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\title{MAT 239: Homework 4}

\begin{document}
	\maketitle
	
	\begin{enumerate}
		\item Let $f:\R^n\to \R$ be defined by $f(x_1,\dots, x_n) = \varepsilon_1 x_1^2+\cdots +\varepsilon_n x_n^2$ where $\varepsilon_i\in \{-1,1\}$. 
		\begin{enumerate}
			\item For $n=2$, sketch $f^{-1}(-1)$, $f^{-1}(0)$, and $f^{-1}(0)$ for the different choices of $\varepsilon_i$.
			\item Show that $0$ is the only critical value of $f$, and conclude that $f^{-1}(1)$ and $f^{-1}(-1)$ are submanifolds. 
			\item Describe $T_{(x_1,\dots, x_n)}(f^{-1}(1))$.
		\end{enumerate}
		
		
		\item Let $SL(n;\R)$ denote the set of $n\times n$ matrices of determinant $1$. 
			\begin{enumerate}
				\item Prove that $SL(n;\R)$ is a manifold of dimension $n^2-1$. [Hint: Show that if $\det(A)\neq 0$, $\det$ is a submersion even when restricted to the set $\{tA\mid t>0\}\subset M_{n\times n}$.]
				\item Let $I$ be the identity matrix. Prove that $T_I(SL(n;\R))$
			\end{enumerate}
		
		\item Let $p:\C\to \C$ be a complex polynomial of one variable $z=x+iy$, $p(z)=z^m+a_{m-1}z^{m-1}+\cdots + a_1z+a_0$. Prove that $p$ has finitely many critical points.
		
		\item Let $L: \C^2\to \C$ be the complex map $L(z_1,z_2) = z_1z_2$ where $z_j=x_j+iy_j$ and we use complex multiplication. Breaking up the function into real components, show that $0\in \C$ is the only critical value, so in particular, $L^{-1}(1)$ is a submanifold. What manifold is $L^{-1}(1)$ diffeomorphic to? What does $L^{-1}(0)$ look like?
		
		\item Consider the function $f:S^2\to \R$ defined by $f(x,y,z)=z$ for $(x,y,z)\in S^2$. Find all the critical points, critical values, and regular values of $f$.
		
		\item Is the intersection of these two submanifolds transverse in $\R^3$? Explain why or why not.
		\begin{enumerate}
			\item $\{z=x^2+y^2 \}$ with $\{z=0\}$ in $\R^3$.
			\item $\{x=0\}$ with $\{z=0\}$ in $\R^3$.
			\item $\{x=y=0\}$ with $\{x=z=0\}$ in $\R^3$.
			\item $\{x=y=0\}$ with $\{x=z=1\}$ in $\R^3$.
			\item $\{x=y=0\}$ with $\{z=x+y\}$ in $\R^3$.
			\item $\{x=y=0\}$ with $\{x=y^2+z^2\}$ in $\R^3$.
		\end{enumerate}
	
		\item Let $f:X\to Y$ be a smooth map transverse to a submanifold $Z\subset Y$. Let $W=f^{-1}(Z)$. (We know that $W$ is a submanifold of $X$.) Prove that $T_xW$ is the preimage of $T_{f(x)}(Z)\subset T_{f(x)}Y$ under the linear map $df_x: T_xX\to T_{f(x)}Y$.

	\end{enumerate}
\end{document}