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\begin{document}

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\section*{Morse theory}

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definitions}{Definitions}\dotfill \pageref*{definitions} \linebreak
\noindent\hyperlink{sketch_of_a_trivial_application}{Sketch of a trivial application}\dotfill \pageref*{sketch_of_a_trivial_application} \linebreak
\noindent\hyperlink{slightly_lesstrivial_example}{Slightly less-trivial example}\dotfill \pageref*{slightly_lesstrivial_example} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak
\noindent\hyperlink{general}{General}\dotfill \pageref*{general} \linebreak
\noindent\hyperlink{morse_complex_and_homology}{[[Morse complex]] and homology}\dotfill \pageref*{morse_complex_and_homology} \linebreak
\noindent\hyperlink{relation_to_supersymmetric_quantum_mechanics}{Relation to supersymmetric quantum mechanics}\dotfill \pageref*{relation_to_supersymmetric_quantum_mechanics} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

\textbf{Morse theory} is the method of studying the [[topology]] of a [[smooth manifold]] $M$ by the study of [[Morse functions]] $M\to\mathbb{R}$ and their associated [[gradient flows]].

Classical Morse theory centered around simple statements like Morse inequalities, concerning just the [[Betti number]]s. It is useful not only for studying manifolds, but also for studying infinite CW-type spaces \emph{homotopically filtered in manifolds}, as by Milnor and Bott (especially \emph{The stable homotopy of the classical groups}) for spaces of paths in a smooth manifold.

Novikov--Morse theory is a variant using [[multivalued functions]]. There is also a [[discrete Morse theory]] for combinatorial cell complexes.

There are some infinite-dimensional generalizations like Floer instanton homology for 3-dimensional manifolds and also the Hamiltonian variant of [[Floer homology]] (and cohomology) for (finite dimensional) [[symplectic manifolds]]. Especially well studied is the case of the [[cotangent bundle]] of a differentiable manifold with its standard symplectic structure; this is sometimes called Floer--Oh homology. Floer homology has been partly motivated by Arnold's conjecture on periodic trajectories in [[classical mechanics]]. The symplectic variant of Floer cohomology is related to [[quantum cohomology]].

Founders of Morse theory were [[Marston Morse]], [[Raoul Bott]] and [[Albert Schwarz]].

\hypertarget{definitions}{}\subsection*{{Definitions}}\label{definitions}

On a smooth manifold $M$, a [[smooth map|smooth]] function $\varphi: M \to \mathbb{R}$ is said to be \emph{Morse} (or \emph{a Morse function}) if

\begin{itemize}%
\item the zero set of $d \varphi$ consists of isolated points, and
\item the [[Hessian]] of $\varphi$ at these points is nondegenerate.

\end{itemize}
The Morse functions on $M$ are dense in most reasonable topologies you could put on $C^{\infty}(M)$. A further condition which is useful in case $M$ is not compact is

\begin{itemize}%
\item if the (closed!) preimage of $( -\infty , \lambda ]$ under $\varphi$ is compact for all $\lambda$, then $\varphi$ is said to be \emph{coercive}, whether or not it is Morse.

\end{itemize}
Together with a (smooth) Riemann structure $g =\langle \cdot,\bullet\rangle$, any real function $\varphi$ on $M$ defines a [[flow]] on $M$ by the equation

\begin{displaymath}
- \langle \dot x, Y_x \rangle = Y_x \varphi = d\varphi(Y_x).
\end{displaymath}
The Morse functions are notable in that the flows they define have isolated fixed-points with trivially linearizable dynamics, and

 no other stable cyles.

When $\varphi$ is Morse and coercive, the unstable manifolds of the fixpoints can be arranged into a [[CW complex]] $C_{unstable} (\varphi,g)$, canonically homeomorphic to $M$. When $M$ is compact, $\varphi$ and $-\varphi$ are automatically both coercive, and $-\varphi$ induces a [[Poincare duality|dual]] CW complexe $C_{stable} (\varphi,g)$. Concretely, \ldots{}

.

\hypertarget{sketch_of_a_trivial_application}{}\subsubsection*{{Sketch of a trivial application}}\label{sketch_of_a_trivial_application}

Let $X \to Y$ be a surjective submersion of compact smooth manifolds, and assume $Y$ is connected. By suitable implicit function theorems, the preimage of any parametrized nonstationary curve $\gamma :(0,1)\to Y$ is a submanifold of $X$, and furthermore the parameter is a Morse function on this submanifold, having \emph{no critical points}. (It is \emph{not} coercive). By a very little more analysis, the Morse gradient flow is therefore a smooth family of homotopy equivalences. A trivial adjustment of the Riemann structure further allows that the Morse flow sends fibers to fibers diffeomorphically, so that in fact the fibers over neighboring points of $Y$ are diffeomorphic. But since $Y$ is connected, this implies that \emph{all} the fibers are diffeomorphic, so that $X\to Y$ is a smooth fiber bundle over $Y$.

\hypertarget{slightly_lesstrivial_example}{}\subsubsection*{{Slightly less-trivial example}}\label{slightly_lesstrivial_example}

The restriction to the [[unit sphere]] in $\mathbb{R}^{n+1}$ of a generic quadratic form is Morse with $2(n+1)$ critical points --- two of each [[bilinear form|index]]; and furthermore this restriction clearly descends to $\mathbb{RP}^n$ as a Morse function with $n+1$ critical points, one of each index. It can be shown that this is indeed the minimal collection of critical points supported by real projective space.

\hypertarget{references}{}\subsection*{{References}}\label{references}

\hypertarget{general}{}\subsubsection*{{General}}\label{general}

\begin{itemize}%
\item [[Raoul Bott]], \emph{\href{http://archive.numdam.org/article/PMIHES_1988__68__99_0.pdf}{Morse theory indomitable}}, \emph{Publications Math\'e{}matiques de L'IH\'E{}S}, 1988, Volume 68, Number 1, Pages 99-114.


\item [[Raoul Bott]], \href{http://www.ams.org/journals/bull/1982-07-02/S0273-0979-1982-15038-8/}{Lectures on Morse Theory, Old and New}, Bull. Amer. Math. Soc. 7 (1982), 331-358.


\item [[Raoul Bott]], The stable homotopy of the classical groups. Ann. of Math. (2) 70 1959 313--337.


\item [[Daniel Freed]], \href{http://www.ams.org/journals/bull/2011-48-04/S0273-0979-2011-01349-0/}{Commentary} on ``Lectures on Morse Theory, Old and New'', Bull. Amer. Math. Soc., 48(4), October 2011, 517--523


\item [[Marco Gualtieri]], \emph{\href{http://www.math.toronto.edu/mgualt/Morse%20Theory/Morse_2009.html}{Course page}}, lecture notes and links.


\item [[Martin Guest]], \href{http://arxiv.org/abs/math/0104155}{Morse theory in the 1990s}.


\item [[Loring Tu]], \emph{Morse theory} \href{http://www.math.harvard.edu/history/bott/bottbio/node9.html}{node} on online bio of R. Bott


\item [[M. M. Postnikov]],     --- .: , 1971



\end{itemize}
\hypertarget{morse_complex_and_homology}{}\subsubsection*{{[[Morse complex]] and homology}}\label{morse_complex_and_homology}

\begin{itemize}%
\item [[John Milnor]], \emph{Lectures on the h-cobordism theorem}, Notes by L. Siebenmann \& J. Sondow, Princeton Univ. Press, 1965.
\item Matthias Schwarz, \emph{Morse homology}, Progress in Mathematics \textbf{111}, 1993

\end{itemize}
There is also a variant due Barannikov, and in more abstract form Viterbo:

\begin{itemize}%
\item S. Barannikov, \emph{The framed Morse complex and its invariants}, Advances in Soviet Math. 21 (1994), 93-115.


\item Fran\c{c}ois Laudenbach, \emph{On an article by S. A. Barannikov}, \href{http://arxiv.org/abs/1509.03490}{arxiv/1509.03490}



\end{itemize}
\hypertarget{relation_to_supersymmetric_quantum_mechanics}{}\subsubsection*{{Relation to supersymmetric quantum mechanics}}\label{relation_to_supersymmetric_quantum_mechanics}

The relation to [[supersymmetric quantum mechanics]] is due to

\begin{itemize}%
\item [[Edward Witten]], \emph{Supersymmetry and morse theory} (\href{http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.jdg/1214437492}{Euclid})

\end{itemize}
Reviews include

\begin{itemize}%
\item [[Gábor Pete]], section 2 of \emph{Morse theory}, lecture notes 1999-2001 (\href{http://www.math.bme.hu/~gabor/morse.pdf}{pdf})

\end{itemize}


\end{document}