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 numbers. It is useful not only for studying manifolds, but also for studying infinite CW-type spaces homotopically filtered in manifolds, as by Milnor and Bott (especially 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.
On a smooth manifold $M$, a smooth function $\varphi: M \to \mathbb{R}$ is said to be Morse (or a Morse function) if
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
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
The Morse functions are notable in that the flows they define have isolated fixed-points with trivially linearizable dynamics, and ...[fixme: less vague?]... 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 dual CW complexe $C_{stable} (\varphi,g)$. Concretely, ….[details].
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 no critical points. (It is 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 all the fibers are diffeomorphic, so that $X\to Y$ is a smooth fiber bundle over $Y$.
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 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.
Raoul Bott, Morse theory indomitable, Publications Mathématiques de L’IHÉS, 1988, Volume 68, Number 1, Pages 99-114.
Raoul Bott, Lectures on Morse Theory, Old and New, Bull. Amer. Math. Soc. 7 (1982), 331-358.
Raoul Bott, The stable homotopy of the classical groups. Ann. of Math. (2) 70 1959 313–337.
Daniel Freed, Commentary on “Lectures on Morse Theory, Old and New”, Bull. Amer. Math. Soc., 48(4), October 2011, 517–523
Marco Gualtieri, Course page, lecture notes and links.
Martin Guest?, Morse theory in the 1990s.
M. M. Postnikov, Введение в теорию Морса — М.: Наука, 1971
There is also a variant due Barannikov, and in more abstract form Viterbo:
S. Barannikov, The framed Morse complex and its invariants, Advances in Soviet Math. 21 (1994), 93-115.
François Laudenbach, On an article by S. A. Barannikov, arxiv/1509.03490
The relation to supersymmetric quantum mechanics is due to
Reviews include