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  \begin{document}


  
  \vskip 1cm

  \centerline{\Large {Partitions of unity and transversality}}
\vskip 1cm
  


\subsection*{Partitions of unity}

\begin{enumerate}[1.]

  \item Let $X$ be a manifold and $A\subset X$ a subset. Suppose that
    a function $f:A\to\R$ is {\it locally smoothly extendable},
    i.e. for every $a\in A$ there is a neighborhood $U_a$ of $a$ and a
    smooth function $f_a:U_a\to\R$ such that $f$ and $f_a$ agree on
    $U_a\cap A$. Show that $f$ {\it smoothly extendable}, i.e. there
    is an open set $U\supseteq A$ and a smooth function $\tilde
    f:U\to\R$ that extends $f$.

  \item Let $X$ be a manifold, $A\subset X$ a closed subset and $U$ an
    open set containing $A$. Let $f:U\to\R$ be smooth. Show that there
    is a smooth function $g:X\to\R$ that agrees with $f$ on $A$. This
    is called the {\it Extension Lemma}.

  \item Prove the following improvement of the smooth approximation
    theorem from class. Let $X$ be a manifold, $\epsilon:X\to
    (0,\infty)$ a continuous function and $f:X\to\R^k$ a continuous
    function. Then there is a smooth function $g:X\to\R^k$ such that
    $\|f(x)-g(x)\|<\epsilon(x)$ for every $x\in X$.

  \item When $U\subset S^n$ is an open set, by a {\it vector field} on
    $U$ we mean a smooth function $v:U\to\R^{n+1}$ such that for every
    $x\in U$ the vector $v(x)\in\R^{n+1}$ is tangent to $x\in S^n$
    (i.e. $v(x)$ and $x$ are perpendicular). Show that for any compact
    subset $K\subset U$ and any vector field $v$ on $U$ there is a
    vector field on all of $S^n$ that agrees with $v$ on $K$.

  

\end{enumerate}

\subsection*{Transversality}

  
  
\begin{enumerate}[resume]

    \item Show that the subspaces of symmetric matrices and of
      skew-symmetric matrices are transverse in the space $M(n)$ of
      all $n\times n$ matrices.

    \item Let $X$ be a manifold. If $A,B\subset X$ are two
      transverse submanifolds and $x\in A\cap B$ show that $T_x(A\cap
      B)=T_x(A)\cap T_x(B)$.

      \item Show that it is possible for two submanifolds $A,B\subset
        X$ to have intersection $A\cap B$ a submanifold with
        $codim(A)+codim(B)=codim(A\cap B)$ even though $A,B$ are {\it
          not} transverse to each other. Here $codim(A)=\dim X-\dim
        A$.

      \item For which values of $a>0$ does the hyperboloid defined by $x^2 + y^2 - z^2 = 1$
intersect the sphere $x^2 + y^2 + z^2 = a$ transversally ? What does the
intersection look like for different values of $a$?

\item Let $f:X\to X$ be a smooth self-map of a manifold $X$. Let
  $$Gr(f)=\{(x,f(x))\in X\times X\}$$ be the graph of $f$ and let
  $\Delta\subset X\times X$ be the diagonal, i.e. the graph of the
  identity. Thus $Gr(f)\cap\Delta$ consists of points $(x,x)\in
  X\times X$ such that $f(x)=x$. Show that $Gr(f)\pitchfork\Delta$ if
  and only if for every fixed point $x$ of $f$ the derivative
  $df_x:T_x(X)\to T_x(X)$ does {\it not} have eigenvalue 1. Hint: The
  tangent space at $(x,f(x))$ of $Gr(f)$ is the graph of
  $df_x:T_x(X)\to T_{f(x)}(X)$ as a subspace of $T_{(x,f(x))}(X\times
  X)=T_x(X)\oplus T_{f(x)}(X)$, which you can prove in charts.





\item Find a counterexample to the Stability Theorem for
  transversality if compactness is not assumed. That is, find
  manifolds $X, P, Y\supset Z$ and a smooth map $F:X\times P\to Y$ so
  that $F_{p_0}\pitchfork Z$ but for every neighborhood $U$ of $p_0\in
  P$ there are points $p\in U$ so that $F_p$ is not transverse to $Z$.
\end{enumerate}
      
\end{document}
