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Nielsen numbers of selfmaps of flat 3-manifolds.

1 Introduction

In the 1920s, J. Nielsen conjectured that for any homeomorphism f : M [right arrow] M of a closed surface M there exists a map g, isotopic to [beta], so that g has exactly N(f) = N(g) fixed points. Here, N(f) is now known as the Nielsen number of f. This homotopy invariant is often a sharp lower bound for the minimal number of fixed points in the homotopy class of f (see e. g. [1, 12]). This conjecture was proven by Jiang [13], Ivanov [11] (for self-homotopy equivalences), and Jiang-Guo [14] using the Nielsen-Thurston classification of surface homeomorphisms. The Nielsen conjecture has been proven for homeomorphisms of manifolds of dimension greater than or equal to 5 [17], and for a large class of 3-manifolds including (after Thurston's geometrization theorem) all irreducible 3-manifolds [16]. Meanwhile, Nielsen numbers of surface maps have been studied using Fox Calculus and other methods of combinatorial group theory. In particular, M. Kelly [18] outlined a method of calculating N(f) for surface homeomorphisms using the work of M. Bestvina and M. Handel based on the theory of train tracks. He also gave algorithms for N(f) for homeomorphisms of certain geometric 3-manifolds [19], including the Seifert manifolds.

The purpose of this work is to make explicit calculation of the Nielsen number of a self homeomorphism of a flat 3-manifold. In particular, for a flat 3-manifold [alpha], we compute

NSH(X) = {N(h) | h [member of] Home(X)}.

Using appropriate group presentations for the fundamental groups of the ten flat 3-manifolds, we further analyze the possible values of N(f) when f is an arbitrary selfmap. In section 2, we recall the ten 3-dimensional flat manifolds by listing their fundamental groups and their presentations. In section 3, we compute the Nielsen number of a self homeomorphism of the first five flat manifolds making use of the automorphisms of the 2-dimensional crystallographic group on which the fundamental group of the flat manifold projects. In section 4, we turn our attention to the remaining cases. In sections 5 and 6, we compute N(f) for arbitrary selfmaps f. For cases 2-5, 9, 10, we use a particular fully invariant subgroup corresponding to the fundamental group of a torus or a Klein bottle that allows us to compute N(f) using fiberwise techniques. We complete the computation of N(f) for the remaining cases using different techniques. In the last section, we determine the flat manifolds for which the Jiang-type condition holds.

2 Flat 3-manifolds and Nielsen numbers

Every isometry of the Euclidean space [R. sup. n] is a rotation followed by a translation. More precisely, the group of isometries Isom([R. sup. n]) is given by the semi-direct product [R. sup. n] x O(n). A subgroup n C Isom([R. sup. n]) is a crystallographic group on [R. sup. n] if [pi] is a discrete uniform subgroup. Moreover, [pi] is called a Bieberbach group if in addition it is torsion free. Given a Bieberbach group [pi], the resulting quotient manifold [R. sup. n] / [pi] is called a flat n-manifold. The group [pi] has a normal maximal abelian subgroup [GAMMA] of finite index and [GAMMA] has rank n. The quotient [PHI] = [pi]/[GAMMA] is called the holonomy group. For more details on flat manifolds, see [3] or [22, Ch. 3].

There are a total of ten flat 3-manifolds whose fundamental groups are listed below, where the first six are orientable and the remaining four are non-orientable.

The following presentations can be found in [22, pp. 117-121].

1. <[[alpha].sub.1], [[alpha].sub.2], [[alpha].sub.3] | [[alpha].sub.j][[alpha].sub.i], 1 [less than or equal to] i, j [less than or equal to] 3> with holonomy [PHI] = {1}.

2. [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] with holonomy [PHI] = [Z.sub.2].

3. [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] with holonomy [PHI] = [Z.sub.3].

4. [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] with holonomy [PHI] = [Z.sub.4].

5. [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] with holonomy [PHI] = [Z.sub.6].

6. [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] with holonomy [PHI] = [Z.sub.2] x [Z.sub.2].

7'. [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] with holonomy [PHI] = [Z.sub.2]. The isomorphism [t.sub.1] [??] [beta], [[alpha].sub.2] [??] t, [[alpha].sub.3] [??] [alpha] gives the following alternate presentation

7. [[pi].sub.1](K) x Z = <[alpha], [beta] | [beta][alpha][[beta].sup.-1] = [[alpha].sup.-1]> x <t> where K is the Klein bottle, with holonomy [PHI] = [Z.sub.2].

8'. [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] with holonomy [PHI] = [Z.sub.2]. The isomorphism [[alpha].sub.2] [??] [([alpha][beta]).sup.2], [[alpha].sub.3] [??] [([alpha][beta]).sup.2]t, [t.sub.1] [??] ([alpha][beta])t gives the following alternate presentation

8. <[alpha], [beta], t | [beta][alpha][[beta].sup.-1] = [[alpha].sup.-1], t[alpha][t.sup.-1] = [alpha], t[beta][t.sup.- 1] = [alpha][beta]> with holonomy [PHI] = [Z.sub.2].

9'. [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] with holonomy [PHI] = [Z.sub.2] x [Z.sub.2]. The isomorphism [[alpha].sub.3] [??] [alpha], [t.sub.2] [??] [beta], [t.sub.1][t.sub.2] [??] t gives the following alternate presentation

9. ([alpha], [beta], t | [beta][alpha][[beta].sup.-1] = [[alpha].sup.-1], t[alpha][t.sup.-1] = [alpha], t[beta][t.sup.- 1] = [[beta].sup.-1]) with holonomy [PHI] = [Z.sub.2] x [Z.sub.2].

10'. [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] with holonomy [PHI] = [Z.sub.2] x [Z.sub.2]. The isomorphism [[alpha].sub.3] [??] [[alpha].sup.-1], [t.sub.2] [??] [beta], [t.sub.1][t.sub.2] [??] t gives the following alternate presentation

10. <[alpha], [beta], t | [beta][alpha][[beta].sup.-1] = [[alpha].sup.-1], t[alpha][t.sup.-1] = [alpha], t[beta][t.sup.-1] = [alpha][[beta].sup.-1]) with holonomy O = [Z.sub.2] x [Z.sub.2].

All of these 10 Bieberbach groups possess natural projections onto some 2-dimensional crystallographic groups. Cases 1 and 7 are straightforward as they project onto [G.sub.1] = Z x Z (torus) and onto [G.sup.3.sub.1] = [[pi].sub.1] (K) (Klein bottle) respectively.

We shall use the notation of the 2-dimensional crystallographic groups as given by R. Lyndon in [21].

Case 2: p : G [right arrow] [G.sub.2] where

[G.sub.2] = ([alpha], [beta], [tau] | [alpha][beta] = [beta][alpha], [[alpha].sup.[tau]] = [[alpha].sup.-1], [[beta].sup.[tau]] = [[beta].sup.-1], [[tau].sup.2] = 1).

and p is given by [[alpha].sub.1] [??] 1, [[alpha].sub.2] [??] [alpha], [[alpha].sub.3] [??] [beta], t [??] [tau].

Case 3: p : G [right arrow] [G.sub.3] where

[G.sub.3] = <[alpha], [beta], [tau] | [alpha][beta] = [beta][alpha], [[alpha].sup.[tau]] = [[alpha].sup.-1] [beta], [[beta].sup.[tau]] = [[alpha].sup.-1], [[tau].sup.3] = 1).

and p is given by [[alpha].sub.1] [??] 1, [[alpha].sub.2] [??] [[beta].sup.-1], [[alpha].sub.3] [??] [alpha], t [??] T.

Case 4: p : G [right arrow] [G.sub.4] where

[G.sub.4] = <[alpha], [beta], [tau] | [alpha][beta] = [beta][alpha], [[alpha].sup.[tau]] = [beta], [[beta].sup.[tau]] = [[alpha].sup.-1], [[tau].sup.4] = 1).

and p is given by [[alpha].sub.1] [??] 1, [[alpha].sub.2] [??] [alpha], [[alpha].sub.3] [??] [beta], t [??] [tau].

Case 5: p : G [right arrow] [G.sub.6] where

and p is given by [[alpha].sub.1] [??] 1, [[alpha].sub.2] [??] [alpha], [[alpha].sub.3] [??] [beta], t [??] [tau].

Case 6: p : G [right arrow] [G.sup.4.sub.2] where

[G.sup.4.sub.2] = <[alpha], [beta], [tau] | [beta][alpha][[beta].sup.-1] = [[alpha].sup.-1], [[alpha].sup.[tau]] = [[alpha].sup.-1], [[beta].sup.[tau]] = [alpha][[beta].sup.-1], [[tau].sup.2] = 1)

and p is given by [t.sub.1] [??] [[beta].sup.-1], [t.sub.2] [??] [tau], [t.sub.3] [??] [tau][beta], [[alpha].sub.1] [??] [[beta].sup.-2], [[alpha].sub.2] [??] 1, [[alpha].sub.3] [??] [alpha].

Case 8: p : G [right arrow] [G.sub.1] = Z x Z = <[tau]> x &lt;b&gt; where p is given by [alpha] - 1, [beta] [??] b, t [??] [tau].

Case 9: p : G [right arrow] [G.sup.2.sub.2] where

[G.sup.2.sub.2] = <[alpha], [beta], [tau] | [beta][alpha][[beta].sup.-1] = [[alpha].sup.-1], [[alpha].sup.[tau]] = [alpha], [[beta].sup.[tau]] = [[beta].sup.-1], [[tau].sup.2] = 1>

and p is given by [alpha] [??] [alpha], [beta] [??] [beta], t [??] [tau].

Case 10: First, the isomorphism [alpha] [??] [alpha], [beta] [??] [beta], t [??] t[beta] gives the group the following presentation

G = <[alpha], [beta], t | [beta][alpha][[beta].sup.-1] = [[alpha].sup.-1], t[alpha][t.sup.-1] = [[alpha].sup.-1], t[beta][t.sup.-1] = [alpha][[beta].sup.-1]>.

p : G [??] [G.sup.4.sub.2] where

[G.sup.4.sub.2] = <[alpha], [beta], [tau] | [beta][alpha][[beta].sup.-1] = [[alpha].sup.-1], [[alpha].sup.[tau]] = [[alpha].sup.-1], [[beta].sup.[tau]] = [alpha][[beta].sup.-1], [[tau].sup.2] = 1)

and p is given by [alpha] [??] [alpha], [beta] [??] [beta], t [??] [tau].

Let [M.sup.n] be a flat manifold with fundamental group [pi]. Then there exists a maximal abelian normal subgroup [GAMMA] such that [pi]/[GAMMA] = [PHI] (the holonomy) is finite. Given a selfmap f : M [right arrow] M, there exist lifts [D.sub.*]f on the [absolute value of ([PHI])]-fold cover [T.sup.n] whose fundamental group is [GAMMA], for each D [member of] [PHI]. There is an averaging formula for the Nielsen number [20] given by

N(f)= 1/[absolute value of ([PHI])] [summation over (D [member of] [PHI])] [absolute value of (det(l - [([D.sub.*]f).sub.#]))]. (2.1)

There is an alternate way of computing N(f) when M is fibered over [S.sup.1]. Consider the fibration [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] where N is a closed surface. Given a fiber-preserving map f : M [right arrow] M inducing [bar.f] : [S.sup.1] [right arrow] [S.sup.1], we can compute N(f) as follows. Let [gamma] = deg [bar.f]. If [gamma] = 1, then [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] so that f is homotopic to a fixed point free map. It follows that f is deformable to be fixed point free and thus N(f) = 0. If [gamma] [not equal to] 1, then N([bar.f]) = [absolute value of (1 - [gamma])]. Without loss of generality, we may assume that [bar.f] has exactly [absolute value of (1 - [gamma])] fixed points each of which is its own fixed point class. The fixed point classes of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] inject into the fixed point classes of f for each [bar.x] [member of] Fix [bar.f]. In fact, we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (2.2)

This fiberwise technique and in particular the formula (2.2) will be useful in section 5 when we compute N(f) for arbitrary selfmaps in most cases.

3 Nielsen numbers of self homeomorphisms: Cases 1 - 5

3.1 Case 1.

This flat manifold is the 3-torus [T.sup.3]. Every homeomorphism f : [T.sup.3] [right arrow] [T.sup.3] induces on the fundamental group a linear map [phi] : [Z.sup.3] [right arrow] [Z.sup.3] and the Nielsen number is N(f) = [absolute value of (det(1 - [phi]))]. It is easy to see that NSH(M) = N [union] {0}.

Next, we use the formula (2.1) to determine the values of the Nielsen numbers of homeomorphisms for Cases 2, 3, 4, and 5.

It is well known that the center Z(G) of a crystallographic group G coincides with the fixed point group [([Z.sup.n]).sup.[PHI]] where [Z.sup.n] is the translation subgroup and [PHI] is the holonomy group. In Case 2, the holonomy [Z.sub.2] is generated by t so that [([Z.sup.3]).sup.[PHI]] is the subgroup of the elements fixed by the automorphism induced by t. From the presentation of G for Case 2, the automorphism induced by t is given by [[alpha].sub.1] [??] [[alpha].sub.1], [[alpha].sub.2] [??] [[alpha].sup.-1.sub.2], [[alpha].sub.3] [right arrow] [[alpha].sup.-1.sub.3]. In other words, the automorphism is given by a diagonal integral matrix which has as eigenvalue with one dimensional eigenspace. We now conclude that Z(G) = <[[alpha].sub.1]>.

For each of the Cases 3, 4, and 5, a similar argument shows that Kerp = <[[alpha].sub.1]> = Z(G). Thus, for every [phi] [member of] Aut(G) for each G in Cases 2-5, [phi] is represented by an array of the form

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

where [kappa] = [+ or -] 1 and A is a 3 x 3 array representing the induced automorphism [bar.[phi]]: G/Z(G) [right arrow] G/Z(G).

Write p to be the array

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Here the columns are the exponents of the generators [[alpha].sub.l], [[alpha].sub.2], [[alpha].sub.3], t of their images under [phi] since every word can be written in the normal form [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Furthermore, [[alpha].sub.1], [[alpha].sub.2], [[alpha].sub.3] generate a maximal abelian normal subgroup [GAMMA] in G so that the lift (or restriction to [GAMMA]) [phi]' is represented by the array

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

If a homeomorphism f has an induced automorphism [phi] on the fundamental group, the averaging formula (2.1) yields

N(f) = 1/[absolute value of ([PHI])] [summation over (0 [less than or equal to] i [less than or equal to] [absolute value of ([PHI]))] [absolute value of (det(l - [theta]([t.sup.i])[phi]'))] (3.1)

where [theta](t) denotes the action of t. In the Cases 2-5, t acts trivially on [[alpha].sub.1] so that [theta]([t.sup.i])[phi]' is also represented by an array of the form

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

for some 2 x 2 array [bar.[A.sub.i]]. Thus, when [kappa] = 1, [absolute value of (det(l - [theta]([t.sup.i])[phi]'))] = 0 for all i, 0 [less than or equal to] i < [absolute value of ([PHI])]. For such homeomorphisms f, we have N(f) = 0. For the rest of this section, we consider automorphisms where [kappa] = -1.

3.2 Case 2.

This group projects onto [G.sub.2]. It follows from [6, 7] that [phi] can be represented by an array of the form

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

with ad - bc = [+ or -] 1. Now the lifts of p are of the form (in fact, matrices)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Let

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

A straightforward calculation using the averaging formula (3.1) shows that if det [bar.A] = -1 then the Nielsen number N(f) = 2 [absolute value of (Tr[bar.A])], where TrX denotes the trace of a matrix X. If det [bar.A] = 1 then N(f) = 2 [absolute value of (Tr[bar.A])] if [absolute value of (Tr[bar.A])] [greater than or equal to] 2 or else N(f) = 4.

Thus, for any homeomorphism f, we have NSH(M) = 2N [union] {0}.

3.3 Case 3.

This group projects onto [G.sub.3]. It follows from [6, 7] that the automorphism A has one of the following two forms:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

The maximal abelian subgroup [GAMMA] is generated by [[alpha].sub.1], [[alpha].sub.2], [[alpha].sub.3] with quotient the holonomy [PHI] = [Z.sub.3]. Moreover, the restriction of [phi] on [GAMMA] is given by the matrix

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Note that [phi]([[alpha].sub.1]) = [[alpha].sup.-1.sub.1]. Since [[alpha].sub.1] = [t.sup.3], it follows that [phi](t) = [[alpha].sup.z.sub.1][[alpha].sup.r.sub.2][[alpha].sup.s.sub.3][t.sup.-1] = [[alpha].sup.z.sub.1][[alpha].sup.r.sub.2][[alpha].sup.s.sub.3][t.sub.2] so that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Now, from [7], we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Now a straightforward calculation shows that det(1 - [phi]') = det(1 - [theta](t)[phi]') = det(1 - [theta]([t.sub.2])[phi]') = 0. Hence such automorphisms also yield N(f) = 0. We conclude that NSH(M) = {0}. Hence, by [16], every homeomorphism of this flat manifold is isotopic to a fixed point free homeomorphism.

3.4 Case 4.

This groups projects onto [G.sub.4]. It follows from [6, 7] that the automorphism A has one of the following two forms:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Note that [phi]([[alpha].sub.1]) = [[alpha].sup.-1.sub.1] Since [[alpha].sub.1] = [t.sup.4], it follows that [phi](t) = [[alpha].sup.z.sub.1][[alpha].sup.r.sub.2][[alpha].sup.s.sub.3][t.sup.3] so that only (ii) can occur. Furthermore, we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Here,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Now a straightforward calculation using the averaging formula shows that N(f) = 0. Thus we conclude that for any homeomorphism f, we have N(f) = or NSH(M) = {0}.

3.5 Case 5.

This group projects onto [G.sub.6]. It follows from [6, 7] that the automorphism A has one of the following two forms:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Note that [phi]([[alpha].sub.1]) = [[alpha].sup.-1.sub.1]. Since [[alpha].sub.1] = [t.sup.6], it follows that [phi](t) = [[alpha].sup.z.sub.1][[alpha].sup.r.sub.2][[alpha].sup.s.sub.3][t.sup.5] so that only (ii) can occur. Furthermore, we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Here,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Now a straightforward calculation using the averaging formula shows that N(f) = 0. Thus we conclude that for any homeomorphism [beta], we have N(f) = 0 or NSH(M) = {0}.

4 Nielsen numbers: remaining cases 6-10

In this section, we compute the Nielsen numbers of self-homeomorphisms of flat manifolds in the remaining 5 cases, 6-10.

4.1 Case 6.

Lemma 4.1. Each element in G can be written as the form [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Proof. By definition of holonomy, the subgroup of G generated by [[alpha].sub.1], [[alpha].sub.2], [[alpha].sub.3] has index 4 in G. Thus, each element of G must be in one of the forms: [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. By using the relation: [t.sub.3][a.sub.j][t.sup.-1.sub.3] = 1, [[alpha].sup.-1.sub.j] = 1, 2. We obtain:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Note that [t.sub.2] = [t.sub.3][t.sub.1]. Lemma 4. says that every group element has such normal form. In particular, a straightforward calculation yields

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Now for any [phi] [member of] Aut(G), using the generators [t.sub.1], [[alpha].sub.2] and [t.sub.3], we can represent [phi] by a 3 x 3 array of the form

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

We now compute [phi]([t.sup.2.sub.1]) under all possible cases for the parities of the pair (a, [epsilon]).
Type      a   [epsilon]    [phi]([t.sup.2.sub.1]) =
                             [phi]([[alpha].sub.1])

(I)    even       even             [t.sup.2a.sub.1]
                              [[alpha].sup.2.sub.2]
                           [t.sup.2[epsilon].sub.3]

(II)   even        odd     [t.sup.2[epsilon].sub.3]

(HI)    odd       even             [t.sup.2a.sub.1]

(IV)    odd        odd    [[alpha].sup.-2b-1.sub.2]


Similarly, we compute [phi]([t.sup.2.sub.3]) under all possible cases for the parities of the pair (r, t).
Type        r      t         [phi]([t.sup.2.sub.3]) =
                               [phi]([[alpha].sub.3])

(I')     even   even                 [t.sup.2r.sub.1]
                               [[alpha].sup.2s.sub.2]
                                     [t.sup.2t.sub.3]

(II')    even    odd                 [t.sup.2t.sub.3]

(III')    odd   even                 [t.sup.2r.sub.1]

(IV')     odd    odd   [[alpha].sup.-2s.sub.-1.sub.2]


If Type (I) occurs, we consider the relation [phi]([t.sub.1][t.sup.2.sub.3][t.sup.-1.sub.1]) = [phi]([t.sup.-2.sub.3]). With a and [epsilon] both even, [phi]([t.sub.1]) = [t.sup.a.sub.1][[alpha].sup.b.sub.2][t.sup.[epsilon].sub.3] lies in the maximal abelian subgroup generated by [[alpha].sub.1], [[alpha].sub.2], [[alpha].sub.3] so that [phi]([t.sub.1]) commutes with [phi]([t.sup.2.sub.3]) = [phi]([[alpha].sub.3]). It follows that [phi]([t.sup.2.sub.3]) = [phi]([t.sup.-2.sub.3]) and so [phi]([t.sub.3]) = 1, a contradiction to the fact that [phi] is an automorphism and [t.sub.3] is a generator. Likewise, if Type (I') occurs then the relation [phi]([t.sub.3][t.sup.2.sub.1][t.sup.- 1.sub.1]) = [phi]([t.sup.-2.sub.1]) leads to [phi]([t.sub.1]) = 1, a contradiction.

Next, we consider the case Type (II) and Type (II'). Then the relation [phi]([t.sub.1][t.sup.2.sub.3][t.sup.-1.sub.1]) = [phi]([t.sup.-2.sub.3]) becomes

[t.sup.r.sub.1][[alpha].sup.s.sub.2][t.sup.t.sub.3][t.sup.2a.sub.1][t.sup.-t.sub.3][[alpha].sup.-s.sub.2][t.sup.- r.sub.1] = [t.sup.-2a.sub.1] [??] [t.sup.2a.sub.1] = [t.sup.-2a.sub.1] [??] a = 0.

This is a contradiction to the assumption that t is odd.

Consider the case Type (III) and Type (III'). Then the relation [phi]([t.sub.1][t.sup.2.sub.3][t.sup.-1.sub.1]) = [phi]([t.sup.-2.sub.3]) becomes

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

This is a contradiction to the assumption that a is odd.

Consider the case Type (IV) and Type (IV'). Then the relation [phi]([t.sub.1][t.sup.2.sub.3][t.sup.-1.sub.1]) = [phi]([t.sup.-2.sub.3]) becomes

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

This is not possible since s is an integer.

Thus, we only need to consider six possible cases below which we compute [phi]([[alpha].sub.2]) = [phi]([t.sup.2.sub.2]) = [phi][([t.sub.3][t.sub.1]).sup.2].
Type                 [phi][([t.sub.3][t.sub.1]).sup.2] =
                  [([t.sup.r.sub.1][[alpha].sup.s.sub.2]
                    [t.sup.t.sub.3][[alpha].sup.b.sub.2]
                         [t.sup.[epsilon].sub.3]).sup.2]

(II) and (III')             [[alpha].sup.-2s-2b-1.sub.2]

(II) and (IV')                    [t.sup.2(r - 1).sub.1]

(III) and (II')           [[alpha].sup.2(s + b)+1.sub.2]

(III) and (IV')           [t.sup.2([epsilon] - t).sub.2]

(IV) and (II')                    [t.sup.2(r - a).sub.1]

(IV) and (III')           [t.sup.2([epsilon] - t).sub.3]


If we denote by [phi]' the restriction of [phi] on the maximal subgroup generated by [[alpha].sub.l], [[alpha].sub.2] and [[alpha].sub.3], then we have the following
Type         (II) and (III')               (II) and (IV')

[phi]    [MATHEMATICAL EXPRESSION     [MATHEMATICAL EXPRESSION
        NOT REPRODUCIBLE IN ASCII]   NOT REPRODUCIBLE IN ASCII]

Type         (III) and (II')              (III) and (IV')

[phi]    [MATHEMATICAL EXPRESSION     [MATHEMATICAL EXPRESSION
        NOT REPRODUCIBLE IN ASCII]   NOT REPRODUCIBLE IN ASCII]

Type          (IV) and (II')              (IV) and (III')

[phi]    [MATHEMATICAL EXPRESSION     [MATHEMATICAL EXPRESSION
        NOT REPRODUCIBLE IN ASCII]   NOT REPRODUCIBLE IN ASCII]


The holonomy [PHI] = [Z.sub.2] x [Z.sub.2] is generated by the images of [t.sub.1] and [t.sub.3]. Their actions of [[alpha].sub.i] are given by the following matrices:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Now a straightforward calculation together with the average formula for N(f), we conclude that in all six cases we have N(f) = 0 or 2 so that NSH(M) = {0, 2} for any homeomorphism.

4.2 Case 7.

The group is [[pi].sub.1](K) x Z. Moreover, we have the following presentation

G = <[alpha], [beta], t | [beta][alpha[][beta].sup.-1] = [[alpha].sup.-1], t[alpha][t.sup.-1] = [alpha], t[beta][t.sup.-1] = [beta]>.

The center of G is Z(G) = <[[beta].sup.2]> x <t>. Let [phi] [member of] Aut(G). Using the generators [alpha], [beta], t, we can represent [phi] by a 3 x 3 array of the form

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Since t [member of] Z(G), [phi](t) [member of] Z(G). It follows that r = 0 and s is even. Now

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and thus c + [(-1).sup.d]c = 0 and so d must be odd. Thus,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Next, we have [phi]([beta][alpha][[beta].sup.-1]) = [phi]([[alpha].sup.-1]). It follows that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

It follows that b = 0. In other words, [phi]([alpha]) = [[alpha].sup.a]. Since [phi] is an automorphism, we have a = [+ or -] 1. Now, we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

From the calculation above, we have [phi]([[beta].sup.2]) = [[beta].sup.4q+2][t.sup.2[delta]]. Now the subgroup generated by [alpha], [[beta].sup.2], t is a maximal abelian subgroup [GAMMA] and the quotient G/[GAMMA] is the holonomy group [PHI] = <[bar.[beta]] | [[bar.[beta]].sup.2] = 1> [congruent to] [Z.sub.2]. The restriction of [phi] on [GAMMA] is given by the matrix

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Since [phi]' is an automorphism, we have det [phi]' = (2q + 1)[gamma] - 4[delta]k = [+ or -] 1. It follows that [gamma] must be odd. The action of [PHI] on [GAMMA] sends [alpha] to [[alpha].sup.-1] and is trivial on [[beta].sup.2] and t.

Thus, it induces another lift [D.sub.*][phi] given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

A straightforward calculation shows that

N(f) = [1/2](0 + 2 [absolute value of (2q([gamma] - 1) - 4[delta]k)]) = [absolute value of (2q([gamma] - 1) - 4[delta]k)] = [absolute value of ([+ or -] 1 - [gamma] - 2q)]

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

with [gamma] an odd integer. In particular, N(f) must be even. In fact, we have NSH(M) = 2N [union] {0}.

The group is [[pi].sub.1](K) x Z. Moreover, we have the following presentation

G = <[alpha], [beta], t | [beta][alpha][[beta].sup.-1] = [[alpha].sup.-1], t[alpha][t.sup.-1] = [alpha], t[beta][t.sup.-1] = [alpha][beta]>.

Note that [alpha], [[beta].sup.2], t generate an index 2 abelian subgroup in G and hence is the maximal abelian subgroup whose quotient group [Z.sub.2] is the holonomy. Let [phi] [member of] Aut(G). Using the generators [alpha], [beta], t, we can represent [phi] by a 3 x 3 array of the form

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Since [phi](t[beta][t.sup.-1]) = [phi]([alpha][beta]), we have

[[alpha].sup.r][[beta].sup.s][t.sup.[gamma]][[alpha].sup.c] [[beta].sup.d][t.sup.[delta]][t.sup.-[gamma]][[beta].sup.- s][[alpha].sup.-r] = [[alpha].sup.a][[beta].sup.b][t.sup.[epsilon]][[beta].sup.d][t.sup.[delta]]. (4.2)

Using the group relations, (4.2) can be rewritten as

[w.sub.1][t.sup.[delta]] = [w.sub.2][t.sup.[epsilon]+[delta]]

where [w.sub.1], [w.sub.2] are words in [alpha] and [beta]. It follows that [epsilon] = 0.

Note that [t.sup.x][beta] = [[alpha].sup.x][beta][t.sup.x] so that [t.sup.x][[beta].sup.y][t.sup.-x] = [([[alpha].sup.x][beta]).sup.y]. Moreover, [[alpha].sup.x][beta][[alpha].sup.x][beta] = [[beta].sup.2]. Since [phi]([beta][alpha][[beta].sup.-1]) = [phi]([[alpha].sup.-1]), we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (4.3)

Case (i): b even

In this case, (4.3) yields

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Case (ii): b odd

In this case, (4.3) yields

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

for some w [??] b = 0 a contradiction since b is odd.

Thus, we conclude that b = 0. Now, [phi] is an automorphism and [phi]([alpha]) = [[alpha].sup.a]. It follows that a = [+ or -] 1.

Since [phi](t[alpha][t.sup.-1]) = [alpha], we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4.4)

Suppose a = -1 so that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where s is even. Now (4.3) yields c - [(-1).sup.d] - c - 1 = 0 so that d must be odd. (Note that d is also odd when a = 1.)

The equality (4.2) becomes

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

It follows that [gamma] must be odd.

Let [phi]' denote the restriction of [phi] on the maximal abelian subgroup generated by [alpha], [[beta].sup.2] and t. A straightforward calculation show that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

The other lift [D.sub.*][phi]' induced by the holonomy action is given by

[D.sub.*][phi]'(w) = [beta][phi]'(w)[[beta].sup.-1].

Again, a straightforward calculation yields

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Thus, if a = 1 then det(1 - [phi]') = 0 while det(1 - [D.sub.*][phi]') = 2[(1 - d) (1 - [gamma]) - [delta]s]. The averaging formula shows that N(f) = [absolute value of (1 - (d + [gamma]) + ([+ or -] 1))] is even. Similarly, if a = -1 then det(1 - [phi]') = 2[(1 - d)(1 - [gamma]) - [delta]s] while det(1 - [D.sub.*][phi]') = 0. Again using the averaging formula yields that N(f) is even. In fact, all even non negative integers can occur as N(f) and hence NSH(M) = 2N [union] {0}

The isomorphism [alpha] [??] [alpha], [beta] [??] [beta], t [??] t[beta] gives the group G the following presentation

G = <[alpha], [beta], t | [beta][alpha][[beta].sup.-1] = [[alpha].sup.-1], t[alpha][t.sup.-1] = [[alpha].sup.-1], t[beta][t.sup.-1] = [[beta].sup.-1]>. (4.5)

This group is the mapping torus [[pi].sub.1](K) [x.sub.[phi]] Z where [phi]([alpha]) = [[alpha].sup.-1] and [phi]([beta]) = [[beta].sup.-1]. Here K denotes the Klein bottle. Using the calculation in [7] and the fact that this group projects onto the group [G.sup.2.sub.2], the normal subgroup [[pi].sub.1](K) is characteristic. In fact, the corresponding flat manifold M is a Klein bottle bundle over the unit circle [S.sup.1]. Given a homeomorphism [beta], it induces the following commutative diagram at the fundamental group level.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Choose a homeomorphism [bar.f] with induced automorphism [bar.[phi]]. Then the following diagram is commutative, up to homotopy.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4.6)

This implies that there is a homotopy [bar.H] : M x [0, 1] [right arrow] [S.sup.1] from p o f to [bar.f] op. The Covering Homotopy Property for the fibration p : M [right arrow] [S.sup.1] yields a homotopy H : M x [0, 1] [right arrow] M covering [bar.H] from f to [??]. It follows that the diagram (4.6) gives rise to the following commutative diagram.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Since [bar.f] is a self homeomorphism of the unit circle, N([bar.f]) = 0 or 2. If N([bar.f]) = 0, it follows that N(f) = 0. Suppose N([bar.f]) = 2. We may assume that [bar.f] has exactly two fixed points at z = 1 and at z = -1. The corresponding fiber maps are f' and f" respectively. It is easy to see that the fixed subgroups Fix [f'.sub.#] and Fix[f".sub.#] are both trivial so that the fixed point classes of f' and of f" inject into the set of fixed point classes of [??] (or f). Since there are only four isomorphism classes of automorphisms of [[pi].sub.1](K), we may assume without loss of generality that the map f' induces the automorphism [alpha] [??] [alpha], [beta] [??] [alpha][[beta].sup.-1] or [beta] [??] [[beta].sup.-1] while f" induces the automorphism [alpha] [??] [alpha], [beta] [??] [alpha][beta] or [beta] [??] [beta]. By computing the Nielsen number of f' and f", we see that N(f') = 2 while N(f") = 0. Hence, we conclude that NSH(M) = {0, 2}.

4.5 Case 10.

This case is similar to Case 9. This group is the mapping torus [[pi].sub.1](K) [x.sub.[phi]] Z where [phi]([alpha]) = [[alpha].sup.-1] and [phi]([beta]) = [alpha][[beta].sup.-1] Thus G has the following presentation

G = <[alpha], [beta], t | [beta][alpha][[beta].sup.-1] = [[alpha].sup.-1], t[alpha][t.sup.-1] = t[beta][t.sup.-1] = [alpha][[beta].sup.-1]). (4.7)

Let [eta] [member of] Aut(G) be given by the following array

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Since [eta]([beta][alpha][[beta].sup.-1]) = [eta]([[alpha].sup.-1]), we have

[[alpha].sup.c][[beta].sup.d][t.sup.[delta]][[alpha].sup.a][[beta].sup.b][t.sup.[epsilon]][t.sup.-[sigma]][[beta].sup.- d][[alpha].sup.-c] = [t.sup.-[epsilon]][[beta].sup.-b][[alpha].sup.-a].

This equality can be rewritten as [w.sub.1][t.sup.[epsilon]] = [w.sub.2][t.sup.-[epsilon]] where [w.sub.i] are words in [alpha] and [beta].

Thus, [epsilon] = 0. Similarly, [eta](t[beta][t.sup.-1]) = [eta]([alpha][[beta].sup.-1]), we have

[[alpha].sup.r][[beta].sup.s][t.sup.[gamma]][[alpha].sup.c][[beta].sup.d][t.sup.[delta]][t.sup.-[gamma]][[beta].sup.-s] [[alpha].sup.-r] = [[alpha].sup.a][[beta].sup.b][t.sup.[epsilon]][t.sup.-[delta]][[beta].sup.-d][[alpha].sup.-c].

This equality can be rewritten as [[??].sub.1][t.sup.[delta]] = [[??].sub.2][t.sup.[epsilon]-[delta]] where [[??].sub.i] are words in [alpha] and [beta].

It follows that [epsilon] = 2[delta] so that [delta] = 0. Since [epsilon] = 0 = [delta], this shows that [[pi].sub.1](K) is characteristic. Now we use the same arguments as in Case 9 to conclude that NSH(M) = {0, 2} for every homeomorphism f of the flat manifold M.

5 Nielsen numbers of arbitrary selfmaps: Cases 2-5, 9, 10

In the previous two sections, with the exception of cases 9 and 10 for which we used fiberwise techniques to compute N(f) for self homeomorphisms, we employed the average formula (3.1) in terms of the Nielsen numbers of the associated lifts to the universal cover [R.sup.3]. For arbitrary selfmaps, it is more manageable to classify these maps up to fiberwise homotopy since for all but two of the ten cases, the flat manifold M fibers over [S.sup.1] with typical fiber N corresponding to a fully invariant subgroup of [[pi].sub.1](M). Thus, we can apply fiberwise techniques. For cases 2-5, N = [T.sup.2] is the 2-torus. For cases 9 and 10, N = K is the Klein bottle.

For each of the cases 2-5, the crystallographic group G is isomorphic to a mapping torus of the form [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] where i = 2, 3, 4, 5 for each case i and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Moreover, the automorphisms [[theta].sub.2], [[theta].sub.3], [[theta].sub.4], [[theta].sub.5] have finite orders of 2, 3, 4, and 6 respectively. Every endomorphism of G will be given by a 3 x 3 array of the form

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where the columns represent the images of [alpha]2, [alpha]3, and t under p in terms of the generators [alpha]2, [alpha]3, t.

The relations defining (i) t[[alpha].sub.2][t.sup.-1] and (ii) t[[alpha].sub.3][t.sup.-1] yield two relations of the form w[t.sup.m] = w'[t.sup.n] where w, w' are words in [[alpha].sub.2], [[alpha].sub.3]. More precisely, we have the following:

Case 2: (i) [w.sub.1][t.sup.[epsilon]] = [w'.sub.1][t.sup.-[epsilon]] and (ii) [w.sub.2][t.sup.[delta]] = [w'.sub.2][t.sup.-[delta]]. It follows that [epsilon] = 0 = [delta].

Case 3: (i) [w.sub.1][t.sup.[epsilon]] = [w'.sub.1][t.sup.-[delta]] and (ii) [w.sub.2][t.sup.[delta]] = [w'.sub.2][t.sub.-[epsilon]-[delta]]. It follows that [epsilon] = 0 = [delta].

Case 4: (i) [w.sub.1][t.sup.[epsilon]] = [w'.sub.1][t.sup.-[delta]] and (ii) [w.sub.2][t.sup.[delta]] = [w'.sub.2][t.sub.-[epsilon]]. It follows that [epsilon] = 0 = [delta].

Case 5: (i) [w.sub.1][t.sup.[epsilon]] = [w'.sub.1][t.sup.-[delta]] and (ii) [w.sub.2][t.sup.[delta]] = [w'.sub.2][t.sub.-[epsilon]+[delta]]. It follows that [epsilon] = 0 = [delta].

Thus every endomorphism of G is of the form

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

so that N = [[pi].sub.1]([[tau].sup.2]) = <[[alpha].sub.2], [[alpha].sub.3] | [[alpha].sub.2][[alpha].sub.3] = [[alpha].sub.3][[alpha].sub.2]> is fully invariant.

For cases 9 and 10, the crystallographic group G is isomorphic to [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] where i = 9, 10 and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Here each of [[theta].sub.9], [[theta].sub.10] is represented by a 2 x 2 array where the columns are the images of [alpha], [beta] under the action [[theta].sub.i].

Case 9: Given an endomorphism [phi], the relation [phi](t[beta][t.sup.-1]) = [phi]([[beta].sup.-1]) yields

[[alpha].sup.r][[beta].sup.s][t.sup.[gamma]][[alpha].sup.c][[beta].sup.d][t.sup.[delta]][t.sup.-[gamma]][[beta].sup.-s] [[alpha].sup.-r] = [t.sup.-[delta]][[beta].sup.-d][[alpha].sup.-c] [??] [w.sub.1][t.sub.[delta]] = [w'.sub.1][t.sup.-[delta]],

for some words [w.sub.1], [w'.sub.1] in [alpha], [beta]. It follows that 5 = 0.

Similarly the relation [phi]([beta][alpha][[beta].sup.-1]) = [phi]([[alpha].sup.-1]) yields

[[alpha].sup.c][[beta].sup.d][[alpha].sup.a][[beta].sup.b][t.sup.[epsilon]][[beta].sup.-d][[alpha].sup.-c] = [t.sup.- [epsilon]][[beta].sup.-b][[alpha].sup.-a] [??] [w.sub.2][t.sup.[epsilon]] = [w'.sub.2][t.sup.-[epsilon]],

for some words [w.sub.2], [w'.sub.2] in [alpha], [beta]. It follows that [epsilon] = 0.

Case 10: Given an endomorphism [phi], similar to Case 9 above, the relation [phi]([beta][alpha][[beta].sup.-1]) = [phi]([[alpha].sup.-1]) yields [epsilon] = 0. Now, the relation [phi](t[beta][t.sup.-1]) = [phi]([alpha][[beta].sup.-1]) yields

[[alpha].sup.r][[beta].sup.s][t.sup.[gamma]][[alpha].sup.c][[beta].sup.d][t.sup.[delta]][t.sup.-[gamma]][[beta].sup.-s] [[alpha].sup.-r] = [[alpha].sup.a][[beta].sup.b][t.sup.-[delta]][[beta].sup.-d][[alpha].sup.-c] [??] [w.sub.1][t.sup.[delta]] = [w'.sub.1][t.sup.-[delta]],

for some words [w.sub.1], [w'.sub.1] in [alpha], [beta]. It follows that [delta] = 0.

Furthermore, for both cases 9 and 10, the relation [phi]([beta][alpha][[beta].sup.-1]) = [phi]([[alpha].sup.-1]) yields

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

This implies that b = 0.

Thus for cases 9 and 10, every endomorphism is of the form

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5.1)

so that N = [[pi].sub.1](K) = <[alpha], [beta] | [beta][alpha][[beta].sup.-1] = [[alpha].sup.-1]> is fully invariant.

We are now ready to compute N(f) for an arbitrary selfmap in the cases 2-5, 9, 10.

5.1 Case 2

Using fiberwise techniques, it follows from (2.2) that the Nielsen number of a selfmap f is given by

N(f) = [[absolute value of (1 - [gamma])] - 1.summation over (i = 0)] [absolute value of (det(I - [[theta].sub.i](B)))].

Here, f induces on the fundamental group the endomorphism given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

with deg [bar.f] = [gamma] where [bar.f] : [S.sup.1] [right arrow] [S.sup.1] is the induced map on the base of the fibration [T.sup.2] [right arrow] M [right arrow] [S.sup.1]. The matrix B is the restriction [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is of order 2.

The relation [phi](t[[alpha].sub.3][t.sup.-1]) = [phi]([[alpha].sup.-1.sub.3]) yields

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (5.2)

This implies that (1) [gamma] is odd or c = 0 and (2) y is odd or d = 0.

Similarly, the relation [phi](t[[alpha].sub.2][t.sup.-1]) = [phi]([[alpha].sup.-1.sub.2]) yields

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (5.3)

This implies that (1) [gamma] is odd or a = 0 and (2) [gamma] is odd or b = 0. Thus, if [gamma] is even then [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and hence N(f) = [absolute value of (1 - [gamma])].

Suppose [gamma] is odd then [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. It follows that det(I - B) = 1 + ad - bc - (a + d) and det(I - [theta]B) = 1 + ad - bc + (a + d).

When [gamma] is odd, [absolute value of (1 - [gamma])] is even. (1) If [absolute value of (1 + ad - bc)] [greater than or equal to] [absolute value of (a + d)] then we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

(2) Otherwise, we have

N(f) = [absolute value of (1 - [gamma])] x [absolute value of (a + d)].

5.2 Case 3

In this case,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and [[theta].sup.3] = I.

The relation [phi](t[[alpha].sub.2][t.sup.-1]) = [phi]([[alpha].sub.3]) yields

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (5.4)

and [phi](t[[alpha].sub.3][t.sup.-1]) = [phi]([[alpha].sup.-1.sub.2][[alpha].sup.-1.sub.3]) yields

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (5.5)

Suppose [gamma] [equivalent] 0 mod 3. Then (5.4) implies that a = c and b = d; (5.5) implies that c = - a - c, d = b - d [??] a = b = c = d = 0. Thus [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and N(f) = [absolute value of (1 - [gamma])].

Suppose [gamma] [equivalent] 1 mod 3. Then (5.4) becomes

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

This implies that -b = c, a - b = d and (5.5) becomes

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

This implies that -d = -a - c, c - d = -b - d. It follows that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] so that det(I - B) = 1 + [[alpha].sub.2] + [b.sup.2] - ab - 2a + b. Moreover, we have [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Now, we have det(I - 0B) = 1 + [[alpha].sub.1] + [b.sup.2] - ab + a + b and det(I - [[theta].sup.2]B) = 1 + [[alpha].sub.2] + [b.sup.2] - ab + a - 2b.

It is straightforward to show that det(I - B), det(I - [theta]B) and det(I - [[theta].sup.2]B) have the same sign. Thus, we conclude that

N(f) = (1 + [[alpha].sub.2] + [b.sup.2] - ab) x [absolute value of (1 - [gamma])].

Suppose [gamma] [equivalent] 2 mod 3. Similar calculations show that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. It follows that det(I - B) = det(I - [theta]B) = det(I - [[theta].sup.2]B) = 1 - [[alpha].sub.2] - [b.sup.2] + ab. Thus,

N(f) = [absolute value of (1 - [[alpha].sub.2] - [b.sup.2] + ab)] x [absolute value of (1 - [gamma])].

5.3 Case 4

In this case,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and [[theta].sup.4] = I.

The relation [phi](t[[alpha].sub.2][t.sup.-1]) = [phi]([[alpha].sub.3]) yields

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5.6)

and [phi](t[[alpha].sub.3][t.sup.-1]) = [phi]([[alpha].sup.-1.sub.2]) yields

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (5.7)

Note that [t.sub.2][[alpha].sub.2][t.sup.-2] = [[alpha].sup.-1.sub.2] and [t.sub.2][[alpha].sub.3][t.sup.-2] = [[alpha].sub.3]. When [gamma] is even, we have [t.sup.[gamma]][[alpha].sub.3][t.sup.-[gamma]] = [[alpha].sub.3]. Thus (5.6) becomes

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

which then implies that b = d and (-1)[([gamma]/2).sub.a] = c. Now (5.7) becomes

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

which then implies that d = -b and [(-1).sup.([gamma]/2)] c = -a. It follows that b = d = 0 and c = a = 0 so that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Hence, we have

N(f) = [absolute value of (1 - [gamma])].

When [gamma] [equivalent] 1 mod 4, (5.6) becomes

[[alpha].sup.r.sub.2][[alpha].sup.s.sub.3][[alpha].sup.a.sub.3][[alpha].sup.-b.sub.2][[alpha].sup.- s.sub.3][[alpha].sup.-r.sub.2] = [[alpha].sup.c.sub.2][[alpha].sup.d.sub.3]

which implies that a = d and -b = c. It follows that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. It is straight-forward to see that det(I - B) = [(1 - a).sup.2] + [b.sup.2], det(I - [theta]B) = [(1 + b).sup.2] + [a.sup.2], det(I - [[theta].sup.2]B) = [(1 + a).sup.2] + [b.sup.2], and det(I - [[theta].sup.3]B) = [(1 - b).sup.2] + [a.sub.2]. Thus

N(f) = [absolute value of (1 - [gamma])] x (1 + [[alpha].sub.2] + [b.sup.2]).

When [gamma] [equivalent] 3 mod 4, (5.6) becomes

[[alpha].sup.r.sub.2][[alpha].sup.s.sub.3][[alpha].sup.-a.sub.3][[alpha].sup.-b.sub.2][[alpha].sup.- s.sub.3][[alpha].sub.-r.sub.2] = [[alpha].sup.c.sub.2][[alpha].sup.d.sub.3]

which then implies that c = -b and d = -a. It follows that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. It is straightforward to see that det(I - B) = 1 - [a.sup.2] + [b.sup.2], det(I - [theta]B) = [(1 + b).sup.2] - [a.sup.2], det(I - [[theta].sup.2]B) = 1 - [a.sup.2] + [b.sup.2], and det(I - [[theta].sup.3]B) = [(1 - b).sup.2] - [a.sup.2]. It is not difficult to see that det(I - B), det(I - [[theta].sup.i]B), for i = 1, 2, 3, are either all non-positive or all non-negative. Thus

N(f) = [absolute value of (1 - [gamma])] x [absolute value of (1 - [a.sup.2] + [b.sup.2])].

5.4 Case 5

In this case,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and [[theta].sup.6] = I.

The relation [phi](t[[alpha].sub.3][t.sup.-1]) [phi]([[alpha].sup.-1.sub.3][[alpha].sub.3]) yields

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5.8)

and [phi](t[[alpha].sub.3][t.sup.-1]) = [phi]([[alpha].sup.-1.sub.2][[alpha].sub.3]) yields

[[alpha].sup.r.sub.2][[alpha].sup.s.sub.3][t.sup.[gamma]][[alpha].sup.c.sub.2][[alpha].sup.d.sub.3][t.sup.-[gamma]] [[alpha].sup.-s.sub.3][[alpha].sup.-r.sub.2] = [[alpha].sup.c-a.sub.2][[alpha].sup.d-b.sub.3]. (5.9)

Suppose [gamma] [equivalent] 0 mod 6. Then (5.8) implies that a = c, b = d and (5.9) implies that c = c - a, d = d - b. It follows that a = b = c = d = 0 and hence [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Suppose [gamma] [equivalent] 1 mod 6. Then (5.8) implies that - b = c, a + b = d and (5.9) implies that - d = c - a, c + d = d - b. It follows that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Suppose [gamma] [equivalent] 2 mod 6. Then (5.8) implies that -a - b = c, a = d and (5.9) implies that -c - d = c - a, c = d - b. It follows that a = b = c = d = 0 and hence [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Suppose [gamma] [equivalent] 3 mod 6. Then (5.8) implies that -a = c, -b = d and (5.9) implies that - c = c - a, - d = d - b. It follows that a = b = c = d = 0 and hence [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Suppose [gamma] [equivalent] 4 mod 6. Then (5.8) implies that b = c, -a - b = d and (5.9) implies that d = c - a, - c - d = d - b. It follows that a = b = c = d = 0 and hence [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Suppose [gamma] [equivalent] 5 mod 6. Then (5.8) implies that a + b = c, -a = d and (5.9) implies that c + d = c - a, -c = d - b. It follows that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Therefore, N(f) = [absolute value of ([gamma] - 1)] if [gamma] [equivalent] 0, 2, 3, 4 mod 6.

If [gamma] [equivalent] 1 mod 6. Then we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

It is easy to see that det(I - B), det(I - [[theta].sup.i]B) for i = 1, ..., 5 are either all nonnegative or all non-positive. It is straightforward to show that (3.1) yields

N(f) = [absolute value of ([gamma] - 1)] x (1 + [a.sup.2] + [b.sup.2] + ab).

If [gamma] [equivalent] 5 mod 6. Then we have

det(I - B) = 1 - [a.sub.2] - [b.sup.2] - ab = det(I - [theta]B) = det(I - [[theta].sup.2]B) = det(I - [[theta].sup.3]B) = det(I - [[theta].sup.4]B) = det(I - [[theta].sup.5]B).

It is straightforward to show that (3.1) yields

N(f) = [absolute value of ([gamma] - 1)] x [absolute value of (1 - [a.sup.2] - [b.sup.2] - ab)].

5.5 Case 9

Every endomorphism is of the form (5.1). Thus, the relation [phi](t[alpha][t.sup.-1]) = [phi]([alpha]) yields

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (5.10)

The relation [phi](t[beta][t.sup.-1]) = [phi]([[beta].sup.-1]) yields

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (5.11)

When [gamma] is even, (5.10) implies that s is even or a = 0. Similarly, (5.11) implies that d = 0 and also s is odd or a = 0. Now, if s is even then c = 0 and if s is odd then a = 0. Thus, these relations yield that [phi] has one of the following form:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

When [gamma] is odd, similar calculations show that [phi] has one of the following form:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

when d is even and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

when d is odd.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Thus, if [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] then (2.2) yields N(f) = [absolute value of ([gamma] - 1)].

If [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] then [absolute value of ([gamma] - 1)] is even, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Now,

(2.2) yields

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (5.12)

In fact, the Nielsen number is given by (5.1) for the following types of endomorphisms:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Finally, for the type [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] odd even with [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

(2.2) yields

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

5.6 Case 10

Every endomorphism is of the form (5.1). Thus, the relation [phi](t[alpha][t.sup.-1]) = [phi]([alpha]) yields

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (5.13)

This implies that s is even or a = 0.

The relation [phi](t[beta][t.sup.-1]) = [phi]([alpha][[beta].sup.-1]) yields

[[alpha].sup.r][[beta].sup.s][t.sup.[gamma]][[alpha].sup.c][[beta].sup.d][t.sup.-[gamma]][[beta].sup.-s] [[alpha].sup.-r] = [[alpha].sup.a][[beta].sup.-d][[alpha].sup.-c]. (5.14)

The relation [phi]([beta][alpha][[beta].sup.-1]) = [phi]([[alpha].sup.-1]) yields

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

This implies that d is odd or a = 0.

Straightforward calculations similar to those in Case 9 show that an endomorphism of G is of one of the following types:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

when s is even or

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

when s is odd.

If [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] then d is even, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. In fact, for all non-negative integer i, we have [[theta].sup.i] B = [[theta].sup.i+2]B. It follows from (2.2) that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (5.15)

Similarly, if [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] then d is even, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

For all non-negative integer i, we have [[theta].sup.i]B = [[theta].sup.i+2]B. Thus, the Nielsen number N(f) is given by (5.15).

If

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

then [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] such that N(f) = [absolute value of ([gamma] - 1)].

Finally, for type

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], ...

such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Since [gamma] is odd, [absolute value of ([gamma] - 1)] is even. Since d is odd, it follows from (2.2) that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

6 Nielsen numbers of arbitrary selfmaps: Remaining Cases 1, 7, 8, and 6

In this section, we compute N(f) for arbitrary selfmaps f on flat 3-manifolds in the four remaining cases. Case 1 is well-known. For case 7 and 8, the flat manifold is a S1-bundle over the torus [T.sup.2] and every self-map is fiber-preserving since the subgroup corresponding to [S.sup.1] is fully-invariant. Moreover, the formula (2.2) is also valid in these situations and therefore can be used to compute N(f). For case 6, we shall use (3.1) for the computation of the Nielsen number.

6.1 Case 1

The corresponding flat manifold is the 3-torus [T.sup.3] with fundamental group [Z.sub.3]. Given a selfmap f inducing an endomorphism [phi] on fundamental group, it is well-known that N(f) = 0 if det(I - [phi]) = 0 and N(f) = [absolute value of (det(I - [phi]))] otherwise.

6.2 Case 7

In this case, G has the following presentation

G = <[alpha], [beta], t | [beta][alpha][[beta].sup.-1] = [[alpha].sup.-1], t[alpha][t.sup.-1] = [alpha], t[beta][t.sup.-1] = [beta]>.

Let [phi] be an endomorphism given by the following 3 x 3 array

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where the columns are the images under [phi] of the generators [alpha], [beta], t. The relation [phi]([beta][alpha][[beta].sup.-1]) = [phi]([[alpha].sup.-1]) yields

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (6.1)

This implies that d is odd or a = 0.

The relation [phi](t[beta][t.sup.-1]) = [phi]([beta]) yields

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (6.2)

Suppose d is even so a = 0. Moreover, [[beta].sup.d] commutes with [alpha] so (6.2) becomes

[[alpha].sup.r][[beta].sup.s][[alpha].sup.c][[beta].sup.-s][[alpha].sup.-r] = [[alpha].sup.c]

This implies that s is even or c = 0. Thus, when d is even, we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Suppose d is odd. Then the relation [phi]([beta][alpha][[beta].sup.-1]) = [phi]([alpha]) yields

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (6.3)

This implies that s is even or a = 0. Now, (6.2) becomes

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Now d is odd, so we have 2r + [(-1).sup.s]c = c. It follows that if s is even then r = 0 and if s is odd then r = c.

Thus, when d is odd, we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

For the cases (i), (ii), (iv), the Nielsen number is N(f) = [absolute value of ((1 - d)(1 - [gamma]) - [[delta].sub.s])]. For case (iii), since d is odd and s is even, [absolute value of ((1 - d)(1 - [gamma]) - [[delta].sub.s])], which is the Nielsen number of the map [bar.f] on the base [T.sup.2], must be even. Since the base torus has fundamental group generated by [beta] and t whereas the fiber [S.sup.1] has fundamental group generated by [alpha], the action of [[pi].sub.1]([T.sup.2]) on the fiber is induced by the relation [beta][alpha][[beta].sup.-1] = It follows that we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

6.3 Case 8

In this case, G has the following presentation

G = <[alpha], [beta], t | [beta][alpha][[beta].sup.-1] = [[alpha].sup.-1], t[alpha][t.sup.-1] = [alpha], t[beta][t.sup.-1] = [alpha][beta]>.

Calculations similar to those in Case 7 show that any endomorphism is of one of the following types:

When d is even, we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

When d is odd, we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

For the cases (i), (ii), (iii), the Nielsen number is N(f) = [absolute value of ((1 - d)(1 - [gamma]) - [[delta].sub.s])].

For case (iv), similar arguments as in Case 7 show that

N(f) = ([absolute value of (l - 2r - [gamma])] + [absolute value of (l + 2r + [gamma])]) x [absolute value of ((1 - d) (l - [gamma]) - [[delta].sub.s])]/2.

6.4 Case 6

In this final case, we make use of the calculations already done in subsection 4.1. For any endomorphism [phi], the restriction [phi]' on the maximal abelian subgroup is of one of the six forms as in (4.1) or [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. For this latter type of endomorphisms, N(f) = 1. We now compute the Nielsen number of a selfmap which induces an endomorphism p given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where the columns are the images under [phi] of the generators [t.sub.1], [[alpha].sub.2], [t.sub.3]. We will make use of the restriction [phi]' of [phi] to the maximal abelian subgroup and [phi]' can be represented by a 3 x 3 matrix where the columns are images under [phi]' of the generators [[alpha].sub.1], [[alpha].sub.2], [[alpha].sub.3].

Suppose [phi]' is of type (II) and (III'), that is, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. It follows that

det(I - [phi]') = (2 + 2s + 2b) (1 - r[epsilon]), det(I - [[theta].sub.1][phi]') = (2s + 2b)(-1 - r[epsilon]), det(I - [[theta].sub.2][phi]') = (2s + 2b)(-1 - r[epsilon]), det(I - [[theta].sub.3][phi]') = (2 + 2s + 2b)(1 - r[epsilon]).

It follows that

N(f) = 1/4 (4 [absolute value of (l + s + b)][absolute value of (l - r[epsilon])] + 4 [absolute value of (s + b)] [absolute value of (1 + r[epsilon])]).

Suppose [phi]' is of type (II) and (IV'), that is, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. It follows that

det(I - [phi]') = 1 - (a - r)[epsilon](2s + 1) = det(I - [[theta].sub.1][phi]') = det(I - [[theta].sub.2][phi]') = det(I - [[theta].sub.3][phi]').

It follows that

N(f) = [absolute value of (1 - (a - r)[epsilon](2s + 1))].

Suppose [phi]' is of type (III) and (II'), that is, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. It follows that

det(I - [phi]') = (1 - a)(1 - t)(-2(s + b)), det(I - [[theta].sub.1][phi]) = (1 - a)(1 + t)(2 + 2(s + b)), det(I - [[theta].sub.2][phi]) = (1 + a)(1 - t)(2 + 2(s + b)), det(I - [[theta].sub.3][phi]) = (1 + a)(1 + t)(-2(s + b)).

It follows that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Suppose p is of type (III) and (IV'), that is, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. It follows that

det(I - [phi]') = (1 - a)(1 - (2s + 1)(t - [epsilon])), det(I - [[theta].sub.1][phi]') = (1 - a)(1 - (2s + 1)(t - [epsilon])), det(I - [[theta].sub.2][phi]') = (1 + a)(1 + (2s + 1)(t - [epsilon])), det(I - [[theta].sub.3][phi]') = (1 + a)(1 + (2s + 1)(t - [epsilon])).

It follows that

N(f) = 1/4 (2 [absolute value of ((l - a)(l - (2s + l)(t - [epsilon])))] + 2 [absolute value of ((1 + a)(l + (2s + l)(t - [epsilon])))]).

Suppose [phi]' is of type (IV) and (II'), that is, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. It follows that

det(I - [phi]') = (1 - (a - r)(2b + 1))(1 - t), det(I - [[theta].sub.1][phi]') = (1 + (a - r)(2b + 1))(1 + t), det(I - [[theta].sub.2][phi]') = (1 - (a - r)(2b + 1))(1 - t), det(I - [[theta].sub.3][phi]') = (1 + (a - r)(2b + 1))(1 + t).

It follows that

N(f) = 1/4 (2 [absolute value of ((l - (a - r)(2b + 1))(1 - t))] + 2 [absolute value of ((1 + (a - r)(2b + 1))(1 + t))]).

Suppose [phi]' is of type (IV) and (III'), that is, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. It follows that

det(I - [phi]')= 1 - r(2b + 1)(t - [epsilon]) = det(I - [[theta].sub.1][phi]') = det(I - [[theta].sub.2][phi]') = det(I - [[theta].sub.3][phi]').

It follows that

N(f) = [absolute value of (1 - r(2b + 1)(t - [epsilon]))].

7 Jiang-type condition

Recall that a space M is of Jiang-type or M satisfies the Jiang-type condition, if for any selfmap f : M [right arrow] M, either L(f) = 0 [??] N(f) = 0 or L(f) [not equal to] 0 [??] N(f) = R(f). Here, L(f), N(f), R(f) denote the Lefschetz, Nielsen, and Reidemeister numbers of f respectively. A group G is said to have property [R.sub.[infinity]] if for all [phi] [member of] Aut(G), R([phi]) = [infinity].

In [5], flat and nilmanifolds whose fundamental groups possess property [R.sub.[infinity]] were constructed. In particular, it was shown that for any n [greater than or equal to] 5, there is a compact nilmanifold of dimension n such that every homeomorphism is isotopic to a fixed point free homeomorphism. This is due to the fact that nilmanifolds are known to be of Jiang-type and by constructing finitely generated nilpotent groups with [R.sub.[infinity]] property, such a nilmanifold has the property that every self homeomorphism f must have N(f) = 0. It is therefore natural to ask whether there exists manifold M that is not of Jiang-type but N(f) = 0 for every self homeomorphism f (see Remark 7.1). In this section, we determine which of the flat 3-manifolds are of Jiang-type.

For Case 1, the 3-torus, it is well-known that the Jiang type condition is satisfied.

For Case 2, the flat manifold is a torus bundle over [S.sup.1]. Consider the fiberwise homeomorphism which induces on the fundamental group of the base the homomorphism given by multiplication by -1 and on the fundamental group of the fiber the automorphism given by the matrix

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

The Lefschetz number of this map restricted to one fiber has value -2 but the Lefschetz number restricted to the other fiber, by routine calculation, is 6. Therefore the indices of the Nielsen classes have different values, i. e., 2 classes have index -1 and 6 classes have index +1. Now, consider a homeomorphism which induces on the fundamental group of the base the homomorphism given by multiplication by -1 and on the fundamental group of the fiber the automorphism given by the matrix

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

The Lefschetz number of this map restricted to one fiber is 0 but the Lefschetz number restricted to the other fiber, by routine calculation, is 4. This implies that the Nielsen number is 4 but the Reidemeister number is infinite. Therefore the Jiang type condition does not hold.

For Cases 3-5, none of these manifolds is of Jiang type. For Case 3 (section 5.2), consider the map inducing [gamma] [equivalent] 1 mod 3 with a = 1 and b = 0 so that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. In this case, det(I - B) = 0 so that R(f) = [infinity]. For Case 4 (section 5.3), consider the map inducing [gamma] [equivalent] 3 mod 4 with a = 0 and b = 1 so that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. In this case, det(I - [[theta].sup.3]B) = 0 so that R(f) = [infinity]. For Case 5 (section 5.4), consider the map inducing [gamma] [equivalent] 1 mod 6 with a = 1 and b = 0 so that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. In this case, det(I - B) = 0 so that R(f) = [infinity]. Thus, we conclude that the Jiang type condition does not hold in general in any of these three cases.

For the remaining Cases 6-10, each of these flat manifolds is not of Jiang-type. For Case 6, one can choose a self-homeomorphism (see section 4.1 and section 6.4) of type (II), (III') with r = [epsilon] = 1, s = 0, b = -1so that N(f) = 2 = [absolute value of (L(f))] but R(f) = [infinity]. For Cases 7-8 (see sections 4.2-4.3), there exist homeomorphisms f so that N(f) = [absolute value of (L(f))] [not equal to] 0 but R(f) = [infinity]. Similarly for Cases 9-10, see sections 4.4-4.5.

For convenience, we summarize our results in the following table:
G            NSH(M)   Jiang Type

1     N [union] {0}          Yes
2    2N [union] {0}           No
3               {0}           No
4               {0}           No
5               {0}           No
6             {0,2}           No
7    2N [union] {0}           No
8           2N  {0}           No
9             {0,2}           No
10            {0,2}           No


Remark 7.1. Based upon our calculations, the flat manifolds in Cases 3-5 have the property that they are not of Jiang-type but every self-homeomorphism has zero Nielsen number while N(f) = [absolute value of (L(f))] (see e. g. [9, 10]) and their fundamental groups have property [R.sub.[infinity]] (see [8]).

References

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[3] L. Charlap, "Bieberbach Groups and Flat Manifolds," Springer, New Yor[alpha], 1986.

[4] H. S. M. Coxeter and W. O. J. Moser, "Generators and relations for discrete groups," Ergebnisse der Mathematik und ihrer Grenzgebiete Fourth edition, Springer-Verlag, Berlin, 14 (1980).

[5] D. Goncalves and P. Wong, Twisted conjugacy classes in nilpotent groups, J. Reine Angew. Math. 633 (2009), 11-27.

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[8] K. Dekimpe and P. Penninckx, The finiteness of the Reidemeister number of morphisms between almost-crystallographic groups. J. Fixed Point Theory Appl. 9 (2011), 257-283.

[9] K. Dekimpe, B. De Roc[alpha], and P. Penninckx, The Anosov theorem for infranil-manifolds with a 2-perfect holonomy group. Asian J. Math. 15 (2011), 539548.

[10] K. Dekimpe, B. De Roc[alpha], and W. Malfait, The Anosov relation for Nielsen numbers of maps of infra-nilmanifolds. Monatsh. Math. 150 (2007), 1-10.

[11] N. Ivanov, Nielsen numbers of self-maps of surfaces, J. Sov. Math. 26 (1984), 1636-1641.

[12] B. Jiang, "Lectures on Nielsen Fixed Point Theory," Contemp. Math. v. 14, Amer. Math. Soc., 1983

[13] B. Jiang, Fixed points of surface homeomorphisms, Bull. (New Series) Amer. Math. Soc. 5 (1981), 176-178.

[14] B. Jiang and J. Guo, Fixed points of surface diffeomorphisms, Pacific J. Math. 160 (1993), 67-89.

[15] B. Jiang and S. Wang, Lefschetz numbers and Nielsen numbers for homeomorphisms on aspherical manifolds, Topology - Hawaii (K. H. Dovermann, ed.), World Sci. Publ. Co., Singapore, 1992, pp. 119-136.

[16] B. Jiang, S. Wang, and Y. -Q. Wu, Homeomorphisms of 3-manifolds and the realization of Nielsen number, Comm. Anal. Geom. 9 (2001), 825-878.

[17] M. Kelly, The Nielsen number as an isotopy invariant, Topology and its Applications 62 (1995), Pages 127-143.

[18] M. Kelly, Computing Nielsen numbers of surface homeomorphisms, Topology 35 (1996), 13-25.

[19] M. Kelly, Nielsen numbers and homeomorphisms of geometric 3-manifolds, Topology Proceedings 19 (1994), 149-160.

[20] S. W. Kim, J. B. Lee, and K. B. Lee, Averaging formula for Nielsen numbers, Nagoya Math. J. 178 (2005), 37-53.

[21] R. Lyndon, "Groups and Geometry," LMS Lecture Note Series 101, Cambridge University Press, 1985 (reprinted with corrections 1986).

[22] J. Wol[beta], "Spaces of Constant Curvature,", Publish or Perish, Inc., Berkeley, 1977.

Dept. de Matematica-IME--USP, Caixa Postal 66.281--CEP 05314-970, Sao Paulo--SP, Brasil; FAX: 55-11-30916183 email:dlgoncal@ime.usp.br

Department of Mathematics, Bates College, Lewiston, ME 04240, U.S.A.; FAX: 1-207-7868331 email:pwong@bates.edu

Institute of mathematics and interdisciplinary science, Capital Normal University, Beijing 100048, China; FAX: 86-10-68900950 email:zhaoxve@mail.cnu.edu.cn

* This work was initiated during the first and second authors' visit to Capital Normal University during June 14-27, 2011. The first author was supported in part by Projeto Tematico Topologia Algebrica Geometrica e Differencial 2008/57607-6. The third author was supported in part by the NSF of China (10931005) and a project of Beijing Municipal Education Commission (PHR201106118).

Received by the editors in January 2013--In revised form in July 2013.

Communicated by Y. Felix.

2010 Mathematics Subject Classification : Primary: 55M20; Secondary: 20H15.
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