The Lumenary · First-year graduate

Differential topology

Name: ______________________

Date: ______________________

You will learn to

State Sard's theorem, prove it for maps between spaces of the same dimension, and use regular values to define the mod 2 and oriented degree of a map between closed manifolds
Prove that the degree is independent of the regular value and invariant under homotopy, using the classification of compact 1-manifolds
Prove the Brouwer fixed-point theorem and the hairy ball theorem using regular values and degree
Compute indices of zeros of vector fields, state the Poincaré–Hopf theorem and check it on spheres and tori
Explain transversality, Morse functions and the difference between homeomorphic and diffeomorphic manifolds

1.
What is the degree of the map $f(z) = \bar z^{\,4}$ from the unit circle $S^1 \subset \C$ to itself?
2.
Find the index at the origin of the planar vector field $v(x, y) = (x^2 - y^2, \, -2xy)$ .
3.
Which of these closed manifolds admit a smooth vector field with no zeros? (Choose all that apply.)
- A.The torus $T^2$
- B.The Klein bottle
- C.The sphere $S^2$
- D.The sphere $S^3$
- E.The projective plane $\R P^2$
- F.The closed orientable surface of genus $2$
4.
Let $h : T^2 \to \R$ be the height function of an upright torus. Which statement is correct?
- A. $h$ has exactly four critical values, and every other real number is a regular value
- B.By Sard's theorem, $h$ has no critical values
- C.Values above the maximum are critical values, since they have no preimages
- D.The level set at a saddle height is a smooth curve, because the saddle is non-degenerate
5.
The rotation of the annulus $A = \{1 \le |z| \le 2\}$ by the angle $\pi$ is continuous and has no fixed point. Which hypothesis of the Brouwer fixed-point theorem does it violate?
- A.Compactness of the space
- B.The space must be homeomorphic to a closed disc
- C.Continuity of the map
- D.The theorem only holds in dimension $1$
6.
In the proof of the hairy ball theorem we use the homotopy $F(x, t) = \cos(\pi t)\, x + \sin(\pi t)\, v(x)$ from the identity to the antipodal map. Why does the proof need $v(x)$ to be a unit tangent vector at every point?
- A.So that $|F(x, t)| = 1$ : the homotopy must stay on the sphere
- B.So that $F$ is smooth
- C.So that $F_1$ is the antipodal map
- D.Because degree is only defined for unit vector fields
7.
A closed orientable surface of genus $2$ stands upright, and $h$ is its height function — a Morse function with one maximum, one minimum and some saddle points. How many saddle points does it have?
8.
The formal section proves critical_values_null : μ (f '' {x | (fderiv ℝ f x).det = 0}) = 0 for a differentiable f : E → E. Which informal statement is this?
- A.The set of critical values of $f$ has measure zero
- B.The set of critical points of $f$ has measure zero
- C. $f$ has no critical values
- D.Almost every point $x$ is a regular point of $f$
9.
The degree of a map $S^1 \to S^1$ , and the index of a planar zero, are winding numbers. Write winding(points): given a list of non-zero complex numbers $w_0, w_1, \dots, w_{n-1}$ sampled along a closed curve (with $w_n = w_0$ ), return the number of times the curve winds anticlockwise round $0$ , by adding up the changes of argument (each reduced to the range $(-\pi, \pi]$ ) and dividing by $2\pi$ . Round to the nearest integer.
Computational — Practice

Answer key — Differential topology

1.
$\deg f = \deg(z \mapsto z^4) \cdot \deg(z \mapsto \bar z) = 4 \cdot (-1) = -4$ . Directly: $f(e^{i\theta}) = e^{-4i\theta}$ , which winds $4$ times clockwise; each of the $4$ preimages of a point is orientation-reversing.
2.
$v = (x - iy)^2 = \bar z^2$ . On the circle $z = e^{i\theta}$ we get $v = e^{-2i\theta}$ , which turns twice clockwise as $\theta$ runs once round. The index is $-2$ . (Check with the lesson's second experiment, which lists $\bar z^2$ .)
3.
A closed manifold has a nowhere-zero vector field exactly when $\chi(M) = 0$ (Poincaré–Hopf for one direction, Hopf's converse for the other). So: torus ( $0$ ), Klein bottle ( $0$ ) and $S^3$ ( $0$ ) do; $S^2$ ( $2$ ), $\R P^2$ ( $1$ ) and the genus- $2$ surface ( $-2$ ) do not. Explicit fields: $\partial_\theta$ on the torus, $\partial_x$ on the Klein bottle square, $iz$ on $S^3$ .
4.
The critical points of $h$ are the maximum, the minimum and the two saddle points, so there are exactly four critical values; every other $y \in \R$ is regular (vacuously so outside $[\min h, \max h]$ ). The level sets at regular values in the image are unions of circles; at a saddle value the level set is singular. Sard's theorem guarantees only that critical values form a null set.
5.
The annulus is compact and the rotation is continuous; what fails is the topology: the annulus is not homeomorphic to a disc (it has non-trivial fundamental group). The proof of Brouwer's theorem builds a retraction of the disc onto its boundary circle; for the annulus, the radial projection $z \mapsto z/|z|$ onto the circle of radius $1$ is a perfectly good retraction onto one boundary circle, and nothing is contradicted.
6.
Homotopy invariance of the degree is for homotopies through maps $S^n \to S^n$ . Since $x \perp v(x)$ , $|F(x,t)|^2 = \cos^2(\pi t) + \sin^2(\pi t)|v(x)|^2$ , which is $1$ exactly when $|v(x)| = 1$ . A nowhere-zero field can be normalised to unit length; a field with a zero cannot, and at the zero $F$ leaves the sphere. This is the formalised lemma rotation_homotopy.
7.
With $c_0 = 1$ , $c_2 = 1$ and $c_1 = s$ saddles, Morse theory (or Poincaré–Hopf for $\nabla h$ ) gives $1 - s + 1 = \chi = 2 - 2 \cdot 2 = -2$ , so $s = 4$ : one saddle at the bottom and one at the top of each hole.
8.
For $f : E \to E$ , $df_x$ is surjective iff $\det df_x \ne 0$ , so {x | (fderiv ℝ f x).det = 0} is the set of critical points and its image f '' {…} is the set of critical values. The theorem says this image has Haar (Lebesgue) measure zero: almost every value is regular. Critical points may fill a set of positive measure.
9.
```
import cmath, math

def winding(points):
    n = len(points)
    total = 0.0
    for k in range(n):
        a, b = points[k], points[(k + 1) % n]
        total += cmath.phase(b / a)
    return round(total / (2 * math.pi))
```
This is correct as long as consecutive samples are close enough that the argument changes by less than $\pi$ between them — a fine enough sampling of a continuous curve avoiding $0$ . The test $z^2(z - \tfrac12 i)$ has zeros of total multiplicity $3$ inside the unit circle, so by the argument principle its image winds $3$ times round $0$ . A finite sample is evidence; the integer is exact because the degree is a homotopy invariant and the samples determine a homotopic polygon.