Higher order Nielsen numbers

Suppose X,Y are manifolds, f,g:X->Y are maps. The well-known Coincidence Problem studies the coincidence set C={x:f(x)=g(x)}. The number m=dimX-dimY is called the codimension of the problem. More general is the Preimage Problem. For a map f:X->Z and a submanifold Y of Z, it studies the preimage set C={x:f(x) in Y}, and the codimension is m=dimX+dimY-dimZ. In case of codimension 0, the classical Nielsen number N(f,Y) is a lower estimate of the number of points in C changing under homotopies of f, and for an arbitrary codimension, of the number of components of C. We extend this theory to take into account other topological characteristics of C. The goal is to find a"lower estimate"of the bordism group Omega_{p}(C) of C. The answer is the Nielsen group S_{p}(f,Y) defined as follows. In the classical definition the Nielsen equivalence of points of C based on paths is replaced with an equivalence of singular submanifolds of C based on bordisms. We let S_{p}^{prime}(f,Y) be the quotient group of Omega_{p}(C) with respect to this equivalence relation, then the Nielsen group of order p is the part of this group preserved under homotopies of f. The Nielsen number N_{p}(f,Y) of order p is the rank of this group (then N(f,Y)=N_{0}(f,Y)). These numbers are new obstructions to removability of coincidences and preimages. Some examples and computations are provided.


Introduction
Suppose X, Y are smooth orientable compact manifolds, dim X = n+m, dim Y = n, m ≥ 0 the codimension, f, g : X → Y are maps, the coincidence set C = Coin(f, g) = {x ∈ X : f (x) = g(x)} is a compact subset of X\∂X.
Consider the Coincidence Problem: "What can be said about the coincidence set C of (f, g)?" One of the main tools is the Lefschetz number L(f, g) defined as the alternating sum of traces of a certain endomorphism on the homology group of Y . The famous Lefschetz coincidence theorem provides a sufficient condition for the existence of coincidences (codimension m = 0): L(f, g) = 0 =⇒ C = Coin(f, g) = ∅, see [1,VI.14], [35,Chapter 7]. Now, what else can be said about the coincidence set? As C changes under homotopies of f and g, a reasonable approach is to try to minimize the "size"of C. In case of zero codimension C is discrete and we simply minimize the number of points in C. The result is the Nielsen number. It is defined as follows. Two points p,q ∈ C belong to the same Nielsen class if (1) there is a path s in X between p and q; (2) f s and gs are homotopic relative to the end-points. A Nielsen class is called essential if it cannot be removed by a homotopy of f, g (alternatively, a Nielsen class is algebraically essential if its coincidence index is nonzero [2]). Then the Nielsen number N (f, g) is the number of essential Nielsen classes. It is a lower estimate of the number of points in C. In case of positive codimension N (f, g) still makes sense as a lower estimate of the number of components of C [36]. However only for m = 0 the Nielsen number is known to be a sharp estimate, i.e., there are maps f ′ , g ′ compactly homotopic of f, g such that C ′ = Coin(f ′ , g ′ ) consists of exactly N (f, g) path components (Wecken property). This minimization is achieved by removing inessential classes through homotopies of f, g.
The Nielsen theory for codimension m = 0 is well developed, for the Fixed Point and the Root Problems [3], [23], [24], and for the Coincidence Problem [4]. However, for m > 0, the vanishing of the coincidence index does not guarantee that the Nielsen class can be removed. Some progress has been made for codimension m = 1. In this case the secondary obstruction to the removability of a coincidence set was considered by Fuller [13] for Y simply connected. Hatcher and Quinn [19] showed that the obstruction to a higher dimensional Whitney Lemma lies in a certain framed bordism group. Based on this result, necessary and sufficient conditions of the removability of a Nielsen class were studied by Dimovski and Geoghegan [9], Dimovski [8] for parametrized fixed point theory, i.e., when f : Y × I → Y is the projection. The results of [9] were generalized by Jezierski [20] for the coincidence problem f, g : X → Y, where X, Y are open subsets of Euclidean spaces or Y is parallelizable. Geoghegan and Nicas [14] developed a parametrized Nielsen theory based on Hochschild homology. For some m > 1, sufficient conditions of the local removability are provided in [32]. Necessary conditions of the global removability for arbitrary codimension are considered by Gonçalves, Jezierski, and Wong [17,Section 5] with N a torus and M a nilmanifold.
In these papers higher order Nielsen numbers are not explicitly defined (except for [8], see the comment in the end of the paper). However they all contribute to the problem of finding the lower estimate of the number of components of C. We extend these results to take into account other topological characteristics of C. In the spirit of the classical Nielsen theory, our goal is to find "lower estimates" of the bordism groups Ω * (C).
The crucial motivation for our approach is the removability results for codimension 1 due to Dimovski and Geoghegan [9] and Jezierski [20]. Consider Theorem 5.3 in [20]. Assume that codimension m = 1, n ≥ 4, X, Y are open subsets of Euclidean spaces. Suppose A is a Nielsen class. Then if f, g are transversal, A is the union of disjoint circles. Define the Pontriagin-Thom map (PT) as the composition where ν is a normal bundle of A, D ⊂ R n is a ball centered at 0 satisfying (f − g)(∂ν) ⊂ R n \D. It is an element of π n (S n−1 ) = Z 2 . Then A can be removed if and only if the following conditions are satisfied: (W1) A = ∂S, where S is an orientable connected surface, f | S ∼ g| S rel A (the surface condition); (W2) the PT map is trivial (the Z 2 -condition). Earlier Dimovski and Geoghegan [9] considered a similar pair of conditions (not independent though) in their Theorem 1.1 and compared them to the codimension 0 case. They write: "...the role of 'being in the same fixed point class' is played by the surface condition (i), while that of the fixed point index is played by the natural orientation. The Z 2 -obstruction is a new feature..." One can use the first observation to define the Nielsen equivalence on the set of of 1-submanifolds of C (here A is Nielsen equivalent to the empty set). However, we will see that the PT map has to serve as the index of the Nielsen class. The index will be defined in the traditional way but with respect to an arbitrary homology theory h * . Indeed in the above situation it is an element of the stable homotopy group π S n+1 (S n ) = Z 2 . More generally, we define the Nielsen equivalence on the set M m (C) of all closed singular m-manifolds in C = Coin(f, g). Two singular m-manifolds p : P → C and q : Q → C belong to the same Nielsen class, p ∼ N q, if 1. ip and iq are bordant, where i : C → N is the inclusion, i.e. there is a map F : W → N extending ip ⊔ iq such that W is a bordism between P and Q; 2. f F and gF are homotopic relative to f p, f q. Then S ′ m (f, g) = M m (C)/ ∼ N is the group of Nielsen classes. Let S a m (f, g) be the group of algebraically essential Nielsen classes, i.e., the ones with non-trivial index. Then the (algebraic) Nielsen number of order m is the rank of S a m (f, g) (these numbers are new obstructions to removability of coincidences). In light of this definition Jezierski's theorem can be thought of as a Wecken type theorem for m = 1.
An area of possible applications of the coincidence theory for positive codimension is discrete dynamical systems (for a related theory of flows see [14]). A dynamical system on a manifold M is determined by a map f : M → M. Then the next position f (x) depends only on the current one, x ∈ M . Suppose now that we have a fiber bundle F → N g − −−−− → M and a map f : N → M. This is a parametrized dynamical system, where the next position f (x, s) depends not only on the current one, x ∈ M, but also the current "state", s ∈ F. The coincidence theory describes the set of all positions and states such that the positions remain fixed, f (x, s) = x. Alternatively, x is the "state" and s is the "input" in a control system. A control system [28, p. 16] is defined as a commutative diagram where N is a fiber bundle over M. The translation along trajectories of this system creates a parametrized dynamical system.
Instead of the Coincidence Problem, throughout the rest of the paper we apply the approach outlined above to the Nielsen Theory for the so-called Preimage Problem considered by Dobrenko and Kucharski [10]. Suppose X, Y, Z are connected CW-complexes, Y ⊂ Z, f : X → Z is a map. The problem studies the set C = f −1 (Y ) and can be easily specialized to the Fixed Point Problem if we put [26]).
Suppose X, Y, Z are smooth manifolds and f is transversal to Y. Then under the restriction dim The paper is organized as follows. Just as for the coincidence problem we define the Nielsen equivalence of singular q-manifolds in C and the group of Nielsen classes where Ω * is the orientable bordism group (Section 2). Next we identify the part of S ′ q (f ) preserved under homotopies of f. The result is the Nielsen group S q (f ), the group of topologically essential classes (Section 3). As we have described above, the Nielsen group is a subgroup of a quotient group of Ω q (C) and, in this sense, its "lower estimate".
The Nielsen number of order p, p = 0, 1, 2, ..., is defined as Clearly the classical Nielsen number is equal to N 0 (f ).
In Section 4 we discuss the naturality of the Nielsen group. In particular we obtain the following Proposition 1.3. Given Z, Y ⊂ Z. Then S * is a functor from the category of preimage problems as pairs (X, f ), f : X → Z, with morphisms as maps k : X → U satisfying gk = f, to the category of graded abelian groups.
For the manifold case, there is an alternative approach to essentiality. In Section 5 the "preimage index" is defined simply as I f = f * : Ω * (C) → Ω * (Y ). It is a homomorphism on S ′ * (f ) and the group of algebraically essential Nielsen classes is defined as S a * (f, Y ) = S ′ * (f, Y )/ ker I f . We show that every algebraically essential class is topologically essential. In Section 6 we consider the traditional index Ind f (P ) of an isolated subset P of C in terms of a generalized homology h * . It is defined in the usual way as the composition where V ⊂ V ⊂ U are neighborhoods of P. Then we show how it is related to I f . In Section 7 we consider some examples of computations of these groups, especially in the setting of the Pontriagin-Thom construction.
In Sections 8 and 9 based on Jezierski's theorem we prove the following Wecken type theorem for codimension 1.
This approach to the study of the homology of the preimage set has a potential application in robust control [22]. The so-called Nyquist map is a map from the "uncertainty space" (i.e., the state input manifold) of the control system to the Riemann sphere. Topological robust control theory tracks the change of the topology of the preimage of the origin under perturbations of this map. In general, robust stability criteria consider "mapping of the uncertainty into a 'performance evaluation' space... and checking whether the image is in the correct subset" [22, p. 20].
To motivate our definitions, in the beginning of each section be will review the relevant part of Nielsen theory for the Preimage Problem following Dobrenko and Kucharski [10], and McCord [26].
All manifolds are assumed to be orientable and compact.

Nielsen Classes.
In Nielsen theory, two points x 0 , (3) f α and β are homotopic relative to {0, 1}. This is an equivalence relation partitioning C into a finite number of Nielsen classes. However, since we want Nielsen classes to form a group, we should think of where N is a p-manifold). Then conditions (1) and (2) express the fact that Recall [6], [33] that two closed p-manifolds N 0 , N 1 are called bordant if there is a bordism between them, i.e., a (p ( (3) f H and G are homotopic relative to S 0 ⊔ S 1 .
We denote the Nielsen class of s ∈ M p (C) by [s] N , or simply [s].
Proposition 2.2. ∼ N is an equivalence relation on M p (C).

Definition 2.3. The group of Nielsen classes of order
The group of Nielsen classes for the coincidence problem will be denoted by S ′ p (f, g). In contrast to the classical Nielsen theory, the elements of Nielsen classes are not points but sets of points. Even in the case of p = 0, one has more to deal with. For example, suppose C = {x, y} and x ∼ N y. The the elements of Another example. Suppose X = Z = S 2 , Y is the equator of Z, f a map of degree 2 such that C = f −1 (Y ) is the union of two circles C 1 and C 2 around the poles. Then S ′ 1 (f ) = Z generated by C 1 ⊔ C 2 . A similar construction applies to X = Z = S n , Y = S n−1 , n ≥ 2, then S ′ n−1 (f ) = Z is generated by the union of two copies of S n−1 .
Let M h p (A, B) denote the semigroup of all homotopy classes, relative to boundary, of maps s ∈ M p (A, B). Consider the commutative diagram: where δ is the boundary map, I is the inclusion. Then we have an alternative way to define the group of Nielsen classes: Then Ω * is a generalized homology [6], [33].
Next we discuss the naturality of this group.
Then by assumption g(x ′ ) = y ′ , so x ′ ∈ C ′ . Therefore the following diagram commutes: The second preimage problem has a diagram analogous to (2.1). Together they provide two opposite faces of a 3-dimensional diagram with other faces supplied by the diagram above. The diagram commutes. Therefore for each s ∈ M p (C), Proposition 2.8. Suppose the following diagram for three preimage problems commutes: . Corollary 2.10. If P is the category of preimage problems as quadruples (X, Z, Y, f ), Y ⊂ Z, f : X → Z, with morphisms as pairs of maps (k, h) satisfying Definition 2.6, then S ′ * is a functor from P to Ab * , the graded abelian groups.

Topologically Essential Nielsen Classes.
In the classical theory, a Nielsen class is called essential if it cannot be removed by a homotopy. More precisely, suppose F : or is empty [10, Corollary 1.5]. Next we say that the Nielsen classes N 0 , N 1 of f 0 , f 1 respectively are in the F -Nielsen relation if there is a Nielsen class N of F such that N 0 , N 1 are the 0-and 1-sections of N. This establishes an "equivalence" relation between some Nielsen classes of f 0 and some Nielsen classes of f 1 . Given a Nielsen class N 0 of f 0 , if for any homotopy there is a Nielsen class of f 1 corresponding to N 0 then N 0 is called essential. In our theory the F -Nielsen relation takes a simple form of two homomorphisms from S ′ . The following result is crucial. Then , M F * should be understood as the set of Nielsen classes of f 0 preserved by F ). Now we say that a class if defined, otherwise we can set s 1 = 0. Thus some classes cannot be reduced to zero by a homotopy and we call them (topologically) essential Nielsen classes. Together (plus zero) they form a group, as follows.
Definition 3.2. The group of (topologically) essential Nielsen classes is defined as (S p (f, Y ) can also be called the Nielsen group of order p, while S ′ p (f, Y ) the pre-Nielsen group.) If f ∼ g then S * (f ) ≃ S * (g). Therefore, ). In this sense, S * (f ) is a "lower estimate" of Ω * (g −1 (Y )) for any g homotopic to f .
The Nielsen number for the coincidence problem is denoted by N p (f, g).
Clearly N 0 (f ) is equal to the classical Nielsen number and provides a lower estimate of the number of path components of f −1 (Y ).
It is easy to verify that this theory is still valid if the oriented bordism Ω * is replaced with the unoriented bordism, or the framed bordism (see examples in Section 7), or bordism with coefficients. In fact, a similar theory for an arbitrary homology theory is valid because every homology theory can be constructed as a bordism theory with respect to manifolds with singularities [5].

Naturality of S * (f ).
Under conditions of Definition 2.6, the homomorphism k * : S * (f ) → S * (g) can be defined as a restriction of k ′ * and the analogues of Propositions 2.7 -2.9 hold. We simplify the situation in comparison to Section 2 by assuming that Z and Y ⊂ Z are fixed. Definition 4.1. Suppose we have another preimage problem g : U → Z connected to the first by map k : X → U such that gk = f. Then the homomorphism induced by k, k * : S * (f ) → S * (g), is defined as the restriction of k ′ * : Proof. For convenience let f = f 0 , g = g 0 , k = k 0 . Suppose G is a homotopy between g 0 and g 1 , K between k 0 and k 1 . Let F = GK, then F is a homotopy between f 0 and f 1 . Let L(t, x) = (t, K(t, x)). Then we have a commutative diagram: where i s : X → {s} × X → I × X and j s : U → {s} × U → I × U, s = 0, 1, are the inclusions. Further, if we add a vertex Z to this diagram we have a commutative pyramid with the other edges provided by f 0 , f 1 , g 0 , g 1 , G, F. Then by naturality of the map induced on S ′ * (Proposition 2.8) we have another commutative diagram: Here the horizontal arrows are injective (Proposition 3.1). Therefore the restriction k ′ 0 * = k ′ 1 * = L ′ * : M F * → M G * is well defined. This conclusion is true for all G, K, so that the restriction k ′ 0 * : ∩ F =GK M F * → ∩ G M G * is well defined. Since S * (f ) is a subset of the former and the latter is S * (g), the statement follows.   Corollary 4.6. If k is a homotopy invariance, gk = f, then S * (f ) = S * (g), i.e., the preimage theory f : X → Z ⊃ Y is "homotopy invariant" (cf. [16]) with respect to X.

5.
The Bordism Index as a Homomorphism on S ′ * (f ). In the classical Nielsen theory, the coincidence index provides an algebraic count of coincidence points. It satisfies the usual properties: (1) Homotopy Invariance: the index is invariant under homotopies of f, g; (2) Additivity: The index over a union of disjoint sets is equal to the sum of the indices over these sets; (3) Existence of Coincidences: if the index is nonzero then there is a coincidence; (4) Normalization: the index is equal to the Lefschetz number; (5) Removability: if the index is zero then a coincidence can be (locally or globally) removed by a homotopy. From the point of view of our approach the additivity property means that we associate an integer to every 0-class, i.e., we have a homomorphism S ′ 0 (f ) → Z =Ω 0 (Y ).
Proposition 5.2. The index is well defined as a homomorphism Of course, I f (z) = 0 =⇒ z = 0. Suppose F : I × X → Z is a homotopy. As before, let f t (·) = F (t, ·) : X → Z, and let i t : X → {t} × X → I × X be the inclusions.

Thus the index I f is preserved under homotopies.
In the classical theory Nielsen classes are sets and the algebraically essential classes are the ones with nonzero index. Similarly, we call z ∈ S ′ q (f ) algebraically essential of I f (z) = 0.
Corollary 5.5. Every algebraically essential class is topologically essential, i.e., z cannot be reduced by a homotopy to the zero p-class, and, therefore, z cannot be "removed" by a homotopy.
We define the group of algebraically essential Nielsen classes as Suppose we have another preimage problem g : U → Z connected to the first by map k : X → U such that gk = f. Then just like in the previous section we define the homomorphism induced by k, k a * : S a * (f ) → S a * (g), as a restriction of k ′ * . Moreover similar properties are satisfied. Thus we have Corollary 5.6. Given Z, Y ⊂ Z. If P(Z, Y ) is the category of preimage problems as pairs (X, f ), f : X → Z, with morphisms as maps k : X → U satisfying gk = f, then S a * is a functor from P(Z, Y ) to Ab * .

The Index of an Isolated Set of Preimages.
From now on we assume that X, Y, Z are smooth manifolds, Y is a submanifold of Z.
Suppose P ⊂ C is an isolated set of preimages. Let U ⊂ V be neighborhoods of P in X such that U ⊂ Int V and V ∩ C = P. In the classical Nielsen theory, the index Ind(f, P ) of P is defined as the image of the generator z of H n+m (X) ≃ Z under the composition Under the restriction dim X +dim Y = dim Z, we are in the classical situation: each class a is an isolated set of preimages A and the index a is defined as the index of A.
In case of a nonzero codimension we can have H n+m (X) = Z, therefore it makes sense to replace in the above definition the generator z with an arbitrary element of H * (X). This turns the index into a graded homomorphism H * (X) → H * (Z, Z\Y ) (which is equal to the Lefschetz homomorphism [31] for the Coincidence Problem). This generality is justified by a number of examples in [15], [31] that show that in order to detect coincidences in a nonzero codimension one may need to take into account all parts of this homomorphism.
A fixed point index with respect to generalized cohomology was considered by Dold [12]. Another example is [27] where the coincidence index is computed in term of cobordism. In addition, we will see in Section 9 that for a nonzero codimension the index expressed in terms of singular homology may be inadequate for removability (some algebraically inessential classes are essential). Therefore under the above restrictions the singular homology H * should be replaced with a generalized homology h * .
The index Ind f (P ; h * ), or simply Ind f (P ), of the set P with respect to h * is the following homomorphism The index does not depend on the choice of U , see [35, p. 189]. The next theorem is proven similarly to Lemmas 7.1, 7.2, 7.4 in [35, p. 190-191] respectively. In particular, if P ∈ Ω 0 (M ) is a subset of C and z = [P ] N , then Im z = P. Suppose f is transversal to Y . Then C is a r-submanifold of X.
Proposition 6.4. Im z is an isolated subset of C and therefore an r-submanifold of X.
Proposition 6.5. If z = 0 ∈ S ′ * (f ) then Im z = ∅. Clearly if Ind f (Im z) = 0 then Im z = ∅. However this does not imply that z is essential. The case of p = 0 is an exception. For convenience we restate the following familiar result. Proposition 6.6. If P ∈ S ′ 0 (f ) and Ind f (P ; h * ) = 0, where h * is an arbitrary homology theory, then P is essential.
The relation between the essentiality of the class and its index is more subtle when p > 0.
Suppose z ∈ S ′ p (f ), z = [s], where s ∈ Ω p (M ). Let P = Im z, then it is an rsubmanifold of C. Let T and T ′ be tubular neighborhoods of C and P respectively such that T ′ is an isolated subset of T. Then the inclusion i : T ′ → T is a bundle map. Suppose s = i * (s ′ ) for some s ′ ∈ Ω p (P ). From the naturality of the Thom isomorphism we have the commutativity of the following diagram: where W is a tubular neighborhood of Y. Then I f (z) = f * (s) = f * i * (s ′ ). Therefore ). Thus we have proven The right hand side can be used for an alternative definition of an algebraically essential class.
Moreover if Ind f (P ; H * ) = 0 then P ∈ S ′ 0 (f ) is essential. Thus for r = 0 we recover the traditional definition of an algebraically essential class.

Some Examples.
Nielsen numbers are hard to compute. Nielsen groups and higher order Nielsen numbers are no different. Below we consider some special cases when the computation is feasible.
Just as before suppose X = Z = S 2 , Y is the equator of Z, f a map of degree 2 such that C = f −1 (Y ) is the union of two circles C 1 and C 2 around the poles. But there is only one generator of S ′ 1 (f ), C = C 1 ∪ C 2 . Also Ind f (C) = 0. Hence N 0 (f ) = 1. This is in fact a "sharp" estimate of the number of components of C (Wecken Property for codimension r = 1) because f is homotopic to the suspension, g, of the degree 2 map of the equator, so that g −1 (Y ) is a circle. The same conclusion applies to X = Z = S n , Y = S n−1 , codimension r = n − 1, n ≥ 2.
For more examples of this nature see [29] and Theorem 1.2 and Section 12 in [9].
In [32] we showed that the cohomology coincidence index I A f g is the only obstruction to removability of an isolated subset A of the coincidence set if any of the three following conditions is satisfied: (1) M is a surface; (2) the fiber of g is acyclic; or (2) the fiber of g is an m-sphere for m = 4, 5, 12 and n large. Of course if the homology index Ind (f,g) (A; H * ) is trivial then so is I A f g . Therefore under these restrictions an algebraically inessential class can be removed. The condition of this proposition is satisfied if we simply assume that f is homotopic to f ′ with f ′ (X) ⊂ Y. For the Coincidence Problem this result takes the following form.
Suppose a parametrized dynamical system F → N f,g − −−−−−→ M, where g is the bundle projection, is generated by a control system. Then it is easy to see that f is homotopic to g and the above theorem can be applied to estimate the "size" of the set of stationary points of the control system.
Proof. Suppose s 0 , s 1 ∈ j * Ω q (C) are bordant in X via H : W → X, i.e., s i : S i → X, ∂W = S 0 ⊔ S 1 , H| Si = s i . Since Y is (q − 1)-connected and is trivial, the classical obstruction theory [1, p. 497] is applied to prove that the map f (s 0 ⊔ s 1 ) : S 0 ⊔ S 1 → Y can be extended to G : W → Y . Further, since Z is (q + 1)-connected, [W, Z] relS0⊔S1 = 0. Therefore G and f H are homotopic relative to S 0 ⊔ S 1 . Thus, if two singular q-manifolds in C are bordant in X, then they are Nielsen equivalent. Now the theorem follows from the above proposition.
The relation between the homotopy class of a map and the preimage of a point is direct in the setting of the Pontriagin-Thom construction [7, p. 196]. For the rest of the section we assume that the Nielsen groups S ′ q (f ), S q (f ) are computed with respect to the framed bordism, i.e., S ′ q (f ) is a quotient group of Ω f r q (C). Let Y = {p}, p ∈ Z = S k , and r ≤ k − 2. Then the conditions of the theorem above are satisfied. Therefore S ′ r (f ) ≃ j * Ω f r r (C). Now, f is homotopic to a map g if and only if C = f −1 (p) is framed bordant to K g = g −1 (p) in X. Let j g : K g → X be the inclusion. Then Thus we have proven the following Then Suppose n ≥ 4 and f is smooth. Then the coincidence problem f, g : X → Y satisfies the Wecken Property of order 1; specifically, g is homotopic to g ′ such that S 1 (f, g) ≃ Ω 1 (Coin(f, g ′ )). Moreover, N 1 (f, g) = rank Ω 1 (Coin(f, g ′ )) is equal to the number of circles in Coin(f, g ′ ) not satisfying (W1).
Proof. Suppose, according to the above proposition, that all 0-classes are circles, A 1 , ..., A s . Suppose also that A 1 , ..., A t satisfy condition (W1) and the rest do not. Let's view A 1 , ..., A t as singular 1-manifolds. Then, first, A i ∼ N ∅ for i = 1, ..., t according to Proposition 8.1. Hence for these i, A i ∈ 0 ∈ S ′ 1 (f, g ′ ), so they don't concern us. Now, suppose A i ∼ N A j for some i > j > t via some surface H. If A i and A j were subsets of different components of H then each would satisfy condition (W1). Therefore H can be assumed connected. But then A i ∪ A j satisfies condition (W1) and, moreover, every pair of points x ∈ A i , y ∈ A j is Nielsen equivalent. Therefore by Proposition 8.2 A i ∪ A j can be further reduced to a single circle. Hence we can assume that each A i , i = t + 1, ..., s, belongs to a different nonzero 1-class. Thus the generators of S ′ 1 (f, g ′ ) are [A i ] N , i = t + 1, ..., s. Now the fact that each of these classes is essential follows from the homotopy invariance of (W1) [20,Lemma 4.2].
A similar result for the Root Problem is easy to prove.
Then the root problem f : X → Z ∋ p satisfies the Wecken Property of order 1 (with respect to framed bordism).
Proof. Just as above assume that C = f −1 (p) is the disjoint union of circles such that A 1 , ..., A t are framed bordant to the empty set and A t+1 , ..., A s are not. Then C is framed bordant to K = A t+1 ∪ ... ∪ A s . Finally S 1 (f, {p}) = Ω f r 1 (K) by Theorem 7.5.
Suppose A is a 1-submanifold of C. Recall condition (W2) the PT map is trivial. The proposition below explains why the PT map should be understood as the coincidence index.
Proposition 9.1. (W2) ⇐⇒ Ind (f,g) (A; π S * ) = 0 (i.e., A is algebraically inessential with respect to π S * ). Proof. Let U ⊂ T be tubular neighborhoods A. We state (W2) as follows: Since n ≥ 4, this is equivalent to the following: where ∆ is the diagonal and d(x, y) = x − y. Now if we apply the stable homotopy functor π S * to the diagram, we have P T * in the upper path and the index of A with respect to h * = π S * in the lower. But d is a homotopy equivalence [11,Lemma VII.4.13,p. 200], and the statement follows.
Observe that the stable homotopy index Ind (f,g) (A; π S * ) is better at detecting essential classes than the traditional index with respect to singular homology. In fact, the latter would not work in the above argument as π S n+1 (S n ) = Z 2 cannot be replaced with H n+1 (S n ) = 0. Secondly, all the Nielsen numbers of higher order in Section 7 would be zero if computed with respect to singular homology.
Proposition 9.2 (Jezierski). Let f, g : X → Y be an admissible map between open subsets of R n+1 , R n respectively, n ≥ 4. Then there are maps f ′ , g ′ compactly homotopic to f, g respectively such that the Nielsen classes satisfying (W1) and (W2) disappear and the remaining ones become circles.
Suppose we are left with the circles A 1 , A 2 , ..., A t each satisfying condition (W1) but not (W2), and A t+1 , A t+2 , ..., A s satisfying (W2) but not (W1). Then each A 1 , A 2 , ..., A t is an (algebraically) essential 0-class (Theorem 6.6). Also each A t+1 , A t+2 , ..., A s is an essential 1-class (Theorem 8.3), therefore an essential 0-class as well. Thus we have proven the following Theorem 9.3. Suppose X, Y are open subsets of R n+1 , R n respectively, n ≥ 4. Then there are maps f ′ , g ′ compactly homotopic to f, g respectively such that Coin(f ′ , g ′ ) has exactly N 0 (f, g) path components, i.e., the coincidence problem f, g : X → Y satisfies the Wecken Property of order 0.
A result of this type is proven by Gonçalves and Wong [18,Theorem 4 (iii)] for the Root Theory and an arbitrary codimension. In the terminology of the present paper their theorem reads as follows: If X, Z are nilmanifolds, p ∈ Z, then there is g homotopic to f such that g −1 (p) has exactly N 0 (f, {p}) components.
Another codimension 1 Wecken type theorem is given by Dimovski [8] for the parametrized fixed point problem: F : I × Y → Y. He defines two independent indices of a Nielsen class V, ind 1 (F, V ) and ind 2 (F, V ), corresponding to conditions (W1), (W2), and then defines a Nielsen number N (F ) as the number of Nielsen classes with either ind 1 (F, V ) = 0 or ind 2 (F, V ) = 0. His Theorem 4.4 (4) reads: If F is homotopic to H such that H has only isolated circles of fixed points and isolated fixed points then the number of fixed points classes of H is bigger than or equal to N (F ). However this does not mean that N (F ) is a lower bound of the number of components of the fixed point set because an examination of the proof of this theorem (Theorems 4.1, 4.2) reveals that only local homotopies, i.e., ones constant outside a neighborhood of the given class, are allowed. This is the reason why there is a direct correspondence between Nielsen classes of two homotopic maps and there is no need for such a construction as the one in Section 3 of the present paper. Also N (F ) can be larger than the estimate provided in the above theorem -Jezierski [20,Example 6.4] gives an example of a Nielsen class that can be removed by a global homotopy but not by a local one.