nLab
Morse theory

Idea

Morse theory is the method of studying the topology of a smooth manifold $M$ by the study of Morse functions $M\to ℝ$ 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 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.

Definitions

On a smooth manifold $M$, a smooth function $\phi :M\to ℝ$ is said to be Morse (or a Morse function) if

• the zero set of $d\phi$ consists of isolated points, and
• the Hessian of $\phi$ at these points is nondegenerate.

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

• if the (closed!) preimage of $(-\mathrm{\infty },\lambda ]$ under $\phi$ is compact for all $\lambda$, then $\phi$ is said to be coercive, whether or not it is Morse.

Together with a (smooth) Riemann structure $g=⟨\cdot ,•⟩$, any real function $\phi$ on $M$ defines a flow on $M$ by the equation

$$-⟨\dot{x},Y_x⟩=Y_x\phi =d\phi (Y_x).$$- \langle \dot x, Y_x \rangle = Y_x \varphi = d\varphi(Y_x).

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

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 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 $Y$.

Slightly less-trivial example

The restriction to the unit sphere in ${ℝ}^{n+1}$ of a generic quadratic form is Morse with $2(n+1)$ critical points — two of each index; and furthermore this restriction clearly descends to ${\mathrm{ℝ\mathbb{P}}}^n$ 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.

References

• 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.

• Loring Tu, Morse theory node on online bio of R. Bott

• M. M. Postnikov, Введение в теорию Морса — М.: Наука, 1971

The relation to supersymmetric quantum mechanics is due to

• Edward Witten, Supersymmetry and morse theory (Euclid)