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\lhead{\tiny Hogde theory, HSE} 
\lfoot{\tiny Issued \firstdate} 
\cfoot{-- \thepage \ -- } \rfoot{\tiny  \sc\version}
\rhead{{\tiny  Misha Verbitsky}}


\begin{document}

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\listok{8}{Hodge theory 8: Cartan's formula and Poincar\'e lemma}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

{\scriptsize
  {\bf Rules:} You may choose to solve only 
``hard'' exercises (marked with !, * and **) 
or ``ordinary'' ones (marked with ! or unmarked),
or both, if you want to have extra stuff to work.
To have a perfect score, a student must obtain
(in average) a score of 10 points per week.

If you have got credit for 2/3 of ordinary problems
or 2/3 of ``hard'' problems, you receive  
$6t$ points, where $t$ is a number depending on the
date when it is done. Passing all ``hard'' 
or all ``ordinary'' problems brings you $10t$ points.
Solving of ``**'' (extra hard) problems is not
obligatory, but each such problem gives you a credit
for 2 ``*'' or ``!'' problems in the ``hard'' set.

The first 3 weeks after giving a handout, $t=1.5$,
between 21 and 35 days, $t=1$, and afterwards, $t=0.7$.
The scores are not cumulative, only the
best score for each handout counts.
}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Lie derivative}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\definition
An associative algebra $A^* = \oplus_{i\in \Z}A^i$
is called {\bf a graded algebra} if for all $a\in A^i$, $b\in A^j$, 
the product $ab$ lies in $A^{i+j}$.
\ed


\definition
Let  $A^* = \oplus_{i\in \Z}A^i$ be a graded algebra over a field.
It is called
{\bf graded commutative}, or {\bf supercommutative},
if $ab = (-1)^{ij} ba$
for all $a\in A^i, b \in A^j$. 
\ed

\remark
Grassmann algebra $\Lambda^* V$
is clearly supercommutative.
\er

%\exercise
%Let  $A^*, B^*$ be graded commutative algebras,
%and $A^* \otimes B^*$ their tensor product,
%with a grading
%$(A^* \otimes B^*)^p := \oplus_{i+j=p} A^i\otimes B^j$,
%and multiplication, defined as
%$a \otimes b \cdot a' \otimes b' = (-1)^{ij}aa' \otimes bb'$,
%where $a'\in A^i, b \in B^j$.
%Prove that it is supercommutative.
%\ez
%
%\exercise
%Let $V, W$ be vector spaces, and
%$A^*:= \Lambda^* V, B^*:=\Lambda^* W$ their Grassmann algebras.
%Prove that  $\Lambda^*(V\oplus W)$ is isomorphic
%to the tensor product $A^* \otimes B^*$,
%defined as above.
%\ez

\definition
Let $A^*$ be a graded commutative algebra, 
and $D:\; A^* \arrow A^{*+i}$ a map which
shifts grading by $i$. 
It is called a
{\bf graded derivation} if
$D(ab) = D(a) b + (-1)^{ij} a D(b)$,
for each $a \in A^j$. 
\ed

\remark
If $i$ is even, graded derivation is just a derivation.
If it is odd, it is called {\bf odd derivation}.
\er

\remark
De Rham differential is an odd derivation, by definition.
\er

\definition
Let $M$ be a smooth manifold, and  $X\in TM$ 
a vector field. Consider an operation of {\bf 
convolution with a vector field} 
$i_X:\; \Lambda^i M \arrow \Lambda^{i-1}M$, 
mapping an $i$-form $\alpha$ to an $(i-1)$-form
$v_1, ..., v_{i-1} \arrow \alpha(X, v_1, ..., v_{i-1})$
\ed

\exercise
Prove that $i_X$ is an odd derivation.
\ez

\exercise[*]
Let $D:\; A^* \arrow A^{*+i}$ be a linear map
such that for all $x\in A$ there exists $N$ such that
$D^N(x)=0$. Prove that $e^D:= 1 + D + \frac
{D^2}{2} + ... + \frac{D^i}{i!} + ... $ is an automorphism
of $A^*$ if and only if $D$ is a derivation.
\ez

\definition
Let $A^*$ be a graded vector space, and 
$E:\; A^*\arrow A^{*+i}$, $F:\; A^*\arrow A^{*+j}$
operators shifting the grading by $i, j$.
Define  {\bf the supercommutator} by the formula
\[ 
  \{E, F\}:= EF - (-1)^{ij} FE.
\]
\ed

\remark
An endomorphism 
which shifts a grading by $i$ is called {\bf even}
if $i$ is even, and {\bf odd} otherwise.
\er


\remark
Supercommutator satisfies 
{\bf graded Jacobi identity,}
\[
\{ E, \{F, G\}\} = \{\{ E, F\}, G\} + (-1)^{\tilde E
\tilde F}  \{ F, \{E, G\}\}
\]
where  $\tilde E$ and $\tilde F$ are 0 if
$E, F$ are even, and 1 otherwise. 
\er

\remark
There is a simple mnemonic rule
which allows one to remember a superidentity,
if you know the commutative analogue.
Each time when in commutative case
two letters $E$, $F$ are exchanged, in supercommutative
case one needs to multiply by $(-1)^{\tilde E\tilde F}$.
\er

\exercise
Let $A^*$ be a graded commutative algebra
and $a\in A$. Denote by $L_a:\; A\arrow A$
the operation of multiplication by $a$:
$L_a(b) = ab$.
Prove that a map $D:\; A^*\arrow A^*$ is a superderivation if and only
if $D(1)=0$ and for each $a\in A^i$, the supercommutator
$\{D, L_a\}$ is equal to $L_b$ for some $b\in A^*$.
\ez 

\exercise[!]
Prove that a supercommutator of superderivations
is again a superderivation.
\ez

\hint
Use the Jacobi identity and apply the previous exercise.
\eh

\definition
Let $B$ be a smooth manifold, and $v\in TM$ a vector field.
An endomorphism $\Lie_v:\; \Lambda^* M \arrow \Lambda^* M$,
preserving the grading is called
{\bf a Lie derivative along $v$} if it satisfies the 
following conditions.
\begin{description}
\item[(i)] On functions $\Lie_v$ is equal to a derivative along $v$.
\item[(ii)] $[\Lie_v, d]=0$
\item[(iii)] $\Lie_v$ is a derivation on the de Rham algebra.
\end{description}
\ed

\exercise
\label{_derivations_gene_Exercise_}
Let $\nu_1, \nu_2:\; \Lambda^*(M) \arrow \Lambda^*(M)$ be
derivations of the de Rham algebra. Suppose that
$\nu_1$ is equal to $\nu_2$ on $C^\infty M=\Lambda^0(M)$ 
and on $d(C^\infty M)$. Prove that $\nu_1=\nu_2$.
\ez

\hint $\Lambda^*(M)$ is generated (multiplicatively)
by $C^\infty M=\Lambda^0(M)$ 
and $d(C^\infty M)$.
\eh

\exercise
Prove that the Lie derivative is uniquely determined
by the properties (i)-(iii).
\ez

\hint
Use the previous exercise.
\eh

\exercise
Prove that $\{d, \{d, E\}\}=0$,
for each $E\in \End(\Lambda^* M)$.
\ez

\hint
Use the graded Jacobi identity.
\eh

\exercise
Prove that $\{d, i_v\}$ commutes with $d$, where
$i_v:\; \Lambda^* M \arrow \Lambda^{*-1}M$ is a convolution
with $v$.
\ez

\hint
Use the previous exercise.
\eh

\exercise[!]
(Cartan formula)
Prove that $\{d, i_v\}$ is a Lie derivative along $v$.
\ez

%\exercise[*]
%Let  $v, v'\in TM$ be two vector fields, and
%$i_{v\otimes v'}:\; \Lambda^* M \arrow \Lambda^{*-2}M$ 
%is a substitution of $v, v'$ into a 2-form, 
%$i_{v\otimes v'}= i_v i_{v'}$.
%Consider an $i$-form $\alpha \in \Lambda^* M$,
%and let $L_\alpha$ be an operator of multiplication by $\alpha$.
%Prove that an operator
%\[
%x\arrow [i_{v\otimes v'}, L_\alpha](x) - L_{i_{v\otimes
%v'}}(\alpha)\wedge x
%\]
%is a derivation.
%\ez


\exercise[*]
Let $\tau:\; \Lambda^*(M) \arrow \Lambda^{*-1}(M)$
be a derivation shifting grading by $-1$.
Prove that there exists a vector field $v\in TM$ such that
$\tau=i_v$, or find a counterexample.
\ez 


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Poincar\'e lemma}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\exercise
Let $t$ be the coordinate function on a real line,
$f(t)\in C^\infty \R$ a smooth function,
and $v:= t \frac d {dt}$ a vector field. Define
\[
R(f)(t):=\int^1_0 \frac{f(\lambda t)}{\lambda} d\lambda
\]
Prove that this integral converges whenever $f(0)=0$, 
and satisfies $\Lie_v R(f)=f$ in this case.
\ez

\exercise
Let $t_1, ..., t_n$ be coordinate functions in $\R^n$,
and  $\vec{r}:=\sum_i t_i \frac d {dt_i}$ a radial vector field.
Consider a function $f\in C^\infty \R^n$ satisfying
$f(0)=0$, and let $x=(x_1, ..., x_n)$ be any point in
$\R^n$. Prove that an integral
\[
R(f)(x):=\int_0^1 \frac{f(\lambda x)}{\lambda} d\lambda
\]
converges, and satisfies $\Lie_{\vec{r}} R(f)=f$.
\ez

\hint
Use the previous exercise.
\eh

\definition
An open subset $U\subset \R^n$
is called {\bf starlike} if for any
$x\in U$ the interval $[0,x]$ belongs to $U$.
\ed

\exercise[!]
Let $U$ be a starlike subset in $\R^n$, and $i>0$.
Construct an operator \[ R:\; \Lambda^i U \arrow \Lambda^i U\]
which satisfies  $\Lie_{\vec{r}} R\alpha=R\Lie_{\vec{r}} \alpha = \alpha$ 
for each $\alpha \in\Lambda^i U$.
\ez

\hint 
Define the integral $R(\alpha)$ as in the previous exercise,
and check that it converges. Prove that $\Lie_{\vec{r}} R(\alpha)=\alpha$.
\eh

\exercise
\label{_R_kernel_Zadacha_}
Prove that any form
$\alpha\in \Lambda^i U$ on a starlike set $U$
satisfying $\Lie_{\vec{r}} \alpha=0$ vanishes if $i>0$.
\ez

\hint
Use the previous exercise.
\eh

\exercise[!]
Prove that $\{R, d\}=0$
\ez

\hint
Check that
\[ \{R, d\} \Lie_{\vec{r}} \alpha=  
Rd \Lie_{\vec{r}} \alpha + dR \Lie_{\vec{r}} \alpha
= - R\Lie_{\vec{r}} d\alpha + d\alpha = 0.
\] 
For any $\beta \in \ker \{R, d\}\cap \Lambda^iM$, 
satisfying $i>0$ or $\beta(0)=0$ for $i=0$, 
solve an equation $\Lie_{\vec{r}}\alpha=\beta$.
\eh

\exercise
Prove that
$\{ d, i_{\vec{r}}\} R(\alpha) =\alpha$, for any $i$-form
$\alpha$ on a starlike set, $i>0$.
\ez

\definition
Let $d$ be de Rham differential.
A form in $\ker d$ is called
{\bf closed}, a form in $\im d$ is called {\bf exact}.
Since $d^2=0$, any exact form is closed.
{\bf The group of $i$-th de Rham cohomology of $M$},
denoted $H^i(M)$, is a quotient of a space of closed
$i$-forms by exact: $H^*(M)=\frac{\ker d}{\im d}$.
\ed

\exercise[!]
Let $\alpha\in \Lambda^i U$ be a closed
$i$-form on a starlike set $U$, with $i>0$.
Prove that $\alpha = d i_{\vec{r}} R(\alpha)$.
\ez

\hint
Use the previous exercise.
\eh

\exercise[!]
(Poincar\'e lemma) Let $U$ be a starlike set.
Prove that $H^i(U)=0$ for each $i>0$, and $H^0(M)=\R$.
\ez

\exercise
Let $\theta$ be a closed odd form, and $d_\theta(x)=dx+\theta\wedge x$
the corresponding operator on $\Lambda^*M$.
Its {\bf cohomology} are defined as 
$H^*(\Lambda^*(M), d_\theta):=\frac{\ker d_\theta}{\im d_\theta}$
\enum
\ite Show that $d_\theta^2=0$.
\ite[*] Let $\theta$ be an exact 1-form. Prove that 
$H^i(\Lambda^*(M), d_\theta)$ are isomorphic to $H^i(M)$.
\ite[*] Let $\theta$ be a closed 1-form. Prove that 
$H^i(\Lambda^*(M), d_\theta)$ are isomorphic to $H^i(M)$,
or find a counterexample.
\ite[*] Let $\theta$ be a closed 3-form. Prove that 
$H^*(\Lambda^*(M), d_\theta)$ are isomorphic to $H^*(M)$,
or find a counterexample.
\ee
\ez



\end{document}
