ON THE UNIQUENESS OF THE FIXED POINT INDEX ON DIFFERENTIABLE MANIFOLDS

The fixed point index enjoys a number of properties whose precise statement may vary in the literature. The prominent ones are those of normalization, additivity, homotopy invariance, commutativity, solution, excision, and multiplicativity (see, e.g., [4, 5, 6, 8, 9, 10]). It is well known that some of the above properties can be used as axioms for the fixed point index theory. For instance, in the manifold setting, it can be deduced from [3] that the first four, provided that the first three are stated as in Section 2, imply the uniqueness of the fixed point index. Actually the result of [3] is not merely confined to the context of (differentiable) manifold: it holds in the framework of metric ANRs. In this more general setting, other uniqueness results based on a stronger version of the normalization property are available for the class of compact maps (see, e.g., [6, Section 16, Theorem 5.1]). Our goal here is to prove that in the framework of finite-dimensional manifolds the fixed point index is uniquely determined by three properties, namely, the Amann-Weisstype properties of normalization, additivity, and homotopy invariance as enounced in Section 2. For this reason, these properties will be collectively referred to as the fixed point index axioms (for manifolds). The fact that in Rm any equation of the type f (x)= x can be written as f (x)− x = 0 shows that in this context the theories of fixed point index and of topological degree are equivalent. Therefore, in this flat case, the uniqueness of the index could be deduced from the Amann-Weiss axioms of the topological degree given in [2]. Here we provide


Introduction
The fixed point index enjoys a number of properties whose precise statement may vary in the literature. The prominent ones are those of normalization, additivity, homotopy invariance, commutativity, solution, excision and multiplicativity (see e.g. [3,5,6,8,9,10]). It is well known that some of the above properties can be used as axioms for the fixed point index theory. For instance, in the manifold setting, it can be deduced from [4] that the first four, provided that the first three are stated as in Section 2, imply the uniqueness of the fixed point index. Actually the result of [4] is not merely confined to the context of (differentiable) manifold: it holds in the framework of metric ANRs. In this more general setting, other uniqueness results based on a stronger version of the normalization property are available for the class of compact maps (see e.g. [5, §16, Theorem 5.1]).
Our goal here is to prove that in the framework of finite dimensional manifolds the fixed point index is uniquely determined by three properties, namely the Amann-Weiss type properties of normalization, additivity and homotopy invariance as enounced in Section 2. For this reason, these properties will be collectively referred to as the fixed point index axioms (for manifolds).
The fact that in R m any equation of the type f (x) = x can be written as f (x) − x = 0 shows that in this context the theories of fixed point index and of topological degree are equivalent. Therefore, in this flat case, the uniqueness of the index could be deduced from the Amann-Weiss axioms of the topological degree given in [2]. Here we provide a simple proof of the uniqueness in R m and we extend this result to the context of finite dimensional manifolds. Some technical lemmas are well known or belong to the folklore. Their proof is given for the sake of completeness.

Preliminaries
Given two sets X and Y , by a local map with source X and target Y we mean a triple g = (X, Y, Γ), where Γ, the graph of g, is a subset of X × Y such that for any x ∈ X there exists at most one y ∈ Y with (x, y) ∈ Γ. The domain D(g) of g is the set of all x ∈ X for which there exists y = g(x) ∈ Y such that (x, y) ∈ Γ; namely, D(g) = π 1 (Γ), where π 1 denotes the projection of X × Y onto the first factor. The restriction of a local map g = (X, Y, Γ) to a subset C of X is the triple Incidentally, we point out that sets and local maps (with the obvious composition) constitute a category.
Whenever it makes sense (e.g. when source and target spaces are manifolds), local maps are tacitly assumed to be continuous.
Throughout the paper M denotes a finite dimensional, smooth, real, Hausdorff, second countable manifold. Given any x ∈ M , I x denotes the identity on the tangent space T x M of M at x.
which, by abuse of terminology, will be referred to as "H is fixed point free on ∂U ".
We shall show that there exists at most one function that to any admissible pair (f, U ) assigns an integer ind(f, U ), called fixed point index of f in U or index of the pair (f, U ), that satisfies the following three axioms.

Homotopy invariance. If H is an admissible homotopy in U , then
ind H(·, 0), U = ind H(·, 1), U . As a consequence of the additivity property and Remark 2.1, one easily gets the following (often neglected) property, which shows that the index of an admissible pair (f, U ) does not depend on the behavior of f outside U .
Let (f, U ) be admissible and let U 1 ⊆ U be open and such that Fix(f, U ) ⊆ U 1 . Then, by the additivity property, Remark 2.1, and localization, one gets Thus, we have the following important property of the fixed point index.
Excision. Given an admissible pair (f, U ) and an open subset and this implies the following property.

The fixed point index for linear maps
In this section we shall prove that, as a consequence of the properties of normalization, additivity and homotopy invariance, the index of an admissible pair The Euclidean norm of a vector v ∈ R m will be denoted by |v|. By L(R m ) we shall mean the normed space of linear endomorphisms of R m , and by GL(R m ) we shall distinguish the group of invertible ones. The identity on R m is represented by the symbol I. It is well known (see e.g. [1]) that the open subset GL(R m ) of L(R m ) has exactly two connected components: Therefore, N(R m ) has two connected components, N + (R m ) and N − (R m ), consisting, respectively, of those A ∈ GL(R m ) for which det(I −A) > 0 and det(I −A) < 0.
Since N + (R m ) and N − (R m ) are open in L(R m ) and connected, they are actually path connected. Consequently, given A ∈ N(R m ), the homotopy invariance implies that ind(A, R m ) depends only on the component of N(R m ) containing A. Therefore, given A ∈ N + (R m ), one has ind(A, R m ) = ind(0, R m ), where 0 is the trivial operator. Thus, by normalization, we get We will prove that ind(A, R m ) = −1 for any A ∈ N − (R m ). As a distinguished representative in N − (R m ) we choose the linear operatorÂ given by   We conclude the section with a technical result regarding linearizable maps. Proof. By definition of differentiability we get where ε : U −p → R m is a continuous map with ε(0) = 0. Thus Since f (p) is nondegenerate, inf |v|=1 I − f (p) v > 0, and this implies that p is an isolated fixed point of f . Let V ⊆ U be any neighborhood of p such that Fix(f, V ) = {p}, and consider the homotopy The above argument shows that in some neighborhood W ⊆ V of p one has Consequently, by excision, Since the affine map H(x, 0) = p + f (p)(x − p) is admissibly homotopic in R m to its linear part x → f (p)x, the homotopy invariance property yields The assertion follows from (3.4) and (3.5).

The uniqueness result
Given a local map f in M and a relatively compact open subset U of M , the pair (f, U ) will be called nondegenerate if f is smooth on U, fixed point free on ∂U , and the Fréchet derivative of f at any fixed point in U is nondegenerate (as in the case of R m , an endomorphism of a vector space is nondegenerate if 1 is not an eigenvalue). Note that, in this case, Fix(f, U ) is necessarily a discrete set, therefore finite, being closed in the compact set U . In particular (f, U ) is an admissible pair.
The following lemma shows that the computation of the fixed point index of any admissible pair can be reduced to that of a nondegenerate pair. Proof. Without loss of generality we may assume that M is embedded in some R k . Thus, because of the ε-Neighborhood Theorem (see e.g. [7]) there exist an open neighborhood Ω of M in R k and a smooth submersion r : Ω → M such that |x − r(x)| = dist(x, M ) for all x in Ω. In particular, M is a retract of Ω. Since V is compact, given δ > 0, the Weierstrass Approximation Theorem implies the existence of a polynomial map f δ : R k → R k such that |f (x) − f δ (x)| < δ for all x ∈ V . Again by the compactness of V , we may assume that δ is such that the homotopy is well defined on V × [0, 1] and fixed point free on ∂V (where ∂V is the boundary of V relative to M ⊆ R k ). Consequently, f is admissibly homotopic in V to the smooth map h := F δ (·, 1). It is enough to prove that h is admissibly homotopic in V to some local map g such that (g, V ) is a nondegenerate pair. Observe first that an admissible pair (g, V ), with g smooth on V and fixed point free on ∂V , is nondegenerate if and only if the graph map x → (x, g(x)) is transversal in V to the diagonal ∆ of M × M . We apply the Transversality Theorem (see e.g. [7]) to the map is an open ball about the origin so small that h(x)+y ∈ Ω for all (x, y) ∈ V × B and the maps x → r(h(x) + y) are all fixed point free on ∂V . This is possible since V is compact and h(x) = x for all x ∈ ∂V .
Since r is a submersion, given any (x, y) ∈ G −1 (∆), the derivative is surjective, and this implies that G is transversal to ∆ in V × B. Consequently, the Transversality Theorem ensures the existence of a pointȳ ∈ B such that the partial map is transversal to ∆ in V . This, as pointed out before, means that any fixed point in V of the smooth map g(x) := r(h(x) +ȳ) is nondegenerate. The conclusion follows by observing that the assumption on B ensures that the homotopy H : V × [0, 1] → M given by H(x, λ) = r(h(x) + λȳ) is fixed point free on ∂V , therefore admissible because of the compactness of V .
We will show that the properties of normalization, additivity and homotopy invariance imply a formula for the computation of the fixed point index that is valid for any nondegenerate pair. Therefore, Lemma 4.1, the excision and the homotopy invariance properties imply the existence of at most one real function on the set of admissible pairs that satisfies the fixed point index axioms. Moreover, since the function defined by this formula is integer valued, so is the fixed point index.
Consequently, there exists at most one function on the set of admissible pairs satisfying the fixed point index axioms, and this function is integer-valued.
Proof. Consider first the case M = R m . Let (f, U ) be a nondegenerate pair in R m and, for any x ∈ Fix(f, U ), let V x be an isolating neighborhood of x. Since Fix(f, U ) is finite, we may assume that the neighborhoods V x 's are pairwise disjoint. The additivity property, Lemma 3.3 and Lemma 3.2 yield Now the uniqueness of the fixed point index on R m follows immediately from Lemma 4.1, taking into account the properties of excision and homotopy invariance.
Let us now consider the general case and denote by m the dimension of M . Let W be an open subset of M which is diffeomorphic to the whole space R m and let ψ : W → R m be any diffeomorphism onto R m . Denote by U the set of all pairs (f, U ) which are admissible and such that U ⊆ W , f (U ) ⊆ W . These pairs may be regarded as admissible in W , and the restriction of the index function to U still satisfies the fixed point index axioms. We claim that for any (f, U ) ∈ U one necessarily has where (for the moment) i denotes the (unique) fixed point index on R m . To show this, denote by V the set of pairs (g, V ) which are admissible in R m and consider the one-to-one correspondence ω : U → V defined by We need to prove that ind = i • ω. Observe that and if two pairs (f, U ) ∈ U and (g, V ) ∈ V correspond under ω, then the sets Fix(f, U ) and Fix(g, V ) correspond under ψ. It is also evident that the function ind • ω −1 satisfies the fixed point index axioms. Thus, i and ind • ω −1 coincide on V, and this implies ind = i • ω, as claimed.
Let now (f, U ) be a given nondegenerate pair in M . Let Fix(f, U ) = {x 1 , . . . , x n } and let W 1 , . . . , W n be n pairwise disjoint open subsets of U such that x j ∈ W j , for j = 1, . . . , n. Since any point of M has a fundamental system of neighborhoods which are diffeomorphic to the whole space R m , we may assume that each W j is diffeomorphic to R m under a diffeomorphism ψ j . For any j, let U j be an open subset of W j such that f (U j ) ⊆ W j . The additivity property yields ind(f, U ) = n j=1 ind(f, U j ), and, by the above claim, we get By the excision property, Lemma 3.2, and the chain rule for the derivative one has for j = 1, . . . , n. Thus ind(f, U ) = n j=1 sign det I xj − f (x j ) .
As in the case when M = R m , the uniqueness of the fixed point index is now a consequence of Lemma 4.1.