Vector Fields and Classical Theorems of Topology

by Daniel Henry Gottlieb

In this talk we prove a collection of classical theorems using the concept of the index of a vector field. They are:

The Intermediate Value Theorem The Fundamental Theorem of Algebra Rouche’s Theorem The Gauss-Lucas Theorem The Gauss-Bonnet Theorem The Brouwer Fixed Point Theorem The Borsuk-Ulam Theorem The Jordan Curve Theorem Gottlieb’s Theorem The Poincare-Hopf Theorem.

The reason for this exercise is to argue that the following equation, which we call the “Law of Vector Fields,” is fated to play a central role in mathematics. These theorems are all easy consequences of the law of vector fields. The proofs are so mechanical that one could say that the Law of Vector Fields is a generalization of each of them.

The Law of Vector Fields is the following: Let M be a compact smooth manifold and let V be a vector field on M so that V (m) ≠ 0 for all m on the boundary ∂M of M. Then ∂M contains an open set ∂_{−}M which consists of all m ∈ ∂M so that V (m) points inside. We define a vector field, denoted ∂ − V on ∂_{−}M, so that for every m ∈ ∂_{−}M we have ∂_{−}V (m) = Projection of V (m) tangent to ∂_{−}M. Under these conditions we have

(1)  Ind V + Ind ∂_{−}V = χ(M)

where Ind V is the index of the vector field and χ(M) is the Euler characteristic of M. ([M], [G_{2}-G_{5}], [P]).

The Law of Vector Fields can be used to define the index of vector fields, so the whole of index theory follows from (1). The definition of index is not difficult, but proving it is well-defined is a little involved [G-S]. The definition proceeds as follows:

a) The index of an empty vector field is zero.

b) If M is a finite set of points and V is defined on all of the M (the vectors are necessarily zero), then Ind(V) = number of points in M. c) If V is a proper vector field on a compact M, by which we mean V has no zeros on

∂M, then we set

Ind V = χ(M) − Ind(∂_{−}V).

d) If V is defined on the closure of an open subset U of a smooth manifold M so that the set of zeros Z is compact and Z ⊂ U, then we say V is a proper vector field. The index Ind V is defined to be Ind(V |M) where M is any compact manifold such that Z ⊂ M ⊂ U.

e) If C is a connected component of Z and C is compact and open in Z define Ind_{C}(V) to be the index of V restricted to an open set containing C and no other zeros of V. A key idea in proving this definition is well-defined is a generalization of the concept

of homotopy which we call otopy. An otopy is what ∂_{−}V undergoes when V undergoes a homotopy. The formal definition is as follows: An otopy is a vector field V defined on the closure of an open set T ⊂ M × I so that V (m, t) is tangent to the slice M × t. The otopy is proper if the set of zeros Z of V is compact and contained in T. The restriction of V to M × 0 and M × 1 are said to be properly otopic vector fields. Proper otopy is an equivalence relation.

The following properties hold for the index:

(2) Let M be a connected manifold. The proper otopy classes of proper vector fields on M are in one to one correspondence via the index to the integers. If M is a compact manifold with a connected boundary, then a vector field V is properly homotopic to W if and only if Ind V = Ind W.

(3) Ind(V |A ∪ B) = Ind(V |A) + Ind(V |B) − Ind(V |A ∩ B)

(4) Ind(V × W) = Ind(V) · Ind(W)

(5) Ind(−V) = (−1)^{dim M}(V)

(6) If V has no zeros, then Ind(V) = 0

(7) Ind(V) = ∑_{C} Ind_{C}(V) for all compact connected components C, assuming Z is the union of a finite number of compact connected components.

For certain vector fields the index is equal to classical invariants. Suppose f : ℝ^{n} → ℝ^{n}. Let M be a compact n submanifold. Define V^{f} by V^{f}(m) = f(m). If f : ∂M → ℝ^{n} − 0, then

(8) Ind V^{f} = deg f.

Suppose f : U → ℝ^{n} where U is an open set of ℝ^{n}. Let V_{f}(m) = m⃗ − f(m)^{⟶}. Then

(9) Ind V_{f} = fixed point index of f on U.

Suppose f : M → N is a smooth map between two Riemannian manifolds. Let V be a vector field on N. Let f^{∗}V be the pullback of V on M. We define the pullback by

⟨f^{∗}V (m), v⃗_{m}⟩ = ⟨V (f(m)), f_{∗}(v⃗_{m})⟩.

Note that for f : M → ℝ and V = \frac{d}{dt}, we have f^{∗}V = gradient f.

Now suppose that f : M^{n} → ℝ^{n} where M^{n} is compact and V is a vector field on ℝ^{n} so that f has no singular points near ∂M and V has no zeros on f(∂M). Then if n > 1

(10) Ind f^{∗}V = ∑ w_{i}v_{i} + (χ(M) − deg N̂)

where N̂ : ∂M → S^{n−1} is the Gauss map defined by the immersion of ∂M if ℝ^{n} under f, and v_{i} = Ind_{c_{i}}(V) for the i^{th} zero of V and w_{i} is the winding number of the i^{th} zero with respect to f : ∂M → ℝ^{n}. That is calculated by sending a ray out from the i^{th} zero and noting where it hits the immersed n − 1 manifold ∂M. At each point of intersection the ray is either passing inward or outward relative to the outward point normal N. Add up these point assigning +1 if the ray is going from inside to outside, and −1 if the ray goes from outside to inside.

§1. The Intermediate Value Theorem:

We have a map f : I → R so that f(0) > 0 and f(1) < 0. We must show that f(c) = 0 for some c. Consider V^{f} on I. Ind(∂_{−}V^{f}) = 2. Hence (1) says Ind V^{f} = −1. Hence (6) implies that V^{f} has a zero, hence f hits 0.

§2. The Fundamental Theorem of Algebra:

The proof will also work for a polynomial in the quaternions, provided the homogeneous polynomial of top degree terms have an isolated zero. Let f(z) = a_{n}z^{n} + a_{n−1}z^{n−1} + ⋯ + a_{0}. Consider V^{f} restricted to a disk about the origin with radius r so large that |a_{n}z^{n}| > |a_{n−1}z^{n−1} + ⋯ + a_{0}|. Consider the homotopy f(z) to g(z) = a_{n}z^{n}, via f_{t}(z) = a_{n}z^{n} + t(a_{n−1}z^{n−1} + ⋯ + a_{0}). The homotopy V^{f_{t}} of the associated vector fields is proper. No zero can appear on the boundary. Then homotopy a_{n}z^{n} to z^{n}. Let h(z) = z^{n}, then contract D through smaller and smaller radii until r = 1. Then Ind(V^{h}) = n ≠ 0 using (1) for example. Since the process is a proper otopy, by (2), Ind(V^{f}) ≠ 0 and so by (6), V^{f} has a zero, so f(z) has a root.

§3. Rouche’s Theorem

If f is analytic on a region M so that |f(z)| > |g(z)| for all z ∈ ∂M, then f(z) has as many zeros as f(z) − g(z), if the zeros are simple. The homotopy f_{t}(z) = f(t) − tg(z) is proper, so Ind V^{f} = Ind V^{f−g}. Now Ind V^{f} = number of simple zeros.

§4. The Gauss-Lucas Theorem

This states that if f(z) is a polynomial, the zeros of f′(z) are contained in the convex hull of the zeros of f(z).

First we study the case of a map f : ℝ^{m} → ℝ^{n} and a vector field V on ℝ^{n}. Suppose x⃗ = (x_{1}, …, x_{m}) ∈ ℝ^{m} represents a point and f_{i} and V_{j} are the components of f and V respectively.

Theorem 1. f^{∗}V (x⃗) = (V_{1}(f(x⃗)), …, V_{n}(f(x⃗)) ( \begin{matrix}\frac{∂f_{1}}{∂x_{1}} & ... & \frac{∂f_{1}}{∂xm} \\ \frac{∂fn}{∂x_{1}} & ... & \frac{∂fn}{∂xm} \end{matrix} ) where (\frac{∂f_{i}}{∂x_{j}} ) is the Ja-

cobian matrix.

Proof. Let σ⃗_{x} = (σ_{1}, …, σ_{m}). Then the defining relation for f^{∗}V (m) given by ⟨f^{∗}V (m), σ⃗⟩ = ⟨V (f(m)), f_{∗}(σ⃗)⟩ yields

⟨f^{∗}V (m), σ⃗⟩ = (V_{1}(f(m)), …, V_{n}(f(m)) ( \begin{matrix}\frac{∂f_{1}}{∂x_{1}} & … & \frac{∂f_{1}}{∂x_{m}} \\ \frac{∂f_{n}}{∂x_{1}} & … & \frac{∂f_{n}}{∂x_{m}} \end{matrix} ) ( σ_{1} ; ⋮ ; σ_{m} ) .

Hence the theorem is proved.

Now suppose f : ℂ → ℂ. Then f(z) can be written as f(z) = u(z) + iv(z).

Theorem 2. If f is a complex analytic function and V is a continuous vector field on ℂ, then f^{∗}V (z) = \overline{f′(z)} · V (f(z)).

Proof. Now f′(z) = \frac{df(z)}{dz} = u_{x} + iv_{x} = v_{y} − iu_{y} since the Cauchy-Riemann equations hold if f is analytic. Hence f′(z) = u_{x}(z) − iu_{y}(z). Now from Theorem 1

f^{∗}V (z) = (V_{1}(f(z)), V_{2}(f(z)) ( \begin{matrix}u_{x} & u_{y} \\ v_{x} & v_{y} \end{matrix} ) |_{z}
= (V_{1}(f(z)), V_{2}(f(z)) ( \begin{matrix}u_{x} & u_{y} \\ −u_{y} & u_{x} \end{matrix} ) |_{z} by Cauchy Riemann
= (u_{x}V_{1}(f(z)) − u_{y}V_{2}(f(z)), u_{y}V_{1}(f(z)) + u_{x}V_{2}(f(z)))|_{z}.

In complex notation we may rewrite the equality above as

f^{∗}V (z) = [u_{x}V_{1}(f(z)) − u_{y}V_{2}(f(z))] + i[u_{y}V_{1}(f(z)) + u_{x}V_{2}(f(z))]
= (u_{x} + iu_{y})(V_{1}(f(z)) + iV_{2}(f(z)))
= \overline{(u_{x} − iu_{y})}(V_{1}(f(z)) + iV_{2}(f(z)))
= \overline{f′(z)}V (f(z)).

Thus f^{∗}V (z) = \overline{f′(z)}V (f(z)). □

Now let V be the vector field given by V (z) = z. Then f^{∗}V (z) = \overline{f′(z)}f(z). Thus f^{∗}V has zeros at the zeros of f(z) and at the zeros of f′(z). If f(z) is a polynomial with zeros at the set a_{1}, …, a_{n}, then

f^{∗}V (z) = \overline{f′(z)}f(z) = |f(z)|^{2} (∑_{i=1}^{n} \frac{z − a_{i}}{|z − a_{i}|^{2}} ) .

Now this vector field cannot have a zero outside of the convex hull of the zeros a_{1}, …, a_{n}. Thus the Gauss-Lucas Theorem holds.

Note that W (z) = \frac{z}{|z|} is the pullback of grad|z| = ∇(|z|). Thus f^{∗}W = ∇|f(z)|. Now since W and V are both pointing in the same directions, we see that f^{∗}V is orthogonal to the level curves |f(z)| = k. So if M_{k} = {z|f(z)| ≤ k}, then

χ(M_{k}) = (# of components of M_{k}) − (# of holes in M_{k})
= (# of zeros of f(z)) − (# of zeros of f′(z) in M_{k}).

This is a theorem of Hurwitz.

§5. The Gauss-Bonnet Theorem

Equation (10) is a generalization of the extrinsic Gauss-Bonnet Theorem. We obtain a proof as follows. The Gauss-Bonnet Theorem as proved by Hopf goes as follows [HO_{1}] [S, P.386]. The curvature integrala of a closed submanifold ∂M of dimension n − 1 in ℝ^{n} is the degree of the Gauss map N̂ : ∂M → S^{n−1} where S^{n−1} is the unit sphere. Then Hopf generalized the Gauss-Bonnet Theorem by showing deg N̂ = \frac{1}{2}χ(∂M) if n is odd. Now if we consider a map f : M^{n} → ℝ^{n} so that (10) holds, and if we let x : ℝ^{n} → ℝ be projection onto the x axis, then (10) becomes

(11) Ind ∇(f ∘ x) = χ(M) − deg N̂

since ∇(f ∘ x) = f^{∗}(∇x) and ∇x has no zeros. Now for odd n, equation (5) states that Ind(∇(f ∘ (−x)) = −Ind ∇(f ∘ x). Hence χ(M) = deg N̂. But χ(M) = \frac{1}{2}χ(∂M). So the Gauss-Bonnet theorem is proven. Note that the argument for closed manifolds M of odd dimension have χ(M) = 0 uses exactly the same path: For closed manifolds Ind V = χ(M) and by (5) Ind(−V) = −Ind(V).

Also note the following 2 corollaries. (a) For n odd, Ind ∇(f ∘ x) = 0. (b) If f is an immersion, then deg N̂ = χ(M). This last is a theorem of Haefliger [Ha]. It follows since if f is an immersion, then ∇(f ∘ x) has no zeros. Hence by (6) Ind ∇(f ∘ x) = 0 and so equation (11) yields deg N̂ = χ(M). [G_{5}].

§6. The Brouwer Fixed Point Theorem

Let f : B → B be a continuous map where B is a unit ball in ℝ^{n}. The Brouwer fixed point theorem asserts that f has a fixed point. We consider f : B → B ⊂ ℝ^{n}. Let V_{f}(m) = m − f(m) be the vector field.

The convexity of B implies V_{f} always points inside B for any m on the boundary. Hence ∂_{−}B = ∂B. So applying (1), we get

Ind V_{f} + χ(S^{n−1}) = χ(B),

hence

Ind V_{f} = 1 − (1 + (−1)^{n−1}) = (−1)^{n} ≠ 0.

Hence V_{f} has a zero, hence f has a fixed point.

The Brouwer fixed point theorem can be greatly generalized. For example, let us say, that f : M → ℝ^{n}, where M is a compact n-dimensional submanifold, is transversal to ∂M if the line segment from m to f(m) is not tangent to ∂M at m for m ∈ ∂M. Then f has a fixed point if χ(M) − ∑ χ(∂M_{i}) ≠ 0, when ∂M_{i} are the components of ∂M so that the line segment from m to f(m) begins by entering M. [G_{5}].

§7. The Borsuk-Ulam Theorem

The key lemma, indeed many people call it the Borsuk-Ulam theorem itself, is that an odd map f : S^{n} → S^{n} has odd degree. Considering S^{n} as the unit sphere in ℝ^{n+1}, we say f is odd if f(−x) = −f(x) for all x ∈ S^{n}.

Using the covering homotopy property for the covering space S^{n} ⟶^{p} ℝP^{n}, we can homotopy f to an odd map f_{1} so that there are only a finite number of pairs of Antipodal Points which are fixed by f_{1}. Now we extend f_{1} : S^{n} → S^{n} to g : B → B by f(rs⃗) = rf_{1}(s⃗) where s⃗ ∈ ∂B and r ∈ [0, 1]. Then V^{g} is a vector field defined by V^{g}(rs⃗) = g(rs⃗) = rf_{1}(s⃗). Now Ind V^{g} = deg g by (8) and deg g = deg f_{1} = deg f. Hence (1) becomes deg f + Ind ∂_{−}V^{g} = 1. Now ∂_{−}V^{g} has zeros exactly at those m ∈ ∂B where f_{1}(m) = −m. But then −m is also a zero of ∂_{−}V^{g}, and the symmetry of f_{1} and the fact, from (1), that index only depends on pointing inside implies that the index at m is equal to the index at −m. Hence Ind ∂_{−}V^{g} is even. Thus deg f = 1 − Ind ∂_{−}V^{g} is odd.

§8. The Jordan Curve Theorem

If S^{n−1} ⊂ ℝ^{n}, then ℝ^{n} − S^{n} is split into two components, the inside and the outside. We will show that if M^{n−1} ⊂ ℝ^{n}, where is a smooth connected submanifold of ℝ^{n}, then ℝ^{n} − M splits into the inside and the outside.

We choose a continuous normal vector field N⃗ on M. Let V be the Electric vector field generate by an electron e. Bring e so close to M that M looks like a hyperplane near e. Move e to the other side of the hyperplane at a speed near that of light along the normal direction. The vector field ∂V, which is V projected onto M, originally and finally has an isolated zero at the foot of the normal. Outside of a small ball about this zero the vector field ∂V does not change. Hence the index of the zero does not change. But the motion changes the zero from ∂_{+}V to ∂_{−}V (or vice versa). Hence Ind(∂_{−}V) changes by ±1. Now suppose there were a path σ from the original position of e to the final position of e which does not cross M. Move e along σ. Then ∂_{−}V undergoes an otopy which changes its index. Hence it is not a proper otopy by (2). Hence a zero must be on the frontier if ∂_{−}V at some time. This can only happen when e is on M, contradicting the statement that σ avoids M.

Now M is the boundary between the component of ℝ^{n} − M which “contains” ∞, and the bounded components. If we put e in one of the bounded components, then Ind V = 1, since the electron is the only defect inside the union of the bounded components of ℝ^{n}−M. Call this union W. If we put e inside another component of W, then the index of this new V_{1} is +1 also. Hence there is a proper homotopy between V and V_{1}. The set of defects of all the V_{t} contains the first and last position of the electron. So there is a small connected open set U containing c and e_{1} which does not intersect M. So e and e′ are in the same path component. So w is connected.

§9. Gottlieb’s theorem

This theorem, named by Stallings in [St], has been considerably generalized, first by Rossett and then by Cheeger and Gromov. I hope the reader will forgive me for calling the theorem by my own name, but I wanted to prove theorems specified by short, commonly known names, using the Law of Vector Fields.

The key lemma in the original proof is: If X is a compact CW -complex so that χ(X) ≠ 0, then G_{1}(X) is trivial. Then if X is a κ(π, 1) we know that G(x) = center of π. Thus we must show that if F : X × S^{1} → X so that F |X × ∗ is the identity and if χ(X) ≠ 0, then F |x × S^{1} is homotopically trivial. The original proof used Nielsen-Wecken fixed point classes [G_{1}]. These were transformed by Stallings [St] into an algebraic setting which is of importance algebraically. The present proof is more elementary and does not need Nielsen-Wecken fixed point theory.

Suppose M is a regular neighborhood of an embedding of X in some ℝ^{n}. Then we have a map F : M × I → M so that F (m, 0) = m and F (m, 1) = m for all m ∈ M. We may adjust F so that F (m, t) ≠ m for all m ∈ ∂M and all t, and so that F_{0} and F_{1} are the identity outside a small collar neighborhood of the boundary.

Now we define the vector field T on M × I by T (m, t) = (m⃗ − \overrightarrow{F (m, t)}, t). Now T is a homotopy on M × I. Let Z be the zeros of T on M × I. Now Z is compact and Z ∩ (∂M × I) is empty. Let U be the open set of M × I so that ∥T (m, t)∥ < ϵ where ϵ is so small that two paths α(t) and β(t) on M are homotopic if the distance between α(t) and β(t) is always less than ϵ. Let W be a path component of U containing m × 0. Then T |U is a proper otopy from T_{0} to T_{1}. Now Ind T_{0} = Ind T_{1} and Ind T_{0} = χ(M) and Ind T_{1} = χ(M). So the path connected W contains M × 0 and M × 1.

We may find a path γ : I → U so that γ(0) = ∗×0 and γ(1) = ∗×1. We can homotopy F : M × S^{1} → M to a G such that G(γ(t), t) = γ(t). Then γ ∼ α · γ where α(t) = G(∗, t). Hence α ∼ 0, which was to be shown.

§10. The Poincare-Hopf Theorem

If M is a close manifold and V is a continuous vector field defined entirely on M, then Ind V = χ(M). This is a special case of (1) because since the boundary ∂M is empty, so is ∂_{−}V and so Ind(∂_{−}V) = 0.

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[M] Marston Morse, Singular points of vector fields under general boundary conditions, Amer. J. Math, 51 (1929), 165–178.

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[Sp] Michael Spivak, A Comprehensive Introduction to Differential Geometry, vol. 5, Publish or Perish, Inc, Wilmington, Delaware, 1979.

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