systems, smooth spaces, manifolds, smooth homotopy types, supergeometry
Differentials
Tangency
The magic algebraic facts
Theorems
Axiomatics
(shape modality $\dashv$ flat modality $\dashv$ sharp modality)
$(\esh \dashv \flat \dashv \sharp )$
dR-shape modality$\dashv$ dR-flat modality
$\esh_{dR} \dashv \flat_{dR}$
(reduction modality $\dashv$ infinitesimal shape modality $\dashv$ infinitesimal flat modality)
$(\Re \dashv \Im \dashv \&)$
fermionic modality$\dashv$ bosonic modality $\dashv$ rheonomy modality
$(\rightrightarrows \dashv \rightsquigarrow \dashv Rh)$
Models
Models for Smooth Infinitesimal Analysis
smooth algebra ($C^\infty$-ring)
differential equations, variational calculus
Chern-Weil theory, ∞-Chern-Weil theory
Cartan geometry (super, higher)
(H-cohomology)
Given a smooth manifold or formal smooth manifold $X$, and given a differential 3-form $H \in \Omega^3(X)$, then forming the wedge product with $H$ is a nilpotent operation, and hence defines the differential of a cochain complex whose entries are the vector spaces $\Omega^{even/odd}(X)$ of all odd-graded or all even-graded differential forms on $X$:
The cochain cohomology of this cochain complex
is also called the H-cohomology of $X$.
Often it is assumed that the 3-form $H$ in def. is closed, because in this case H-cohomology serves as an approximation to the corresponding $H$-twisted de Rham cohomology (prop. below). But of course for def. to make sense in itself, $H$ need not be closed.
It is immediate to consider this more generally:
(H-cohomology in graded-commutative algebra)
Let $\mathcal{A}$ be a graded commutative algebra in characteristic zero and let $H \in \mathcal{A}$ be an odd-graded element. This implies that consecutive multiplication with $H$
is a chain complex.
The cochain cohomology of this cochain complex
is the H-cohomology of $\mathcal{A}$.
(H-cohomology of weakly decomposable forms is non-vanishing)
Let $\mathcal{A}$ be a $\mathbb{N}$-graded-commutative algebra. and let $H \in \mathcal{A}$ be of odd degree $deg(H) = 2 n +1$
If $H$ is weakly decomposable in that it is in the ideal generated by the elements of degree 1
This statement and the following proof are due to Sackel 18.
First consider the following:
Claim: Assuming that the H-cohomology of $H$ did vanish, then for every sequence of natural numbers $1 \leq i_1 \lt \ldots \lt i_{\ell} \leq k$, there would exist elements $\gamma_{i_1\cdots i_{\ell}} \in \mathcal{A}$ of degree $deg(\gamma_{i_1\cdots i_{\ell}}) = k-3-\ell$ (vanishing if $\ell \gt k-3$) such that the products of $\alpha$-s with indices not contained in this sequence
(hats indicate that the corresponding element is omitted) would satisfy
Observe that this claim immediately implies the statement by contradiction: For the case $\ell = k$ the left hand side of (2) is $1$ by definition (1), while its right-hand side would vanish, because the corresponding elements $\gamma$ are in negative degree. This is a contradiction, implying that the assumption of vanishing H-cohomology must be wrong.
Hence it is now sufficient to prove the above claim. We proceed by induction on $\ell$:
For $\ell = 0$ we have
and so by the assumption of vanishing $H$-cohomology, there exists $\gamma_{\emptyset}$ with
This is (2) for $\ell = 0$.
So assume now that there is $1 \leq \ell \leq k$ such that (2) holds for all $\ell \lt \ell_0$. Then
where in the third line we used the induction assumption.
By vanishing of $H$-cohomology, this implies (2) for $\ell = \ell_0$, thus finishing the proof of the claim, by induction.
(H-cohomology of odd symplectic form)
Let $X$ be a finite dimensional graded manifold in degrees 0 and 1 and equipped with a symplectic form $\omega$ of odd coordinate degree (e.g. the symplectic form on a Poisson Lie algebroid when regarded as a symplectic Lie n-algebroid).
Then every H-cohomology-class (def. ) for $H = \omega$ has a unique representative which on any local coordinate chart of Darboux coordinates $(x^i, p_i)_{i \in \{1, \cdots, n\}}$ is of the form
The terminology “$H$-cohomology” is used in
The case of H-cohomology for $H = \omega$ an odd symplectic form (as on a Poisson Lie algebroid regarded as a symplectic Lie n-algebroid) is considered in
The general non-vanishing of H-cohomology is pointed out in
Last revised on March 5, 2018 at 08:06:05. See the history of this page for a list of all contributions to it.