Coincidence classes in nonorientable manifolds

In this article we studied Nielsen coincidence theory for maps between manifolds of same dimension without hypotheses on orientation. We use the definition of semi-index of a class, we review the definition of defective classes and study the appearance of defective root classes. We proof a semi-index product formula type for lifting maps and we presented conditions such that defective coincidence classes are the only essencial classes.


Introduction
Nielsen coincidence theory was extended ( [Dobreńko & Jezierski] and [Jezierski 2]) to maps between nonorientable topological manifolds using the notion of semi−index (a non negative integer) for a coincidence set.
We consider maps f, g : M → N between manifolds without boundary of the same dimension n, we define h = (f, g) : M → N × N, then using microbundles (see [Jezierski 2] for details) we can suppose that h is in a transverse position.
Let w be a path satisfying the Nielsen relation between x, y ∈ Coin(f, g). We choose a local orientation γ 0 of M in x and denote by γ t the translation of γ 0 along w(t).
Definition 1.1. [Jezierski 2,1.2] We will say that two points x, y ∈ Coin(f, g) are R-related (xRy) if and only if there is a path w establishing the Nielsen relation between them such that the translation of the orientation h * γ 0 along a path in the diagonal ∆(N) ⊂ N × N homotopic to h • w in N × N is opposite to h * γ 1 . In this case the path w is called graph-orientation-reversing.
Since (f, g) is transverse, Coin(f, g) is finite. Let A ⊂ Coin(f, g), then A can be represented as A = {a 1 , a 2 , · · · , a s ; b 1 , c 1 , · · · , b k , c k } where b i Rc i for any i and a i Ra j for no i = j. The elements {a i } i of this decomposition are called free.
Definition 1.2. In the above situation the semi − index of the pair is the number of free elements of this decomposition of A.
In [Dobreńko & Jezierski] and [Jezierski 2] we can find the proof of the fact that this definition does not depend on the decomposition of A. Moreover, if U ⊂ M is an open subset we can extend this definition to a semi-index of a pair on the subset U (|ind|(f, g; U)).
In [Jezierski 1] the authors studied when a coincidence point x ∈ Coin(f, g) satisfy xRx. These points can only appear in the nonorientable case, they will be called self-reducing points. They represent a new situation (see [Jezierski 1,Example 2.4]) that never occurs in the orientable case or in the fixed point context.
x ∈ Coin(f, g) and let H ⊂ π 1 (M), H ′ ⊂ π 1 (N) denote the subgroups of orientation-preserving elements. We define: there exists a loop α based at x such that f • α ∼ g • α and exactly one of the loops α or f • α is orientation-preserving).
Definition 1.6. A coincidence class A is called defective if A contains a self-reducing point.
Lemma 1.7. ([Jezierski 1, 2.3]) If a Nielsen class A contains a selfreducing point (i.e. A is defective) then any two points in this class are R-related, and thus:

The root case
In [Brown & Schirmer] we can find a different approach to extending the Nielsen root theory to the nonorientable case. Using the concept of orientation-true 1 map they classified maps between manifolds of the same dimension in three types (see also [Olum] and [Skora]): (1) Type I: f is orientation-true.
(2) Type II: f is not orientation-true but does not map an orientationreversing loop in M to a contractible loop in N.
(3) Type III: f maps an orientation-reversing loop in M to a contractible loop in N. Further, a map f is defined to be orientable if it is of Type I or II, and nonorientable otherwise.
For orientable maps they describe an Orientation Procedure ( [Brown & Schirmer,2.6]) for root classes, using local degree with coefficients in Z. For maps of Type III the same procedure is only possible with coefficients in Z 2 . They then defined the multiplicity of a root class, that is an integer for orientable maps and an element of Z 2 for maps of Type III Now if we consider the root classes for a map f to be the coincidence classes of the pair (f, c) where c is the constant map, we have: (i) If f is orientable, no root class of f is defective.
(ii) If f is of Type III, then all root classes of f are defective.
Proof: By lemma 1.5, a coincidence class C of the pair (f, c) is defective if and only if there exists a point x ∈ C and a loop α in x such that f • α ∼ 1 and α is orientation-reversing. This proves the first statement. Now let f be a Type III map. Then there exists a loop α ∈ π 1 (M, x 0 ) such that α is orientation-reversing and f • α ∼ 1. If x is a root of f choosing a path β from x to x 0 we have that γ = β −1 αβ is a loop in x such that γ is orientation-reversing and f • γ ∼ 1. Then all roots of f are self-reducing points.
In fact [Brown & Schirmer,Lemma 4.1] shows the equality between the multiplicity of a root class and its semi-index.
Theorem 2.3. Let M and N be manifolds of the same dimension such that M is nonorientable and N is orientable. If f : M → N is a map then all essential root classes of f are defective.
Proof: There are no orientation-true maps from a nonorientable to an orientable manifold. If f is a Type II map then by [Brown & Schirmer,Lemma 3.10] deg(f ) = 0 and f has no essential root classes. The result follows by proposition 2.2.
Using the ideas of Proposition 2.2 we can also state: Lemma 2.4. Let f, g : M → N be two maps between manifolds of the same dimension. If there exist a coincidence point x 0 and a graphorientation-reverse loop α in x 0 such that f (α) is in the center of π 1 (N), then all coincidence points of the pair (f, g) are self-reducing points.
Proof: We can suppose that all coincidences of the pair (f, g) have image on the same point in N. If x 1 ∈ Coin(f, g) we choose a path β from x 1 to x 0 and take the loop γ = β −1 • α • β at x 1 . Since α belongs to the center of π 1 (N), γ is a graph-orientation-reverse loop at x 1 .
Corollary 2.5. Let f, g : M → N be two maps between manifolds of the same dimension such that f # (π 1 (M)) is contained in the center of π 1 (N). If (f, g) has a defective class then all classes of (f, g) are defective.

Covering maps
Let M and N be compact, closed manifolds of the same dimension; f, g : M → N be two maps such that Coin(f, g) is finite. and p : M → M and q : N → N be finite coverings such that there exist lifts f , g : M → N of the pair f, g.
Corollary 3.2. Let x 0 ∈ Coin( f , g) and x 0 = p( x 0 ). Then p −1 (x 0 ) ∩ Coin( f , g) have exactly #Coin( f * ,x 0 , g * ,x 0 ) elements. Lemma 3.3. Let x 0 and x ′ 0 be two coincidences of the pair ( f , g) such that p( x 0 ) = p( x ′ 0 ) = x 0 , and let γ the unique element of D( M ) such that γ( x 0 ) = x ′ 0 . The points x 0 and x ′ 0 are in the same coincidence class of ( f , g) if and only if there exists γ ∈ π 1 (M, x 0 ) such that: Proof: (⇒) If x 0 and x ′ 0 are in the same coincidence class of ( f , g), there exists a path β from x 0 to x ′ 0 that realizes the Nielsen relation, (i.e. f • β ∼ g • β).
Corollary 3.4. In lemma 3.3, if the points x 0 and x ′ 0 are in the same coincidence class of ( f , g) then x 0 R x ′ 0 if and only if sign( f * ,x 0 (γ)) · sign(γ) = −1. In this case, x 0 is a self-reducing coincidence point.
Theorem 3.7. Let M and N be compact, closed manifolds of the same dimension, f, g : M → N be two maps such that Coin(f, g) is finite, and p : M → M and q : N → N be finite coverings such that there exist lifts f , g : M → N of the pair (f, g). If C is a coincidence class of the pair ( f , g) then C = p( C) is a coincidence class of the pair (f, g) and Proof: The fact that C = p( C) is a coincidence class of the pair (f, g) is known. We choose a point x 0 ∈ C. Since Coin(f, g) is finite we can suppose that a decomposition C = {x 1 , x 2 , · · · , x s ; c 1 , c ′ 1 , c 2 , c ′ 2 , · · · , c n , c ′ n } is such that each x i is free, and for all pairs c j , c ′ j we have c j Rc ′ j . Now we choose paths {α i } i , 2 ≤ i ≤ s; {β j } j and {γ j } j , 1 ≤ j ≤ n (see figure 1) such that: • α i is a path in M from x 1 to x i that realizes the Nielsen relation.
• β j is a path in M from x 1 to c j that realizes the Nielsen relation.
• γ j is a graph-orientation-reversing path in M from c j to c ′ j .
PSfrag replacements Figure 1. The class C and the chosen paths.
For each element {x k 1 } of p −1 (x 1 )∩Coin( f , g) (by Corollary 3.2 there were #Coin( f * ,x 1 , g * ,x 1 ) such elements) we take lifts { α k i } i,k , { β k j } j,k and { φ k j } j,k of the paths {α i } i , {β j } j and {γ j • β j } j respectively, starting at x k 1 . Using Corollary 3.5 and Lemma 3.6 at the point x 1 ∈ C we obtain that the set p −1 (C)∩Coin( f , g) is the union of #Coin( f * ,x 1 , g * ,x 1 ) copies of the class C and these copies can be divided in #Coin( f * ,x 1 , g * ,x 1 ) #jx 1 (Coin(f # ,g # )x 1 ) disjoint coincidence classes of the pair ( f , g). Each one of these classes contains #j x 1 (Coin(f # , g # ) x 1 ) copies of the class C.
The result follows by Corollary 3.4.

Two folded orientable covering
Let M and N be manifolds of same dimension with M nonorientable and N orientable; f, g : M → N be two maps such that Coin(f, g) is finite, and p : M → M be the two-fold orientable covering of M. We define f , g : M → N by f = f • p and g = g • p.
Under the above conditions, if C is a coincidence class of the pair (f, g) then p −1 (C) ⊂ Coin( f , g) is such that: (1) p −1 (C) can be divided in two disjoint sets C and C ′ , of the same cardinality, such that p( C) = p( C ′ ) = C.
(2) If x 1 , x 2 ∈ C (or C ′ ) then x 1 and x 2 are in the same coincidence class of ( f , g) (3) C and C ′ are in the same coincidence class of the pair ( f , g) if and only if C is defective.
Proof: We need only to take q : N → N as the identity map in the Corollary 3.2, Corollary 3.4 and Lemma 3.6.
Corollary 4.2. Under the hypotheses of lemma 4.1 we have: (1) If C is not defective then C and C ′ are two coincidence classes of the pair ( f , g) with ind( f , g, C) = −ind( f , g, C ′ ) and |ind|( f , g, C) = |ind(f, g, C)|.
(2) If C is defective then C ∪ C ′ is a unique coincidence class of the pair ( f , g) with ind( f , g, C ∪ C ′ ) = 0.
Proof: The first part is a consequence of lemma 4.1, and the second can be proved using the fact that M is the double orientable covering of M. It is useful to remember that the pair ( f , g) is a pair of maps between orientable manifolds. (1) L( f , g) = 0; (2) N( f , g) is even; (3) N(f, g) ≥ N ( f , g) 2 ; (4) If N( f , g) = 0 then all coincidence classes with nonzero semiindex of the pair (f, g) are defective.
Proof: We need only to see that p(Coin( f , g)) = Coin(f, g), and that in the pair ( f , g) we "lose" the defective classes.

Applications
Theorem 5.1. Let f, g : M → N be two maps between compact closed manifolds of the same dimension such that M is nonorientable and N is orientable. Suppose that N is such that for all orientable manifolds M ′ of the same dimension of N and all pairs of maps f ′ , g ′ : M ′ → N we have L(f ′ , g ′ ) = 0 ⇒ N(f ′ , g ′ ) = 0. Then all coincidence classes with nonzero semi-index of the pair (f, g) are defective.
Proof: The hypotheses on N are enough to show, using the notation of the proof of theorem 4.1, that N( f , g) = 0. So by corollary 4.3, all coincidence classes with nonzero semi-index of the pair (f, g) are defective.
We note that the hypotheses on the manifold N in theorem 5.1, in dimension greater then 2, are equivalent to the converse of Lefschetz theorem. In dimension 2 these hypotheses are not equivalent but necessary for the converse of Lefschetz theorem.
Remark 5.2. The following manifolds satisfy the hypotheses on the manifold N in the theorem 5.1: (