Introduction. Let F = (F 1 , . . . , F k ) : R n → R k , n − k > 0, be a C 1 -map such that W = F −1 (0) is compact and rank[DF (x)] ≡ k at every x ∈ W . From the implicit function theorem W is an (n − k)-dimensional C 1 -manifold.

POLONICI MATHEMATICI LVI.3 (1992)

On topological invariants of vector bundles

by Zbigniew Szafraniec (Gda´ nsk)

Abstract. Let E → W be an oriented vector bundle, and let X(E) denote the Euler number of E. The paper shows how to calculate X(E) in terms of equations which describe E and W .

Introduction. Let F = (F 1 , . . . , F k ) : R ⁿ → R ^k , n − k > 0, be a C ¹ -map such that W = F ⁻¹ (0) is compact and rank[DF (x)] ≡ k at every x ∈ W . From the implicit function theorem W is an (n − k)-dimensional C ¹ -manifold.

Let G 1 , . . . , G s : R ⁿ → R ^m , where m = s + n − k, be a family of C ¹ -vector functions, and assume that the vectors G 1 (x), . . . , G s (x) are linearly inde- pendent for every x ∈ W . Define

E = {(x, y) ∈ W × R ^m | y ⊥ G _i (x), i = 1, . . . , s}

= n

(x, y) ∈ W × R ^m   

X y j G ^j _i (x) = 0, i = 1, . . . , s o .

Clearly E is an (n − k)-dimensional vector bundle over W . In particular, if s = k and G i = grad F i then E becomes T W . Later we shall describe how to orient W and E.

Let X(E) be the Euler number of the bundle E (see [1], Chapter 5.2).

The problem is how to calculate X(E) in terms of F and G 1 , . . . , G s . Let S R = {(x, λ) ∈ R ⁿ × R ^s | kxk ² + kλk ² = R ² }, and let H : R ⁿ × R ^s → R ^m × R ^k be the map given by

H(x, λ) =  X ^s

i=1

λ i G i (x) , F (x)  , where λ = (λ 1 , . . . , λ s ).

Take R > 0 such that W ⊂ {x ∈ R ⁿ | kxk < R}. It is easy to see that H|S R : S R → R ^m × R ^k − {0}. Since n + s = m + k, the topological degree

1991 Mathematics Subject Classification: 57R20, 57R22.

Key words and phrases: vector bundles, characteristic classes.

deg(H|S R ) of the map H|S R is well defined. We shall prove (see Theorem 4) that

X(E) = (−1) ^n(s+k)+k deg(H|S R ) .

As a corollary we get a formula (see Theorem 5) which expresses the Euler characteristic χ(W ) in terms of F . A very similar formula has been proved in [2]. The advantage of the present work is that it is usually easy to find the appropriate value of R. The same is not necessarily true in [2].

2. Preliminaries. We assume that every space R ⁿ , n > 0, has the canonical orientation corresponding to its canonical ordered basis.

Let F = (F 1 , . . . , F k ) : R ⁿ → R ^k be a C ¹ -map as above. For each x ∈ W there is a natural inclusion T x W ⊂ R ⁿ . Vectors ξ k+1 , . . . , ξ n ∈ T x W are said to be positively oriented if grad F 1 (x), . . . , grad F k (x), ξ k+1 , . . . , ξ n form a positively oriented basis in R ⁿ . From now on we assume W to be equipped with this orientation.

Let G 1 , . . . , G s : R ⁿ → R ^m , m = s + n − k, and the vector bundle E over W be as in the introduction. If E(x) is the fibre of E over x ∈ W then there is a natural inclusion E(x) ⊂ R ^m . Vectors v s+1 , . . . , v m ∈ E(x) are said to be positively oriented if G 1 (x), . . . , G s (x), v s+1 , . . . , v m form a positively oriented basis in R ^m . So E is an (n − k)-dimensional oriented vector bundle.

Let

E ⁰ = {(x, y) ∈ W × R ^m | y ∈ span(G ₁ (x), . . . , G s (x))} . Then E ⁰ is a trivial vector bundle over W such that E ⊕ E ⁰ is trivial.

Let p : W → E be a C ¹ -section of E such that p(x) = 0, for some x ∈ W . There are C ¹ -sections v s+1 , . . . , v m : U → E defined in some open neighbourhood U of x in W such that v s+1 (x), . . . , v m (x) are linearly in- dependent and positively oriented in E(x). The sections v s+1 , . . . , v m de- fine a trivialization of E over U , and thus there are unique C ¹ -functions t s+1 , . . . , t m : U → R such that p = P m

i=s+1 t i v i over U . Let (x k+1 , . . . , x n ) be a positively oriented coordinate system in some neighbourhood of x in W .

Definition. ind(p, x) = sign ∂(t s+1 , . . . , t m )

∂(x k+1 , . . . , x n ) (x).

One can prove that the definition of ind(p, x) does not depend on the

choice of v s+1 , . . . , v m and (x k+1 , . . . , x n ). Note that if the section p is

transversal to the zero-section at x then ind(p, x) is the index of p at x

(see [1], Chapter 5.2).

Let P = (P ¹ , . . . , P ^m ) : R ⁿ → R ^m be a C ¹ -vector function. There are sections p : W → E, p ⁰ : W → E ⁰ such that P |W = p + p ⁰ . Let H : R e ⁿ × R ^s → R ^m × R ^k be given by

H(x, λ) = e



P (x) +

s

X

d=1

λ d G d (x), F (x)

 .

Lemma 1. A point x ∈ R ⁿ is in p ⁻¹ (0) ⊂ W if and only if there is a unique λ ∈ R ^s such that e H(x, λ) = 0.

P r o o f. (⇒) If p(x) = 0 then P (x) = p ⁰ (x) ∈ E ⁰ (x), where E ⁰ (x) is the fibre of E ⁰ over x. The vectors G 1 (x), . . . , G s (x) form a basis in E ⁰ (x), and thus there is a unique λ = (λ 1 , . . . , λ s ) ∈ R ^s such that

P (x) +

s

X

d=1

λ d G d (x) = 0 . Since x ∈ W = F ⁻¹ (0), we get e H(x, λ) = 0.

(⇐) Clearly x ∈ W , P (x) = p(x) + p ⁰ (x) ∈ span(G 1 (x), . . . , G s (x)), and so p(x) = 0.

From now on we assume that x ∈ p ⁻¹ (0). Let λ ∈ R ^s be as in Lemma 1.

Since n + s = m + k, the derivative matrix D e H(x, λ) is a square matrix.

Lemma 2. ind(p, x) = (−1) ^n(s+k)+k sign det[D e H(x, λ)].

P r o o f. We can find a coordinate system (x 1 , . . . , x n ) in R ⁿ such that

(1) ∂F i

∂x j

(x) = 0 , for every 1 ≤ i ≤ k, j ≥ k + 1. Let

A =  ∂F i

∂x j

(x)



1≤i,j≤k

.

From (1) and from the fact that rank[DF (x)] = k we deduce that det[A] 6= 0.

For x = (x 1 , . . . , x n ) ∈ R ⁿ we write x = (x ⁰ , x ⁰⁰ ), where x ⁰ = (x 1 , . . . , x k )

∈ R ^k , x ⁰⁰ = (x k+1 , . . . , x n ) ∈ R ^n−k . From the implicit function theorem there is a germ of a C ¹ -function ψ = (ψ 1 , . . . , ψ k ) : (R ^n−k , x ⁰⁰ ) → (R ^k , x ⁰ ) such that

(2) F i (ψ(x ⁰⁰ ), x ⁰⁰ ) ≡ 0 , 1 ≤ i ≤ k .

Since graph ψ = W in some neighbourhood of x, we can treat (x k+1 , . . . , x n ) as a coordinate system in some neighbourhood of x in W . From (1)

(3) the coordinate system (x k+1 , . . . , x n ) is positively oriented if and only

if det[A] > 0.

From (1), (2), for every 1 ≤ i ≤ k, j ≥ k + 1 we have

∂

∂x j

[F i (ψ(x ⁰⁰ ), x ⁰⁰ )](x ⁰⁰ ) = ∂F i

∂x 1

(x) ∂ψ 1

∂x j

(x ⁰⁰ ) + . . . + ∂F i

∂x k

(x) ∂ψ k

∂x j

(x ⁰⁰ ) = 0 . The matrix A is non-singular, and therefore

(4) ∂ψ i

∂x j

(x ⁰⁰ ) = 0 , for 1 ≤ i ≤ k , j ≥ k + 1 .

There are C ¹ -vector maps V s+1 , . . . , V m : R ⁿ → R ^m defined in some neighbourhood of x such that V s+1 (x), . . . , V m (x) form a positively oriented basis in E(x). Write G d = (G ¹ _d , . . . , G ^m _d ), 1 ≤ d ≤ s, and V d = (V _d ¹ , . . . , V _d ^m ), s + 1 ≤ d ≤ m. Since s < m, after an orientation preserving change of coordinates in R ^m we may assume that

(5) G ⁱ _d (x) = δ di , for 1 ≤ d ≤ s , V _d ⁱ (x) = δ di , for s + 1 ≤ d ≤ m , where δ di is the Kronecker delta.

There are C ¹ -functions T 1 , . . . , T m : R ⁿ → R defined in a neighbourhood of x such that

P =

s

X

d=1

T d G d +

m

X

d=s+1

T d V d .

Since p(x) = 0, we have (T 1 (x), . . . , T s (x)) = −λ and T s+1 (x) = . . . = T m (x) = 0. Let θ : (R ^n−k , x ⁰⁰ ) → (W, x) be given by θ(x ⁰⁰ ) = (ψ(x ⁰⁰ ), x ⁰⁰ ), and let p ⁱ = P ⁱ ◦ θ, t _i = T i ◦ θ, g ⁱ _d = G ⁱ _d ◦ θ and v _d ⁱ = V _d ⁱ ◦ θ. Then

(6) (t 1 (x ⁰⁰ ), . . . , t s (x ⁰⁰ )) = −λ , t s+1 (x ⁰⁰ ) = . . . = t m (x ⁰⁰ ) = 0 .

Let Z : R ⁿ → R be a C ¹ -function and let z : R ^n−k → R be given by z = Z ◦ θ. From (4) we have

(7) ∂z

∂x j

(x ⁰⁰ ) =

k

X

i=1

∂Z

∂x i

(x) ∂ψ i

∂x j

(x ⁰⁰ ) + ∂Z

∂x j

(x) = ∂Z

∂x j

(x) , for k + 1 ≤ j ≤ n.

Take i ∈ {s + 1, . . . , m}, j ∈ {k + 1, . . . , n}. Then p ⁱ =

s

X

d=1

t d g _d ⁱ +

m

X

d=s+1

t d v ⁱ _d , and therefore, from (5) and (6),

∂p ⁱ

∂x j

(x ⁰⁰ ) = −

s

X

d=1

λ d

∂g _d ⁱ

∂x j

(x ⁰⁰ ) + ∂t i

∂x j

(x ⁰⁰ ) ,

and so, from (7), we have

∂t i

∂x j

(x ⁰⁰ ) = ∂P ⁱ

∂x j

(x) +

s

X

d=1

λ d

∂G ⁱ _d

∂x j

(x) .

Let m ij be the above expression, and let M = [m ij ] s+1≤i≤m,k+1≤j≤m . From (1) and (5) it is easy to see that the derivative matrix D e H(x, λ) has the form





? ? I

? M 0

A 0 0



 ,

where I is the s × s identity matrix, so det[D e H(x, λ)] = (−1) ^n(s+k)+k × det[M ] det[A]. By (3),

ind(p, x) = (−1) ^n(s+k)+k sign det[D e H(x, λ)] .

3. Main theorem. Let H : R ⁿ × R ^s → R ^m × R ^k be given by H(x, λ) =

 X ^s

i=1

λ i G i (x), F (x)

 . Lemma 3. H ⁻¹ (0) = W × {0}.

P r o o f. If (x, λ) ∈ H ⁻¹ (0) then F (x) = 0, i.e. x ∈ W . By our assump- tion, the vectors G 1 (x), . . . , G s (x) are linearly independent, and so λ = 0.

Let B R = {(x, λ) | kxk ² + kλk ² < R ² } and S _R = ∂B R . Since W is compact, by the above lemma there is R > 0 such that H ⁻¹ (0) ⊂ B R . Hence H|S R : S R → R ^m × R ^k − {0}. Let deg(H|S _R ) be the topological degree of H|S R .

Theorem 4. X(E) = (−1) ^n(s+k)+k deg(H|S R ).

P r o o f. Let D R = {x ∈ R ⁿ | kxk < R}. For each C ¹ -map P : R ⁿ → R ^m there are sections p : W → E, p ⁰ : W → E ⁰ such that P |W = p + p ⁰ . For each ε > 0 we can choose P so that

(1) sup

x∈D

kP (x)k < ε ,

(2) if p(x) = 0 then ind(p, x) 6= 0, i.e. p is transversal to the zero-section.

Let e H = e H(x, λ) = (P (x) + P s

i=1 λ i G i (x), F (x)) . From (1) and Lemma

3 we can show (using Cramer’s rule) that e H ⁻¹ (0) lies close to W × {0} and

thus, for small ε, e H ⁻¹ (0) ⊂ B R . The manifold W is compact and so, from

(2) and Lemma 1, e H ⁻¹ (0) is finite, say e H ⁻¹ (0) = {(x ¹ , λ ¹ ), . . . , (x ^m , λ ^m )}.

Then p ⁻¹ (0) = {x ¹ , . . . , x ^m } and according to the definition of X(E) (see [1], Chapter 5.2) and Lemma 2

X(E) =

m

X

j=1

ind(p, x ^j ) = (−1) ^n(s+k)+k

m

X

j=1

sign det[D e H(x ^j , λ ^j )] .

Clearly the last sum equals deg( e H|S R ), and since H|S R and e H|S R are ho- motopic for ε small enough, we conclude that

X(E) = (−1) ^n(s+k)+k deg(H|S R ) .

Clearly T W = {(x, y) ∈ W × R ⁿ | y ⊥ grad F _i (x), i = 1, . . . , k}. It is well known that X(T W ) = χ(W ), where χ(W ) is the Euler characteristic of W . Let H : R ⁿ × R ^k → R ⁿ × R ^k be given by

H(x, λ) =  X ^k

i=1

λ i grad F i (x), F (x)  .

As above, there is R > 0 such that H ⁻¹ (0) ⊂ B R and so we have a contin- uous map H|S R : S R → R ⁿ × R ^k − {0}. As a consequence of Theorem 4 we have

Theorem 5. χ(W ) = (−1) ^k deg(H|S R ).

A very similar version of the above theorem has been proved in [2].

Example 1. Let W = S ² = {x ∈ R ³ | x ² ₁ + x ² ₂ + x ² ₃ − 1 = 0}, let G = G(x) = (3 + x 1 x 2 − x ² ₃ , x 1 x 2 − x 2 , x 1 − x 2 x 3 ), and let E 1 = {(x, y) ∈ S ² × R ³ | y ⊥ G(x)}. Then

H = H(x, λ)

= (3λ + x 1 x 2 λ − x ² ₃ λ, x 1 x 2 λ − x 2 λ, x 1 λ − x 2 x 3 λ, x ² ₁ + x ² ₂ + x ² ₃ − 1) and R = 2. Thanks to a computer program written by Marek Izydorek and S lawomir Rybicki from the Mathematical Department of the Technical University of Gda´ nsk we have been able to calculate that deg(H|S 2 ) = 0, so X(E) = 0.

Example 2. Let G = G(x) = (3x 1 + x 1 x ² ₂ , 3x 2 + x 2 x 3 , 3x 3 ), and let E 2 = {(x, y) ∈ S ² × R ³ | y ⊥ G(x)}. Then

H = H(x, λ) = (3x 1 λ + x 1 x ² ₂ λ, 3x 2 λ + x 2 x 3 λ, 3x 3 λ, x ² ₁ + x ² ₂ + x ² ₃ − 1) and R = 2. As above we have calculated that deg(H|S 2 ) = −2, so X(E) = 2.

The author wants to express his gratitude to Marek Izydorek and S la-

womir Rybicki for their cooperation.

References

[1] M. W. H i r s c h, Differential Topology , Springer, New York 1976.

[2] Z. S z a f r a n i e c, The Euler characteristic of algebraic complete intersections, J. Reine Angew. Math. 397 (1989), 194–201.

INSTITUTE OF MATHEMATICS UNIVERSITY OF GDA ´NSK WITA STWOSZA 57 80-952 GDA ´NSK, POLAND

Re¸ cu par la R´ edaction le 5.10.1990

R´ evis´ e le 4.11.1991