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\title{Smooth Manifolds Fall 2022 - Homework 9}

\begin{document}

\maketitle


\begin{problem}
Show that if $M_i$ are compact, connected, oriented manifolds of the same dimension for $i = 1,2,3$, and $f : M_1 \to M_2$ and $g : M_2 \to M_3$ are $C^\infty$ functions, then $\deg(g \of f) = \deg(g)\deg(f)$.
\end{problem}

\begin{problem}
Show that if $M$ and $N$ are compact, connected and oriented manifolds of the same dimension, and $F : M \to N$ has $\deg(F) = 0$, then $F$ has a critical point.
\end{problem}


\begin{problem}
Let $A$ be an $n \times n$ square matrix with integer entries, $\T^n = \R^n / \Z^n$ and $F_A : \T^n \to \T^n$ be defined by

\[ F_A([x]) =[Ax]\]

where the notation $[x]$ denotes the equivalence class $x + \Z^n$. 

\begin{enumerate}
\item Show that $F_A$ is well-defined.
\item Compute $\deg(F_A)$ by computing the signed number of preimages of a regular value of $F_A$.
\item Compute $\deg(F_A)$ by computing $\int F_A^*\omega$, where $\omega$ is the standard $n$-form $dx_1 \wedge \dots \wedge dx_n$.
\end{enumerate}
\end{problem}

{\color{red} Non-graded.}

\begin{problem}
Let $\Gamma$ be a countable group acting properly discontinuously on an oriented manifold $M$ by diffeomorphisms. Show that the quotient manifold $M / \Gamma$ is orientable if and only if every $\gamma \in \Gamma$ is orientation-preserving.
\end{problem}

\begin{problem}
Let $\omega$ be a top form on a $C^\infty$ manifold $M$ and $\varphi_t : M \to M$ be a flow on $M$ generated by a vector field $X$. Show that $\varphi_t^*\omega = \omega$ for all $t \in \R$ if and only if $\iota_X\omega$ is closed. Use this to find a condition for a flow generated by a vector field $X = \sum f_i \frac{\del}{\del x_i}$ to preserve the standard top form $\omega = dx_1 \wedge \dots \wedge dx_n$ on $\R^n$.
\end{problem}

\begin{problem}
Let $f : M \to M$ be an orientation-preserving diffeomorphism of an oriented compact manifold $M$. Fix a non-vanishing top form $\omega_0$. Show that there is a unique $C^\infty$ function $\lambda : M \to \R$ such that $f^*\omega = e^\lambda \omega$, and $f$ preserves {\it some} non-vanishing top form if and only if there exists a $C^\infty$ function $h : M \to \R$ such that

\[  \lambda = h \of f - h. \]
\end{problem}

\begin{problem}
If $G$ is a connected, compact Lie group, consider the squaring map $s(g) = g^2$.

\begin{enumerate}[(a)]
\item Show that if $G  = \T^n$, then $\deg(s) = 2^s$. Check this formula both by computing the degree at a regular value, as well as using a volume form.
\item Show that if $G$ is the set of unit quaternions, then $\deg(s) = 2$ [{\it Hint}: Show that if $g \not= \pm 1$, then there is a unique 1-parameter subgroup passing through $g$ and that any square root must belong to it]
\end{enumerate}
\end{problem}

\begin{remark}
If you know about the structure of compact Lie groups, you may try the following more difficult exercise: $\deg(s) = 2^r$, where $r$ is the maximal dimension of a connected abelian subgroup of $G$.
\end{remark}




\end{document}




