Seminaire: Poincare-Hopf Theorem

WANG Hong

November 9, 2012

Poincare Hopf Theorem Let M be a compact manifold and w a smooth vectorfield on M with isolated zeros. If M has a boundary then w is required to point outwardat all boundary points. The sum

∑ι of the indices at the zeros of such a vector field is

equal to the Euler numberχ(M) =

∑mi=0(−1)irankHi(M).

In particular this index sum is a topological invariant of M : it does not depend onthe particular choice of vector field.

1 Some definitions

Critical value A smooth map f : M → N , M, N manifolds dimension m, n. Set ofcritical points C = {x ∈M |dfx : TMx → TNf(x), rank dfx < n}. f(C) := set of criticalvalues. N − f(c) = set of regular values.

Sard Theorem (admit) Let f : U → Rn be a smooth map, defined on an open setU ⊂ Rm, and let C = {x ⊂ U | rank dfx < n}. Then f(c) ⊂ Rn has Lebesgue mesurezero.

If f : M → N , and dim M = dim N . Regular values exist and dense.

Oriented manifolds An orientation on an n−dimensional manifold is given by a nowherevanishing differential n−form. i.e. ∃{U,ψU}, a chart of manifold M, and ∀U ∩ V 6= 0,ψV ◦ ψ−1U : Rn → Rn, |d(ψV ◦ ψ−1U )| > 0 .

Orientation for boudary Each orientation for M determines an orientation for ∂M : For∀x ∈ ∂M , (v1, ..., vm) positively oriented basis for TMx, s.t. v1, ..., vm ∈ T∂Mx and v1is an outward vector.

Let M, N be oriented n−dimentional manifolds without boundary and let :f : M → Nbe a smooth map. M is compact and N is connected. For any regular value y ∈ N define:deg(f ; y) =

∑x∈f−1(y) sign dfx.

deg(f ; y) is a locally constant function of y.It is defined on a dense open subset of N.

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2 Brower degree

Theorem A : The integer deg(f ; y) does not depend on the choice of regular value y.

Theorem B : If y is smoothly homotopic to g. Then deg f = deg g.

Lemma 2.1 M = ∂X . X compact oriented. M oriented as ∂. If f : M → N extends toa smooth map F : X → N , then deg(f ; y) = 0 for every regular value y.

Proof : First suppose y is a regular value for F. Then the compact 1-manifold F−1(y)is a finite union of arcs and circles(Implicit function theorem), with only boundary pointsof the arcs lying on M = ∂X. Let A ⊂ F−1(y) be one of these arcs. ∂A = {x} ∪ {y}.We will show that sign dfa + sign dfb = 0.

The idea is to show that orientations for X and N determine an orientation for A(while its two boundary points lie on M),then A goes inward at one boundary point, andoutward at the other point. As M is oriented as boundary related to an outward vector,we have sign dfa + sign dfb = 0.

Here we define orientation for A as follows:v1(x) denote the positively oriented unit vector tangent to A at x if:1). dFx(v1) = 0 (v1 tangent to A.2). dFx(v2, ..., vm+1) is a positively oriented basis for TNy and v1, v2, ..., vm+1 is a

positively oriented basis for TXx.v1(x) is a smooth function and points out at one boundary pont (say b), and inward

at the other boundary point (say a). Thus sign dfa = −1, sign dfb = 1. Adding up overall such arcs A, we have deg(f ; y) = 0.

More generally suppose y0 is a regular value for f , but not for F , as deg(f ; y) isconstant within some neighborhood U of y0, choose a regular value y for F within U ,then deg(f ; y0) = deg(f ; y) = 0.]

Lemma 2.2 The degree deg(g; y) is equal to deg(f ; y) for any common regular value y.Proof Consider a smoothy homotopy: F = [0, 1] × M → N . f(x) = F (0, x),

g(x) = F (1, x). [0, 1] × M can be oriented as a product. 1 × M with the correctorientation, 0×M with the wrong orientation. Then deg(g; y)− deg(f ; y) = 0.]

If y and z are both regular values for f : M → N . Choose a diffeomorphism(existenceof h will be talked about later for those who have interest) h : N → N that carries y toz and is isotopic(connected ’through homeomorphisms’) to the identity. By definition,deg(f ; y) = deg(h◦f, h(y)) = deg(h◦f, z), and by isotopy and Lemma 2.2, deg(h◦f ; z) =deg(f ; z). This completes the proof of Theorem A and Theorem B.]

In order to proof the existence of h, we need the following:

Homogeneity Lemma Let y and z be arbitrary interior points of the smooth, connectedmanifold N , Then there exists a diffeomorphism h : M → N that is smoothly isotopicto the identity and carries y into z.

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Proof As N is connected, we could consider locally, it is enough to proof this lemmafor N = Rn. The idea is to fix the area out of unit ball Bn, and construct a vector fieldwhose flow send 0 to an arbitrary point in Bn, and tends to zero near Sn−1.

ψ : Rn → R who satisfies1) ψ(x) > 0 for ‖x‖ < 1.2) ψ(x) = 0 for ‖x‖ ≥ 1.(ex:ψ = exp(−(1− ‖x‖2)−1).For ∀c ∈ Sn−1. dxi

dt = ciψ(xi, ..., xn). For ∀x̄ ∈ Rn, ∃!x = x(t), x(0) = x̄, Ft(x̄) :=x(t).

1). Ft(x̄) is defined for all t and x̄ in Bn, and depends smoothly on t and x̄.2).F0(x̄) = x̄.3).Fs+t = Fs ◦ Ft(x̄).∀y ∈ Bn, let c = y

‖y‖ . As ψ(x) > 0 for ‖x‖ < 1, flow Ft travels by direction c with a

nonzero speed in B, so for some time t, Ft(0) = y.]

3 Vector fields and the Euler number

Index of a vector field Consider an open set U ⊂ Rm and a smooth vector field v :U → Rm with an isolated zero at the point z ∈ U . v̄(x) := v(x)

‖v(x)‖ maps a small spherecentered at z into the unit sphere. The degree of this mapping is called the index τ ofv at the zero z, note Indz(v), and Ind(v) =

∑z,v(z)=0 Indz(v).

Remark: Index of a vector field is counted only at its zeros, because at nonzeropoints, v̄(x) could extends to the whole ball, and Lemma 2.1 tells us deg(v̄(x), z) = 0.

To define the concept of index for a vector field w on an arbitary manifold, we needthe following lemma:

Lemma 3.1 Suppose that the vector field v on U corresponds to v′ = df ◦ v ◦ f−1 on U ′

under a diffeomorphism f : U → U ′. Then the index of v at an isolated zero z is equalto the index of v′ at f(z).

According to definition, index of a vector field at one zero is a local property. So it’senough to consider the case in Rm.

Lemma 3.2 Any orientation preserving diffeomorphism f of Rm is smoothly isotopic tothe identity.

Proof suppose f(0) = 0.f(x) =

∑xigi(x) for gi(x) =

∫ 10

∂f∂xi

(tx)dt.

Let F (x, t) = f(xt)t , therefore F (x, 0) = df0(x).

Thus F (x, t) gives a isotopy between f and linear map df0(x), as SL(R,m) is pathconnected, df0(x) is isotopic to id.]

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For |dfy > 0|, use Lemma 3.2. If |dfy| < 0, it’s enough to consider a reflection ρ, andv′ = ρ ◦ ρ−1, and observe that Indy(v) = Indρ(y)(v

′).]

Let M be a compact manifold and w0, w1 smooth vector fields on M with isolatedzeros(w0 6= λw1). If M has a boundary, then wi is required to point outward at allboundary points.

Lemma 3.3 Ind(w0) = Ind(w1).Consider smooth vector fields wt = tw0 + (1− t)w1, t ∈ [0, 1]. Let S = {t|Ind(wt =

Ind(w0}, we will show that S is open and closed.S is open: for wt0 with finite isolated zeros zi, and for each zi choose a sufficient

small sphere Si to calculate Indzi(wt0). ∃ε s.t. ∀t, |t − t0| < 0, the zeros of wt (maybemore than zeros of wt0) lie in Si. According to Lemma 2.1, Ind(wt0) = Ind(wt).

S is closed: if {tk} ⊂ S, and t is the limit, then the same argument shows thatInd(wtk) = Ind(wt) for some k great.

Thus S = [0, 1] and specially, Ind(v0) = Ind(v1).]

Vector Bundle A real vector bundle consists of:1) topological spaces X(base space and E total space2) a continuous subjection πE → X (bundle projection)3) ∀x ∈ X, π−1(x) has a finite-dimensional real vector space structurewhere the following compatibility condition is satisfied: ∀x ∈ X, ∃U ⊂ X, k a natural

number, and a homeomorphism: ψ : U ×Rk → π−1(U) such that for all x ∈ U .a) (π ◦ ψ)(x, v) = x for all vectors v in Rk, andb) v → ψ(x, v) is a isomorphism between Rk and π−1(x).(U,ψ) is called local trivalisation of the vector bundle, and x→ kx is a fuction locally

constant, so if E is connected, kx is equal to a constant k, and E is said to be a vectorbundle of rank k.

Generalization on vector bundle A smooth vector field w on a compact manifold M , couldbe seen as a section of its tangent bundle TM . More generally, consider a rank n vectorbundle over a compact oriented manifold without boundary M , dimM = rankE, and asmooth section s : M → E. In the same way, we could define index of a smooth sectionwith isolated zeros, named IndE(s). Here IndE(s) is defined on zeros of s, as ToxE

∼=TxM ⊕ Ex. And the same argument as in Lemma 3.3, we have IndE(s1)=IndE(s2),∀s1, s2 smooth sections with isolated zeros.

Property Given a smooth vector bundle F → Y , and a smooth map f : X → Y ,dimX =dimY = rankF , one could define a ”pullback” vector bundlef∗F on X, that is the fiberover a point x ∈ X is essentially just the fiber over f(x) ∈ Y . And for any smoothsection s : Y → F , with isolated zeros, we have Indf∗F (s ◦ f) = deg(f) ∗ IndF (s).

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