Critical Homeomorphisms

Why This Matters

In algebraic topology, understanding how continuous maps behave at special points unlocks deep insights into the structure of spaces themselves. Critical homeomorphisms sit at the intersection of topology and analysis, connecting the smooth structure of manifolds to their topological invariants. You're being tested on your ability to recognize how local behavior at critical points—where derivatives vanish or fail to be invertible—translates into global topological information about the entire space.

This topic ties directly to Morse theory, one of the most powerful tools for decomposing manifolds into understandable pieces. The concepts here—critical point classification, stability under perturbation, gradient flows, and index calculations—appear repeatedly in exam questions that ask you to analyze how topology changes as you move through a function's level sets. Don't just memorize definitions—know what each concept tells you about the relationship between a function's critical behavior and the manifold's topology.

---

Foundations: What Makes a Point Critical

Before diving into applications, you need to understand the basic machinery. A critical point occurs where the differential of a smooth map fails to be surjective—intuitively, where the function "stalls" and local changes become degenerate.

Definition of Critical Homeomorphisms

• Continuous structure-preserving maps—these are homeomorphisms between topological spaces that exhibit special behavior at isolated critical points where smoothness conditions break down
• Differentiable manifold context—critical homeomorphisms are typically studied on smooth manifolds where we can take derivatives and identify where the differential $df$ fails to be invertible
• Bridge between analysis and topology—they connect the calculus-based notion of "derivative equals zero" to topological properties of the underlying space

Critical Points and Their Significance

• Points where $\nabla f = 0$—critical points are locations where all partial derivatives vanish simultaneously, signaling potential extrema or saddle behavior
• Local extrema and saddle points—these points indicate local maxima, local minima, or saddle points, each carrying distinct topological meaning
• Global structure from local data—the count and type of critical points encode information about the manifold's Euler characteristic and homology groups

---

Classification and Local Analysis

The power of critical point theory comes from our ability to classify points based on local behavior. The Hessian matrix—the matrix of second partial derivatives—tells us everything we need to know about non-degenerate critical points.

Classification of Critical Points

• Hessian matrix $H_f = \left(\frac{\partial^2 f}{\partial x_i \partial x_j}\right)$—this symmetric matrix encodes second-order behavior and determines the critical point's type
• Index of a critical point—the number of negative eigenvalues of the Hessian; index 0 means minimum, index $n$ means maximum, other values indicate saddle points
• Morse Lemma application—near a non-degenerate critical point, the function looks like $f(x) = f(p) - x_1^2 - \cdots - x_k^2 + x_{k+1}^2 + \cdots + x_n^2$ where $k$ is the index

Local Behavior Near Critical Points

• Taylor expansion analysis—the second-order Taylor series $f(x) \approx f(p) + \frac{1}{2}(x-p)^T H_f (x-p)$ reveals whether the point is a max, min, or saddle
• Non-degeneracy condition—a critical point is non-degenerate when $\det(H_f) \neq 0$, meaning the Hessian has no zero eigenvalues
• Topological stability—non-degenerate critical points are isolated and persist under small perturbations, making them tractable for topological analysis

---

Morse Theory Connections

Morse theory transforms the study of critical points into a systematic method for understanding manifold topology. The key insight: passing through a critical value changes the topology of sublevel sets in a predictable way.

Relation to Morse Theory

• Morse functions—smooth functions $f: M \to \mathbb{R}$ with only non-degenerate critical points; these are generic (most functions are Morse after small perturbation)
• Handle attachment—crossing a critical point of index $k$ corresponds to attaching a $k$-handle to the sublevel set, changing its homotopy type
• Morse inequalities—if $c_k$ = number of index-$k$ critical points and $b_k$ = $k$-th Betti number, then $c_k \geq b_k$ with equality in the alternating sum (Euler characteristic)

Connection to Gradient Flows

• Negative gradient field $-\nabla f$—defines a dynamical system where points flow toward lower values of $f$, with critical points as equilibria
• Stable and unstable manifolds—the stable manifold of a critical point $p$ consists of all points flowing toward $p$; these manifolds organize the topology
• Morse-Smale condition—when stable and unstable manifolds intersect transversely, the gradient flow gives a cell decomposition of the manifold

---

Stability and Invariants

Understanding when critical points persist under perturbation—and what topological information they encode—is essential for applications.

Stability of Critical Homeomorphisms

• Structural stability—a critical homeomorphism is stable if nearby functions (in $C^2$ topology) have the same number and types of critical points
• Persistence under perturbation—non-degenerate critical points persist; small changes shift their location but don't create or destroy them
• Bifurcation phenomena—at degenerate critical points, perturbations can split one critical point into several or annihilate pairs of critical points

Role in Studying Topological Invariants

• Euler characteristic formula—$\chi(M) = \sum_{k=0}^{n} (-1)^k c_k$ where $c_k$ counts critical points of index $k$; this is independent of the choice of Morse function
• Betti number bounds—critical point counts give upper bounds on homology ranks, with perfect Morse functions achieving equality
• CW-complex structure—Morse theory provides a systematic way to build the manifold as a cell complex, revealing its homotopy type

---

Concrete Examples

Seeing these ideas in action solidifies understanding. The examples below illustrate how critical point analysis reveals topology.

The Cubic Function $f(x) = x^3 - 3x$

• Two critical points—setting $f'(x) = 3x^2 - 3 = 0$ gives critical points at $x = \pm 1$, with $f''(1) = 6 > 0$ (minimum) and $f''(-1) = -6 < 0$ (maximum)
• One-dimensional Morse theory—the sublevel sets $\{x : f(x) \leq c\}$ change from empty to an interval to two intervals to one interval as $c$ increases
• Index calculation—the minimum has index 0, the maximum has index 1, and $\chi(\mathbb{R}) = 1 - 1 = 0$ is undefined (non-compact), but restrictions to intervals work

Height Function on the Torus

• Four critical points—a vertical torus has one minimum (bottom), one maximum (top), and two saddles (inner and outer circles at middle height)
• Index distribution—one index-0, two index-1, one index-2 critical point; check: $\chi(T^2) = 1 - 2 + 1 = 0$ ✓
• Handle decomposition—start with a disk (0-handle), attach two 1-handles (creating genus), cap with a 2-handle (closing the surface)

Applications in Manifold Theory

• Existence of Morse functions—every smooth manifold admits a Morse function; this is foundational for using critical point methods
• Classification results—Morse theory helps prove that surfaces are classified by genus, and provides tools for higher-dimensional classification
• Invariant construction—critical point data feeds into Floer homology, Morse homology, and other sophisticated invariants used in modern topology

---

Quick Reference Table

---

Self-Check Questions

1. Compare and contrast the height function on a sphere versus a torus: how do the numbers and indices of critical points reflect the difference in their Euler characteristics?

2. If a Morse function on a closed surface has exactly one minimum, one maximum, and four saddle points, what is the genus of the surface? Which topological invariant lets you calculate this?

3. Two critical points both satisfy $\nabla f = 0$, but one has Hessian with eigenvalues $(2, 3)$ and the other has eigenvalues $(2, -3)$. How do their indices differ, and what does this mean for the local topology?

4. Explain why non-degenerate critical points are "stable" while degenerate ones are not. What happens to a degenerate critical point under a generic small perturbation?

5. (FRQ-style) Given a Morse function on a compact manifold with $c_0 = 1$, $c_1 = 3$, $c_2 = 3$, and $c_3 = 1$, calculate the Euler characteristic. What can you conclude about the Betti numbers $b_1$ and $b_2$?