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\title{MAT 239: Homework 6}

\begin{document}
	\maketitle
	
	\begin{enumerate}
		\item Let $C_{p,q}$ be the curve in the torus $T^2=\R^2/\Z^2$ given by the points
			$$C_{p,q} = \{ (qt,pt) \mid t\in \R \}.$$
			Orient $C_{p,q}$ by the image of the positive $\partial_t$ vector under the differential of the immersion $\phi(t) = (qt,pt)$.
			\begin{enumerate}
				\item Calculate the oriented intersection number $I(C_{1,0},C_{p,q})$.
				\item Calculate the oriented intersection number $I(C_{0,1},C_{p,q})$.
				\item Calculate the oriented intersection number $I(C_{3,5},C_{p,q})$.
				\item (Ungraded) Can you find a general formula for $I(C_{m,n},C_{p,q})$?
			\end{enumerate}
		
		\item Recall that the \emph{degree} of a map $f:X\to Y$ where $\dim X=\dim Y$ is the oriented intersection number of $f$ with a point $\{y\}$. For a positive integer $m$ prove that
			\begin{enumerate}
				\item $f_m:S^1\to S^1$ defined by $f_m(z) = z^m$ (viewing $z\in S^1\subset \C$ as a complex number) has degree $m$.
				\item $g_m:S^1\to S^1$ defined by $g_m(z) = \overline{z}^m$ has degree $-m$ (where the bar denotes complex conjugation).
			\end{enumerate}
		
		\item The antipodal map $a:S^k\to S^k$ is defined by $a(x)=-x$.
			\begin{enumerate}
				\item Compute the degree of $a$ in terms of $k$.
				\item Prove that the antipodal map is homotopic to the identity if and only if $k$ is odd. Hint: To construct the homotopy when $k$ is odd consider the following homotopy in the $k=1$ case:
				$$\left[\begin{array}{cc} \cos(\pi t) & -\sin(\pi t)\\ \sin(\pi t) & \cos(\pi t) \end{array}  \right]$$
			\end{enumerate}
		
		\item Euler characteristic and vector fields. Let $\Delta\subset X\times X$ be the diagonal submanifold of points $\{(x,x)\}$.
			\begin{enumerate}
				\item Show that the tangent bundle is
				$$T_{(x,x)}(\Delta)=\{(v,v)\in T_xX\times T_xX \cong T_{(x,x)}(X\times X)\}$$ 
				and the normal bundle is
				$$N_{(x,x)}(\Delta;X\times X) = \{(v,-v)\in T_xX\times T_xX \cong T_{(x,x)}(X\times X) \}.$$
				\item Prove that the map $\Psi: TX\to N(\Delta;X\times X)$ defined by $\Psi(x,v) = ((x,x),(v,-v))$ is a diffeomorphism. By the tubular neighborhood theorem, this implies there is a neighborhood of $X$ in $TX$, a neighborhood of $\Delta$ in $X\times X$, and a diffeomorphism between these neighborhoods which sends $X$ to $\Delta$ in the usual way $x\mapsto (x,x)$.
				\item Let $Z$ denote the zero section $Z=\{(x,0)\in TX\}$. Prove that the self-intersection number of the diagonal $I(\Delta,\Delta)$ in $X\times X$ (i.e. the Euler characteristic of $X$) is equal to the self-intersection number of the zero section $I(Z,Z)$ in $TX$. (Orient $TX$ by locally in each coordinate chart specifying a basis for $T_{(x,v)}(TX)$ of the form $(\partial_{x_1},\dots, \partial_{x_n},\partial_{y_1},\dots, \partial_{y_n})$ where $(x_1,\dots, x_n)$ give coordinates on $X$ and $(y_1,\dots, y_n)$ are the corresponding induced coordinates on $T_xX$.
				\item Let $V:X\to TX$ be any vector field on $X$. Prove that $I(V,Z)=I(Z,Z)$ by constructing a homotopy from $V$ to the inclusion of the zero section. Conclude that the Euler characteristic of $X$ is equal to $I(V,Z)$ for any vector field $V$ on $X$.
			\end{enumerate}
		
		\pagebreak
		
		\item Morse theory and Euler characteristic.
			\begin{enumerate}
				\item Let $f_k(x_1,\dots, x_n)= -x_1^2-\cdots -x_k^2+x_{k+1}^2+\cdots x_n^2$ be the standard model for an index $k$ Morse critical point. Calculate the gradient vector field 
				$$\nabla f =\left\langle \frac{\partial f}{\partial x_1},\cdots, \frac{\partial f}{\partial x_n}\right\rangle.$$
				\item Let $Z$ denote the zero section $Z =\{((x_1,\dots, x_n),(0,\dots, 0))\in T\R^n\}$. Prove that $\nabla f$ is transverse to $Z$.
				\item Calculate $I(\nabla f, Z)$ in terms of $k$.
				\item Suppose $f:X\to \R$ is a Morse function. By the Morse Lemma, for every critical point of $f$, there exist local coordinates defined near that point such that $f$ looks like one of the standard models in part (a). The corresponding value of $k$ is called the \emph{index} of the critical point of $f$. Let $k(p)$ denote the index of a critical point $p\in Crit(f)$. Prove that 
				$$I(\nabla f, Z) = \sum_{p\in Crit(f)} (-1)^{k(p)}.$$
				(By the previous problem, this computes the Euler characteristic of $X$. Note that a key idea in Morse theory is that a Morse function on $X$ induces a cell decomposition where critical points of index $k$ are in bijection with cells of dimension $k$. This recovers the definition of Euler characteristic as an alternating sum of the number of cells.)
			\end{enumerate}
		
		\item Let $X$ be a manifold with boundary, $Y$ a manifold, and $Z$ a submanifold. Fix orientations on $X$, $Y$, and $Z$. Let $f:X\to Y$ be a smooth map such that $f$ and $\partial f$ are transverse to $Z$. Then $W:=f^{-1}(Z)$ is a submanifold with boundary of $X$. Let $A$ denote the boundary of $f^{-1}(Z)$. Show that the following two induced orientations on $A$ differ by $(-1)^{\codim Z}$:
		\begin{itemize}
			\item First give $W=f^{-1}(Z)$ the preimage orientation induced by $f$. Then give $A$ the boundary orientation induced as the boundary of $W$.
			\item First give $\partial X$ the boundary orientation induced as the boundary of $X$. Then give $A$ the preimage orientation induced as the preimage of $\partial f: \partial X\to Y$.
		\end{itemize}
		Hint: At a fixed point $p\in A$, show that 
		\begin{enumerate}
			\item you can choose the same vector $n_p$ to serve as an outward pointing vector in $T_p(W)$ to $A$, and as an outward pointing vector in $T_p(X)$ to $\partial X$.
			\item you can choose the same basis $(v_1,\dots, v_k)$ for the normal space $N_p(A;\partial X)$ to serve also as a basis for $N_p(W;X)$.
		\end{enumerate}
		Then name a basis $(w_1,\dots, w_\ell)$ for $T_pA$ and an oriented basis $(u_1,\dots, u_m)$ for $T_{f(p)}Z$, and use the definitions of induced orientations to determine what condition ensures that $(w_1,\dots, w_\ell)$ represents the above specified orientation in each of the two cases.
		
	\end{enumerate}

\end{document}