Manifolds Unfolded

Mathematics often advances by discovering unexpected bridges. One of the most beautiful such bridges connects calculus—the study of functions and derivatives—with topology, the study of shape.
This bridge is called Morse Theory, introduced by Marston Morse.

The central idea is surprisingly simple, yet profound:

The global shape of a space can be understood by studying the behavior of functions on the space.

This post explains that idea carefully—from intuition to precise mathematics—without shortcuts or vague metaphors.

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1. The Core Question

Suppose you are given a smooth surface or space. You want to understand its shape, more precisely, its topology (e.g. how many holes it has, is it connected or not etc.).

Differential Topology answers these questions, but often in abstract language.
Morse theory asks instead:

Can we recover this information using ordinary calculus?

The answer is yes, provided we choose the right kind of function.

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2. Functions as Probes of Geometry

Let $M$ be a smooth manifold. Choose a smooth function:$f:M\to \mathbb R$ which assigns a real number to each point of $M$ in a smooth fashion.
Nothing topological has happened yet—we are still doing calculus.

Now define, for each real number $a$,$$M_a = \{ x \in M \mid f(x) \le a \}.$$called sublevel sets.

As $a$ increases, the sets $M_a$​ grow.
The key question is:

When does the shape of $M_a$​ actually (topologically) change?

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3. Regular Points and Critical Points

At a point $p \in M$, the derivative of $f$ measures how the function changes in nearby directions.

• If the derivative is nonzero, then near $p$, the function behaves like a slope.
• If the derivative is zero, then the function is locally flat.

Definition (Critical Point)

A point $p \in M$ is a critical point of $f$ if the differential $df_p:T_pM\to \mathbb R$ of $f$ vanishes at $p$.

The corresponding value $f(p)$ is called a critical value.

Key fact:
If $a$ is not a critical value, then the sublevel sets $M_a$​ and $M_{a+\varepsilon}$​ have the same topology, for small enough $\epsilon$​.

So topology only changes at critical values.

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4. Non-Degeneracy: Why Second Derivatives Matter

Not all critical points are equally useful.

At a critical point $p$, consider the matrix $H_p(f)$ of second partial derivatives $\frac{\partial^2f}{\partial x^i\partial x^j}$ (the Hessian) with respect to any local coordinate chart about $p$ (Note that, the Hessian can be defined in a coordinate-independent way at a critical point $p$).
If the Hessian matrix is invertible, the critical point is called non-degenerate. Otherwise, it is called degenerate.

This non-degeneracy condition ensures that the critical point is stable and isolated.

Definition (Morse Function)

A smooth function is a Morse function if all its critical points are non-degenerate.

A deep but important fact is:

Morse functions are abundant (generic)—almost any smooth function can be slightly adjusted to become Morse.

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5. The Morse Lemma: Local Normal Form

The power of Morse theory begins with a precise local statement.

Morse Lemma

If $p$ is a non-degenerate critical point of $f$, then there exist local coordinates near $p$ such that$$f(x) = f(p) – x_1^2 – \cdots – x_k^2 + x_{k+1}^2 + \cdots + x_n^2.$$

No approximation. No remainder term. This equality is exact in suitable coordinates.

The integer $k$ is called the index of the critical point.

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6. What the Index Means

The index counts how many independent directions near $p$ cause the function to decrease in a neighborhood of that critical point.

• Index $0$: local minimum (the function increases in every direction around the point)

• Index $n$: local maximum (the function decreases in every direction around the point)

• Intermediate index: saddle point (the function increases in some directions, while decreases in some other directions)

This number is not a picture—it is an invariant determined by second derivatives.

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7. How Topology Changes

Now comes the central theorem.

As the parameter $a$ crosses a critical value corresponding to a critical point of index $k$, the topology of $M_a$​ changes in a precise way:

A k-dimensional cell is attached to the space.

This means:

• Index 0 → a new connected component appears
• Index 1 → a tunnel is formed
• Index 2 → a cavity is filled
• Higher indices → higher-dimensional analogues

Nothing else ever happens.

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8. A Concrete Example: The Torus

Take a vertically oriented torus $\mathbb T^2\subset \mathbb R^3$ and define $f$ to be the height function.

As height increases:

1. A minimum appears → one component forms
2. First saddle → a tunnel opens

3. Second saddle → another tunnel opens

4. Maximum → a surface caps everything off

From these four critical points alone, we recover the torus’s topology.

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9. Why Morse Theory Matters

Morse theory is not a trick. It is a philosophy:

• Local information (derivatives) determines global structure (topology)

This idea underlies modern geometry, dynamical systems, and even quantum field theory.

Whenever a space is complicated, one searches for the right function on it.

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10. Final Perspective

Morse theory teaches a profound lesson:

To understand a space, do not stare at it—let a function explore it.

Which echoes with Felix Klein‘s philosophy:

Geometry is dictated by the functions on those geometric objects.

as geometry in its basic level, is topology.

With nothing more exotic than derivatives and quadratic forms, Morse theory reveals the hidden architecture of space.