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Nielsen-Reidemeister indices for multivalued maps.

1 Introduction

Let p : [~.X] [right arrow] X be a finite (not necessarily connected) n-fold cover of a compact ENR (Euclidean Neighbourhood Retract) X and suppose that f : [~.X] [right arrow] X is a continuous map. Then we can define a multivalued map F : X [??] X by

F (x) = {f([~.x]) | [~.x] [member of] [~.X], p ([~.x]) = x} (x [member of] X).

Thus F (x) is a finite non-empty set with cardinality at most n: #F(x) [less than or equal to] n. If #F(x) = n for all x, the multivalued map F is an n-valued map in the sense of Schirmer [16]. In any case, following Brown [4] we can associate with the pair (f, p) an n-valued map [~.F] : [~.X] [??] [~.X] by setting

[~.F] ([~.x]) = {[~.y] [member of] [~.X] | p([~.y])=f([~.x])} ([~.x] [member of] [~.X]).

(To be accurate, Brown considered only the case of a map f that factors through p as a composition [~.X] [right arrow] X [right arrow] X.)

A systematic study of the fixed point theory of the pair (f, p), by analogy with the theory of Vietoris fractions (see [13, Section 19]), was undertaken in [8], using the notation 'f/p ' for the pair considered as a fraction. The primary object of study, which we shall call the fixed-point set of f/p, is the coincidence set of the pair.

Definition 1.1. The fixed-point set of f/p is the closed subspace

Fix(f/p) = {[~.x] [member of] [~.X] | f([~.x]) = p([~.x])}

of [~.X]. It projects by p onto the fixed subspace Fix(F) = {x [member of] X | x [member of] F (x)} of the multivalued map F.

The basic fixed-point index of f/p was constructed in [8], and called there the homotopy Lefschetz index, as an element

h-L(f/p) [member of] [[omega].sub.0](h-Fix(f/p))

of the stable homotopy group of the homotopy fixed-point set of f/p.

Definition 1.2. The homotopy fixed-point set of f/p is the subspace

h-Fix(f/p) = {([~.x],[alpha]) | [~.x] [member of] [~.X], [alpha] : [0,1] [right arrow] X, [alpha](0) = p ([~.x]), [alpha](1) =f([~.x])},

of [~.X] x map([0,1], X). Thus, each element is given by a point [~.x] of [~.X] and a (continuous) path from p([~.x]) to f([~.x]) in X. We write [pi] : h-Fix(f/p) [right arrow] [~.X] for the projection to the first factor. The fixed-point set is included as a subspace of the homotopy fixed-point set, Fix(f/p) [??] h-Fix(f/p), by mapping to [~.x] to ([~.x],[alpha]) where [alpha] is the constant path at p([~.x]) = f([~.x]).

Now [[omega].sub.0](h-Fix(f/p)) is just the direct sum [[direct sum].sub.[gamma]] Z[[gamma]], of copies of Z indexed by the components [gamma] of h-Fix(f/p), and we can decompose h-L(f/p) as

[mathematical expression not reproducible],

where [l.sub.[gamma]] [member of] Z is zero for all but finitely many components [gamma]. Classically, when p is the identity [~.X] = X [right arrow] X, this is the Reidemeister trace of f, and the Nielsen number counts the components [gamma] such that [l.sub.[gamma]] = 0. We make the same definition for the fraction f/p.

Definition 1.3. The Nielsen number N(f/p) of f/p is the number of components [gamma] of h-Fix(f/p) such that [l.sub.[gamma]] is non-zero.

It is a consequence of [8, Proposition 3.5] that, when F is an n-valued map, N(f/p) is equal to Schirmer's Nielsen index N(F). The main purpose of this note is to show that, in general, N(f/p) is equal to the Nielsen number N([~.F]) of the n-valued map [~.F] studied by Brown. This will be deduced in Section 3 from the commutativity property of the fixed-point index.

In Section 2, expanding a brief account in [7, Section 5], we give a definition of the fixed-point index following the pattern of Dold's construction [10,11,12] of the index for single-valued maps on ENRs, treating the finite cover p : [~.X] [right arrow] X as the 0-dimensional special case of a fibrewise smooth manifold p : [~.X] [right arrow] X over a compact ENR X with each fibre a closed manifold of some fixed dimension m.

Section 4 computes the index in a particular example involving projective spaces.

Notation. Given a real vector bundle [xi] over a space X and a subspace U of X, we shall use the superscript notation U[xi] for the Thom space of the restriction [xi] | U of the vector bundle [xi] to the subspace U. Similar notation is used for virtual vector bundles. In particular, the Thom space [U.sup.-[xi]] of the negative -[xi] | U is realized, by a trivialization [xi] [direct sum] [eta] [??] X x [R.sup.k] for some vector bundle [eta] over X, as the desuspension [[summation].sup.-k]([U.sup.[eta]]) of the Thom space of [eta] | U.

2 Construction of the fixed-point index

We now fix a compact ENR X and a fibrewise smooth manifold p : [~.X] [right arrow] X with each fibre a closed smooth manifold of dimension m. Its fibrewise tangent bundle, which is an m-dimensional real vector bundle over [~.X], will be denoted by [tau](p).

An account of general fibrewise manifolds (or manifolds over a base) can be found in [1, Section 1] or [9, Part II, Section 11]. Many examples arise as follows. Suppose that G is a compact Lie group, M is a closed G-manifold of dimension m and P [right arrow] X is a principal G-bundle. Then [~.X] = P [x.sub.G] M [right arrow] X is a fibrewise manifold. Its fibrewise tangent bundle is P [x.sub.G] [tau]M [right arrow] P [x.sub.G] M, where [tau]M [right arrow] M is the tangent bundle of M. Every finite n-fold covering space arises in this way, with G a finite group acting on a finite set M of cardinality n.

Consider a (continuous) map f : [~.X] [right arrow] X. The fixed-point set and homotopy fixed-point set of f/p

Fix (f/p) [??] h-Fix (f/p) [[pi].arrow] [~.X]

are defined as in Definitions 1.1 and 1.2. There are associated multivalued maps F : X [??] X and [~.F] : [~.X] [??] X, given by

F (x) = f([p.sup.-1] (x)) and [~.F] ([~.x]) = [p.sup.-1] (f([~.x])), (x [member of] X, [~.x] [member of] [~.X]).

Suppose that U [??] [~.X] is an open subspace such that U [intersection] Fix(f/p) is compact. We shall first construct a topological Lefschetz index

t-L(f/p | U) [member of] [[omega].sub.0]([U.sup.-[tau](p)])

in the stable homotopy group of the Thom space of the restriction -[tau](p) | U of the virtual bundle -[tau](p) to the subspace U.

We choose a fibrewise smooth embedding j : [~.X] [right arrow] X x F, over X, for some Euclidean space F. The (fibrewise) normal bundle [nu] of j satisfies [tau](p) [direct sum] [nu] = [~.X] x F (up to homotopy). Using the Riemannian metric (on F) we can construct a fibrewise tubular neighbourhood of j over X: D([nu]) [right arrow] Xx F. (Here D([nu]) is the closed unit disc bundle in an appropriately scaled metric.) We also need to choose an embedding i : X [right arrow] X' [??] E of the ENR X as a retract of an open subspace X' of a Euclidean space E, with a retraction r : X' [right arrow] X. Then construct the pullback p' : [~.X]' = r*[~.X] [right arrow] [~.X]' of p : X [right arrow] X and the corresponding tubular neighbourhood j : D([nu]') [right arrow] X' x F, where [nu]' = r*[nu], of [~.X]'. The map f : [~.X] [right arrow] X extends to a map f' = i [??] f [??] r : [~.X]' [right arrow] X'.

Let U' denote the open subset [r.sup.-1](U) of [~.X]'. Then U' [intersection] Fix(f'/p) = U [intersection]Fix(f/p) is compact. To avoid complicating the notation, we shall regard D([nu]), using i and j, as a subspace of E [direct sum] F and X' as a subspace of E.

The Lefschetz index will be represented by a pointed map

[E.sup.+] [conjunction] [F.sup.+] [right arrow] [E.sup.+] [conjunction] [U.sup.[nu]],

where the superscript '+' means the one-point compactification so that [E.sup.+] and [E.sup.+] [conjunction] [F.sup.+] = [(E [direct sum] F).sup.+] are spheres with basepoint at infinity. The Thom space [U.sup.[nu]] is the topological quotient of the disc bundle D([nu]|U) by the unit sphere bundle S([nu]|U).

By the compactness of U' [intersection] Fix(f'/p'), we may choose an open neighbourhood V in [~.X]' with compact closure [bar.V] such that

U' [intersection] Fix (f'/p') [??]. V [??]. [bar.V][??]. U'

and then a real number [member of] > 0 such that ||p'([~.x]) - f([~.x]) || >> [member of] for all [~.x] [member of] [bar.V] - V.

Now we can write down an explicit pointed map

[phi] : [E.sup.+] [conjunction] [F.sup.+] = [(E [direct sum] F).sup.+] [right arrow] [E.sup.+] [conjunction] [U.sup.[nu]] = [E.sup.+] [conjunction] (D([nu] |U)/S([nu]|U))

as follows. A point v in the closed subspace D([nu]'|[bar.V]) of [(E [direct sum] F).sup.+] lies in the fibre of [nu] over some point x [member of] [bar.V]: v [member of] D([[nu].sub.[~.X]]). We define

[phi](v) = [[c.sub.[member of]](p'([~.x]) - f'([~.x])),r(v)],

where c[member of] : E [right arrow] [E.sup.+] is given by

[mathematical expression not reproducible]

Notice that, if [~.x] [member of] [bar.V] - V, then ||p([~.x]) - f([~.x]) || >> [member of], so that [phi](v) = *. This means that [phi] takes the value * on the boundary S([nu]'| [bar.V]) [union] D([nu]'|[bar.V] - V) of D([nu]'| [bar.V]) and we can extend [phi] over [(E [direct sum] F).sup.+] to take the value * on the closed complement of the open unit disc bundle B([nu]' | V).

Forming the class of [phi] as a stable map from [F.sup.+] to [U.sup.[nu]], we obtain the topological Lefschetz index

t-L(f/p | U) [member of] [[omega].sub.0]([U.sup.-[tau](p)]) = [[omega].sup.0]{[F.sup.+]; [U.sup.[nu]]}.

(Here we use the notation [[omega].sup.0]{A; B} for the group of stable maps from a pointed space A to a pointed space B.)

To construct the homotopy Lefschetz index, we choose U to be an open neighbourhood of Fix(f/p) such that the straight line segment joining p([~.x]) to f([~.x]) lies in X' for all [~.x] [member of] U (that is, (1 - t)ip([~.x]) + ti f([~.x]) [member of] X' for 0 << t << 1). Then we have a map

U [right arrow] h-Fix (f/p) : [~.x] [right arrow] ([~.x], [alpha]), where [alpha](t) = r((1 - t)ip([~.x]) + tif([~.x])),

extending the inclusion of Fix(f/p) in h-Fix(f/p).

We define the homotopy Lefschetz index, or Nielsen-Reidemeister index,

h-L(f/p) [member of] [[[~.[omega]].sub.0]](h-Fix(f/p)-.sup.[[pi]].sup.[tau](p))

of f/p to be the image of t-L(f/p | U) under the induced map

[[~.[omega].sub.0]][([U.sup.-[pi]]*.sup.[tau](p)]) [right arrow] [[~.[omega].sub.0][(h-Fix[(f/p)-.sup.[pi]]*.sup.[tau]].sup.(p.sup.)]).

It determines the global topological Lefschetz index of f/p

t-L(f/p) [member of] [[~.[omega].sub.0]] ([X.sup.-[tau](p)]),

which is defined as [pi]*(h-L(f/p)).

It is, of course, necessary to verify that the classes so constructed are independent of the choices made. This is best done by placing the definition in the wider context of fibrewise maps and simultaneously establishing the homotopy invariance of the index. When p is the identity map 1 : [~.X] [right arrow] X, so that f is a map X [right arrow] X, the construction reduces to Dold's definition of the fixed-point index of a single-valued map as described, for example, in [9, 6]: we take F to be the zero vector space and j to be the identity map 1 : X [right arrow] X = X x 0. The verification proceeds as in this special case, and the details will be omitted here.

It is clear from the construction that the index h-L(f/p) vanishes if the fixed-point set Fix(f/p) is empty. The standard properties of the Lefschetz index (localization at the fixed-point set, additivity, homotopy invariance, multiplicativity) also follow essentially as in the classical case. Commutativity, which is more subtle, will be the subject of the next section. In the remainder of this section we look at two special features of the theory for multivalued maps.

Trivial bundles. We consider, first, the case in which the bundle p : [~.X] [right arrow] X is trivial. Suppose that p is the projection X x M [right arrow] X, where M is a closed smooth manifold of dimension m. The fibrewise tangent bundle [tau](p) is the pullback of the tangent bundle [tau] M of M.

From f, which is now a map f : X x M [right arrow] X, we can construct a fibrewise map

[f.sup.#] : X x M [right arrow] X x M, (x,y) [right arrow] (f(x,y),y)

over the compact manifold M, that is, a family of maps [f.sup.#.sub.y] : X [right arrow] X parametrized by y [member of] M: [f.sup.#.sub.y] (x) = f(x, y).

The fibrewise fixed-point set

[Fix.sub.M]([f.sup.#]) = {(x,y) [member of] X x M | [f.sup.#.sub.y](x) = x}

and homotopy fixed-point set h-[Fix.sub.M]([f.sup.#]), defined as

{((x,y),[alpha]) | (x,y) G X x M, [alpha] : [0,1] [right arrow] X, [alpha](0) = x, [alpha](1) = [f.sup.#.sub.y](x)},

are transparently the same as the fixed-point sets Fix(f/p) and h-Fix(f/p) of f/p. We shall show, by comparing the definitions, that the index h-L(f/p) of the fraction coincides with the fibrewise fixed-point index h-[L.sub.M]([f.sup.#]) of the fibrewise map.

The fibrewise homotopy Lefschetz index h-[L.sub.M]([f.sup.#]) is an element of the group

[[omega].sup.0.sub.M]{M x S ; h-[Fix.sub.M][([f.sup.#]).sub.+M]}

of fibrewise stable maps over M from M x [S.sup.0] to the fibrewise pointed space obtained by adjoining a disjoint basepoint to each fibre of h-[Fix.sub.M]([f.sup.#]) [right arrow] M. The two indices are related by the Poincare-Atiyah duality isomorphism

[mathematical expression not reproducible].

(See, for example, [7, Proposition 4.1] and the references given there.)

Proposition 2.1. Suppose that p : [~.X] = X x M [right arrow] X is trivial, as described in the text. Then the fixed-point index h-L(f/p) of f/p is equal to the image under the duality isomorphism [[lambda].sub.M]

[mathematical expression not reproducible]

of the fibrewise fixed-point index h-[L.sub.M]([f.sup.#]) of the fibrewise map [f.sup.#] determined by f.

Outline proof. This will be verified by following through the explicit geometric definitions. We choose an embedding of X as a retract r : X' [right arrow] X of an open subspace X' [??] E of a Euclidean space E, a smooth embedding of M in a Euclidean space F, with normal bundle [nu], and a tubular neighbourhood D([nu]) [right arrow] F. This allows us to treat X and M as subspaces of E and F, respectively.

Suppose that U [??] X x M is an open neighbourhood of Fix(f/p) = Fix([f.sup.#]). Let V be in an open neighbourhood of Fix (f/p) inX'xM such that [bar.V] is compact and contained in [(r x 1).sup.-1]U. There is some [member of] > 0 such that ||x - f(r(x),y) || >> [member of] for all (x, y)

[member of] [bar.V] - V.

The topological fibrewise Lefschetz index t-[L.sub.M]([f.sup.#] | U) is a stable map M x [S.sup.0] [right arrow] [U.sub.+M] over M. (See, for example, [9, Part II, Section 6].) It is represented by the map

E+ x M [right arrow] (E+ x M) [[conjunction].sub.M] [U.sub.+M]

sending (x,y) [member of] [bar.V] to [[c.sub.[member of]](x - f(r(x),y)), (r(x),y)] and a point (x,y) in the complement of V in [E.sup.+] x M to the basepoint over y [member of] M. (Here, again, [U.sub.+M] is the fibrewise pointed space U [??] M obtained by adjoining a basepoint in each fibre.

The duality isomorphism [[lambda].sub.M] is constructed in three steps by taking the smash product over M with the identity map [mathematical expression not reproducible] on the fibrewise one-point compactification of [nu] over M to get a fibrewise stable map

[mathematical expression not reproducible],

then collapsing fibrewise basepoints to a single point to get a map of pointed spaces

[E.sup.+] [conjunction] [M.sup.[nu]] [right arrow] [E.sup.+] [conjunction] [U.sup.[nu]],

and finally composing with the product of the Pontryagin-Thom map [F.sup.+] [right arrow] [M.sup.[nu]] with the identity on [E.sup.+] to produce an explicit map

[E.sup.+] [conjunction] [F.sup.+] [right arrow] [E.sup.+] [conjunction] [U.sup.[nu]].

This is exactly the map defining t-L(f/p | U).

Of course, the bundle [~.X] [right arrow] X is locally trivial and there is a similar local result. Suppose that U [??] [~.X] is an open set such that U [intersection] Fix(f/p) is a component N of the fibre M of p at a point [x.sub.0] [member of] X. Choose a local trivialization [p.sup.-1](W) = W x M[right arrow] W of [~.X][right arrow]X over some open neighbourhood W of [x.sub.0]. To keep the notation simple, we shall identify [p.sup.-1](W) with W x M. By replacing U by a smaller neighbourhood of N we may assume that it has the form U = V x N, where V [??] W is an open neighbourhood of [x.sub.0], and that f(U) [??] W. Then f determines a fibrewise map [f.sup.#] : V x N [right arrow] W x N over N with fixed-point set Fix([f.sup.#]) = {[x.sub.0]} x N.

The argument outlined above expresses t-L(f/p | U) as the image of the fibrewise topological Lefschetz index t-[L.sub.N]([f.sup.#] | U) under the duality isomorphism

[mathematical expression not reproducible].

Because X is an ENR, there is a smaller open neighbourhood [V.sub.0] [??] V of [x.sub.0] inside V such that the inclusion [V.sub.0] [right arrow] V is homotopic, through a homotopy inside V that fixes [x.sub.0], to the constant map at [x.sub.0]. It follows that t-[L.sub.N]([f.sup.#] | U) is the image of the fibrewise Lefschetz index

LN([f.sup.#] | U) [member of] [[omega].sup.0](N) = [[omega].sup.0]{[N.sub.+]; [S.sup.0]} = [[omega].sub.0]([N.sup.-[tau]N])

under the map induced by the inclusion of [S.sup.0] as {[x.sub.0]}+ in V+. We shall refer to this class [L.sub.N]([f.sup.#] | U) as a local index. It contributes to the index h-L(f/p) through the homomorphism

k : [[~.[omega]].sub.0]([N.sup.-[tau]N]) [right arrow] [[~.[omega]].sub.0](h-Fix[(f/p).sup.-[[pi]]*.sup.[[tau]].sup.(p)])

induced by the inclusion k : N [right arrow] Fix(f/p) [right arrow] h-Fix(f/p).

Proposition 2.2. Consider a general (so locally trivial) fibrewise manifold p : [~.X] [right arrow] X. Suppose that Fix (f/p) is the disjoint union of a finite number of connected components [N.sub.i], i [member of] I, of the fibres over points [x.sub.i] [member of] X. Let [k.sub.i] : [N.sub.i] [right arrow] Fix (f/p) [right arrow] h-Fix (f/p) denote the inclusion and let [mathematical expression not reproducible] be the local Lefschetz index described above. Then

h-L(f /p) = [summation over (i[member of]I)] ([k.sub.i])*([L.sub.i]).

Proof. It suffices to show that, for any neighbourhood U of Fix(f/p), the index t-L(f/p | U) is the sum of terms ([k.sup.U.sub.i])*([L.sub.i]), (i [member of] I), where [k.sup.U.sub.i] : [N.sub.i] [right arrow] U is the inclusion. This follows, by the additivity of the index, from the discussion above applied to disjoint neighbourhoods [U.sub.i] of [N.sub.i] in U.

Corollary 2.3. Suppose that p : [~.X] [right arrow] X is a finite n-fold cover and that #F(x) = n for all x [member of] X. Then h-L(f/p) coincides with the fixed-point index of the n-valued map F defined by Schirmer in [16].

Proof. Schirmer's definition proceeds by reduction, through a homotopy, to the case in which Fix(f/p) is a finite set. In that case, we can apply Proposition 2.2 to express the index as a sum of local Lefschetz indices as in [16]. The assertion then follows from the homotopy invariance of the index.

Smooth fibre bundles. We consider next the case in which X is a closed manifold and p : [~.X] [right arrow] X is a smooth fibre bundle. The fixed-point set Fix(f/p) is just the coincidence set {[~.x] [member of] [~.X] | p([~.x]) = f([~.x])} of p and f, and we shall show that h-Fix (f/p) is exactly the homotopy coincidence index, in the terminology of [7], of p and f. We form the trivial bundle E = [~.X] x X [right arrow] [~.X] over [~.X] with a preferred null section z, z(x) = (x,p(x)), and sections, s (x) = (x, f(x)), associated withf. The pullback [nu] = z*[[tau].sub.[~.X]]E of the fibrewise tangent bundle of E over [~.X] is identified with the tangent bundle [tau]X of X.

The coincidence set is called in [7] the null set Null(s) = {[~.x] [member of] [~.X] | s([~.x]) = z([~.x])} of the section s. The homotopy null set h-Null(s), defined in [7, Definition 2.3] as the space of pairs ([~.x], [alpha]) where [~.x] [member of] [~.X] is a point of the base and [alpha] is a path in the fibre over [~.x] from z([~.x]) to s([~.x]), is exactly h-Fix(f/p).

The homotopy Euler index of s is constructed in [7, Definition 2.4] as an element

h-[gamma](s) G [[omega].sup.0.sub.X] {X x [S.sup.0]; h-Null[(s).sup.[pi]*.sub.X.sup.[nu]]},

which is the (asymmetric) homotopy coincidence index of p and f. Again we have a duality isomorphism

[mathematical expression not reproducible].

Now the tangent bundle [tau][~.X] of the total space [~.X] of the bundle is identified, up to homotopy, with p*[tau]X [direct sum] [tau](p). This allows us to substitute -[pi]*[tau](p) for [pi]*[nu]-[pi]*[tau][~.X].

Proposition 2.4. Suppose p : [~.X] [right arrow] X is a smooth fibre bundle, as described in the text. Then h-L(f/p) is the image under the isomorphism

[mathematical expression not reproducible]

of the homotopy coincidence index h-[gamma](s) of p and f.

Proof. This is again established by a careful comparison of the two definitions. See [7, Proposition 5.4].

When f is smooth and transverse to p, that is, when p x f: [~.X] [right arrow] X x X is transverse to the diagonal, we can use Koschorke's definition of the coincidence index from [14, Section 4] to give a geometric description of h-L(f/p). The set Fix(f /p) is then an m-dimensional closed submanifold, N say, of [~.X], with tangent bundle [tau]N identified with the restriction of [tau](p).

Corollary 2.5. Suppose that p : [~.X] [right arrow] X is a smooth fibre bundle and that f : [~.X] [right arrow] X is smooth and transverse to p. Then the inclusion of N = Fix (f /p) in h-Fix(f/p)

[[~.[omega]].sub.0]([N.sup.-[tau]N]) [right arrow] [[~.[omega]].sub.0](h-Fix[(f/p).sup.-[[pi]]*.sup.[tau](p)])

maps the fundamental class [N] of N to h-L(f/p).

Proof. The interpretation of [[lambda].sub.[~.X]] ([gamma](s)) as the class represented by N is explained in [7, Proposition 4.6].

3 Commutativity

We begin by introducing an informal category of fractions in which the objects are compact ENRs and a morphism a/p from X to Y is a pair (a, p) consisting of a fibrewise manifold p : [~.X] [right arrow] X, with fibres closed manifolds of some dimension, m say, and a map a : [~.X] [right arrow] Y. (To be formal, a morphism should be specified by such a pair up to equivalence, so that the morphisms form a set.) Associated with the fraction a / p is the multivalued map A : X [??] Y given by

A(x) = {a([~.x]) | x [member of] [~.X], p([~.x]) = x} (that is, a([p.sup.-1](x))).

Composition is defined as follows. Suppose that Z is another compact ENR and that b/q is a morphism from Y to Z prescribed by a fibrewise manifold q : [~.Y] [right arrow] Y with fibres of dimension n and a map b : [~.Y] [right arrow] Z. We form the pullback

a*[~.Y] = {([~.x], [~.y]) [member of] [~.X] x [~.Y] | a([~.x]) = q([~.y])},

together with maps r : a*[~.Y] [right arrow] X, specifying a fibrewise manifold of dimension m + n, and c : a*[~.Y] [right arrow] Z given by r([~.x],[~.y]) = p([~.x]) and c([~.x],[~.y]) = b([~.y]). The composition b/q [??] a/p is defined to be c/r, as illustrated in the following diagram.

[mathematical expression not reproducible]

If B : Y [??] Z and C : X [??] Z are the multivalued maps associated with b/q and c/r, then C = B [??] A, that is,

C(x) = [[union].sub.y[member of]A](x) B(y).

The identity morphism on X is the fraction 1/1 given by the fibrewise manifold 1 : [~.X] = X [right arrow] X of dimension 0 and the identity map 1 : [~.X] = X [right arrow] X.

The index was defined in Section 2 for endomorphisms in this category. Turning to a discussion of commutativity, we now suppose that Z = X so that we can form the compositions b/q [??] a/p, which is an endomorphism of X, and a/p [??] b/q, an endomorphism of Y. There is also an associated endomorphism of X x Y given by the fibrewise manifold pxq: [~.X] x [~.Y] [right arrow]X x Y of dimension m + n and the map [tau] [??] (a xb) : [~.X] x [~.Y] [right arrow] X x Y, ([~.x], [~.y]) [right arrow] (b ([~.y]),a([~.x])), where [tau]:Y x X [right arrow] X x Y interchanges the two factors.

Proposition 3.1. There are homotopy equivalences

[mathematical expression not reproducible]

under which the fixed-point indices h-L(b/q [??] a/p), h-L([tau] [??] (a x b)/(p x q)) and h-L(a/p [??] b/q) coincide.

Proof. The homotopy fixed-point set

h-Fix(b/q [??] a / p) = {([~.x], [~.y], [alpha]) | [~.x] [member of] [~.X], [~.y] [member of] [~.Y], [alpha] : [0,1] [right arrow] X, [alpha](0) = p([~.x]), [alpha](1) = b([~.y]), a([~.x]) = q([~.y])}

is included in

h-Fix([tau] [??] (a xb)/(p x q)) = {([~.x], [~.y], [alpha],[beta]) | [~.x] [member of] [~.X], [~.y] [member of] [~.Y], [alpha] : [0,1] [right arrow] X, [beta] : [0,1] [right arrow] Y, [alpha](0) = p([~.x]), [alpha](1) = b([~.y]), [beta](0) = q([~.y]), [beta](1) = a([~.x])}

as the subspace of quadruples ([~.x], [~.y], [alpha], [beta]) with [beta] a constant path. Both contain the fixed subspace (abbreviated for the purposes of this proof to 'Fix')

Fix = Fix(b/q [??] a/p) = Fix([tau] [??] (a x b)/(p x q)) = {([~.x], [~.y]) | a([~.x]) = q([~.y]), b([~.y]) = p([~.x])} [??] [~.X] x [~.Y]

with both [alpha] and [beta] constant.

Since q : [~.Y] [right arrow] Y is a fibration, we can choose a lifting function

l : {([~.y],[beta]) | [~.y] [member of] [~.Y], [beta] : [0,1] [right arrow] Y, [beta](0) = q([~.y])} [right arrow] map([0,1], [~.Y])

such that [~.[beta]] = l([~.y], [beta]) is a path lifting [beta] with [~.[beta]](0) = [~.y].

Following [8, Proposition 3.10], we construct a deformation retraction of h-Fix([tau] [??] (a x b)/(p x q)) to the subspace h-Fix(b/q [??] a/p):

([~.x], [~.y],[alpha],[beta]) [right arrow] ([~.x], [~.[beta]] (t),[alpha]t,[beta]t), (0 << t << 1),

where [~.[beta]] = l([~.y],[beta]) and

[mathematical expression not reproducible]

and [beta]t (u) = [beta](t + (1 - t)u) for 0 << u << 1.

This establishes that inclusion

h-Fix(b/q [??] a / p) [right arrow] h-Fix([tau] [??] (a x b)/(p x q))

is a homotopy equivalence. Notice that the pullback of the fibrewise tangent bundle [tau](p x q) restricts to the pullback of the fibrewise tangent bundle of a* [~.Y] [right arrow] X: the fibre at ([~.x], [~.y],[alpha], [beta]) is equal to [[tau].sub.[~.x]] (p) [direct sum] [[tau].sub.[~.y]](q).

To prove commutativity, we shall essentially follow Dold's argument in the classical case [10] and reduce to working on subspaces of Euclidean spaces.

It suffices to show that, for any 'small' open neighbourhood U of the fixed-point set Fix in [~.X] x [~.Y], the inclusion

[[~.[omega]].sub.0][((U [intersection] a*[~.Y]).sup.-([tau](p)[direct sum][tau](q))]) [right arrow] [~.[omega]] 0([U.sup.-([tau](p)[direct sum][tau](q))])

maps t-L(b/q [??] a/p | U [intersection] a*[~.Y]) to t-L([tau] [??] (a x b)/(p x q) | U).

As in the construction of the index, we first choose embeddings X [right arrow] X' [right arrow] [E.sub.X] and Y [??]* Y' [??]* [E.sub.Y] of X and Y as retracts r : X' [right arrow] X and s : Y' [right arrow] Y of open subspaces of Euclidean spaces [E.sub.X] and [E.sub.Y], and form the pullbacks [~.X]' = r*[~.X], [~.Y]' = s*[~.Y]. Then a and b are extended, using the retractions, to maps a' : [~.X]' [right arrow] Y and b' : [~.Y]' [right arrow] X. Thus

(a')*[~.Y] = {([~.x],[~.y]) [member of] [~.X]' x [~.Y] | a'([~.x]) = q([~.y])} [??][~.X]' x [~.Y].

Now choose an open neighbourhood V of

Fix = {([~.x],[~.y]) [member of] [~.X]' x [~.Y] | a'([~.x]) = q([~.y]), b([~.y]) = p'([~.x])}

in (a')*[~.Y] such that [bar.V] is compact and contained in U' = [(r x s).sup.-1] (U).

Writing [D.sub.[delta]]([E.sub.Y]) for a closed disc of radius [delta] > 0 centred at 0 in [E.sub.Y], we can use the compactness of Y to choose [delta] such that y + w [member of] Y' for all (y, w) [member of] Y x [D.sub.[delta]]([E.sub.Y]) = Z, say. Then we have a map

(y,w) [RIGHT ARROW]*y+w:Z = YX [D.sub.[delta]](EY) [right arrow] Y'

and can pull back q' : [~.Y]' [right arrow] Y' to get a fibrewise manifold [~.Z] [right arrow] Z over Z. Since [D.sub.[delta]](EY) is contractible, there is a fibrewise equivalence

[theta] : [~.Y] x [D.sub.[delta]](EY) [right arrow] [~.Z]

of fibrewise manifolds over Y x [D.sub.[delta]](EY) = Z with [theta] ([~.y], 0) = ([~.y], 0) for [~.y] [member of] [~.Y].

The equivalence [theta] provides a family of diffeomorphisms

[[theta].sub.y,w] : [[~.Y].sub.y] [right arrow] [[~.Y].sub.y+w], (y, w) [member of] Z,

such that q'([[theta].sub.y,w]([~.y])) = y + w and [[theta].sub.y,0]([~.y]) = y. It is convenient to be slightly imprecise and abbreviate [[theta].sub.y,w]([~.y]) to [theta]([~.y], w). This allows us to write down a crucial map

[psi] : [bar.V] x [D.sub.[delta]] ([E.sub.y]) [right arrow] [~.X]' x [~.Y]', (([~.x],[~.y]),w) [right arrow] ([~.x], [theta]([~.y],w)).

Notice that, for ([~.x],[~.y]) [member of] Fix, [psi] maps (([~.x],[~.y]),0) to ([~.x],[~.y]). The image of the open neighbourhood V x [B.sub.[delta]](EY) of Fix x {0} is an open neighbourhood W of Fix in [~.X]' x [~.Y]' with closure [bar.W] equal [psi]([bar.V] x [D.sub.[delta]] ([E.sub.Y])).

We shall use the neighbourhood V for the description of the topological Lefschetz index of b/q [??] a/p and the neighbourhood W for [tau] [??] (a x b)/(p x q).

The index t-L(b/q [??] a/p) | U [intersection] a*[~.Y] is determined by the map

g : [bar.V] [right arrow] [E.sub.X], ([~.x],[~.y]) [right arrow] p'([~.x]) - b([~.y]),

and the index t-L([tau] [??] (a x b)| U) by the map

[h.sub.0] : [bar.W] [right arrow] [E.sub.X] x [E.sub.Y]

taking [psi] (([~.x], [~.y]), w), for (([~.x],[~.y]),w) [member of] [bar.V] x [D.sub.[delta]] ([E.sub.Y]), to

(p' ([~.x]) - b' ([theta]([~.y],w)),q' ([theta]([~.y],w)) - a' ([~.x])) = (p'([~.x]) - b' ([theta]([~.y],w)),w).

In order to relate [h.sub.0] to g, we deform this map by a homotopy [h.sub.t] : [bar.W] [right arrow] [E.sub.X] x [E.sub.Y]:

[psi](([~.x],[~.y]),w) [right arrow] (p'([~.x]) - (1 - t)b' ([theta]([~.y],w)) - tb([~.y]),w), 0 << t << 1,

which is nowhere zero on the boundary [bar.W] - W. (For, if w = 0, we have [theta]([~.y], w) = [~.y], so that p'([~.x]) - (1 - t)b' ([theta]([~.y],w)) - tb([~.y]) = p'([~.x]) - b([~.y]), which is non-zero on [bar.V] - V.) By compactness, there is an [member of] > 0 such that ||[h.sub.t](([~.x],[~.y]),w)|| >> [member of] for all points (([~.x],[~.y]),w) in [bar.W] - W.

We can thus use the map [h.sub.1], taking [psi](([~.x], [~.y]), w) to (g([~.x], [~.y]), w), rather than [h.sub.0], to realize the index.

To complete the construction we need to choose fibrewise embeddings [~.X] [right arrow] X x [F.sub.p] : [~.x] [right arrow] ([~.x],[j.sub.p]([~.x])), [~.Y] [right arrow] Y x [F.sub.q] : [~.y] [right arrow] ([~.y],[j.sub.q]([~.y])), and tubular neighbourhoods D([[nu].sub.p]) [??] X x [F.sub.p], D([[nu].sub.q]) [??] Y x [F.sub.q], which we pull back to tubular neighbourhoods of [~.X]' and [~.Y]'.

The map [h.sub.1] and the embedding of the disc bundle D([[nu].sub.p] [direct sum] [[nu].sub.q]|[bar.W]) into ([E.sub.X] [direct sum] [E.sub.Y]) x ([F.sub.p] [direct sum] [F.sub.q]) determine the index of [tau] [??] (a x b)/(p x q).

Using [theta] we can pull back the embedding of [~.Y]' into Y' x [F.sub.q] to an embedding of (a')*[~.Y] x [D.sub.[delta]](EY) into [~.X]' x [D.sub.[delta]](EY) x [F.sub.q] over [~.X]' x [D.sub.[delta]](EY). Combining this with the embedding of [~.X]' into X' x [F.sub.p], we obtain a fibrewise embedding of (a')*[~.Y] x [D.sub.[delta]](EY) into [~.X]' x [D.sub.[delta]]([E.sub.Y]) x ([F.sub.p] [direct sum] [F.sub.q]) over X' x [D.sub.[delta]]([E.sub.Y]). Then we can include X' into [E.sub.X] and D([E.sub.Y]) into [E.sub.Y] to embed (a')*[~.Y] x [D.sub.[delta]]([E.sub.Y]) into ([E.sub.X] [direct sum] [E.sub.Y]) x ([F.sub.p] [direct sum] [F.sub.q]). A tubular neighbourhood of this embedding and the map g on [bar.V] then determine the index of (b/q) [??] (a/p), or, to be precise, g x 1 on [bar.V] x [D.sub.[delta]]([E.sub.Y]) determines the index of ((b/q) [??] (a/p)) x (z/1), where z/1 is the fraction given by the identity and zero maps 1, z : [D.sub.[delta]]([E.sub.Y]) [right arrow] [D.sub.[delta]]([E.sub.Y]).

The maps [h.sub.1] on [bar.W] and g x 1 on [bar.V] x [D.sub.[delta]](EY) correspond via [psi]. However, the embeddings into ([E.sub.X] [direct sum] [E.sub.Y]) x ([F.sub.p] [direct sum][F.sub.q]) do not correspond: they map (([~.x], [~.Y]), w) [member of] [bar.V] x [D.sub.[delta]](EY) to (p'([~.x]), a'([~.x]) +w, [j.sub.p]([~.x]), [j.sub.p]([~.y])) and (p'([~.x]), w, [j.sub.p]([~.x]), [j.sub.p]([~.y])), respectively. But the two are connected, as in the classical proof [10], by a second homotopy:

(([~.x],[~.y]),w) [right arrow] (p'([~.x]), (1 - t)a'([~.x]) + w, [j.sub.p]([~.x]), [j.sub.q]([~.y])), 0 << t << 1.

We conclude that the topological index of [tau] [??] (a x b) / (p x q) on [bar.W] coincides with the topological index of (b/q [??] a/p) x (z/1) on [bar.V] x [D.sub.[delta]](EY). But taking the product with the index of z/1 amounts to suspending by the identity map on [E.sup.+.sub.Y]. So this completes the proof of the commutativity property of the index.

Another, rather more elementary, proof for the special case of finite covers can be found in [8].

Let us now revert to the setting of Section 2 in which p : [~.X] [right arrow] X is a fibrewise manifold and f : [~.X] [right arrow] X is a map. We shall apply Proposition 3.1 to the case in which Y is equal to [~.X], q : [~.Y] = [~.X] [right arrow] Y = [~.X] and a : [~.X] [right arrow] Y = [~.X] are both the identity on [~.X], and b : [~.Y] = [~.X] [right arrow] X is the given map f. Then b/q [??] a/p = f/1 [??] 1/p is just f/p. The composition a/p [??] b/q = 1/p [??] f/1 is given by the fibrewise manifold

f*[~.X] = {([~.x],[~.y]) [member of] [~.X] x [~.X] | f([~.x]) = p([~.y])} [right arrow] [~.X], ([~.x],[~.y]) [right arrow] [~.x]

and map ([~.x],[~.y]) [right arrow] [~.y] : f*[~.X] [right arrow] [~.X], and it determines the multivalued map [~.F] : [~.X] # [~.X] given by [~.F] ([~.x]) = {[~.y] [member of] [~.X] | f([~.x]) = p([~.y])}.

Now suppose that p : [~.X] [right arrow] X is a finite n-fold cover. The equality h-L(1/p [??] f/1) = h-L(f/1 [??] 1/p) = h-L(f/p), together with Corollary 2.3 applied to the fraction 1/p [??] f/1, demonstrates that Schirmer's fixed-point index of the n-valued map [~.F] constructed by Brown coincides with the fixed-point index of f/p. Equality of the Nielsen numbers is an immediate consequence.

Proposition 3.2. Suppose that p : [~.X] [right arrow] X is an n-fold covering space and that f : [~.X] [right arrow] X is a map. Then the Nielsen number as defined by Schirmer, N([~.F]), of the n-valued map [~.F] : [~.X] [??] [~.X], [~.F] ([~.x]) = [p.sup.-1](f([~.x])), is equal to theNielsen number, N(f/p), of f/p.

4 Projective spaces

We shall apply Proposition 2.2 to compute the fixed-point indices in some examples involving projective spaces.

First of all, we take X to be the quaternionic projective space H[P.sub.n] = HP([H.sup.n+1]) of 1-dimensional (left) H-subspaces of [H.sup.n+1] (n >> 1), X to be the sphere [S.sup.4n+3] = S([H.sup.n+1]) and p : [~.X] [right arrow] X to be the principal Sp(1)-bundle S ([H.sup.n+1]) [right arrow] HP([H.sup.n+1]). The fibrewise tangent bundle [tau](p) is trivial with fibre the Lie algebra Ri [direct sum] Rj [direct sum] Rk of Sp(1) [??] H. In coordinates, p([z.sub.0],...,[z.sub.n]) = [[z.sub.0],...,[z.sub.n]], where [z.sub.i] [member of] H, [summation][|[z.sub.i]|.sup.2] = 1.

We shall look at the case in which f admits a lift to a map [~.f] : [~.X] [right arrow] [~.X].

Proposition 4.1. Let p : [~.X] [right arrow] X be the principal Sp(1)-bundle [S.sup.4n+3] [right arrow] H[P.sub.n], where n >> 1, and suppose that f : [~.X] [right arrow] X is equal top [??] [~.f], where [~.f] : [S.sup.4n+3] [right arrow] [S.sup.4n+3] is a map of degree d. Then the topological Lefschetz index t-L(f/p) in

[[omega].sub.3]([S.sup.4n+3]) = [[omega].sub.3](*) = (Z/24Z)[nu],

where [nu] is the standard generator represented by the framed manifold Sp(1) with the left invariant framing, is equal to (1 + nd)[nu].

Proof. Let r, s >> 1 be positive integers such that r - s = d. We shall make the computation using the specific map f given by

[mathematical expression not reproducible],

where [a.sub.i] [member of] R and 1 < [a.sub.1] < [a.sub.2] <... < [a.sub.n]. It is elementary to check that the fixed-point set is the union of n + 1 fibres [N.sub.i], i = 0,...,n, over the points [x.sub.i] = [[e.sub.i]], where [e.sub.0],..., [e.sub.n] is the standard basis of [H.sup.n+1].

The fixed submanifold [N.sub.0] is non-degenerate. To calculate the local index [L.sub.0] [member of] [[omega].sup.0]([N.sub.0]) we can, therefore, linearize and reduce to consideration of the map, [f.sub.0.sup.#] say,

S(H) x [H.sup.n] [right arrow] [H.sup.n] : ([z.sub.0], ([v.sub.1],..., [v.sub.n])) [right arrow] ([a.sub.1][z.sub.0.sup.1-d] [v.sub.1],...,[a.sub.n][z.sub.0.sup.1-d] [v.sub.n]).

The index is determined by 1 - [f.sub.0.sup.#]:

S(H) x ([H.sup.n] - {0}) [right arrow] [H.sup.n] - {0}, ([z.sub.0], ([v.sub.i])) [right arrow] ((1 - [a.sub.i][z.sub.0.sup.1-d])[v.sub.i]),

which is homotopic to the map [l.sub.0] taking ([z.sub.0], ([v.sub.i])) to ([z.sub.0.sup.1-d] [v.sub.i]). Thus, [L.sub.0] is the image under the J-homomorphism

J : K[O.sup.-1] ([N.sub.0]) [right arrow] [[omega].sup.0][([N.sub.0]).sup.x]

(to the group of units in the stable cohomotopy ring [[omega].sup.0]([N.sub.0])) of the class determined by [l.sub.0]. The relevant groups are K[O.sup.-1]([S.sup.3]) = (Z/2Z) [direct sum] Z and [[omega].sup.0]([S.sup.3]) = Z [direct sum] (Z/24Z)[nu]. The class [[l.sub.0]] is equal to (0,n(d - 1)) [member of] K[O.sup.-1]([S.sup.3]) (with an appropriate choice of signs for the generators) and maps under J to 1 + n(d - 1)[nu]. (Care is needed to distinguish the generators [nu] and - [nu].)

A similar, but easier, computation for [L.sub.i], i > 0, shows that [L.sub.i] = 1 [member of] [[omega].sup.0]([N.sub.i]).

The composition of the duality isomorphism determined by the left invariant framing of [S.sup.3] = Sp(1) and the homomorphism induced by projection to a point *:

[mathematical expression not reproducible]

maps 1 to [nu] and [nu] to [nu]. Hence the sum of the n + 1 local indices [L.sub.i] maps to (1 + n(d - 1))[nu] + n[nu] = (1 + nd)[nu] [member of] [[omega].sub.3](*). By Proposition 2.2, this is equal to the global index in [[omega].sub.3]([S.sup.4n+3]) = [[omega].sub.3](*).

(For the degree zero case d = 0, we can also take f to be the constant map at [[e.sub.0]]. The fixed-point set is a single fibre [N.sub.0] at [x.sub.0] and [L.sub.0] = 1. The single term [L.sub.0] gives the global index as [nu] [member of] [[omega].sub.3](*). This confirms the sign of the generator [nu] in the general calculation above.)

In this example, the homotopy fixed-point index contains no more information than the topological index. For a map f with S(H[e.sub.0]) [??] Fix(f/p), the projection [pi] : h-Fix(f/p) [right arrow] S([H.sup.n+]) is locally fibre homotopy trivial with fibre [PHI] = [OMEGA](HP([H.sup.n+1]),[x.sub.0]) and is actually a product S(H[e.sub.0]) x [PHI] over S(H[e.sub.0]). The map [PHI] [right arrow] Sp(1) coming from the fibration induces an isomorphism [[pi].sub.i]([PHI]) [right arrow] [[pi].sub.i](Sp(1)) for i < 4n + 2. It follows that [[omega].sub.3](h-Fix(f/p)) = [[omega].sub.3](Sp(1)) = (Z/24Z) [direct sum] Z and, because the inclusion S(H[e.sub.0]) [right arrow] S([H.sup.n+1]) is null homotopic, that h-L(f/p) lies in the Z/24Z summand.

Remark 4.2. If d [not equal to] 1, the map [~.f] : S([H.sup.n+1]) [right arrow] S([H.sup.n+1]) has a fixed point and thus h-Fix(f/p) is non-empty. If d = 1, but n + 1 is not divisible by 24, the computation of the index shows that h-Fix (f/p) is non-empty, that is, that there exists a point v [member of] S([H.sup.n+1]) such that [~.f](v) [member of] Hv. (Note, however, that there may be a map f : [~.X] [right arrow] X with Fix(f/p) = [empty set] that does not admit a lift [~.f]. For example, if n = 1, so that H[P.sub.4] = [S.sup.4], the antipodal involution on [S.sup.4] provides a map f with no fixed points.) It is, therefore, natural to ask whether, when n + 1 is a multiple of 24, there is some map [~.f] of degree 1 such that [~.f](v) [??] Hv for all v [member of] S([H.sup.n+1]). Certainly there is no example in which f is of the form f(v) = A(v)/||A(v)|| for some non-singular R-linear endomorphism of [H.sup.n+1]. (For, if Av [??] Hv for all non-zero v, we have a family of maps [A.sub.t] : S(H) [right arrow] GL([R.sup.4(n+1])), [A.sub.t](z)v = cos([pi]t/2)zv + sin([pi]t/2)Av, 0 << t << 1, with [A.sub.1] constant, but [A.sub.0] representing a non-trivial element in [[pi].sub.3] [(GL([R.sup.4].sup.(n+1]))) [??] Z. See [17].)

There is a similar result for complex projective spaces.

Proposition 4.3. Let p : [~.X] [right arrow] X be the principal U(1)-bundle [S.sup.2n+1] [right arrow] C[P.sub.n], where n >> 1, and suppose that f : [~.X] [right arrow] X is equal top [??] [~.f], where [~.f] : [S.sup.2n+1] [right arrow] [S.sup.2n+1] is a map of degree d. Then the topological Lefschetz index t-L(f/p) in

[[omega].sub.1]([S.sup.2n+1]) = [[omega].sub.1](*) = (Z/2Z)[eta],

where [eta] is the Hopf element represented by U(1) with the left invariant framing, is equal to (1 + nd)[eta].

In view of Proposition 2.4, these results on quaternionic and complex projective spaces provide examples of coincidence indices in codimension 3 and 1. The corresponding problem for real projective spaces is much simpler.

Proposition 4.4. Let p : [~.X] [right arrow] X be the principal O(1)-bundle [S.sup.n] [right arrow] R[P.sub.n], where n >> 1, and suppose that f : [~.X] [right arrow] X is equal to p [??] [~.f], where [~.f] : [S.sup.n] [right arrow] [S.sup.n] is a map of degree d. Then h-Fix (f/p) is the disjoint union of two components:

h-Fix(f/p) = h-Fix([~.f]) [??] h-Fix(-[~.f])

and the homotopy Lefschetz index h-L(f/p) in

[[omega].sub.0] (h-Fix (f/p)) = [[omega].sub.0] (h-Fix ([~.f])) [direct sum] [[omega].sub.0] (h-Fix (-[~.f])) = Z [direct sum] Z

is equal to (1 + [(-1).sup.n]d, 1 - d).

The Nielsen number is thus: N(f/p) = 0 if d = 1 and n is odd; N(f/p) = 1 if either d = 1 and n is even, or d = -1 and n is odd; and otherwise N (f/p) =2.

Proof. As in the other cases, we could write down an explicit map [f.sup.#] with Fix(f/p) = Fix([~.f]) [??] Fix(-[~.f]) finite and sum the local indices. But the sums on the two components must give the classical Lefschetz numbers of [~.f] and -[~.f], which can be computed as traces in rational cohomology: L([~.f]) = 1 + [(-1).sup.n] d, L(-[~.f])=1 + [(-1).sup.n]([(-1).sup.n+1]d).

Remark 4.5. In this section we were able to compute a non-trivial stable homotopy index in dimension m = 3, but in general the stable homotopy indices are likely to be difficult to calculate if the dimension m is greater than zero. More tractable indices can be obtained by taking the Hurewicz image of the stable homotopy index in ordinary homology. In particular, using the Thom isomorphism for [tau] (p), we get indices in mod 2 homology h-[L.sup.H] (f/p) [member of] [H.sub.m](h-Fix(f/p); [F.sub.2]) and t-[L.sup.H](f/p) [member of] [H.sub.m]([~.X]; [F.sub.2]). There is even a trace formula for t-[L.sup.H](f/p) in [H.sub.m] ([~.X]; [F.sub.2]):

for a [member of] [H.sup.m]([~.X]; [F.sub.2]), <a, t-[L.sup.H](f/p)> = [summation over (i)][(-1).sup.i] tr [A.sub.i] [member of] [F.sub.2],

where [A.sub.i] is the composition

[mathematical expression not reproducible]

involving the Umkehr map p! for the fibrewise manifold. (Compare [15, Section 2].)

References

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[5] R. F. Brown and K. Kolahi, Nielsen coincidence, fixed point and root theories of n-valued maps. J. Fixed Point Theory Appl. 14 (2013), 309-324.

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[8] M. C. Crabb, Lefschetz indices for n-valued maps. J. Fixed Point Theory Appl. 17 (2015), 153-186.

[9] M. C. Crabb and I. M. James, Fibrewise homotopy theory. Springer, Berlin, 1998.

[10] A. Dold, Lectures on Algebraic Topology. Springer, Berlin-Heidelberg-New York, 1972.

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[12] A. Dold, The fixed point transfer of fibre-preserving maps. Math. Zeit. 148 (1976), 215-244.

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M. C. Crabb

Received by the editors in October 2016 - In revised form in December 2016.

Communicated by K. Dekimpe, D.L. Goncalves and P. Wong.

2010 Mathematics Subject Classification : Primary 54H25, 55M20, Secondary 55P42, 55R70.

Key words and phrases : Lefschetz index, Reidemeister trace, Nielsen number, multivalued map, coincidence index.

Institute of Mathematics, University of Aberdeen, Aberdeen AB24 3UE, UK

email: m.crabb@abdn.ac.uk
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Author:Crabb, M.C.
Publication:Bulletin of the Belgian Mathematical Society - Simon Stevin
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Date:Oct 1, 2017
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