Morse Theory

Prerequisites: Manifold, Calculus. Full path (3 pages)

• Intuitive
• Concrete
• Abstract

Intuitive

Stand a doughnut (a torus) upright on a table, like a tyre on its edge, and slowly fill the room with water. Watch the shape of the part of the doughnut that is under water as the level rises. Most of the time the wet region only grows, without changing its shape in any essential way. Its shape changes only at a few special heights where the surface is horizontal. At the bottom point a small wet disk appears. At the lower inner edge of the hole the wet region closes into a band around the tyre. At the upper inner edge the band closes over the hole, and the wet region is the whole doughnut except a small disk around the top. At the very top point that last disk is filled in. These special points are the minima, saddles and maxima of the height function.

Morse theory turns this observation into a method. From the list of critical points of a function on a space, and what kind each one is, we can read off how the space is put together and get information about its shape that does not depend on the particular function. For the doughnut we find one minimum, two saddles and one maximum. Counting minima and maxima with a plus sign and saddles with a minus sign gives zero, the Euler characteristic of the torus, for any such function on it. On a sphere the height function has only a minimum and a maximum, and the same count gives two.

In physics the same ideas appear whenever a potential or an action is studied through its stationary points: equilibria of a potential energy, classical paths as stationary points of an action, and tunnelling between minima.

• A visual introduction to Morse theory

Concrete

Let $M$ be a smooth $n$-dimensional manifold and $f:M\to\mathbb{R}$ a smooth function. A point $p$ is critical if $df(p)=0$, that is, $\partial f/\partial x^i=0$ in local coordinates. At a critical point the Hessian $$H_{ij}=\frac{\partial^2 f}{\partial x^i\partial x^j}(p)$$ is well defined, independent of coordinates up to the transformation of a symmetric bilinear form. The critical point is nondegenerate if $\det H\neq0$, and its index $\lambda$ is the number of negative eigenvalues of $H$. A Morse function is one whose critical points are all nondegenerate. Morse functions exist on every manifold, and any smooth function can be turned into one by an arbitrarily small perturbation.

The Morse lemma states that near a nondegenerate critical point of index $\lambda$ there are coordinates in which $$f = f(p) - x_1^2-\dots-x_\lambda^2 + x_{\lambda+1}^2+\dots+x_n^2.$$ So index $0$ is a local minimum, index $n$ a local maximum, and the indices in between are saddles with $\lambda$ downhill directions.

Example: the upright torus. Take the height function on a torus standing on its edge. It has four critical points: the bottom ($\lambda=0$), the bottom and top of the inner circle of the hole ($\lambda=1$ each, saddle points), and the top ($\lambda=2$). On a round sphere the height function has the bottom ($\lambda=0$) and the top ($\lambda=2$).

Let $c_\lambda$ be the number of critical points of index $\lambda$ on a compact manifold and $b_\lambda$ the Betti numbers, the number of independent $\lambda$-dimensional holes. Then $$\sum_\lambda(-1)^\lambda c_\lambda = \chi(M) = \sum_\lambda(-1)^\lambda b_\lambda,$$ and the weak Morse inequalities hold, $c_\lambda\ge b_\lambda$. For the torus, $b_0=1$, $b_1=2$, $b_2=1$, so the height function above has the smallest possible number of critical points. The inequalities also work in the other direction: knowing the topology of $M$, they give a lower bound on the number of critical points of any Morse function on it, for example the number of equilibrium positions of a mechanical system with configuration space $M$ whose equilibria are all nondegenerate.

Abstract

For a Morse function $f$ on a compact manifold $M$ write $M^a=f^{-1}((-\infty,a])$. The two basic theorems are:

1. If $f^{-1}([a,b])$ contains no critical point, then $M^a$ and $M^b$ are diffeomorphic, and $M^a$ is a deformation retract of $M^b$. The flow of $-\nabla f/|\nabla f|^2$ (for any Riemannian metric) pushes the level set $f=b$ down to $f=a$.
2. If $f^{-1}([a,b])$ contains exactly one critical point $p$, in its interior, and $p$ has index $\lambda$, then $M^b$ is homotopy equivalent to $M^a$ with a $\lambda$-cell attached, and diffeomorphic to $M^a$ with a $\lambda$-handle $D^\lambda\times D^{n-\lambda}$ attached.

Hence $M$ has the homotopy type of a CW complex with one $\lambda$-cell for each critical point of index $\lambda$. This gives the strong Morse inequalities $$\sum_{k=0}^{\lambda}(-1)^{\lambda-k}c_k\ \ge\ \sum_{k=0}^{\lambda}(-1)^{\lambda-k}b_k\quad\text{for each }\lambda,$$ with equality for $\lambda=n$, which is the Euler characteristic identity.

A sharper statement is the Morse complex. Choose a metric such that the stable and unstable manifolds of the gradient flow intersect transversally (the Morse-Smale condition). Let $C_\lambda$ be the free abelian group generated by the critical points of index $\lambda$, and define $\partial:C_\lambda\to C_{\lambda-1}$ by counting, with signs determined by orientations, the gradient flow lines from a critical point of index $\lambda$ to one of index $\lambda-1$. Then $\partial^2=0$ and the homology of $(C_*,\partial)$ is isomorphic to the singular homology of $M$. The Morse inequalities follow from this by linear algebra.

Witten gave a physical derivation using supersymmetric quantum mechanics on $M$. Deform the exterior derivative to $d_t=e^{-tf}d\,e^{tf}$, which has the same cohomology as $d$, and consider the Laplacian $\Delta_t=d_td_t^*+d_t^*d_t$, which plays the role of a supersymmetric Hamiltonian. For large $t$ it contains the potential $t^2|df|^2$, and its low-lying eigenstates in degree $\lambda$ localize near the critical points of index $\lambda$. Perturbation theory gives $c_\lambda$ approximate zero modes in degree $\lambda$, while the exact zero modes, the harmonic forms, number $b_\lambda$. The difference is caused by tunnelling between critical points whose index differs by one, and the tunnelling paths (instantons) are the gradient flow lines that define the Morse differential.

Morse theory also applies to infinite-dimensional spaces of paths. Geodesics between two points are critical points of the energy functional on the path space, and the Morse index theorem states that the index of a geodesic equals the number of points in its interior that are conjugate to the starting point, counted with multiplicity. Floer homology extends the Morse complex to action functionals whose critical points have infinite index and coindex, such as the symplectic action, whose critical points are periodic orbits of Hamiltonian systems.

Path to this page

1. Calculus
2. Linear Algebra
3. Manifold

Authors: Jakob Schwichtenberg