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\begin{document}
\pagestyle{fancy} \lhead{Differential Topology, Assignment 1}
\rhead{Due date: September 26, 2013} \cfoot{} 
% \begin{quote}
% \emph{Treat this
% as a take-home exam.  Don't discuss problem sets until after they are
% handed in.  If you need clarification or help, ask me after class.  }
% \end{quote}
\section{Extensions}
\noindent Let $U$, $W$ be finite-dimensional vector spaces. An
\emph{extension} of $W$ by $U$ is a vector space $V$, together with
linear maps $\alpha,\beta$ such that
\begin{equation}
  \label{eq:1}
  \xymatrix{0\ar[r] & U \ar[r]^\alpha & V\ar[r]^\beta & W\ar[r] & 0}
\end{equation}
is an exact sequence of linear maps, meaning that the kernel of each
map is equal to the image of the previous map in the sequence. 

\begin{exercise}A \emph{splitting} $s$ of the sequence~\eqref{eq:1} is a map $s:W\to V$ such that
  $\beta s = \id_W$.
\begin{enumerate}[a)]
\item Prove that such a splitting exists.  
\item Prove that a splitting $s$ induces a map $t:V\to U$ such that
  $s\beta + \alpha t = \id_V$. As a result, show that $t\alpha =
  \id_U$ and that this gives an isomorphism between $V$ and $U\oplus W$.
\item Prove that the set of splittings of~\eqref{eq:1} is an affine
  space modeled on $\Hom(W, U)$.

  Recall that an affine space $A$ modeled on a vector space $V$ is a
  set $A$ of ``points'' together with a transitive and faithful action
  of the group $(V,+)$ of ``translations''.  Essentially, $A$ is a
  copy of $V$ without a distinguished origin. 
\end{enumerate}
\end{exercise}

\begin{exercise}
Let $X,Y$ be finite-dimensional vector spaces and let $R\in\Hom(X,Y)$. 
As described in class, this gives rise to two exact sequences
\begin{align}
    \xymatrix{0\ar[r] & \Ker R\ar[r] & X\ar[r] & \Im R\ar[r] & 0}\label{eq31}\\
    \xymatrix{0\ar[r] & \Im R\ar[r] & Y\ar[r] & \Cok R\ar[r] & 0}\label{eq32},
 \end{align}
where $\Cok R$ is the cokernel of $R$, defined by the quotient $Y/\Im R$.
Prove that if bases for $\Ker R$, $\Im R$, and $\Cok R$ are chosen, and if splittings
of~\eqref{eq31} and~\eqref{eq32} are chosen, then this defines bases for $X$ and $Y$, 
and that relative to these bases, $R$ has a matrix with a block form given by
\begin{equation}
  \label{eq:4}
  R = \begin{pmatrix}
    \mathbf{1}_{r\times r} & 0\\
    0 & 0
  \end{pmatrix},
\end{equation}
where $r$ is the rank of $R$ and $\mathbf{1}_{r\times r}$ is an identity matrix of size $r$.



\end{exercise}


\begin{exercise}
  Let $(V,\alpha,\beta)$ and $(V',\alpha',\beta')$ be two extensions
  of $W$ by $U$.  A morphism of extensions from $V$ to $V'$ is a
  linear map $\psi:V\to V'$ rendering the following diagram commutative.
  \begin{equation}
    \begin{aligned}
    \label{eq:2}
    \xymatrix@R=1ex{ 
      &V\ar[rd]^\beta\ar[dd]^-{\psi} & \\
       U \ar[ru]^\alpha\ar[rd]_{\alpha'} &  & W\\
       &V'\ar[ru]_{\beta'} & 
      }
    \end{aligned}
  \end{equation}
  \begin{enumerate}[a)]
  \item Prove that any such morphism is an isomorphism.
  \item Prove that any two extensions of $W$ by $U$ are isomorphic.
  \item Choose bases for $W$ and $U$, and concatenate to form a basis
    for the trivial extension $V = U\oplus W$. Describe the matrix of
    a general automorphism of the extension $V$.
\end{enumerate}

\end{exercise}

\section{Koszul sequences}
Let $V$ be a vector space of dimension $n$, and fix
  $\xi\in V^*\setminus\{0\}$.  Then $\xi$ defines an operator $e_\xi$ on
  $\wedge^\bullet V^*$ defined by 
\begin{equation}
  \label{eq:4}
  e_\xi(\sigma) = \xi\wedge \sigma.
\end{equation}
Note that $e_\xi$ sends $\wedge^k V^*$ to $\wedge^{k+1}V^*$; in other words
it is an operator of degree $+1$. 

Now choose $v\in V$ and define an operator $i_v$ on $\wedge^\bullet
V^*$ as follows: on $\wedge^0 V^*$ it is zero, and on $\wedge^1 V^*$
it is given by the pairing between $V$ and $V^*$, so that for
$\alpha\in\wedge^1 V^*=V^*$, we have
\begin{equation}
  \label{eq:6}
  i_v \alpha = \alpha(v).
\end{equation}
To complete the definition of $i_v$, we require that it is a graded
derivation of $\wedge^\bullet V^*$, which means that it satisfies the
following Leibniz rule:
\begin{equation}
  \label{eq:8}
  i_v(\alpha\wedge \beta) = (i_v\alpha)\wedge\beta + (-1)^\alpha \alpha\wedge i_v\beta.
\end{equation}
The operator defined in this way is called ``the interior product by
$v$'' and is of degree $-1$.

\begin{exercise}\mbox{}

\begin{enumerate}[a)]

\item Prove that the following sequence is an exact sequence:
\begin{equation}
  \label{eq:5}
  \xymatrix{0\ar[r] & \wedge^0 V^*\ar[r]^-{e_\xi} & \wedge^1 V^*\ar[r]^-{e_\xi} & \cdots\ar[r]^-{e_\xi} & \wedge^{n-1}V^*\ar[r]^-{e_\xi} & \wedge^n V^*\ar[r] & 0}.
\end{equation}

\item Prove that $i_vi_v = 0$.

\item Prove that the following
sequence is an exact sequence:
\begin{equation}
  \label{eq:9}
  \xymatrix{0 & \wedge^0 V^*\ar[l]& \wedge^1 V^*\ar[l]^-{i_v} & \cdots\ar[l]^-{i_v} & \wedge^{n-1}V^*\ar[l]^-{i_v} & \wedge^n V^*\ar[l]^{i_v} & 0\ar[l]}.
\end{equation}

\item Compute the operator $i_v e_\xi + e_\xi i_v$.  It should be an operator of degree $0$, of course. 
\item Fix $\xi$ and $v$ as above, but with $\xi(v)=1$.  Prove that any k-form $\alpha\in \wedge^k V^*$ can be
decomposed as follows:
\begin{equation}
  \label{eq:7}
  \alpha = \xi \wedge \beta + \gamma,
\end{equation}
for a unique pair $(\beta,\gamma)$ such that $i_v\beta= i_v\gamma = 0$.
\end{enumerate}
\end{exercise}


\end{document}
