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\title{MAT 239: Homework 6}

\begin{document}
	\maketitle
	
	\begin{enumerate}
		
		\item Let $f:X\to Y$ and $g:Y\to Z$ be smooth maps between manifolds, and suppose $X$ is compact. Further suppose that $\dim(X)+\dim(W)=\dim(Z)$. Assume $g$ is transverse to a submanifold $W$ of $Z$ so $g^{-1}(W)$ is a submanifold of $Y$. Prove that
		$$I_2(f,g^{-1}(W)) = I_2(g\circ f,W).$$
		
		\item Let $\Delta \subset S^1\times S^1$ be the diagonal submanifold $\Delta = \{(e^{i\theta},e^{i\theta}) \subset S^1\times S^1 \}$. We will calculate $I_2(\Delta,\Delta)$.
			\begin{enumerate}
				\item Show that $f:S^1\to S^1\times S^1$ defined by $f(e^{i\theta}) = (e^{i\theta}, e^{i(\theta+\varepsilon)})$ is homotopic to the inclusion of the diagonal $i:S^1\to S^1\times S^1$ $i(e^{i\theta}) = (e^{i\theta},e^{i\theta})$.
				\item Show that for $0<\varepsilon<2\pi$, the above defined map $f$ is transverse to $\Delta$.
				\item Calculate $I_2(\Delta,\Delta)$ by counting the number of points in $f^{-1}(\Delta)$ mod 2.
			\end{enumerate}
		
		\item If $f:X\to Y$ is homotopic to a constant map and $\dim X>0$, show that for any submanifold $Z\subset Y$ such that $\dim X+\dim Z=\dim Y$, $I_2(f,Z)=0$.
		
		\item Let $f_0,f_1:X\to \R^n$ be two smooth maps. The \emph{straight line homotopy} from $f_0$ to $f_1$ is defined by $F:X\times I\to \R^n$
		$$F(x,t) = (1-t)f_0(x)+tf_1(x).$$
		(Note, it is important that the target space is $\R^n$ so we have the vector space operations of scalar multiplication and addition.)
			\begin{enumerate}
				\item Check that $F$ is indeed a smooth homotopy from $f_0$ to $f_1$.
				\item Prove that every map $f:X\to \R^n$ is homotopic to a constant map.
				\item Conclude using the previous problem that for any map $f:X\to \R^n$ with $\dim X>0$ and any submanifold $Z\subset \R^n$ with $\dim X+\dim Z=n$ that $I_2(f,Z)=0$.
			\end{enumerate}
		
		\item Two compact submanifolds $K,L\subset M$ are cobordant in $M$ if there exists a compact submanifold with boundary $W$ in $M\times I$ (where $\partial W = W\cap \partial (M\times I)$) such that $W\cap M\times\{0\} = K\times \{0\}$ and $W\cap M\times\{1\} = L\times \{1\}$. (Note that this means $\dim K= \dim L = \dim W-1$.) Prove that if $K$ and $L$ are cobordant in $M$ then for every submanifold $Z$ of $M$ such that $\dim K+\dim Z=\dim M$, $I_2(K,Z) = I_2(L,Z)$.
		
		\item So far we have defined the (mod 2) intersection number between two submanifolds, or between a map and a submanifold of its target space. In this problem we will define the (mod 2) intersection number between two maps. Suppose $X$ and $Z$ are compact manifolds, and $f:X\to Y$ and $g:Z\to Y$ are smooth maps into a manifold $Y$. Suppose further that $\dim X+\dim Z = \dim Y$. Let $\Delta = \{(y,y)\in Y\times Y\}$ be the diagonal submanifold. Define
		$$I_2(f,g):= I_2(f\times g,\Delta).$$
			\begin{enumerate}
				\item Check that the dimensions add up appropriately for $I_2(f\times g,\Delta)$ to make sense.
				\item Prove that $I_2(f,g)=I_2(f',g)=I_2(f,g')$ if $f$ is homotopic to $f'$ and $g$ is homotopic to $g'$.
				\item Prove that $I_2(f,g) = I_2(g,f)$. (Hint: Use the map $s:Y\times Y\to Y\times Y$ defined by $s(a,b)=(b,a)$, and the first exercise.)
				\item In the case that $Z$ is a submanifold of $Y$ and $g$ is the inclusion map $g=i:Z\to Y$, check that $I_2(f,i) = I_2(f,Z)$. (Check that the new definition gives the same answer as the old definition.)
				\item Use the previous part to prove that if $X,Z$ are compact submanifolds of $Y$ then $I_2(X,Z)=I_2(Z,X)$.
			\end{enumerate}
		
	\end{enumerate}
\end{document}