A DE MOIVRE LIKE FORMULA FOR FIXED POINT THEORY

Daniel H. Gottlieb

1. Introduction.

Suppose V is a vectorfield on a compact manifold M with or without boundary ∂M. There is a formula, originally due to Marston Morse [Mo] in 1929, which is not as widely known as it should be. It was rediscovered in 1968 by C. Pugh [P] and by the author [G] in 1985. The formula related the index of the vectorfield V with the index of a vectorfield on part of the boundary. This formula, like de Moivre’s e^{iθ} = cos θ + i sin θ, contains a large amount of information.

So let V be a continuous vectorfield on M with no zeros on ∂M. Then

(1)  Ind(V) + Ind(∂_{−}V) = χ(M).

Here χ(M) denotes the Euler-Poincare number of M, and ∂_{−}V is a vectorfield defined on part of ∂M as follows. Let ∂_{−}M be the subset of ∂M containing all m so that V (m) points inward. Then ∂_{−}V is the vectorfield on ∂_{−}M given by first restricting V to ∂M, and then projecting V | ∂M, using the outward pointing vectorfield N, onto its component field tangent to ∂M. This is denoted ∂V. Then ∂_{−}V = ∂V | ∂_{−}M.

Observe that the vectorfield ∂_{−}V is defined on a space of one dimension lower than V. Thus the formula may be thought of as an inductive definition of index. For example, if M = I and V is a vectorfield on I, then Ind V = 1 − Ind ∂_{−}V. Now ∂_{−}M consists of two points, one point, or no points. Correspondingly, we let the corresponding Ind (∂_{−}V) equal 2, 1, or 0 respectively. Then the respective Ind V equal -1, 0, or 1. In the case M is a circle we have Ind (∂_{−}V) = 0 since ∂S^{1} = ∂_{−}S^{1} = ∅, so Ind V = 0. Thus we have defined the index for all connected 1manifolds, and since the sum of the indices over disjoint manifolds is the index of the union of the vectorfields, we can calculate the index for one dimension. Similarly, the information in one dimension permits the calculation in two dimensions. There is a technical difficulty in that ∂_{−}M is not a compact manifold, but the formula (1) is morally a definition.

Thus one should expect that (1) should imply all formulas involving the index of a vectorfield. Also a systematic searching of vectorfield phenomena might yield new results when viewed in conjunction with (1). In the rest of this report we will massage (1) and obtain some well known results. Also we will produce a formula relating the Euler-Poincare number, the Lefschetz number and the coincidence number. This leads to an apparently unknown generalization to immersions of a famous theorem of Hopf: If α : M^{n} → ℝ^{n} is an immersion, then the Gauss map on ∂M must have degree equal to χ(M).

Typeset by AMS-TEX

2. Massaging the formula.

a) If ∂_{−}M is empty, then Ind (∂_{−}V) = 0 so Ind (V) = χ(M). This is Hopf’s theorem for closed manifolds or when V | ∂M always points outwards.

b) For nice enough vectorfields, the generic case, we get a sequence of manifolds M ⊇ ∂_{−}M ⊇ ∂_{−}∂_{−}M ⊇ ∂_{−}^{3}M … where ∂_{−}∂_{−}M is the set of points on the boundary of ∂_{−}M on which ∂V pointsinto ∂_{−}M, and etc. Then we can eliminate the index of the smaller dimensional vectorfields and show that

(2)  Ind V = χ(M) − χ(∂_{−}M) + χ(∂_{−}^{2}M) − …

Proving similar versions of (2) was the goal of [Mo] and [P].

c) On the other hand, we can try and eliminate the Euler-Poincare number. To do that, we first define ∂_{+}M and ∂_{0}M by letting ∂_{+}M be the manifold where V | ∂M points outward and ∂_{0}M is the set where V | ∂M is tangent to ∂M. Thus ∂M = ∂_{−}M ⋃ ∂_{+}M ⋃ ∂_{0}M. Then let ∂_{+}V = ∂V | ∂M_{+} and ∂_{0}V = ∂V | ∂M_{0}. Now Ind ∂_{0}V = 0 since ∂_{0}V has no zeros. So

(3)  Ind (∂V) = χ(∂M) = Ind (∂_{−}V) + Ind (∂_{+}V)

Thus for even dimensional M, since ∂M is odd dimensional and so χ(∂M) = 0, we have:

(4)  Ind (∂_{−}V) = −Ind (∂_{+}V).

For odd dimensional M we have

(5)  Ind (∂_{−}V) = 2χ(M) − Ind (∂_{+}V).

Combining (5) and (1) in the odd dimensional case we get

(6)  Ind V = \frac{1}{2}(Ind (∂_{+}V) − Ind (∂_{−}V)).

In the even dimensional case there is no such formula, so the theory is different in even and odd dimensions. This formula was used by Sue Goodman [Gd] in the case V is transversal to ∂M. Then Ind V = \frac{1}{2}(χ(∂_{+}M) − χ(∂_{−}M)).

d) Now let us write (1) using Ind (∂_{+}V). Using (4) and (5) we get

(7)  Ind V = (−1)^{n}χ(M^{n}) + Ind (∂_{+}V).

which is Morse’s version of the formula, [Mo]

e) To prove that Ind (−V) = (−1)^{n}Ind (V) is instructive. Here n denotes the dimension of M. We assume that the formula is true for vectorfields on (n − 1) - dimensional spaces. Then Ind (∂_{−}(−V)) = Ind (−∂_{+}V) = (−1)^{n−1}(∂_{+}V). Using this and (1) and (3) we have

Ind (−V) = χ(M) − Ind (∂_{−}(−V))
= χ(M) − (−1)^{n−1}Ind (∂_{+}V)
= χ(M) + (−1)^{n}(χ(∂M) − Ind (∂_{−}V))

If n is even then

Ind (−V) = χ(M) + (0 − Ind (∂_{−}V)
= Ind V

If n is odd

Ind (−V) = χ(M) − [2χ(M) − Ind (∂_{−}V)]
= −(χ(M) − Ind (∂_{−}V)) = −Ind V

f) Let V be the vectorfield on ℝ^{n} given by V (\overrightarrow{x}) = A\overrightarrow{x} where A is a linear transformation. If detA ≠ 0, then \overrightarrow{0} is the only zero of V. We will show, using (1), the well known fact

(8)  Ind (V) = \frac{detA}{| detA |}.

Proof. We let M = \{\overrightarrow{x} ∥ \overrightarrow{x} ∥≤ 1\}. Then on M we see that Ind V +Ind (∂_{−}V) = 1. Now A is homotopic through linear transformations A_{t} such that det(A_{t}) ≠ 0, to a diagonal matrix D. The corresponding vectorfields V_{t} all have a single zero at \overrightarrow{0} and so the index of V_{t} on M does not change. Now D can be chosen to be either the identity I or the diagonal matrix J with −1 in the (1, 1) position and 1’s down the main diagonal. In the case of detA > 0, I may be chosen for D and the vectorfield V_{1} : V_{1}(\overrightarrow{x}) = \overrightarrow{x} always points outward on ∂M. Thus ∂_{−}M is empty and so ind (∂_{−}V_{1}) = 0. Hence Ind V = 1. For the case det(A) < 0, the vectorfield is V_{1}(x_{1}, …, x_{n}) = (−x_{1}, x_{2}, …, x_{n}). Then ∂_{−}M consists of the set of vectors \overrightarrow{x} such that \overrightarrow{x} · J\overrightarrow{x} < 0. That is the set \{(x_{1}, …, x_{n}) ∈ ∂M | x_{1}^{2} > \frac{1}{2}\}. The boundary of ∂_{−}M are the spheres given by x_{1} = ±\frac{1}{\sqrt{2}}. Now if \overrightarrow{x} is on the boundary of ∂_{−}M we see that \overrightarrow{x} · J\overrightarrow{x} = 0. Now consider for ϵ > 0

(\overrightarrow{x} + ϵJ\overrightarrow{x}) · J(\overrightarrow{x} + ϵJ\overrightarrow{x}) =
\overrightarrow{x} · J\overrightarrow{x} + ϵ\overrightarrow{x} · (J^{2}\overrightarrow{x} + J\overrightarrow{x} · J\overrightarrow{x}) + ϵ^{2}J\overrightarrow{x} · J^{2}\overrightarrow{x} =
ϵ(\overrightarrow{x} · \overrightarrow{x} + J\overrightarrow{x} · J\overrightarrow{x}) > 0.

Thus J\overrightarrow{x} is tangent to the sphere at \overrightarrow{x} and points out of ∂_{−}M. Hence by (1), Ind ∂_{−}V = χ(∂_{−}M). But ∂_{−}M can be deformed to S^{0} = {(1, 0, …, 0), (−1, 0, …, 0)}. So χ(∂_{−}M) = 2. Hence from (1)

Ind V_{1} + 2 = 1

so Ind V = −1.

g) Let f : ℝ^{n} be given by f(\overrightarrow{x}) = A\overrightarrow{x} where A is a linear transformation. If A has no eigenvalue λ = 1 then f has a single fixed point at \overrightarrow{0} and the Lefschetz number

(9)  Λ_{f} = \frac{det(I − A)}{| det(I − A) |}

Proof. The fixed point index of f at \overrightarrow{0} is the index of the vectorfield V (\overrightarrow{x}) = \overrightarrow{x} − f(\overrightarrow{x}) = (I − A)\overrightarrow{x}.

Comparing (f) and (g) we see that if the linear transformation A has a unique zero and fixed point, the vectorfield index and the fixed point index agree if and only if A has an even number of eigenvalues between zero and one for even n, and in the case of odd n, they agree if and only if there are an odd number of eigenvalues between zero and one.

3. e^{iπ} + 1 = 0

De Moivre’s formula has the famous special case which relates the five most common constants of calculus. Similarly formula (1) implies an equation which relates the three most common algebraic invariants of fixed point theory, the Euler-Poincare number, the Lefschetz number and the coincidence number.

If ∂M^{n} ⊂ ℝ^{n+1}, recall that the Gauss map N̂ : ∂M → S^{n} is given by translating the “outward pointing” unit normal vector at m, denoted N(m) to the origin. Similarly for any nonzero vectorfield V on ∂M we define the Gauss map V̂ : ∂M → S^{n} by translating the unit vector V (m)/ ∥ V (m) ∥ to the origin.

Now suppose that f : M^{n} → M^{n} ⊂ ℝ^{n+1} is a map such that f has no fixed points on ∂M. Let V be the vectorfield

V (m) = (m − f(m))_{m}

Then

(10)  Λ_{f} + Λ_{(−V̂),N̂} = χ(M)

where Λ_{f} is the Lefschetz number of f and Λ_{(−V̂),N̂} is the coincidence number of the Gauss maps for V and N.

Equation (10) follows from (1) and the following observation. We use the description of coincidence numbers found in the last chapter of Vick [V]. There we see that a coincidence number Λ_{f,g} is defined which is equal to the sum of all the coicidence numbers of all the coincidence points of f, g : M_{1}^{n} → M_{1}^{n} where M_{1}^{n} and M_{i}^{n} are closed oriented manifolds. Then Vick shows that

(11)  Λ_{f,g} = ∑_{i}(−1)^{i}trace(g^{!}f^{∗})_{i}

where g^{!} : H^{i}(M_{1}, ℤ) → H^{i}(M_{2}, ℤ) is the Umkehr map. Now it is not hard to show that

(12)  Ind (∂_{−}V) = Λ_{−V̂ ,N̂}

Since Ind V = Λ_{f}, (12) and (1) imply (10).

4. Normal degree.

Let α : M^{n} → ℝ^{n+1} be a smooth immersion of a closed oriented manifold. Let V be a vectorfield on ℝ^{n+1} with no zeros on α(M). Then

(13)  Ind (∂_{−}V) = deg(N̂) − deg(V̂)

where N̂ is the Gauss map of the unit normal chosen so that the orientation of M^{n} and ℝ^{n+1} agree and ∂_{−}V consists of those vectors of V such that N(m)·V (m) < 0. This follows from (12) and (11).

We define the winding number W (p, α) at p ∈ ℝ^{n+1} of α : M → ℝ^{n+1} of p ∉ α(M) as follows. Consider a ray ρ starting at p. Assume that ρ intersects α(M) transversally. For each m so that α(m) meets ρ, assign +1 if ρ and N(m) point to the same “side” at m and assign −1 if they point to opposite sides. Then W (p, x) is defined to be the sum of these ±1’s. This W (p, x) is equal to the degree of the map

π ∘ α : M^{n} →^{α} ℝ^{n+1} − p →^{ϕ} S^{n}

Now suppose that V has a finite number of zeros at points p_{i} ∉ χ(M). For each p_{i} there is a local index Ind (p_{i}, V). Then we have the following formula

(14)  ∑_{i} W (p_{i}, α) · Ind (p_{i}, V) + Ind (∂_{−}V) = deg(N̂)

Proof. Since M is a closed oriented manifold immersed in ℝ^{n+1}, we see that M must be the boundary of an oriented manifold W. Now the embedding α : ∂W ↪ ℝ^{n+1} extends to a smooth map f : W → ℝ^{n+1} for which the p_{i} are regular values. Then the vectorfield V may be considered as a map V : ℝ^{n+1} → ℝ^{n+1}. Then deg(V ∘ f) = ∑_{i} W (p_{i}, α) · Ind (p_{i}, V) because this is the local degree calculated

using the preimage of 0^{→}. But deg(V ∘ f) = deg(V̂) where V̂ : ∂W →^{α} (ℝ^{n+1} − p) → ℝ^{n+1} − 0 →^{ϕ} S^{n}. Then (l4) follows from (13).

Equation (l4) affords a means of studying the degree Gauss map of an immersion This is called the normal degree and is a classical topic begun by Hopf in [H1] and continued by Milnor in [Mi]. Equations (1) and (14) combine to give the following result.

Theorem:. If α : M^{n+1} → ℝ^{n+1} is an immersion, then the degree of the Gauss map N̂ : ∂M → S^{n} is equal to the χ(M).

Proof. Since α is an immersion, the vectorfield V on ℝ^{n} pulls back to a vectorfield V ∗ on M. Now Ind V^{∗} = ∑_{i} W (p_{i}, α)Ind (p_{i}, V), so (1) and (l4) imply that degN̂ = χ(M).

Hopf orginally proved this in [H2] for α an embedding. This is a famous and important theorem. Also Hopf showed in [H2] that degN̂ = \frac{1}{2}χ(∂M) for even immersions. The case of ∂M = S^{1} is known and the immersions are classified.

Bibliography

[Go] Goodman, Sue E., Closed leaves in foliation of codimension one, Comment Math. Helv. 50 (1975), 383-388.

[G] Gottlieb, Daniel Henry, A de Moivre formula for fixed point theory, ATAS de 5o. Encontro Brasiliero de Topologia, Universidade de Saõ Carlos, S.P. Brazil (to appear).

[H1] Hopf, Heinz, Über die curvatura integra geschlossener Hyperflächen, Math Ann. 95 (1925/26), 340-367.

[H2] Hopf, Heinz, Vectorfelder in n-dimensionalen Mannigfaltigkeiten, Math. Ann. 96 (1926/l927), 225-250.

[Mi] Milnor, John, On the immersion of n-manifolds in (n + 1)-space, Comment. Math. Helv. 30 (1956), 275-284.

[Mo] Morse, Marston, Singular points of vectorfields under general boundary conditions, American Journal of Mathematics 51 (1929), 165-178.

[P] Pugh, Charles C., A generalized Poincare index formula, Topology 7 (1968), 217-226.

[V] Vick, James W., Homology Theory, Academic Press, New York.