Basics of Morse Theory and the Poincaré-Hopf Theorem

2026-05-15T08:00:00.000Z

Basics of Morse Theory and the Poincaré-HopfTheorem

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 toCohomology (by Madsen and Tornehave) to prove the Poincaré-Hopftheorem.

0. Preliminaries

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

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

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

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

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

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

1. Introduction

Diagram 1.1: Torus

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

As the height increases, thetopology of changes asfollows:

.
is empty.
.
. Attaching a 0-cell
.
.Attaching a 1-cell
.
. Attaching a1-cell
.
(the full torus).Attaching a 2-cell

Diagram 1.3

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

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

2. Definitions andLemmas

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

Definition 6 (Non-degenerate critical point). Acritical point is callednon-degenerate if and only if the Hessian matrixevaluated at is non-singular,i.e.,

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

Proof. By the fundamental theorem of calculus and the chainrule, 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 localcoordinate system in a neighborhood of with for all , such that the identity holds throughout . Here, the integer is called theindex of at, which equals the negative indexof 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 diagonalHessian matrix at :

Step 1: Express as aquadratic 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 andevaluating at the origin gives: Thus, the matrix isnon-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 swapcoordinates and .
If but, we apply a rotation and , whichconverts 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 ofcoordinates:

is a validcoordinate 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 byinduction. ◻

Corollary 1 (Corollary 2.3). Non-degeneratecritical points are isolated.

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

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

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 bythe directional derivative:

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

Proof. The proof relies on solving the correspondingordinary 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 acoordinate chart , where .

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

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

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

Remark 1. The condition that vanishes outside a compact setcannot be omitted.
Counterexample: Let the manifold be the open interval , andconsider the constant vector field (i.e., ).

Solving the ODE yields .
Thus, the flow is given by . However, if we start at and let , thepoint reaches . Thesolution 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 amanifold . Let and suppose that the set , consisting of all with , is compact, andcontains no critical points of .Then is diffeomorphic to . Furthermore, is a deformation retract of , so that the inclusion map is a homotopyequivalence.

(Proof omitted.)

Theorem 2 (Theorem 3.2). Let be a smooth function,and let be a non-degeneratecritical point with index .Setting , suppose that iscompact, and contains no critical point of other than , for some . Then, for allsufficiently small , theset has the homotopytype of with a -cell attached.

Proof of Theorem3.2

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:

Diagram 3.1

Let be a smooth function satisfying:

Denote and . We define a new function :

Assertion 1 (1). The region coincideswith 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 . Wecompute the differential of : By our construction of : Since the coefficients are strictlynon-zero , requires: Hence, the origin is the only critical point. ◻

Assertion 3 (3). The region is adeformation retract of .

Proof. Since ,we have . Because the latter iscompact by assumption, the set is alsocompact.

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

Diagram 3.3

Assertion 4 (4). is adeformation retract of .

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

Diagram 3.4

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

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

Diagram 3.5

Theorem 3 (Theorem 3.5). If is a differentiable function on amanifold with no degeneratecritical points, and if each iscompact, then has the homotopytype 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 thesphere to . Then the identity map of extends to a homotopy equivalence

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

Proof of Theorem3.5

Case 1: iscompact.

Let be the critical values of .
Since each is compact, thesequence has no clusterpoint. We proceed by induction on the critical values.

For , the set.
Assume that for some , has the homotopy type of aCW-complex. Let be the homotopy equivalence, where is a CW-complex. Let be the smallest critical value suchthat .
By Theorem 3.1, for sufficiently small , there is a deformationretraction (and thus a homotopy equivalence):
By Theorem 3.2 and 3.3, is obtained by attachingcells to : where the attaching maps are .
The composition map maps the boundary into . By CellularApproximation, this continuous map is homotopic to a cellularmap: where .
Because the attaching maps are homotopic, by Lemma3.6 and Lemma 3.7, the resulting spaces arehomotopy equivalent. Therefore: Since theboundaries are attached to the correct lower-dimensional skeleta of, this new space is strictly avalid CW-complex.

This completes the inductive step.

Case 2: is notcompact.
(Proof omitted.)

6. Manifolds inEuclidean Space

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

Let be amanifold of dimension .

Definition 9. We define the space : ( is an -dimensional manifold, and can bedifferentiably 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 afocal point of if is a focal point of for some .

Theorem 4 (Theorem 6.1, Sard). If and are differentiable manifolds having acountable basis, of the same dimension, and is of class , then the image of the set ofcritical 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 focalpoint . By Sard’sTheorem, the set of focal points has measure . ◻

Local Coordinates and Fundamental Forms:
Let be localcoordinates. 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 . Thusthere are at most focal points of along , each being counted with its propermultiplicity.

Proof. Let vectorfields be the unit vectors of orthogonal directions. Themap iswritten as: The partialderivatives are: Since are linearly independent (non-degenerate basis). Let the Jacobian matrixof be . is degenerate the block matrix is degenerate, where: The degeneracy condition simplifies to: Using theorthogonality condition , taking the derivative yields . Substitutingthis: ◻

Distance Squared Function:
Now for a fixed point , we consider the function . Taking the first derivative:Then is a critical point of isperpendicular 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 ofmeasure ) the function has no degeneratecritical points.

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

Proof.

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

◻

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

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

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

-thBetti number of .
.

And when , we notethe Euler characteristic of :

Theorem 6 (Theorem 5.2, Weak Morse Inequalities).If denotes the numberof 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, forany smooth vector field on , the total index is a topologicalinvariant: (where is aGauss 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 inFrom Calculus to Cohomology, the local index of the vectorfield at the zero point is determined by the sign of the determinant ofits Jacobian:

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

Corollary 4 (Corollary 6.8). Any bounded smoothfunction can beuniformly approximated by a smooth function which has no degenerate criticalpoints. Furthermore can be chosenso that the -th derivatives of on the compact set uniformly approximate the correspondingderivatives 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 isextremely large, and . By Theorem 6.6, we can choose such that the distance squared function isnon-degenerate.

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

Step 4: Convergence Analysis.
Recall that . Sincethe 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. ◻

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