Morse Theory — Mathematics Knowledge System

Overview

Morse theory studies the topology of smooth manifolds through the critical points of smooth real-valued functions. A Morse function is a smooth function f: M → ℝ whose critical points (where df = 0) are all non-degenerate (the Hessian is non-singular). As the level set f⁻¹(c) rises through the manifold, its topology changes only at critical values — and the change is precisely determined by the index of the critical point (the number of negative eigenvalues of the Hessian). This gives a handle decomposition of M and leads to the Morse inequalities relating critical point counts to Betti numbers. Morse theory is the source of many deep results in topology, surgery theory, and Floer homology.

Intuition

Imagine a torus T² standing vertically on a table and a height function f (the height above the table). As you rise from the bottom, the level set f⁻¹(c) starts empty, then becomes a circle (near the bottom circle of the torus), then splits into two circles, merges back into one, and disappears at the top. Each topological change — each 'event' — corresponds to a critical point of f (where the level set changes topology). At the bottom, a minimum (index 0); at the inner top of the hole, a saddle (index 1); at the outer equator, another saddle (index 1); at the top, a maximum (index 2). The Morse inequalities count these critical points and relate them to the topology of T².

Formal Definition

Definition

Let M be a smooth n-manifold and f: M → ℝ a smooth function. A point p ∈ M is a critical point if df_p = 0. The Hessian H_p(f) is the symmetric bilinear form on T_pM defined (in local coordinates) by the matrix of second derivatives. The critical point p is non-degenerate if H_p(f) is non-singular. The index λ(p) is the number of negative eigenvalues of H_p(f).

df_p = 0 \iff p \text{ is a critical point}

Critical point condition

H_p(f) = \left(\frac{\partial^2 f}{\partial x^i \partial x^j}(p)\right)

Hessian matrix at a critical point

\lambda(p) = \text{number of negative eigenvalues of } H_p(f)

Morse index

C_k = \#\{\text{critical points of } f \text{ with index } k\}

Number of index-k critical points

C_k \geq b_k, \quad \sum_{k=0}^n (-1)^k C_k = \chi(M) = \sum_{k=0}^n (-1)^k b_k

Morse inequalities and Euler characteristic identity

Properties

Morse functions are generic

\text{Morse functions are dense in } C^\infty(M, \mathbb{R}) \text{ (every smooth function can be perturbed to a Morse function)}

Topology changes only at critical levels

\text{If } [a,b] \text{ contains no critical values, then } f^{-1}([a,b]) \cong f^{-1}(a) \times [0,1]

Handle attachment at critical points

\text{When crossing a critical value of index } \lambda, \text{ a } \lambda\text{-handle } D^\lambda \times D^{n-\lambda} \text{ is attached}

Theorems

Theorem 1: Morse Lemma

\text{Near a non-degenerate critical point } p \text{ of index } \lambda, \text{ there exist local coordinates in which } f = f(p) - x_1^2 - \cdots - x_\lambda^2 + x_{\lambda+1}^2 + \cdots + x_n^2.

Theorem 2: Morse Inequalities

C_k \geq b_k(M; \mathbb{Z}), \quad \sum_{k=0}^n (-1)^k C_k = \chi(M)

Theorem 3: Handle Decomposition

\text{Every compact smooth manifold M admits a CW decomposition with one cell of dimension } \lambda \text{ for each critical point of index } \lambda \text{ of any Morse function.}

Theorem 4: Weak Morse Inequalities

C_k \geq b_k \text{ for all } k, \text{ and } C_0 \geq 1, C_n \geq 1 \text{ for compact connected manifolds.}

Worked Examples

1
Parametrise T² by (θ, φ) → ((R + r cos θ)cos φ, (R + r cos θ)sin φ, r sin θ).
2
The height function is f = z = r sin θ. Critical points occur where df = 0:
$\frac{\partial f}{\partial \theta} = r\cos\theta = 0 \implies \theta = 0, \pi$
3
And ∂f/∂φ = 0 is automatically satisfied for all φ since f is independent of φ. Wait — f = r sin θ is independent of φ. Critical circles arise at θ = 0, π. For isolated critical points we need a slight perturbation or take a tilted torus.
4
For a generic height function on T² (torus tilted so all critical points are isolated), there are 4 critical points: 1 minimum (index 0), 2 saddles (index 1), 1 maximum (index 2).
$C_0 = 1,\; C_1 = 2,\; C_2 = 1$
5
Verify Euler characteristic: χ(T²) = C_0 − C_1 + C_2 = 1 − 2 + 1 = 0. Correct.
$\chi(T^2) = 0$

✓ Answer

A generic height function on T² has 1 minimum, 2 saddles, 1 maximum, satisfying χ = 1−2+1 = 0.

Practice Problems

Mediumfree response

A compact manifold M has a Morse function with C_0 = 1, C_1 = 3, C_2 = 3, C_3 = 1. What is χ(M)?

Hardproof writing

Prove that every compact smooth manifold admits at least 2 critical points for any Morse function.

Common Mistakes

Common Mistake

Every smooth function on a compact manifold is a Morse function

Morse functions have non-degenerate critical points (non-singular Hessian). Functions with degenerate critical points (like f(x)=x³ at x=0) are not Morse. However, Morse functions are generic: any smooth function can be perturbed slightly to become Morse.

Common Mistake

The Morse inequalities give equalities C_k = b_k

The Morse inequalities are inequalities: C_k ≥ b_k. Equality holds for a perfect Morse function (one that minimises the total number of critical points), but in general there can be excess critical points that cancel in homology.

Quiz

Historical Background

Morse theory was developed by Marston Morse in his 1934 book The Calculus of Variations in the Large, building on earlier work by Birkhoff and Poincaré on geodesics. Morse was motivated by the problem of counting geodesics on Riemannian manifolds. John Milnor's 1963 Morse Theory (Princeton Annals lecture notes) gave the modern clean exposition and remains the definitive introduction. Andreas Floer generalised Morse theory to infinite-dimensional settings in 1988, creating Floer homology, which led to spectacular results in symplectic geometry and 4-manifold topology (Donaldson's work). Witten's 1982 paper gave a supersymmetric quantum mechanics interpretation of Morse theory.

1934
Morse publishes The Calculus of Variations in the Large, founding Morse theory
Marston Morse
1963
Milnor's Morse Theory lecture notes give the modern exposition
John Milnor
1982
Witten interprets Morse theory via supersymmetric quantum mechanics
Edward Witten
1988
Floer generalises Morse theory to infinite dimensions, creating Floer homology
Andreas Floer

Summary

A Morse function f: M → ℝ has only non-degenerate critical points; the Morse index counts negative Hessian eigenvalues.
The Morse Lemma gives canonical local coordinates: f = f(p) − x₁² − ... − xˡ² + x_{λ+1}² + ... + xₙ² near each critical point.
Topology of level sets changes only at critical values; crossing a critical value of index λ attaches a λ-handle.
Morse inequalities: C_k ≥ b_k and Σ(−1)^k C_k = χ(M) — relating critical points to topology.
Floer homology (1988) extends Morse theory to infinite dimensions, revolutionising symplectic and 4-manifold topology.

References

BookMilnor, J. — Morse Theory, Princeton University Press, 1963 (Annals of Mathematics Studies 51)
BookMatsumoto, Y. — An Introduction to Morse Theory, American Mathematical Society, 2002
WebsiteWikipedia — Morse theory
WebsitenLab — Morse theory