\classheader{2013-01-07}

The last time I taught this course was in 2002. Back then, differential topology was taught first. Here is what I covered back then, together with some book recommendations:

\begin{itemize}
\item For basics (e.g., the definition of differentiable manifold), I recommend Chapter 1 of Frank Warner's \textit{Foundations of Differentiable Manifolds and Lie Groups}.

\item I also covered the notion of ``general position''. A good reference for this is Chapter 1 of Milnor's \textit{Topology from the Differentiable Viewpoint}. The book by Guillemin and Pollack is also good for this.

\item Hirsch's \textit{Differential Topology} is good for everything.

\item The final main topic I covered last time was de Rham cohomology, the Poincar\'e lemma, etc.
\end{itemize}

\subsection*{General Position}

Given $f_0:M\to N$, how beautiful can a perturbation $f$ (of $f_0$) get?

For example, consider the map $f_0:\S^1\to\R^2$ with the following graph:
\begin{center}
\begin{tikzpicture}[scale=0.8]
\draw [thick,postaction={decorate},decoration={markings,mark=at position 0.55 with {\arrow[>=angle 90,scale=0.8]{>}},mark=at position 0.95 with {\arrow[>=angle 90,scale=0.8]{>}}}] plot [smooth cycle, tension=1] coordinates {(0,0) (-1,2) (-2,0) (-1,-2)};
\draw [thick,postaction={decorate},decoration={markings,mark=at position 0.45 with {\arrow[>=angle 90,scale=0.8]{>>}},mark=at position 0.9 with {\arrow[>=angle 90,scale=0.8]{>>}}}] plot [smooth cycle, tension=1] coordinates {(0,0) (-2,1) (-4,0) (-2,-2)};
\end{tikzpicture}
\end{center}
The focus of our attention is at the point of self-tangency. We have two options: we can make the curve avoid itself,
\begin{center}
\begin{tikzpicture}
\draw[thick,postaction={decorate},decoration={markings,mark=at position 0.35 with {\arrow[>=angle 90,scale=0.8]{>>}},mark=at position 0.85 with {\arrow[>=angle 90,scale=0.8]{>>}}}] (0,0) to[bend right=20] (0,1);
\draw[thick,postaction={decorate},decoration={markings,mark=at position 0.25 with {\arrow[>=angle 90,scale=0.8]{>}},mark=at position 0.85 with {\arrow[>=angle 90,scale=0.8]{>}}}] (0.4,0) to[bend right=40] (0.4,1);
\end{tikzpicture}
\end{center}
or we can make it intersect itself in a non-tangent way,
\begin{center}
\begin{tikzpicture}
\draw[thick,postaction={decorate},decoration={markings,mark=at position 0.25 with {\arrow[>=angle 90,scale=0.8]{>>}},mark=at position 0.85 with {\arrow[>=angle 90,scale=0.8]{>>}}}] (0,0) to[bend right=80,distance=2cm] (0,1);
\draw[thick,postaction={decorate},decoration={markings,mark=at position 0.25 with {\arrow[>=angle 90,scale=0.8]{>}},mark=at position 0.85 with {\arrow[>=angle 90,scale=0.8]{>}}}] (0.9,-0.2) to[bend right=40] (0.9,1.2);
\end{tikzpicture}
\end{center}
We might also consider a map $f_0:\S^1\to\R^3$ that intersects itself:\vspace{-0.1in}
\begin{center}
\begin{tikzpicture}
\node (y) at (0,0) {};
\path[thick] (y.center) edge[loop,distance= 1.7cm,out=330,in=30] node[pos=0.2,auto,swap] {} (y.center);
\path[thick] (y.center) edge[loop,distance= 1.7cm,out=150,in=210] node[pos=0.8,auto,swap] {} (y.center);
\end{tikzpicture}
\end{center}\vspace{-0.1in}
But because we are working in $\R^3$, there is no reason it should intersect itself at all; we can just perturb the curve at the intersection appropriately.
\begin{center}
\begin{tikzpicture}
\node (y) at (0,0) {};
\path[thick,shorten >=0.1in] (y.center) edge[loop,distance= 1.7cm,out=330,in=30] node[pos=0.2,auto,swap] {} (y.center);
\path[thick,shorten >=0.1in] (y.center) edge[loop,distance= 1.7cm,out=150,in=210] node[pos=0.8,auto,swap] {} (y.center);
\end{tikzpicture}
\end{center}
This is an illustration of the Whitney theorem; given an $n$-dimensional manifold, we can always find an embedding in $\R^k$ for $k\geq 2n+1$, and an immersion for $k=2n$. This result requires Sard's theorem. These theorems fall under the heading of general position.

Here are some other results I plan to talk about:

\begin{itemize}
\item Given a map $f_0:M\to N$ and a closed submanifold $A\hookrightarrow N$, the preimage $f_0^{-1}(A)$ may not be a submanifold of $M$, but after an appropriate perturbation, $f^{-1}(A)$ is a submanifold.
\item A closed submanifold $A\hookrightarrow N$ of codimension $r$ defines a cohomology class $\theta(A)\in H^r(N;\Z/(2))$. If its normal bundle is oriented, we actually get a cohomology class $\theta(A)\in H^r(N;\Z)$.
\item Let $f:M\to N$ be a map. If $f$ is transverse to $A$, then $f^{-1}(A)$ is also a submanifold and $\theta(f^{-1}(A))\in H^r(M;\Z/(2))$ and $f^*(\theta(A))=\theta(f^{-1}(A))$. If $M$ and $A$ are both submanifolds of $N$ that meet transversally, then $M\cap A$ is a submanifold of $N$ and $\theta(A)\cupprod\theta(M)=\theta(A\cap M)$.
\end{itemize}

This whole course is repeated applications of three statements, two of which you've seen already:
\begin{itemize}
\item Inverse Function Theorem
\item Existence and Uniqueness of Solutions to ODEs
\item Sard's Theorem
\end{itemize}

\subsection*{Differentiable Manifolds}

As you should all know, a topological manifold is a space which is locally homeomorphic to $\R^n$.

We can't consistently decide when a function on a manifold is differentiable. To fix this problem, we give a manifold a differentiable structure. The usual definition of a differentiable structure on a topological manifold is via coordinate charts. However, if you work with complex algebraic geometry, a more natural definition is with sheaves. You can find the definition of sheaf, subsheaf, etc. in Warner's book, or in Gunning's \textit{Lectures on Riemann Surfaces}.

Let $\Omega\subset\R^n$ be an open set. A function $f:\Omega\to\R$ is $k$-times continuously differentiable, and we say that $f$ is $C^k$, when the following holds:
\begin{itemize}
\item[(a)] If $k=0$, $f$ is continuous, and
\item[(b)] if $k>0$, then $\partial_i f(x)$ are defined for all $1\leq i\leq n$ and all $x\in\Omega$, and $\partial_i f:\Omega\to \R$ is $C^{k-1}$.
\end{itemize}
We say that $f$ is $C^\infty$ when $f$ is $C^k$ for all $k$. We can define the notions of $C^k$, $C^\infty$, and $C^\omega$ manifolds (the notation $C^\omega$ means real analytic, i.e., the Taylor series equals the function). For any topological space $X$, we will write $C^0(X)$ for the set of real-valued continuous functions on $X$, and $C^0_X$ for the sheaf of such functions on $X$.

Given an open $\Omega\subset\R^n$, we can consider the set $R=C^k(\Omega)$ of $C^k$ functions on $\Omega$ (or $C^\infty(\Omega)$ or $C^\omega(\Omega)$, respectively). Observe that 
\begin{itemize}
\item[(1)] $R$ is a ring under pointwise addition and multiplication, 
\item[(2)] all constant functions are in $R$, and
\item[(3)] $R$ is a subsheaf of the sheaf of continuous functions $C_\Omega^0$; that is, for all $\Omega'\subseteq\Omega$ we have $R(\Omega')\subseteq C^0(\Omega')$, and the restriction of an element of $R(\Omega)$ to $\Omega'$ is an element of $R(\Omega')$.
\end{itemize}
Thus, in fact, $R$ is a sheaf of $\R$-algebras.

Given a pair $(M,R)$ where $M$ is a topological space and $R$ is a subsheaf of $C_M^0$, we say that a pair $(U,\psi)$ is a coordinate chart when $U\subset M$ is open and $\psi:U\to\R^m$ is a homeomorphism onto its image $\psi(U)$, which we require to be open in $\R^m$.

We say that a coordinate chart $(U,\psi)$ is admissible if, for all open $U'\subset U$ and $h\in C^0(U')$,
\[h\in R(U')\iff h\circ (\psi|_{U'})^{-1}\in C^\infty(\psi(U')).\]
(Note that the map $\psi|_{U'}:U'\to\psi(U')$ is a homeomorphism because $\psi$ was a homeomorphism of $U$ onto its image.) Lastly, we say that $M$ is a $\Cinfty$ manifold if it is covered by admissible coordinate charts. The definitions of $C^k$ and $C^\omega$ manifolds are analogous.

{\color{myred}{\rule{\textwidth}{0.02in}}\vspace{-0.1in}\bf
\begin{center}
$\blacktriangledown$\quad What is the redundancy, exactly?\quad $\blacktriangledown$
\end{center}}
This definition actually has a redundancy in the case of $C^\infty$ manifolds. This is because for $C^\infty$ manifolds, we have partitions of unity available to us.

\vspace{-0.1in}{\color{myred}{\rule{\textwidth}{0.02in}}}


But I'm using the more general definition because other objects, such as real analytic or complex analytic manifolds, do not have partitions of unity.

Let's review:
\begin{enumerate}
\item We've defined the notion of a coordinate chart for any topological space $M$.
\item Given a pair $(M,R)$ where $R\subseteq C^0_M$, we defined what it means for a chart to be admissible.
\item $(M,R)$ is a $C^\infty$ manifold if $M$ is covered by admissible charts.
\end{enumerate}

When discussing a given $\Cinfty$ manifold $M$, we will refer to its sheaf as $C_M^\infty$.

