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\begin{document}
\pagestyle{fancy} \lhead{Differential Topology, Assignment 4}
\rhead{Due date: Friday Nov 22, 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*{Degree}
% Let $M$ be a compact $n$-manifold and $f:M\to N$ a smooth map to an
% $n$-manifold $N$. The degree mod 2, $\deg_2(f)\in\ZZ/2\ZZ$ is defined as the
% cardinality of $f^{-1}(y)$ modulo $2$, for $y$ any regular value of
% $f$.  We first show that $\deg_2(f)$ is independent of $y$ and that it
% depends only on the homotopy class of $f$.



% \begin{exercise}
%   Let $X$ be a manifold with boundary and let $F:X\to N$ be a smooth
%   map.  Now choose $q\in N$ to be a regular value for both $F$ and
%   $\partial F = F|_{\partial X}$. Prove that $F^{-1}(q)$ is a
%   submanifold with boundary, and that this boundary is given by
%   $F^{-1}(q)\cap \partial X$.  Note that a submanifold with boundary
%   is defined to be locally modeled on the inclusion of half-spaces
%   $H^k\subset H^n$ given by
%   \[
%   (x_1,\ldots, x_k)\mapsto (0,\ldots,0,x_1,\ldots, x_k),
%   \]
%   where $x_k\geq 0$. To illustrate this result, provide one explicit
%   example of the situation in question.
% \end{exercise}

% \begin{exercise}
%   Let $g$ be a map smoothly homotopic to $f$, meaning that there
%   exists a smooth map
%   \begin{equation}
%     \label{eq:1}
%     F:[0,1]\times M\to N
%   \end{equation}
%   with $H(0,-)=f$ and $H(1,-)=g$. Choose a value $y\in N$ which is
%   regular for both $f$, $g$.  By studying the preimage $F^{-1}(y)$,
%   conclude that $\#f^{-1}(y)=\#g^{-1}(y)\pmod 2$. [First prove under the
%   assumption that $y$ is regular for $F$]
% \end{exercise}

% \begin{exercise}
%   Let $\gamma:[0,1]\to N$ be a smooth path with $\gamma(0)=y_0$ and
%   $\gamma(1)=y_1$.  Prove there exists a vector field $V$ on $N$ whose
%   flow $\Phi_t:N\to N$ is a diffeomorphism which moves $y_0$ to $y_t =
%   \gamma(t)$, for all $t\in [0,1]$.
% \end{exercise}

% \begin{exercise}
%   Any two regular values $y_0,y_1\in N$ of $f$ may be joined by a smooth path
%   $\gamma$. Use the results above to show that $\#f^{-1}(y_0)=\#f^{-1}(y_1)\pmod 2$.
% \end{exercise}

% % \vspace{2em}
% % \noindent An orientation on an n-dimensional real vector space $V$ is
% % an equivalence class of nonzero elements in $\wedge^n V$, where $v,v'$
% % are equivalent if $v'=\lambda v$ with $\lambda\in\RR_{> 0}$.  An
% % $n$-manifold $M$ is \emph{orientable} if $\wedge^n TM$ is isomorphic to the trivial
% % line bundle

% \pagebreak


\section*{Morse theory of the Grassmannian}

Recall that $\Gr(2,4)$ is the Grassmannian of 2-dimensional complex linear
subspaces $\Lambda\subset \CC^4$. We have a natural Hermitian inner product on $\CC^4$ given by 
\begin{equation}
  \label{eq:2}
  \IP{x,y} = x\cdot \overline{y} = \sum x_i\overline{y_i}.
\end{equation}
The symmetry group of this inner product is the unitary group
 \begin{equation}
 U(4) = \{X\in \GL(4,\CC)\ :\ XX^*=\id\},
 \end{equation}
 where $X^* = \overline{X}^\top$. 
\begin{exercise}
  Prove that $U(4)$ acts transitively on $\Gr(2,4)$ and determine the stabilizer subgroup 
  associated to the point $\Lambda_0 = \mathrm{Span}(e_1,e_2)$.
\end{exercise}

\noindent The Lie algebra of $U(4)$,
 i.e. the tangent space at the identity element, is the real vector
 space (of what dimension?) of skew-adjoint matrices
 \begin{equation}
 \uu(4) = \{X\in \End(\CC^4)\ :\ A + A^* = 0\}.
 \end{equation}
 This is called a Lie algebra because the commutator $[A_1,A_2]$ of
 two elements of $\uu(4)$ is also in $\uu(4)$.  There is a natural
 embedding of $\Gr(2,4)$ inside $\uu(4)$ which goes as follows:

 Any $\Lambda\in \Gr(2,4)$ has an orthogonal complement $\Lambda^\bot$
 using the Hermitian inner product, and from the decomposition of any
 element $x\in \CC^4$ as $x = x_\Lambda + x_{\Lambda^\bot}$ with
 $x_\Lambda\in \Lambda$ and $x_{\Lambda^\bot}\in\Lambda^\bot$, we can define
a projection operator $P_{\Lambda}:\CC^4\to \Lambda$ by 
\begin{equation}
  \label{eq:3}
  P_\Lambda : x\mapsto x_{\Lambda}.
\end{equation}

\begin{exercise}
  Prove that $\iota:\Lambda\mapsto iP_{\Lambda}$ is an embedding of
  $\Gr(2,4)$ into $\uu(4)$, and that for $g\in U(4)$, we have
  \begin{equation}
  \iota(g\cdot \Lambda) =g \iota(\Lambda)g^{-1},
  \end{equation}
  where on the left hand side we have the usual action of $g$ on a
  subspace of $\CC^4$ and on the right hand side we have the
  conjugation action of $U(4)$ on $\uu(4)$, known as the adjoint
  action.  Conclude that $\iota(\Gr(2,4))$ consists of a single
  adjoint orbit $\mathcal{O}$ and describe the point
  $\iota(\Lambda_0)$ in this orbit.
\end{exercise}


\noindent Let $\hh\subset \uu(4)$ be the diagonal skew-adjoint
matrices (a 4-dimensional real subspace).  Elements in $\hh$ with
distinct eigenvalues are called \emph{regular} elements of $\hh$, and
the others are called \emph{singular}.  Regular elements have the
following nice property: any element of $\uu(4)$ which commutes with a
regular diagonal element must itself be diagonal.



\begin{exercise}
  Let $\hh\subset \uu(4)$ be the diagonal matrices (a 4-dimensional
  real subspace). Determine the intersection of $\Gr(2,4)\cong\mathcal{O}$ with $\hh$.  
\end{exercise}

\noindent The real Lie algebra $\uu(4)$ has a Euclidean inner product 
\begin{equation}
  \label{eq:4}
  \IP{A,B} = \mathrm{Tr}(AB^*) = -\mathrm{Tr}(AB).
\end{equation}
This inner product is invariant under the adjoint action of $U(4)$, and also satisfies 
\[
\IP{[A,B],C} + \IP{B,[A,C]} = 0, 
\]
for all $A,B,C\in \uu(4)$.
We define a Morse function on the Grassmannian as follows: Choose real
numbers $a_1>\cdots>a_4$, so that $Z = \mathrm{diag}(ia_1,\ldots
ia_4)$ is a regular element in $\hh$.  Then we define 
\begin{equation}
  \label{eq:5}
  f(\Lambda) = \IP{\iota(\Lambda), Z}.
\end{equation}

\begin{exercise}
  Prove that $f$ is the restriction of the square of the distance
  function on $\uu(4)$ to the adjoint orbit
  $\mathcal{O}\cong\Gr(2,4)$: use the polarization identity to prove
  that
  \begin{equation}
    \label{eq:6}
    f(\Lambda) = -\tfrac{1}{2}||\iota(\Lambda) - Z||^2 + C,
  \end{equation}
  for some constant $C$ independent of $\Lambda$.

\end{exercise}

\begin{exercise}
  Use the above results to determine the critical points of $f$; a
  fact which will be useful is the following: the tangent space to
  $\mathcal{O}$ at a point $X\in\mathcal{O}$, translated to the origin
  of $\uu(4)$, coincides with the image of the infinitesimal adjoint
  action:
  \begin{equation}
    T_{X}\mathcal{O} = \{[X,Y]\ :\ Y\in\uu(4)\}.
  \end{equation}
\end{exercise}


\begin{bonus}
Determine the Morse indices of the critical points of $f$.
\end{bonus}
\begin{bonus}
Determine the descending manifolds of $f$.
\end{bonus}






\end{document}
