Basics of Morse Theory and the Poincaré-Hopf Theorem

We mainly focus on Morse Theory by J. Milnor. In Chapter 6, we will use Theorem 11.27 and Lemma 11.20 from From Calculus to Cohomology (by Madsen and Tornehave) to prove the Poincaré-Hopf theorem.

0. Preliminaries

Definition 1 (Homotopy). Let be continuous maps. We say is homotopic to , denoted by , if there exists a continuous map such that:

Definition 2 (Homotopy Equivalence). Two spaces and are homotopy equivalent if there exist continuous maps and such that:

Definition 3 (Deformation Retract). Let . A deformation retract is a continuous map such that:

Lemma 1 (Pasting Lemma). Let , where and are closed subsets of . If and are continuous maps such that , then the map defined by is continuous on .

Definition 4 (CW-complex). A CW-complex is defined by induction on its skeleta:

: A discrete set of points (0-cells).
: Obtained by attaching -cells to the -skeleton via continuous attaching maps . Formally,

1. Introduction

Let be the height function above the plane. We denote the sublevel set by .

As the height increases, the topology of changes as follows:

.
is empty.
.
. Attaching a 0-cell
.
. Attaching a 1-cell
.
. Attaching a 1-cell
.
(the full torus). Attaching a 2-cell

Intuitively, near the critical points , the function can be approximated locally as:

Whenever we cross a critical value , it turns out that the topological change is completely determined by attaching a -cell to , where is the number of negative signs in the local quadratic form.

2. Definitions and Lemmas

Definition 5 (Critical point). A point is called a critical point of a smooth function , if for a local coordinate system in a neighborhood of , we have

Definition 6 (Non-degenerate critical point). A critical point is called non-degenerate if and only if the Hessian matrix evaluated at is non-singular, i.e.,

Lemma 2 (Lemma 2.1). Let be a function in a convex neighborhood of in , with . Then for some suitable functions defined in , satisfying .

Proof. By the fundamental theorem of calculus and the chain rule, we have: Therefore, we can simply define . ◻

Lemma 3 (Lemma 2.2, The Morse Lemma). Let be a non-degenerate critical point for . Then there exists a local coordinate system in a neighborhood of with for all , such that the identity holds throughout . Here, the integer is called the index of at , which equals the negative index of inertia of the Hessian matrix .

Proof. Without loss of generality, assume is the origin , and . If the lemma holds, the function takes the form , which gives a diagonal Hessian matrix at :

Step 1: Express as a quadratic form.
By Lemma 2.1, we can write in some neighborhood of . Since is a critical point, . Applying Lemma 2.1 again to each , we obtain .

Substituting this back, we get . By defining , we may assume without loss of generality that is symmetric.

Assertion: . Taking the first derivative yields: Taking the second derivative and evaluating at the origin gives: Thus, the matrix is non-singular.

Step 2: Imitate the diagonalization process.
By induction, suppose there exist coordinates in a neighborhood of such that where is symmetric.

Ensuring that : If , we can perform a linear change of coordinates to fix it:

If for , we simply swap coordinates and .
If but , we apply a rotation and , which converts the cross term into square terms, ensuring the new .

Now, let , which is a smooth and non-zero function in a smaller neighborhood . We define a new set of coordinates:

is a valid coordinate system: The Jacobian matrix has the block form: Evaluated at the origin, . Thus, , making it a valid diffeomorphism by the Inverse Function Theorem.

By completing the square, the function in a neighborhood becomes: which successfully isolates the -th term. The proof follows by induction. ◻

Corollary 1 (Corollary 2.3). Non-degenerate critical points are isolated.

Proof. By the Morse Lemma, there exists a neighborhood of where . Setting the gradient to zero yields . Thus, (the origin) is the only critical point in . ◻

Definition 7 (1-parameter group of diffeomorphisms). A 1-parameter group of diffeomorphisms of a manifold is a map , such that:

For each , the map defined by is a diffeomorphism of onto itself.
For all .

Definition 8 (Vector field generated by the group). For any smooth, real-valued function , the vector field generated by the group is defined by the directional derivative:

Lemma 4 (Lemma 2.4). A smooth vector field on which vanishes outside of a compact set generates a unique 1-parameter group of diffeomorphisms of .

Proof. The proof relies on solving the corresponding ordinary differential equation (ODE).

Step 1: Set up the ODE.
We need to find the integral curve satisfying: with the initial condition . (Here, the derivative means for any smooth function ).

Step 2: Local coordinate representation.
For any , choose a coordinate chart , where .

Step 3: Local existence and uniqueness.
In local coordinates, the vector field equation becomes a standard system of first-order ODEs: By the Picard-Lindelöf theorem, this system admits a unique smooth solution locally for .

Step 4: Global extension via compactness.
Since the vector field vanishes outside the compact set , we only need to worry about points inside . We can cover by a finite number of such neighborhoods , each with a guaranteed survival time . Let .

Since is strictly positive, the map is well-defined for all . For any arbitrary time , we can extend the flow globally by composing the map a finite number of times: This composition preserves smoothness and the group law (), ensuring that is a globally well-defined diffeomorphism. ◻

Remark 1. The condition that vanishes outside a compact set cannot be omitted.
Counterexample: Let the manifold be the open interval , and consider the constant vector field (i.e., ).

Solving the ODE yields .
Thus, the flow is given by . However, if we start at and let , the point reaches . The solution blows up in finite time, meaning cannot be defined for all .

3. Homotopy Type in Terms of Critical Values

Theorem 1 (Theorem 3.1). Let be a smooth real valued function on a manifold . Let and suppose that the set , consisting of all with , is compact, and contains no critical points of . Then is diffeomorphic to . Furthermore, is a deformation retract of , so that the inclusion map is a homotopy equivalence.

(Proof omitted.)

Theorem 2 (Theorem 3.2). Let be a smooth function, and let be a non-degenerate critical point with index . Setting , suppose that is compact, and contains no critical point of other than , for some . Then, for all sufficiently small , the set has the homotopy type of with a -cell attached.

Proof of Theorem 3.2

Setup:
Choose a local coordinate system in a neighborhood of such that: Consider a closed disk , defined by .

Let be the -cell attached, given by:

Let be a smooth function satisfying:

Denote and . We define a new function :

Assertion 1 (1). The region coincides with the region .

Proof.

If , then .
If , then:

Thus, the sublevel sets coincide. ◻

Assertion 2 (2). The critical points of are the same as those of .

Proof. Set . We compute the differential of : By our construction of : Since the coefficients are strictly non-zero , requires: Hence, the origin is the only critical point. ◻

Assertion 3 (3). The region is a deformation retract of .

Proof. Since , we have . Because the latter is compact by assumption, the set is also compact.

Suppose contains a critical point. By Assertion 2, it can only be (the origin). But at : This violates the assumption that it lies in . Thus, it contains no critical points. According to Theorem 3.1, the region is a deformation retract. ◻

Assertion 4 (4). is a deformation retract of .

Let . We construct the deformation retraction by distinguishing three cases (further details omitted).

Remark 2 (Remark 3.3). More generally suppose that there are non-degenerate critical points with indices in . Then a similar proof shows that has the homotopy type of .

Remark 3 (Remark 3.4). A simple modification of the proof of 3.2 shows that the set is also a deformation retract of .

Theorem 3 (Theorem 3.5). If is a differentiable function on a manifold with no degenerate critical points, and if each is compact, then has the homotopy type of a CW-complex, with one cell of dimension for each critical point of index .

Lemma 5 (Lemma 3.6, Whitehead). Let and be homotopic maps from the sphere to . Then the identity map of extends to a homotopy equivalence

Lemma 6 (Lemma 3.7). Let be an attaching map. Any homotopy equivalence extends to a homotopy equivalence

Proof of Theorem 3.5

Case 1: is compact.

Let be the critical values of .
Since each is compact, the sequence has no cluster point. We proceed by induction on the critical values.

For , the set .
Assume that for some , has the homotopy type of a CW-complex. Let be the homotopy equivalence, where is a CW-complex. Let be the smallest critical value such that .
By Theorem 3.1, for sufficiently small , there is a deformation retraction (and thus a homotopy equivalence):
By Theorem 3.2 and 3.3, is obtained by attaching cells to : where the attaching maps are .
The composition map maps the boundary into . By Cellular Approximation, this continuous map is homotopic to a cellular map: where .
Because the attaching maps are homotopic, by Lemma 3.6 and Lemma 3.7, the resulting spaces are homotopy equivalent. Therefore: Since the boundaries are attached to the correct lower-dimensional skeleta of , this new space is strictly a valid CW-complex.

This completes the inductive step.

Case 2: is not compact.
(Proof omitted.)

6. Manifolds in Euclidean Space

Whitney Embedding Theorem: Any smooth -manifold can be differentiably embedded in .

Let be a manifold of dimension .

Definition 9. We define the space : ( is an -dimensional manifold, and can be differentiably embedded in ).

Define the endpoint map by:

A point is a focal point of if:

, where .
The Jacobian matrix of at is degenerate (singular).

The point will be called a focal point of if is a focal point of for some .

Theorem 4 (Theorem 6.1, Sard). If and are differentiable manifolds having a countable basis, of the same dimension, and is of class , then the image of the set of critical points has measure in .

Corollary 2 (Corollary 6.2). For almost all , the point is not a focal point of .

Proof. Consider the map . is a focal point . By Sard’s Theorem, the set of focal points has measure . ◻

Local Coordinates and Fundamental Forms:
Let be local coordinates. The embedding function is given by:

Then we can naturally get:

The first fundamental form:
The second fundamental form:

We let , like the unitized curvature grid in . Then are the eigenvalues of .

Lemma 7 (Lemma 6.3). The focal points of along are precisely the points , where . Thus there are at most focal points of along , each being counted with its proper multiplicity.

Proof. Let vector fields be the unit vectors of orthogonal directions. The map is written as: The partial derivatives are: Since are linearly independent (non-degenerate basis). Let the Jacobian matrix of be . is degenerate the block matrix is degenerate, where: The degeneracy condition simplifies to: Using the orthogonality condition , taking the derivative yields . Substituting this: ◻

Distance Squared Function:
Now for a fixed point , we consider the function . Taking the first derivative: Then is a critical point of is perpendicular to at .

Taking the second derivative (Hessian): Let (where is the unit normal vector, is distance). Since at the critical point, :

Then we have:

Lemma 8 (Lemma 6.5). The point is a degenerate critical point of if and only if is a focal point of .

Theorem 5 (Theorem 6.6). For almost all (all but a set of measure ) the function has no degenerate critical points.

Corollary 3 (Corollary 6.7). On any manifold there exists a differentiable function, with no degenerate critical points, for which each is compact.

Proof.

Embedding to (as a closed subset).
Set a point outside , apart from focal points. Define .
. Being the intersection of a closed set and a bounded closed ball, is compact.

◻

Application 1 (1). A differentiable manifold has the homotopy type of a CW-complex. This follows from the above corollary and Theorem 3.5.

Application 2 (2, Poincaré-Hopf Index Theorem). On a compact manifold there is a vector field such that the sum of the indices of the critical points of equals , the Euler characteristic of .

Proof. 1. Topological Definitions:
In Chapter 5, we define:

-th Betti number of .
.

And when , we note the Euler characteristic of :

Theorem 6 (Theorem 5.2, Weak Morse Inequalities). If denotes the number of critical points of index on the compact manifold , then , and

(Proof omitted.)

2. Vector Field Index:
According to Theorem 11.27 in From Calculus to Cohomology, for any smooth vector field on , the total index is a topological invariant: (where is a Gauss map defined in Thm 11.27).
So, we can selectively choose a specific vector field: , where is a Morse function.

3. Local Index Calculation:
In every neighborhood of any non-degenerate critical point of , by the Morse Lemma: The gradient vector field is: Clearly, .

The Jacobian matrix of the vector field (which is the Hessian of ) at is: From Lemma 11.20 in From Calculus to Cohomology, the local index of the vector field at the zero point is determined by the sign of the determinant of its Jacobian:

4. Conclusion:
So the sum of the indices of the critical points of is: ◻

Corollary 4 (Corollary 6.8). Any bounded smooth function can be uniformly approximated by a smooth function which has no degenerate critical points. Furthermore can be chosen so that the -th derivatives of on the compact set uniformly approximate the corresponding derivatives of , for .

Proof. Step 1: Choose some embedding ( is a bounded and closed subset). We can specifically choose the embedding such that .

Step 2: Choose a reference point , where is extremely large, and . By Theorem 6.6, we can choose such that the distance squared function is non-degenerate.

Step 3: Define a new function : Since is just a linear transformation of , is also perfectly non-degenerate. We expand the expression:

Step 4: Convergence Analysis.
Recall that . Since the embedding is bounded, on the compact set for all . When we let and : (The error terms vanish, meaning uniformly approximates ).
Also, for the derivatives, taking the limits yields: This completes the proof. ◻