Exterior Algebra¶

Let $V$ be a real vector space of dimension $n$ .

Definition, Alternating algebraic forms:

For each $k$ , we define $\Alt^k V$ as the space of alternating $k$ -linear maps $V \times \cdots \times V \rightarrow \mathbb{R}$ .

Note

$\Alt^0 = \mathbb{R}$ ,
$\Alt^1 = V^{*}$ is the dual space of $V$ (the space of covectors)

Definition, Exterior product:

For $\omega \in \Alt^j$ and $\eta \in \Alt^k$ , their exterior (wedge) product is given by:

$(\omega \wedge \eta ) (v_1, \cdots, v_{j+k}) = \sum_{\sigma} (\mathrm{sign}~ \sigma) \omega (v_{\sigma(1)}, \cdots, v_{\sigma(j)}) \eta (v_{\sigma(j+1)}, \cdots, v_{\sigma(j+k)})$

for all $v_i \in V$ . Where the sum is over all permutations $\sigma$ of $\{ 1,\cdots,j+k \}$ , for which $\sigma(1)< \cdots <\sigma(j)$ and $\sigma(j+1)< \cdots <\sigma(j+k)$ .

Note

The exterior product is bilinear, associative,
anti-commutative: $\eta \wedge \omega = (-1)^{jk} \omega \wedge \eta$ for all $\omega \in \Alt^j$ and $\eta \in \Alt^k$ .

Definition, Grassmann Algebra:

Grassmann Algebra is defined by:

$\Alt V := \bigoplus_k \Alt^k V$

This is a anti-commutative graded algebra. Also called Exterior Algebra of $V^{*}$

In the case of $V=\mathbb{R}^n$ , we have:

$\Alt V^0 \sim \mathbb{R}$ ,
$\Alt V^1 \sim \mathbb{R}^n$ ,
$\Alt V^{n-1} \sim \mathbb{R}^n$ , using Riesz representation theorem,
$\Alt V^n \sim \mathbb{R}$ , using the map $v \longmapsto \det(v,v_1,\cdots,v_{n-1})$ .

Basis¶

Let $v_1,\cdots,v_n$ be a basis of $V$ and $\mu_1,\cdots,\mu_n$ the associated dual basis for $V^*$ ( $\mu_i(v_j) = \delta_{ij}$ ).

For any increasing permutations $\sigma, \rho : \{ 1,\cdots,k \} \longrightarrow \{ 1,\cdots,n \}$ , we have:

$\mu_{\sigma(1)} \wedge \cdots \wedge \mu_{\sigma(k)} (v_{\rho(1)}, \cdots, v_{\rho(k)}) = \chi_{\sigma,\rho}$

thus the $\binom {n}{k}$ algebraic $k$ -forms $\mu_{\sigma(1)} \wedge \cdots \wedge \mu_{\sigma(k)}$ , form a basis for $\Alt^k V$ and $\dim \Alt^k V = \binom {n}{k}$ .

Definition, Interior product:

Let $\omega$ be a $k$ -form, and $v \in V$ . The interior product of $\omega$ and $v$ is the $(k-1)$ -form $\omega \lrcorner v$ defined by:

$\omega \lrcorner v (v_1,\cdots,v_{k-1}) = \omega (v,v_1,\cdots,v_{k-1})$

We have for $\omega \in \Alt^k V$ , $\eta \in \Alt^l V$ and $v \in V$ :

$(\omega \wedge \eta) \lrcorner v = (\omega \lrcorner v)\wedge \eta + (-1)^k \omega \wedge (\eta \lrcorner v)$

Definition, Inner product:

If $V$ is has an inner product, then $\Alt^k V$ is endowed with an inner product given by:

$(\omega , \eta) = \sum_{\rho} \omega (v_{\rho(1)}, \cdots, v_{\rho(k)}) \eta (v_{\rho(1)}, \cdots, v_{\rho(k)}), ~~~\forall \omega, \eta \in \Alt^k V.$

where the sum is over increasing sequences $\rho : \{ 1,\cdots,k \} \longrightarrow \{ 1,\cdots,n \}$ , and $v_1, \cdots,v_n$ is any orthonormal basis.

Orientation and Volume form¶

Todo

add Orientation and Volume form

Definition, Pullback:

A linear transformation of vector spaces $L: V \rightarrow W$ induces a transformation $L^{*}: \Alt W \rightarrow \Alt V$ , called the pullback, and given by:

$L^{*} \omega (v_{1}, \cdots, v_{k}) = \omega (L v_{1}, \cdots, L v_{k}),~~~~~ \forall \omega \in \Alt^k W,~~~ v_{1}, \cdots, v_{k} \in V$

The pullback acts contravariantly: if $U \xrightarrow{~K~} V \xrightarrow{~L~} W$ then,

$\Alt W \xrightarrow{~K^{*}~} \Alt V \xrightarrow{~L^{*}~} \Alt U$
$L^{*} (\omega \wedge \eta) = L^{*} \omega \wedge L^{*} \eta$

Let V be a subspace of W. For the inclusion $\imath_V : V \longrightarrow W$ , we can define its pullback $\imath_V^{*}$ : this is a surjection of $\Alt W$ onto $\Alt V$ .

If W has an inner product and $\pi_V : W \longrightarrow V$ is the orthogonal projection. We can define its pullback $\pi_V^{*}$ : this an injection of $\Alt V$ onto $\Alt W$ .

Let us consider the composition : $W$ shortstack{ $\pi_V$ \ $\longrightarrow$ } $V$ shortstack{ $\imath_V$ \ $\longrightarrow$ } $W$ , and its pullback $\pi_V^* \imath_V^*$ .

Definition, The tangential and normal parts:

$\pi_V^* \imath_V^*$ associates for each $\omega \in \Alt^k$ its tangential part $\omega_{\parallel}$ with respect to $V$ :

$(\pi_V^* \imath_V^* \omega) (v_1,\cdots,v_k) = \omega (\pi_V v_1, \cdots, \pi_V v_k), ~~~~~\forall v_1,\cdots,v_k \in W.$

$\omega - \pi_V^* \imath_V^* \omega$ associates for each $\omega \in \Alt^k$ its normal part $\omega_{\perp}$ with respect to $V$ .

The tangential part of $\omega$ vanishes if and only if the image of $\omega$ in $\Alt^k V$ vanishes.

Let $V$ be an oriented inner product space, with volume form $\mbox{vol}$ . Let $\omega \in \Alt^k V$ . We can define a new linear map $L_{\omega}$ as the composition of $\Alt^{n-k} V \longrightarrow \Alt^n V$ such as:

$\mu \longmapsto \omega \wedge \mu$

and the canonical isomorphism of $\Alt^n V$ onto $\mathbb{R}$ , and using the Riesz representation theorem, there exists an element $\star \omega \in \Alt^{n-k} V$ such that : $L_{\omega} (\mu) = (\star \omega , \mu)$ , i.e.:

$\omega \wedge \mu = (\star \omega , \mu) \mbox{vol}, ~~~\omega \in \Alt^{k}, ~\mu \in \Alt^{n-k}$

Definition, The Hodge star operation:

The linear map which maps $\Alt^k V$ onto $\Alt^{n-k} V$ $\omega \longmapsto \star \omega$ is called the Hodge star operator.

If $e_1,\cdots,e_n$ is any positively oriented orthonormal basis, and $\sigma$ a permutation, we have

$\omega(e_{\sigma(1)}, \cdots, e_{\sigma(k)}) = (\mathrm{sign} \sigma) \star \omega(e_{\sigma(k+1)}, \cdots, e_{\sigma(n)})$

$\star \star \omega = (-1)^{k(n-k)} \omega, ~~~\forall \omega \in \Alt^k V$ , thus the Hodge star is an isometry.
$(\star \omega)_{\parallel} = \star (\omega_{\perp})$ and $(\star \omega)_{\perp} = \star (\omega_{\parallel})$
the image of $\star \omega$ in $\Alt^k V$ vanishes if and only if $\omega_{\perp}$ vanishes.

$\begin{tabular}{|c|l|} \hline $\Alt^0 \mathbb{R}^3 \cong \mathbb{R}$ & $c \leftrightarrow c$ \\ % \hline $\Alt^1 \mathbb{R}^3 \cong \mathbb{R}^3$ & $u_1 \diff x_1 + u_2 \diff x_2 + u_3 \diff x_3 \leftrightarrow u$ \\ % \hline $\Alt^2 \mathbb{R}^3 \cong \mathbb{R}^3$ & $u_3 \diff x_1 \wedge \diff x_2 - u_2 \diff x_1 \wedge \diff x_3 + u_1 \diff x_2 \wedge \diff x_3 + \leftrightarrow u$ \\ % \hline $\Alt^3 \mathbb{R}^3 \cong \mathbb{R}$ & $c \diff x_1 \wedge \diff x_2 \wedge \diff x_3 \leftrightarrow c$ \\ \hline \end{tabular}$

$\begin{tabular}{|c|l|} \hline $ \wedge : \Alt^1 \mathbb{R}^3 \times \Alt^1 \mathbb{R}^3 \longrightarrow \Alt^2 \mathbb{R}^3$ & $\times : \mathbb{R}^3 \times \mathbb{R}^3 \longrightarrow \mathbb{R}^3$ \\ $ \wedge : \Alt^1 \mathbb{R}^3 \times \Alt^2 \mathbb{R}^3 \longrightarrow \Alt^3 \mathbb{R}^3$ & $\cdot : \mathbb{R}^3 \times \mathbb{R}^3 \longrightarrow \mathbb{R}$ \\ \hline \end{tabular}$

$\begin{tabular}{|c|l|} \hline $ L^* : \Alt^0 \mathbb{R}^3 \longrightarrow \Alt^0 \mathbb{R}^3 $ & $\id : \mathbb{R} \longrightarrow \mathbb{R}$ \\ $ L^* : \Alt^1 \mathbb{R}^3 \longrightarrow \Alt^1 \mathbb{R}^3 $ & $L^T : \mathbb{R}^3 \longrightarrow \mathbb{R}^3$ \\ $ L^* : \Alt^2 \mathbb{R}^3 \longrightarrow \Alt^2 \mathbb{R}^3 $ & $(\det L )L^{-1} : \mathbb{R}^3 \longrightarrow \mathbb{R}^3$ \\ $ L^* : \Alt^3 \mathbb{R}^3 \longrightarrow \Alt^3 \mathbb{R}^3 $ & $(\det L) : \mathbb{R} \longrightarrow \mathbb{R}$ ~~~($c \longmapsto c \det L$) \\ \hline \end{tabular}$

$\begin{tabular}{|c|l|} \hline $ \lrcorner v : \Alt^1 \mathbb{R}^3 \longrightarrow \Alt^0 \mathbb{R}^3 $ & $v \cdot : \mathbb{R}^3 \longrightarrow \mathbb{R}$ \\ $ \lrcorner v : \Alt^2 \mathbb{R}^3 \longrightarrow \Alt^1 \mathbb{R}^3 $ & $v \times : \mathbb{R}^3 \longrightarrow \mathbb{R}^3$ \\ $ \lrcorner v : \Alt^3 \mathbb{R}^3 \longrightarrow \Alt^2 \mathbb{R}^3 $ & $v : \mathbb{R} \longrightarrow \mathbb{R}^3$ ~~~($c \longmapsto c v$) \\ \hline \end{tabular}$

$\begin{tabular}{|c|l|} \hline inner product on $\Alt^k \mathbb{R}^3$ induced & dot product on $\mathbb{R}$ and $\mathbb{R}^3$ \\ by dot product on $\mathbb{R}^3$ & \\ $\volume = \diff x_1 \wedge \diff x_2 \wedge \diff x_3$ & $(v_1,v_2,v_3) \longmapsto \det(v_1|v_2|v_3)$ \\ \hline \end{tabular}$

$\begin{tabular}{|c|l|} \hline $ \star : \Alt^0 \mathbb{R}^3 \longrightarrow \Alt^3 \mathbb{R}^3 $ & $\id : \mathbb{R} \longrightarrow \mathbb{R}$ \\ $ \star : \Alt^1 \mathbb{R}^3 \longrightarrow \Alt^2 \mathbb{R}^3 $ & $\id : \mathbb{R}^3 \longrightarrow \mathbb{R}^3$ \\ \hline \end{tabular}$

Exterior Calculus on manifolds and Differential forms¶

Let $\Omega$ be a smooth manifold, of dimension $n$ .

$\forall x \in \Omega$ we denote by $T_x \Omega$ the tangent space. This is a vector space of dimension $n$ ,
tangent bundle $\{ (x,v), ~~ x \in \Omega, v \in T_x \Omega \}$ ,
Applying the exterior algebra to the tangent spaces, we obtain the exterior forms bundle, whose elements are pairs $(x,\mu)$ with $x \in \Omega$ and $\mu \in \Alt^k T_x \Omega$ .
a differential $k$ -form $\omega$ is a section of this bundle. This is a map which associates to each $x \in \Omega$ an element $\omega_x \in \Alt^k T_x \Omega$ ,
if the map $\mathcal{L}_{\omega}^k : x \longmapsto \omega_x (v_1(x), \cdots, v_k(x))$ is smooth (whenever $v_i$ are smooth), we say that $\omega$ is a smooth differential $k$ -form,
we define $\Lambda^k(\Omega)$ the space of all smooth $k$ -forms on $\Omega$ ,
$\Lambda^0(\Omega) = \mathcal{C}^{\infty}(\Omega)$ ,
if the map $\mathcal{L}_{\omega}^k$ is $\mathcal{C}^{m}(\Omega)$ , we define differential $k$ -forms with less smoothness $\mathcal{C}^{m} \Lambda^k (\Omega)$ .

Let $\Omega$ be a smooth manifold, of dimension $n$ .

Exterior product:

if $\omega \in \Lambda^k(\Omega)$ and $\eta \in \Lambda^j(\Omega)$ , we may define $\omega \wedge \eta$ as $(\omega \wedge \eta)_x = \omega_x \wedge \eta_x$ and the Grassmann algebra $\Lambda(\Omega) := \bigoplus_k \Lambda^k(\Omega)$

Differential forms can be differentiated and integrated, without recourse to any additional structure, such as a metric or a measure.

Exterior differentiation:

For each $\omega \in \Lambda^k(\Omega)$ , can define the $(k+1)$ -form $\diff \omega \in \Lambda^{k+1}(\Omega)$ , such as:

$\diff\omega_x(v_1,\cdots,v_{k+1}) = \sum_{j=1}^{k+1} (-1)^j \partial_{v_j} \omega_x(v_1,\cdots,\hat{v_j},\cdots,v_{k+1})$

where the hat is used to indicated a suppressed argument.

This defines a graded linear operator of degree $+1$ , of $\Lambda(\Omega)$ onto $\Lambda(\Omega)$ .

We have the following properties:

$\diff \circ \diff = 0$
$\diff (\omega \wedge \eta) = \diff \omega \wedge \eta + (-1)^k \omega \wedge \diff \eta, ~~\forall \omega \in \Lambda^k(\Omega), \eta \in \Lambda^j(\Omega)$ ,
(Pullback) let $\phi$ be a smooth map of $\Omega$ onto $\Omega^{\prime}$ . Then $\phi^*(\omega \wedge \eta) = \phi^*(\omega) \wedge \phi^*(\eta)$ and $\phi^* (\diff \omega) = \diff (\phi^* \omega)$ ,
(Interior product) the interior product of a differential $k$ -form $\omega$ with a vector field $v$ ,
we obtain a $(k-1)$ -form by : $(\omega \lrcorner v)_x := \omega_x \lrcorner v_x$ ,
(Trace operator) the pullback $i_{\partial \Omega}^*$ of $i_{\partial \Omega}$ is the trace operator $\trace$

Integration:

If is an oriented, piecewise smooth -dimensional submanifold of , and is a continuous -form, then th integral is well defined :
- [0-forms] can be evaluated at points,
- [1-forms] can be integrated over directed curves,
- [2-forms] can be integrated over directed surfaces,
(Inner product) The $L^2$ -inner product of two differential $k$ -forms on an oriented Riemannian manifold $\Omega$ is defined as :

$(\omega,\eta)_{L^2 \Lambda^k} = \int_{\Omega} (\omega_x,\eta_x) \volume = \int \omega \wedge \star \eta$

The completion of $\Lambda^k(\Omega)$ in the corresponding norm defines the Hilbert space $L^2 \Lambda^k(\Omega)$ .

We have the following results:

(Integration) if $\phi$ is an orientation-preserving diffeomorphism, then

$\int_{\Omega} \phi^* \omega = \int_{\Omega^{\prime}} \omega, ~~~ \forall \omega \in \Lambda^n(\Omega^{\prime})$

Theorem, Stokes theorem:

If $\Omega$ is an oriented $n$ -manifold with boundary $\partial \Omega$ , then

$\int_{\Omega} \diff \omega = \int_{\partial \Omega} \trace \omega, ~~~ \forall \omega \in \Lambda^{n-1}(\Omega)$

Theorem, Integration by parts:

If $\Omega$ is an oriented $n$ -manifold with boundary $\partial \Omega$ , then

$\int_{\Omega} \diff \omega \wedge \eta = (-1)^{k-1} \int_{\Omega} \omega \wedge \diff \eta + \int_{\partial \Omega} \trace \omega \wedge \trace \eta, ~~~ \forall \omega \in \Lambda^{k}(\Omega), \eta \in \Lambda^{n-k-1}(\Omega)$

Sobolev spaces of differential forms¶

As for the classical case, we can define the Sobolev spaces as:

$H^s \Lambda^k(\Omega)$ is the space of differential $k$ -forms such that $\mathcal{L}_{\omega}^k \in H^s(\Omega)$ .
$H \Lambda^k(\Omega) = \{ \omega \in L^2 \Lambda^k(\Omega),~~ \diff \omega \in L^2 \Lambda^{k+1}(\Omega) \}$ . The associated norm is :

$\| \omega \|_{H \Lambda^k}^2 = \| \omega \|_{H \Lambda}^2 := \| \omega \|_{L^2 \Lambda^k}^2 + \| \diff \omega \|_{L^2 \Lambda^{k+1}}^2$

$H \Lambda^{0}(\Omega)$ coincides with $H^1 \Lambda^{0}(\Omega)$ ,
$H \Lambda^{n}(\Omega)$ coincides with $L^2 \Lambda^{n}(\Omega)$ ,
for $0 < k < n$ , we have $H^1 \Lambda^k(\Omega) \subset H \Lambda^k(\Omega) \subset L^2 \Lambda^k(\Omega)$ , strictly.

$\begin{tabular}{|c|c c c c c|} \hline $k$ & $\Lambda^k$ & $H \Lambda^k$ & $\diff \omega$ & $\int_f \omega$ & $\kappa \omega$ \\ \hline & & & & & \\ 0 & $\mathcal{C}^{\infty}$ & $H^1$ & $\nabla \omega$ & $\omega(f)$ & $0$ \\ 1 & $\mathcal{C}^{\infty}(\mathbb{R}^3)$ & $H(\rots,\mathbb{R}^3)$ & $\rots \omega$ & $\int_f \omega \cdot t \diff \mathcal{H}_1$ & $x \longmapsto x \cdot \omega(x)$ \\ 2 & $\mathcal{C}^{\infty}(\mathbb{R}^3)$ & $H(\divs, \mathbb{R}^3)$ & $\divs \omega$ & $\int_f \omega \cdot n \diff \mathcal{H}_2$ & $x \longmapsto x \times \omega(x)$ \\ 3 & $\mathcal{C}^{\infty}$ & $L^2$ & $0$ & $\int_f \omega \diff \mathcal{H}_3$ & $x \longmapsto x \omega(x)$ \\ & & & & & \\ \hline \end{tabular}$

Cohomology and De Rham Complex¶

The De Rham complex is the sequence of spaces and mappings

$0 \xrightarrow{\quad} \Lambda^0(\Omega) \xrightarrow{~\diff~} \Lambda^1(\Omega) \xrightarrow{~\diff~} \cdots \xrightarrow{~\diff~} \Lambda^n(\Omega) \xrightarrow{\quad} 0$

Since, $\diff \circ \diff = 0$ , we have

$\mathcal{R}(\diff : \Lambda^{k-1}(\Omega) \longrightarrow \Lambda^k(\Omega)) \subset \mathcal{N}(\diff : \Lambda^{k}(\Omega) \longrightarrow \Lambda^{k+1}(\Omega))$

If $\Omega$ is an oriented Riemannian manifold, we have the following cohomology:

$0 \xrightarrow{\quad} H \Lambda^0(\Omega) \xrightarrow{~\diff~} H \Lambda^1(\Omega) \xrightarrow{~\diff~} \cdots \xrightarrow{~\diff~} H \Lambda^n(\Omega) \xrightarrow{\quad} 0$

The coderivative operator $\delta : \Lambda^{k}(\Omega) \longrightarrow \Lambda^{k-1}(\Omega)$ is defined as:

$\star \delta \omega = (-1)^k \diff \star \omega,~~~ \omega \in \Lambda^k(\Omega)$

we have

$(\diff \omega , \eta ) = (\omega , \delta \eta ) + \int_{\partial \Omega} \trace \omega \wedge \trace \eta, ~~~ \forall \omega \in \Lambda^{k}(\Omega), \eta \in \Lambda^{k+1}(\Omega),$

$\delta$ is a graded linear operator of degree $-1$ .
$\delta$ is the formal adjoint of $\diff$ whenever $\omega$ or $\eta$ vanishes near the boundary.
we define the spaces

$H^* \Lambda^k(\Omega) = \{ \omega \in L^2 \Lambda^k(\Omega),~~ \delta \omega \in L^2 \Lambda^{k-1}(\Omega) \}.$

we have $H^* \Lambda^k(\Omega) = \star H \Lambda^{n-k}(\Omega)$ .

we obtain the dual complex

$0 \xleftarrow{\quad} H^* \Lambda^0(\Omega) \xleftarrow{~\delta~} H^* \Lambda^1(\Omega) \xleftarrow{~\delta~} \cdots \xleftarrow{~\delta~} H^* \Lambda^n(\Omega) \xleftarrow{\quad} 0$

Cohomology with boundary conditions¶

Let $\Lambda_0^k(\Omega)$ be the subspace of $\Lambda^k(\Omega)$ of smooth $k$ -forms with compact support. We have $\diff \Lambda_0^k \subset \Lambda_0^{k+1}$ .

The De Rham complex with the compact support is

$0 \xrightarrow{\quad} \Lambda^0_0(\Omega) \xrightarrow{~\diff~} \Lambda^1_0(\Omega) \xrightarrow{~\diff~} \cdots \xrightarrow{~\diff~} \Lambda^n_0(\Omega) \xrightarrow{\quad} 0$

Recall that the closure of $\Lambda_0^k(\Omega)$ in $H \Lambda^k(\Omega)$ is

$H_0 \Lambda^k(\Omega) = \{ \omega \in H \Lambda^k(\Omega),~~ \trace \omega =0\}.$

The $L^2$ version of the last complex is

$0 \xrightarrow{\quad} H_0 \Lambda^0(\Omega) \xrightarrow{~\diff~} H_0 \Lambda^1(\Omega) \xrightarrow{~\diff~} \cdots \xrightarrow{~\diff~} H_0 \Lambda^n(\Omega) \xrightarrow{\quad} 0$

Definition, Harmonic forms:

The harmonic $k$ -forms are the differential $k$ -forms that verify the differential equations

$\left\{ \begin{aligned} \diff \omega &=& 0,\\ \delta \omega &=& 0,\\ \trace \star \omega &=& 0.\\ \end{aligned} \right.$

this defines the following space,

$\mathfrak{H}^k (\Omega) = \{ \omega \in H \Lambda^k(\Omega) \cap H_0^* \Lambda^k(\Omega),~~\diff \omega = 0, \delta \omega = 0 \}$

We can also define the following space,

$\mathfrak{H}_0^k (\Omega) = \{ \omega \in H_0 \Lambda^k(\Omega) \cap H^* \Lambda^k(\Omega),~~\diff \omega = 0, \delta \omega = 0 \}$

As we can see, $\star \mathfrak{H}^k (\Omega) = \mathfrak{H}_0^{n-k} (\Omega)$ .

Proposition, Poincaré duality:

There is an isomorphism between the $k$ th De Rham cohomology space and the $(n-k)$ th cohomology space with boundary conditions.

Homological Algebra and Hilbert complexes¶

Homological Algebra¶

A cochain complex is a sequence of vector spaces and linear maps

$k$ -cocycles $\mathfrak{Z}^k := \mathcal{N}(d_k)$ ,
$k$ -coboundaries $\mathfrak{B}^k := \mathcal{R}(d_{k-1})$ ,
$k$ -cohomology $\mathcal{H}^k(V) := \mathfrak{Z}^k / \mathfrak{B}^k$ ,
we say that the sequence is exact, if the cohomology vanishes (i.e. $\forall~k,~~ \mathcal{H}^k(V) = \{0\}$ ),
Given two cochain complexes $V,V^{\prime}$ , a cochain map $f =(f_k)$ (such as $\diff^{\prime}_k f_k = f_{k+1} \diff_k$ )

$\begin{array}{ccccccccc} \cdots & \longrightarrow & V_{k-1} & \mbox{\shortstack{$\diff_{k-1}$ \\ $\longrightarrow$}} & V_{k} & \mbox{\shortstack{$\diff_k$ \\ $\longrightarrow$}} & V_{k+1} & \longrightarrow~\cdots \\ & & \downarrow f_{k-1} & & \downarrow f_{k} & & \downarrow f_{k+1} & & \\ \cdots & \longrightarrow & V_{k-1}^{\prime} & \mbox{\shortstack{$\diff_{k-1}^{\prime}$ \\ $\longrightarrow$}} & V_{k}^{\prime} & \mbox{\shortstack{$\diff_k^{\prime}$ \\ $\longrightarrow$}} & V_{k+1}^{\prime} & \longrightarrow~\cdots \end{array}$

$f_k$ maps $k$ -cochains to $k$ -cochains and $k$ -coboundaries to $k$ -coboundaries, thus induces a map $\mathcal{H}^k(f) : \mathcal{H}^k(V) \longrightarrow \mathcal{H}^k(V^{\prime})$ .

Let $V^{\prime} \subset V$ be two cochain complexes,

The inclusion $\imath_V$ is a cochain map and thus induces a map of cohomology $\mathcal{H}^k(V^{\prime}) \longrightarrow \mathcal{H}^k(V)$ ,
If there exists a cochain projection of $V$ onto $V^{\prime}$ , (this leads to $\pi \circ \imath = \id_{V^{\prime}}$ ) so $\mathcal{H}^k(\pi) \circ \mathcal{H}^k(\imath) = \id_{\mathcal{H}^k(V^{\prime})}$ .

$\begin{array}{ccccccc} \cdots & \longrightarrow & V_{k-1} & \mbox{\shortstack{$\diff_{k-1}$ \\ $\longrightarrow$}} & V_{k} & \longrightarrow~\cdots \\ & & \pi_{k-1} \downarrow \uparrow \imath & & \pi_{k} \downarrow \uparrow \imath & & \\ \cdots & \longrightarrow & V_{k-1}^{\prime} & \mbox{\shortstack{$\diff_{k-1}$ \\ $\longrightarrow$}} & V_{k}^{\prime} & \longrightarrow~\cdots \end{array}$

Thus, $\mathcal{H}^k(\imath)$ is injective and $\mathcal{H}^k(\pi)$ is surjective. Hence, if one of the cohomology spaces $\mathcal{H}^k(V)$ vanishes, then so does $\mathcal{H}^k(V^{\prime})$

Cycles and boundaries of the De Rham complex¶

$k$ -cocycles

$\mathfrak{Z}^k = \{ \omega \in H\Lambda^k(\Omega),~~ \diff \omega = 0 \}, ~~~ \mathfrak{Z}^{*k} = \{ \omega \in H^*\Lambda^k(\Omega),~~ \delta \omega = 0 \},$

$\mathfrak{Z}_0^k = \{ \omega \in H_0\Lambda^k(\Omega),~~ \diff \omega = 0 \}, ~~~ \mathfrak{Z}_0^{*k} = \{ \omega \in H_0^*\Lambda^k(\Omega),~~ \delta \omega = 0 \},$

$k$ -coboundaries

$\mathfrak{B}^k = \diff H\Lambda^{k-1}(\Omega), ~~~ \mathfrak{B}^{* k} = \delta \Lambda^{k+1}(\Omega),$

$\mathfrak{B}_0^k = \diff H_0\Lambda^{k-1}(\Omega), ~~~ \mathfrak{B}_0^{* k} = \delta \Lambda_0^{k+1}(\Omega),$

each of the spaces of cycles is closed in $\mathcal{H} \Lambda^k(\Omega)$ ( $\mathcal{H}^* \Lambda^k(\Omega)$ ), as well in $L^2 \Lambda^k(\Omega)$ .
each of the spaces of boundaries is closed in $L^2 \Lambda^k(\Omega)$ .
let $\perp$ denotes the orthogonal complement in $L^2 \Lambda^k(\Omega)$ ,

$\mathfrak{Z}^{k \perp} \subset \mathfrak{B}^{k \perp} = \mathfrak{Z}_0^{* k} , ~~~ \mathfrak{Z}^{* k \perp} \subset \mathfrak{B}^{* k \perp} = \mathfrak{Z}_0^{k}$

$\mathfrak{Z}_0^{k \perp} \subset \mathfrak{B}_0^{k \perp} = \mathfrak{Z}^{* k} , ~~~ \mathfrak{Z}_0^{* k \perp} \subset \mathfrak{B}_0^{* k \perp} = \mathfrak{Z}^{k}$

The Hodge decomposition¶

There are two Hodge decompositions, with different boundary conditions,

$L^2 \Lambda^k(\Omega) = \underbrace{\mathfrak{B}^{k}}_{\mathfrak{Z}_0^{* k\perp}} \oplus \underbrace{\mathfrak{H}^{k} \oplus \mathfrak{B}_0^{* k}}_{\mathfrak{Z}_0^{* k}=\mathfrak{B}^{k\perp}} = \overbrace{\mathfrak{B}^{k} \oplus \mathfrak{H}^{k}}^{\mathfrak{Z}^{k}=\mathfrak{B}_0^{* k\perp}} \oplus \overbrace{\mathfrak{B}_0^{* k}}^{\mathfrak{Z}^{k\perp}}$
$L^2 \Lambda^k(\Omega) = \underbrace{\mathfrak{B}_0^{k}}_{\mathfrak{Z}^{* k\perp}} \oplus \underbrace{\mathfrak{H}_0^{k} \oplus \mathfrak{B}^{* k}}_{\mathfrak{Z}^{* k}=\mathfrak{B}_0^{k\perp}} = \overbrace{\mathfrak{B}_0^{k} \oplus \mathfrak{H}_0^{k}}^{\mathfrak{Z}_0^{k}=\mathfrak{B}^{* k\perp}} \oplus \overbrace{\mathfrak{B}^{* k}}^{\mathfrak{Z}_0^{k\perp}}$

Summary¶

$\begin{tabular}{|c||c|c|c|c|} \hline $\omega^k \in \Lambda^k(\Omega)$ & $k=0$ & $k=1$ & $k=2$ & $k=3$ \\ \hline $\diff \omega^k$ & $\Grad u$ & $\Curl \uu$ & $\Div \uu$ & $-$ \\ $\delta \omega^k$ & $-$ & $-\Div \uu$ & $\Curl \uu$ & $-\Grad u$ \\ $\mathfrak{i}_{\boldsymbol{\beta}} \omega^k$ & $-$ & $\boldsymbol{\beta} \cdot \uu$ & $\uu \times \boldsymbol{\beta}$ & $u \boldsymbol{\beta}$ \\ $\mathfrak{j}_{\boldsymbol{\beta}} \omega^k$ & $u \boldsymbol{\beta}$ & $-\uu \times \boldsymbol{\beta}$ & $\boldsymbol{\beta} \cdot \uu$ & $-$ \\ $L_{\boldsymbol{\beta}} \omega^k$ & $\boldsymbol{\beta} \cdot \Grad u$ & $\Grad \left(\boldsymbol{\beta} \cdot \uu \right) + \left(\Curl \uu \right) \times \boldsymbol{\beta}$ & $\Curl \left(\uu \times \boldsymbol{\beta} \right) + \boldsymbol{\beta} \Div \uu$ & $\Div \left( u \boldsymbol{\beta} \right)$ \\ $\mathcal{L}_{\boldsymbol{\beta}} \omega^k$ & $-\Div \left( u \boldsymbol{\beta} \right)$ & $-\Curl \left(\uu \times \boldsymbol{\beta} \right) - \boldsymbol{\beta} \Div \uu$ & $-\Grad \left(\boldsymbol{\beta} \cdot \uu \right) - \left(\Curl \uu \right) \times \boldsymbol{\beta}$ & $-\boldsymbol{\beta} \cdot \Grad u$ \\ \hline $\tr \omega^k$ & $u(\xx)$ & $\uu(\xx) \times \nn(\xx)$ & $\uu(\xx) \cdot \nn(\xx)$ & $-$ \\ \hline \hline $H \Lambda^k(\Omega)$ & $\Hgrad$ & $\Hcurl$ & $\Hdiv$ & $\Ltwo$ \\ $V_k$ & $\Vgrad$ & $\Vcurl$ & $\Vdiv$ & $\Vltwo$ \\ \hline \end{tabular}$

References