Nielsen ﬁxed point theory for partially ordered sets · Nielsen ﬁxed point theory for partially ordered sets Peter Wong Department of Mathematics, Bates College, Lewiston, ME

Topology and its Applications 110 (2001) 185–209

Nielsen fixed point theory for partially ordered sets

Peter Wong

Department of Mathematics, Bates College, Lewiston, ME 04240, USA

Received 12 December 1997; received in revised form 27 February 1999

Abstract

In this paper, we introduce a Nielsen type numberN∗(f,P) for any selfmapf of a partiallyordered setP of spaces. This Nielsen theory relates to various existing Nielsen type fixed pointtheories for different settings such as maps of pairs of spaces, maps of triads, fibre preserving maps,equivariant maps and iterates of maps, by exploring their underlying poset structures. 2001 ElsevierScience B.V. All rights reserved.

Keywords:Nielsen fixed point theory; Relative maps; Fibre-preserving maps; Equivariant maps;Iterated maps; Maps of triads

AMS classification:Primary 55M20, Secondary 58F20; 57S99

1. Introduction

The celebrated Lefschetz–Hopf fixed point theorem asserts that the nonvanishing ofthe Lefschetz numberL(f ) of a selfmapf :X→X on a compact connected polyhedronguarantees the existence of a fixed point for every mapg homotopic tof . The converse,however, does not hold true in general. The Nielsen numberN(f ) of f , a more subtlehomotopy invariant, measures thesizeof the fixed point set in the sense thatN(f ) 6MF[f ] =min{# Fixg | g ∼ f }. A classical result of Wecken states thatN(f ) is indeed asharp lower bound in the homotopy class off for many spaces, in particular, for compactmanifolds of dimension at least 3. Thus, in this case, the vanishing ofN(f ) is sufficient todeform a mapf to be fixed point free. Among the central problems in Nielsen fixed pointtheory (or the theory of fixed point classes) are (1) the computation ofN(f ) and (2) therealization ofN(f ), i.e., whenN(f )=MF[f ].

E-mail address:[email protected] (P. Wong).

0166-8641/01/$ – see front matter 2001 Elsevier Science B.V. All rights reserved.PII: S0166-8641(99)00182-0

186 P. Wong / Topology and its Applications 110 (2001) 185–209

In an attempt to compute the Nielsen number, Brown [5] employed fibrewise techniquesto obtain, under certain conditions, a product formulaN(f ) = N(fb) · N(f ) for a fibre-preserving mapf , in terms of possibly simpler Nielsen numbersN(fb) andN(f ) of thefibre (overb) and of the base, respectively. Since then, many researchers have improvedand applied the Nielsen theory for fibre-preserving maps (see, e.g., [9,14–16,29,43]). Forapplications, new settings such as Nielsen theory for compact maps and that for compactlyfixed maps, introduced by Brown [6] and Fadell and Husseini [10], respectively, havealso emerged asNielsen type fixed point theories. While major advancements in classicalNielsen theory have been made, it was the advent of the relative Nielsen theory, introducedby Schirmer in 1986 [32], that led to many new Nielsen type theories and hence newNielsen type invariants, some of which are closely related. (See also the excellent surveyarticle [33] by Schirmer.) However, many of these invariants were introduced in an ad hocmanner for a particular setting.

In [32], Schirmer considered relative mapsf : (X,A)→ (X,A) of a compact polyhedralpair (X,A) and she introduced the so-called relative Nielsen number which is defined tobe

N(f ;X,A) :=N(f )−N(f,fA)+N(fA),wherefA = f |A :A→A andN(f,fA) denotes the number of essential fixed point classesof f that contain an essential fixed point class offA. It was shown thatN(f ;X,A) hasthe same properties as the classical Nielsen number. In addition to the usual hypotheseson X and onA, if A can beby-passedin X, thenN(f ;X,A) can be realized in therelative homotopy class off . It turns out that this by-passing condition is equivalent tothe codimension hypothesis required to perform stepwise deformation in [12]. Similar by-passing hypotheses will be used to ensure that the Nielsen type numbers for posets areindependent of the choice of the admissible ordering (see Theorems 4.1 and 5.1). Whilethe computation of the relative Nielsen number is certainly more difficult than computingthe ordinary one, the definition ofN(f ;X,A) has the same geometric flavor as the classicalone in the sense that it is defined in terms ofessentialfixed point classes only.

Subsequent work of Schirmer and of others further investigated the relative Nielsentheory and introduced new Nielsen type numbers for new settings (see [7,19–21,34,44–48]). One of them is the Nielsen number of a triad [35]. Letf : (X,A1,A2)→ (X,A1,A2)

be a map of a triad consisting of a compact polyhedronX and two subpolyhedraA1 andA2 with X =A1∪A2. The Nielsen type number off [35] is defined to be

N(f ;A1∪A2) :=N(f1;A1,A0)+N(f2;A2,A0)−N(f0)− IJ(f1, f2),

wherefi : Ai → Ai, i = 0,1,2, A0 = A1 ∩ A2 and IJ(f1, f2) denotes the number ofinessentially joined pairs of essential fixed point classes off1 andf2. The computation ofthis number has been carried out in [35] in the case whereX is the double of a manifold,the suspension of a polyhedron, a manifold with a handle attached and the connected sumof two manifolds. This Nielsen type number, like the relative Nielsen number, is basedupon the inclusion–exclusion principle. One can see immediately that the situation fortriads is much more complicated than that in the relative case. Moreover, the definition of

P. Wong / Topology and its Applications 110 (2001) 185–209 187

N(f ;A1 ∪A2) involves the termIJ(f1, f2) which is defined using lifts together with thework of Zhao [44] since inessential (therefore possibly empty) classes must be taken intoconsideration (for the definition, see [35] or Section 4).

Relative Nielsen theory was employed in [14] to describe the relationship among theNielsen numbersN(f ),N(fb) and N(f ) for a fibre-preserving mapf :p → p of afibrationp with base mapf and fibre map (overb) fb. Thus, this relates the fibrewiseNielsen theory initiated in [5] to the relative theory. In fact, the Nielsen type number forfibre preserving mapsNF (f,p) defined in [14] and in [16] can be regarded as a relativeNielsen number.

Techniques from relative Nielsen theory are also useful in the Nielsen theory forequivariant maps (see [37,38]) and for iterated maps (see [37,40]). For aG-spaceX, whereG is a compact Lie group, theG-action induces a natural stratification onX according tothe isotropy types onX. Therefore, a natural partial ordering on these subspaces is given sothat everyG-map onX is automatically an order preserving map on this partially orderedset (see Section 6). In the periodic point case, for any fixed positive integern, one canregard then-periodic points off as the fixed points of aG-mapgf under theG-action onthen-fold product ofX whereG= Zn.

The purpose of this paper is to give a systematic study of the inter-relations among someof the existing Nielsen type fixed point theories and their respective Nielsen type invariants.We propose a combinatorial approach to Nielsen theory by exploring the underlying posetstructure for these settings. More precisely, we introduce a Nielsen type theory for anorder preserving mapf on a partially ordered setP and we define a Nielsen type numberN(f,P). Under certain conditions, we show that this number coincides with some of theexisting ones in the Nielsen theory for relative maps, fibre-preserving maps and maps oftriads for appropriate partially ordered setsP . Moreover, we introduce new Nielsen typeinvariants forG-orbits ofG-maps and for periodic orbits of maps of a fixed period. Likethe relative Nielsen number of Schirmer, this Nielsen type number is defined geometricallyin terms ofessentialfixed point classes only.

This paper is organized as follows. In Section 2, we review the basic properties of theclassical Nielsen number and some terminology from combinatorics. For any admissibleordering of a partially ordered set of spaces, we introduce in Section 3 a Nielsen typenumberN(f,P;4) which has the usual properties of the Nielsen number. The relationshipbetweenN(f,P;4) and the Nielsen number of a triad is discussed in Section 4. For certainposetsP , we give conditions in Section 5 for whichN(f,P;4) is independent of thechoice of the admissible ordering, in which case, we writeN(f,P) instead. To furtherincorporate equivariant Nielsen theory into this unified Nielsen theory, we generalizethe definition ofN(f,P;4). In Section 6, we show that the fibrewise Nielsen typenumber is a special case of this Nielsen type number for appropriateP . We show thatN∗(f,P) :=max{N(f,P;4)} yields an equivariant Nielsen type number forG-orbits offixed points for a suitable partially ordered setP and hence a new Nielsen type invariantfor periodic orbits. In Section 7, we discuss the converse of the Lefschetz Theorem inthe context of selfmaps of posets. We end this paper with some concluding remarks inSection 8.


2. Nielsen numbers and posets

In this section, we review the basic properties of the classical Nielsen number (see [3,22]and [27] for details) and some basic notions from combinatorics (see [36]).

Let X be a compact ANR (absolute neighborhood retract) andf :X→ X a selfmap.There exists a unique integer valuedfixed point indexsatisfying the homotopy, additivityand commutativity axioms (see [3, Ch. 5]). Let Fixf = {x | f (x)= x} be the fixed point setof f . Suppose Fixf 6= ∅. Two fixed pointsx andy are said to belong to the samefixed pointclass(x ∼ y) if, and only if, there exists a pathα : [0,1]→X such thatα(0)= x, α(1)= yandα is homotopic tof ◦ α (write α ∼ f ◦ α) relative to the endpoints. A fixed pointclassF is essentialif the fixed point index, denoted byi(f,F), is nonzero. Denote byF(f )= Fixf/∼ the weighted set of (nonempty) fixed point classes (equivalence classes)of f with weight being the fixed point index. TheNielsen numberof f is defined to be thenumber of essential fixed point classes off and is denoted byN(f ).

Nielsen’s original description of fixed point classes is via the covering space approach.SupposeX is connected. Denote byX the universal cover ofX and byp : X→ X itscovering map. For any liftf : X→ X of f (i.e.,f ◦ p = p ◦ f ), p(Fix f )⊂ Fixf . Thenthe fixed point set off can be partitioned intofixed point classes(possibly empty) ofthe formp(Fix f ), i.e., Fixf =⋃f p(Fix f ). We denote byFPC(f ) the weighted set offixed point classes off (see [22]) and it is easy to see that there is an index preservinginjection F(f ) 7→ FPC(f ). To distinguishemptyfixed point classes, one must use thebijection betweenFPC(f ) and the weighted set of lifting classes (see [22, p. 44]) off ,i.e., conjugacy classes (in the group of deck transformations) of lifts off to the universalcover. In the case whereX is not connected,FPC(f )=⊔c FPC(f |Xc) where the disjointunion ranges over the connected componentsXc of X.

We now list the basic properties of the classical Nielsen numberN(f ).

Lower Bound. For anyf :X→X, N(f )6 # Fixf where#Z denotes the cardinality ofthe setZ.

Homotopy. If f :X→X andg :X→X are homotopic, thenN(f )=N(g).

Commutativity. Let f :X→ Y and g :Y → X be maps between compact ANRs. ThenN(f ◦ g)=N(g ◦ f ).

Homotopy Type. Given the following homotopy commutative diagram

Xf

h

X

h

Yg

Y

whereh :X→ Y is a homotopy equivalence between two compact ANRs. ThenN(f ) =N(g).


Under fairly weak hypotheses, the Nielsen number can be realized, as the followingtheorem indicates.

Theorem 2.1 (Minimum Theorem [23]).Suppose thatX is a connected finite polyhedronwith no local separating points and is not a surface of negative Euler characteristic, thenfor any selfmapf :X→ X, there exists a mapg :X → X homotopic tof such thatN(f )=N(g)= # Fixg.

We now turn to combinatorics from which we will recall the notion of partially orderedsets and Möbius inversion. Apartially ordered setor simplyposetis a setP together witha relation6 satisfying

(i) x 6 x,∀x ∈P ;(ii) x 6 y andy 6 x ⇒ x = y;(iii) x 6 y andy 6 z⇒ x 6 z.

A chainis a poset in which for any two elementsx andy, eitherx 6 y or y 6 x. LetC bea chain of a posetP . If C is finite, then thelengthof C is defined to be(C)= #C− 1 andtherank of a finite posetP is given by

`(P)=max{`(D) |D is a chain andD is a subposet ofP

}.

A principal order idealgenerated byx is defined by〈x〉 = {y ∈ P | y 6 x}. A posetPis said to have a0 if there is an element0 ∈ P such that06 x, ∀x ∈ P . Similarly, Phas a1 if 1∈ P andx 6 1, ∀x ∈ P . Wheneverx 6 y, the interval [x, y] is defined to bethe set[x, y] = {z ∈ P | x 6 z 6 y}. The posetP is calledlocally finite if every intervalis finite. Suppose thatx andy belong to a posetP . A least upper bound(greatest lowerbound, respectively) ofx andy is an upper bound (lower bound, respectively)z, i.e.,x 6 zandy 6 z (z 6 x andz 6 y, respectively), such that every upper (lower) boundw of xandy satisfiesz6 w (w 6 z, respectively). Alattice is a posetL for which every pair ofelements has a least upper bound and a greatest lower bound. If a least upper bound ofx

andy exists, then it is unique and is denoted byx ∨ y. Similarly, we letx ∧ y denote thegreatest lower bound ofx andy if it exists.

Example 2.2. Let (X,A) be a pair whereX is a compact connected ANR andA =⋃k−1i=1 Ai hask − 1 connected components each of which is a compact ANR. We form

a posetP = {X,A1, . . . ,Ak−1} with 6 induced by inclusion and the rank`(P) is equal to1. Note thatP =P ∪ {∅} is a lattice with0= ∅ and1=X.

Example 2.3. Consider a triad(X,A1,A2) whereX,A1 andA2 are compact ANRs sothatA0 = A1 ∩ A2 andAi ⊂ X for i = 0,1,2. Then the posetP = {X,A1,A2,A0} is alattice under inclusion with0=A0 and1=X. Moreover, (P)= 2.

Example 2.4. Letp :E→ B be a Hurewicz fibration such thatE,B andF , a typical fibre,are connected compact ANRs. Letξ = {b1, . . . , bk} ⊂ B, Fi = p−1(bi) for i = 1, . . . , kandFξ =⋃k

i=1Fi . Then(E,Fξ ) can be given a poset structure as in Example 2.2.


Example 2.5. Given a finite system of fibrations

E1p1

B1=E2p2 . . . pk−1

Bk−1=Ek pk ∗

F1 F2 . . . Fk−1

whereFi,Ei andBi are connected compact ANRs. For any 16 i < j 6 k−1, we have thatpj ◦ · · · ◦pi :Ei→Bj is a fibration and the set{Ei,p−1

i (p−1i+1(· · ·p−1

j (ξj ) · · ·))} is a poset

under inclusion with1=Ei for anyξj = {xj1, . . . , xjk(j)}, xj` ∈Bj for `= 1, . . . , k(j). Itis easy to see that every one of these posets has rank 1.

Such examples occur, for instance, in fibering a nilmanifold [9]. That is,E1 is anilmanifold (homogeneous space of a nilpotent Lie group); theBi ’s are nilmanifolds andtheFi ’s are tori.

Example 2.6. Let G be a topological group. For any closed subgroupH of G, denoteby (H) the conjugacy class ofH in G. The setC(G) consisting of conjugacy classesof closed subgroups ofG can be given a partial ordering as follows. For(H), (K) ∈C(G), (H) 6 (K) if, and only if, H is conjugate inG to a subgroup ofK. The posetC(G) has0= (1) and1= (G) where 1 is the trivial subgroup. LetX be aG-space, i.e.,there exists a continuous functionΦ :G × X→ X such thatΦ(1, x) = x, ∀x ∈ X andΦ(g1,Φ(g2, x))= Φ(g1g2, x), ∀x ∈ X andg1, g2 ∈G. For any closed subgroupH , welet

XH = {x ∈X | hx(=Φ(h,x))= x, ∀h ∈H},

XH ={x ∈X |Gx =H

}and X(H) = {x ∈X | (Gx)> (H)}

whereGx = {g ∈G | gx = x} is the isotropy subgroup atx. The collection{XH | (H) ∈C(G)} forms a poset under inclusion with1= X(1). If XG is nonempty then{XH } is alattice with0=XG. Moreover, the partial ordering onC(G) induces a partial ordering onthe collection{X(H)}, i.e., if (H)6 (K) thenX(K) ⊂X(H).

Example 2.7. Let X be a compact ANR andn be a fixed positive integer. Denote byYnthen-fold product ofX. Then the cyclic group of ordern, denoted byZn, acts onYn via(x1, . . . , xn) 7→ (xn, x1, . . . , xn−1). For anym|n, Yn/m can be embedded intoYn by

(x1, . . . , xn/m) 7→ (

m copies︷︸︸︷x1, . . . , x1, . . . ,

m copies︷︸︸︷xn/m, . . . , xn/m).

Note thatYn/m coincides with the fixed point setYZmn . Moreover, since cyclic groups

are Abelian, each subgroup inZn has itself as its conjugacy class so thatYZmn = Y (Zm)n .

Thus, Example 2.6 yields a latticeL = {Yn/m|m|n} of subspaces ofYn partially orderedby inclusion with0= Y1 and1= Yn. If n is prime, then (L)= 1. Writen=∏k

i=1pαii in

the primary factorization ofn wherepi ’s are distinct primes andαi ’s are positive integers.Then`(L) is equal tok, the number of distinct prime divisors ofn.


LetP be a locally finite poset. TheMöbius functionµP onP is defined recursively by(1) µP (x, x)= 1, ∀x ∈ P ;(2) µP (x, y)=−

∑x6z6y µP (x, z), ∀x 6 y.

If, in addition, every principal order ideal inP is finite then we have the followingformula (see [36, 3.7.1]).

Möbius Inversion Formula. Let f,g :P→C. Then

g(x)=∑y6x

f (y) ∀x ∈P⇔ f (x)=∑y6x

µP (y, x)g(y) ∀x ∈P .

Example 2.8. Let P = N partially ordered by divisibility, i.e.,a 6 b if, and only if, a|b.The Möbius function is given by

µN(a, b)= (−1)t if b/a is a product oft distinct primes,

0 otherwise

andµN(a, b)= µ(b/a) the classical number theory Möbius function.For anyd ∈N, ϕ(d)= #{j ∈N | 16 j 6 d and(j, d)= 1}. The functionϕ is the Euler

totient function. Thus, for anyn ∈N, n=∑d |n ϕ(d) and by the Möbius Inversion Formula,we have

ϕ(n)=∑d |nµ(n/d)d.

3. Nielsen type numbers for posets

While the classical Nielsen number is defined to be the number of essential fixedpoint classes, the relative Nielsen number [32] is defined combinatorially, based upon aninclusion–exclusion principle. In this section, we introduce Nielsen type numbers definedcombinatorially, in the same spirit as in the relative Nielsen number.

Let P be a finite poset consisting of a compact ANRX and a collection of subspacesA1, . . . ,Ak−1 each of which is a compact ANR. We put a partial ordering onP such thatAj 6 Ai if, and only if, Aj ⊆ Ai . In addition, we letAk = X and assume that for any16 i, j 6 k, if Ai ∩Aj 6= ∅ thenAi ∩Aj ∈ P . Thus,P has a1=X and a0=⋂k

i=1Ai (ifit is nonempty). We can relabel the indices such thatAj 6 Ai ⇒ j 6 i. Such an orderingA1, . . . ,Ak−1 is called anadmissible ordering. By a mapf :P→ P ′ between two suchfinite posets we shall mean an order preserving mapf :X→ Y such thatfi = f |Ai :Ai→Bi for all i = 1, . . . , k whereP = {X,A1, . . . ,Ak−1} andP ′ = {Y,B1, . . . ,Bk−1}. LetNP(k) be the category of such finite posetsP and (partially) order preserving mapsf :P→ P ′. A homotopyin this category is an order preserving mapH :P × [0,1]→ P ′such thatH |X× [0,1] andH |Ai × [0,1] are ordinary homotopies. Note thatH is a mapin NP(k) sinceP × [0,1] can be given the partial ordering induced by that onP , i.e.,


A× [0,1]6 B × [0,1] if, and only if,A 6 B in P . For anyP ∈ NP(k) and a selfmapf :P→ P , we define the minimal number of fixed points in the homotopy class off to be

MFP [f ] =min{# Fixg | g ∼P f

}.

Given any P ∈ NP(k) and an admissible orderingA1, . . . ,Ak−1, we define for16 i 6 k,

Xi ={x ∈X | x ∈Aj, for somej 6 i

}=⋃j6i

Aj

so thatXk =Ak =X. Note that each of these subspaces is a compact ANR and we have afiltration of subspacesX1 ⊆ · · · ⊆Xk = X. For 16 i 6 k, we letf (i)= f |Xi :Xi→ Xi

and

N(f (i);X6i

) = #{F ∈ F(f (i)) | i(f (i),F) 6= 0 andF contains

an essentialG ∈ F(f (j)) for somej < i}.

For a given admissible ordering4 onA1, . . . ,Ak−1,Ak = X, theNielsen type numberof f :P→P is defined to be

N(f,P;4)=k∑i=1

[N(f (i))−N(f (i);X6i)

].

This number depends upon the choice of the admissible ordering as we illustrate in thefollowing

Example 3.1. Let A0 be the figure-eight space andA1 be obtained by attaching a 2-diskto the left-hand circle ofA0.

The subspaceA2 is obtained by attaching two 2-spheres at the pointsx and z,respectively, as shown in the following figure.


ConsiderX =A1 ∪A2 and the two admissible orderings: (41) A0,A1,A2,X and (42)A0,A2,A1,X.

Let f0 :A0→ A0 be the map as in Jiang’s example in [24] so thatf0 flips the left-hand circle and it takes the generatorsα 7→ α−1, β 7→ α−1β2. The mapf0 has three fixedpoints x, z and y of index+1,0 and−1 respectively. Note thatx and y are Nielsenequivalent as fixed points off0 which has two inessential fixed point classes. The mapf1 :A1→A1 is the obvious extension off0 by extending the flip to the 2-disk. Finally, themapf2 :A2→A2 extendsf0 by being the identity map on each of the two 2-spheres.

Note thatN(f0)= 0 sincef0 has no essential fixed point classes;N(f1)= 1 sinceA1

is homotopy equivalent to a circle andf1 sends the only generatorβ to β2. OnA2, z doesnot belong to the fixed point class containingx andy and these two classes are essentialso thatN(f2)= 2. Finally, the composite mapf :X→X of the triad hasN(f )= 1 sinceL(f ) 6= 0 andx andz are Nielsen equivalent as fixed points.

Then, for (41),

N(f (1)

)= 0; N(f (2)

)−N(f (2);X62)= 1;

N(f (3)

)−N(f (3);X63)= 0

and

N(f,P;41)= 1

whereas for (42), we have

N(f (1)

)= 0; N(f (2)

)−N(f (2);X62)= 2;

N(f (3)

)−N(f (3);X63)= 0

and

N(f,P;42)= 2.

Note that the homomorphismπ1(X − A1) → π1(X) induced by inclusion isnotsurjective, i.e.,A1 cannot be by-passed inX. The dependence of the choice of theadmissible ordering in this example lies in the fact that the two essential fixed point classesof f2 belong to thesameessential fixed point class off in X.

We shall see in Section 5 when this Nielsen number can be independent of the admissibleordering.

We now prove the basic properties ofN(f ), as listed in section two, forN(f,P;4).

Theorem 3.2 (Lower bound).LetP ∈NP(k). For anyf :P→P and for any admissibleordering4 onP , N(f,P;4)6 # Fixf .

Proof. Let X1⊆ · · · ⊆Xk =X be the filtration induced by the given admissible ordering4. Letm(i)= [N(f (i))−N(f (i);X6i )]. If m(i) > 0 then we let

Fi1, . . . ,Fim(i)


be the essential fixed point classes off (i) that do not contain any essential fixed pointclasses off (r) for any r < i. Moreover, we can choosem(i) fixed pointsxi1, . . . , xim(i) ,one from each of these classes. Forj > i them(j) fixed points{xjs } chosen in a similarway are distinct from anyxir ’s that have been chosen before. Thus,{xir | 16 i 6 k,16r 6m(i)} ⊆ Fixf and

N(f,P;4)=k∑i=1

m(i)6 # Fixf. 2Theorem 3.3 (Homotopy).Suppose thatg ∼P f :P → P . Then for any admissibleordering onP , N(f,P;4)=N(g,P;4).

Proof. Let H :P × [0,1] → P be the homotopy withH0 = f andH1 = g. Fix somei, 16 i 6 k. Suppose thatF ∈ F(f (i)) such thati(f (i),F) 6= 0, and for somej, Aj < Ai ,there exists aG ∈ F(f (j)) with i(f (j),G) 6= 0 andF containsG. ThenF is H|Xi -related(see [22, I.3.10]) to an essentialF′ ∈ F(g(i)) andG isH|Xj -related to an essentialG′ ∈ F(g(j)). Moreover,F′ containsG′. Hence,N(f (i);X6i ) = N(g(i);X6i ). Theassertion now follows from the homotopy invariance of the ordinary Nielsen number.2Theorem 3.4 (Commutativity).Letf :P→P ′ andg :P ′ →P be order preserving mapsbetween two posetsP,P ′. Given admissible orderings4,4′ onP andP ′, respectively,such thatf ◦ g andg ◦ f are selfmaps inNP(k), N(f ◦ g,P ′;4′)=N(g ◦ f,P;4).

Proof. The equality follows from the commutativity property of the fixed point index andthe 1–1 correspondence between the fixed point classes off ◦ g andg ◦ f . 2Theorem 3.5 (Homotopy type).Given the following homotopy commutative diagram

Ph

f Ph

P ′gP ′

whereh :P→ P ′ is a homotopy equivalence between two posets inNP(k). Let h be thehomotopy inverse ofh. For any admissible orderings4 and4′ such thath ◦ g ◦ h andg ◦ h ◦ h are selfmaps ofP andP ′ in NP(k), we haveN(f,P;4)=N(g,P ′;4′).

Proof. Sincef ∼P h◦g ◦h andg ∼P g ◦h◦ h, we haveN(f,P;4)=N(h◦g ◦h,P;4)andN(g,P ′;4′)=N(g ◦ h ◦ h,P ′;4′). By Theorem 3.4,N(h ◦ g ◦ h,P;4)=N(g ◦ h ◦h,P ′;4′). 2Convention. If N(f,P;4) is independent of the choice of the admissible ordering, wewriteN(f,P)=N(f,P;4).

We also define

N∗(f,P) := max{N(f,P;4) |4 is an admissible ordering onP

}.


It is straightforward to show thatN∗(f,P) is a Nielsen type number satisfying all the usualproperties of the classical Nielsen number.

4. Maps of triads

It is clear from the definition thatN(f,P;4) reduces to the ordinary Nielsen numberN(f ) if k = 1, i.e., Ai = ∅, and that it reduces to the relative Nielsen number [32]N(f ;X,A) whenk = 2. In this section, we discuss the relationship between the Nielsennumber of a triad introduced by Schirmer in [35] and the Nielsen type numberN(f,P;4)whereP is the triad. Throughout this section,P will denote the triad(X,A1,A2) withA0=A1 ∩A2 andX =A1∪A2.

To define the Nielsen type number for a triad, one needs to define the notion ofcontainmentof fixed point classes in terms of lifts in the sense of [44]. LetA ⊂ Xbe a subpolyhedron of a polyhedronX. Assume thatX andA are both connected. LetηA : A→ A,η : X → X denote the universal covers ofA and of X, respectively andlet jA :A→ X be the inclusion map. Choose base pointsa0 ∈ A, a0 ∈ η−1

A (a0) andx0 ∈ η−1(jA(a0)). Let jA : A→ X be the unique lift ofjA with jA(a0) = x0. Thus thefollowing diagram commutes

A

ηA

jAX

ηX

AjA

X

Moreover, given a selfmapf : (X,A)→ (X,A) and a liftfA : A→ A of fA, there existsa unique liftf : X→ X of f such that the following commutes

AjA

fAA

jA

Xf

X

Xf

X

A

jAfA

A

jA

This gives rise to a well-defined function from the set of conjugacy classes of liftsof fA to that off . Hence, it defines a functioniFPC

A,X : FPC(fA)→ FPC(f ) by sending

ηA Fix(fA) to ηX Fix(f ). Equivalently, by identifyingFPC(f ) with the lifting classes off , iFPC

A,X (fA) = f . We say that a fixed point classFA of fA is containedin a fixed point

classF of f if iFPCA,X (fA)= f wherefA andf are the lifts corresponding to the classesFA

andF, respectively.


Let (X,A1,A2) be a triad consisting of a compact polyhedronX and two subpolyhedraA1 andA2 such thatX = A1 ∪ A2. Given a selfmapf : (X,A1,A2)→ (X,A1,A2), aNielsen type number forf is defined as follows [35]. Two fixed point classesF1 andF2

areinessentially joinedif (i) there existsF0 such thatF0 is contained in bothF1 andF2,andi(f0,F0)= 0, whereFj ∈ FPC(fi), i = 0,1,2 andf0 :A0→A0,A0= A1 ∩A2; (ii)if F′0 ∈ FPC(f0) is contained in bothF1 andF2 then i(f0,F′0) = 0. The pair(F1,F2) iscalled aninessentially joined pair of essential fixed point classes off1 andf2 if (F1,F2)

is an inessentially joined pair andFj is essential forj = 1,2. Then theNielsen number ofthe triad is defined as

N(f ;A1∪A2)=N(f1;A1,A0)+N(f2;A2,A0)−N(f0)− IJ(f1, f2),

where N(f1;A1,A0) and N(f2;A2,A0) denote the relative Nielsen numbers andIJ(f1, f2) is the number of inessentially joined pairs of essential fixed point classes off1 andf2.

As we saw in Example 3.1,N(f,P;4)may vary depending on the admissible ordering.When the subspaces can be by-passed, we obtain the following

Theorem 4.1. Let P = {X,A1,A2,A0} with A0 = A1 ∩ A2 andX = A1 ∪ A2. SupposethatA0 can be by-passed inAi for i = 1,2 andAj can be by-passed inX for j = 1,2.ThenN(f,P;4) is independent of the choice of the admissible ordering.

Proof. There are two admissible orderings with the following filtrations:

(41) : X1=A0, X2=A0 ∪A1, X3=A1∪A2=X;(42) : X1=A0, X2=A0 ∪A2, X3=A1∪A2=X.

Let f :P→ P be an order preserving map. SinceA0 is a compact ANR, there exists aneighborhoodU of A0 in A1 and a homotopyH of f1 relative toA0 so thatH(·,0)= f1

andH(x,1)= f1 ◦ r(x) wherer :U→ A0 is a retraction. Thus,i(f0,F)= i(f1,F ∩A1)

for any fixed point classF of f0. Then, we extend the homotopy to all ofX so that withoutloss of generality, we may assume that for a fixed point classF of f ,

i(f,F∩Aj)= i(fj ,F∩Aj) for j = 0,1,2.

Now letF1, . . . ,Fm be the essential fixed point classes off0, G1, . . . ,Gn be the essentialfixed point classes off1 that do not contain any essential fixed point classes off0 andH1, . . . ,Hq be the essential fixed point classes off2 that do not contain any essential fixedpoint classes off0.

It is clear thatFi ∩Gj = ∅ for all i = 1, . . . ,m; j = 1, . . . , n. Similarly, Fi ∩Hj = ∅for all i = 1, . . . ,m; j = 1, . . . , q . Let F ∈ F(f ) be a fixed point class containingGj .Suppose thatF also containsHk for somek, 16 k 6 q . ThenF cannot contain any otherHi with i 6= k because the by-passing condition ofA1 in X implies thatHi andHk belongto the same fixed point class off2 in A2 and thusi andk must be the same. Similarly,if F ∈ F(f ) containsHj andGi thenF contains no otherGj with i 6= j . Let ` be thenumber of pairs(Gi ,Hj ) so thatGi andHj belong to the same fixed point class off .


LetF be an essential fixed point class off that do not contain any essential fixed pointclasses off0 and off1. ThenF does not contain any of theFi ’s or theGj ’s. ThenF mustcontain someHk and there are at mostq suchF ’s. On the other hand, ifHi is not oneof the ` classes ofHk ’s each of which is Nielsen equivalent to someGj in X, thenHiis contained in an essentialF ′ ∈ F(f ) with i(f,F ′) = i(f2,Hk). Thus there are exactly(q − `) suchF ’s. Similarly, there are exactly(n− `) essential fixed point classes off thatdo not contain any essential fixed point classes off0 and off2.

Therefore, we have for the first admissible ordering

N(f,P;41)=m+ n+ (q − `)and for the second one

N(f,P;42)=m+ q + (n− `).Hence, the two admissible orderings yield the same Nielsen type number.2Remark 4.2. In Example 3.1,N(f ;A1 ∪A2)= 1 whereasN∗(f,P)= 2. Sincef musthave at least two fixed points,N∗(f,P) yields a better lower bound forMFP [f ] thanN(f ;A1 ∪ A2). Note thatN(f,P;4) > N(f ;A1 ∪ A2). On the other hand, by-passingconditions yield the following

Theorem 4.3. If A0 can be by-passed inAi andAi can be by-passed inX for i = 1,2,then

N(f ;A1∪A2)>N(f,P).

Proof. Consider the admissible ordering which yields the filtrationX1 = A0, X2 =A0 ∪ A1, X3 = X. Let m,n andq be as in the proof of Theorem 4.1 and let` be thenumber of pairs of the form(Gi ,Hj ) whereGi andHj belong to the same fixed pointclass off . Let k be the number ofHj that belong to the same fixed point class off assomeGi . Under the by-passing conditions,k = ` andN(f (3))−N(f (3);X63)= q − k.Furthermore,N(f,P;4) is independent of the choice of the admissible ordering so thatN(f,P;4)=N(f,P). If (Gi ,Hj ) is one of the pairs, it need not be inessentially joined,i.e.,Gi andHj may belong to the same fixed point class off but do not contain any fixedpoint class off0. It follows thatIJ(f1, f2)6 `. Hence,

N(f,P) = m+ n+ (q − k)6 (m+ n)+ (m+ q)−m− IJ(f1, f2)

= N(f ;A1∪A2). 2Next, we give conditions under which equality of the two Nielsen type numbers holds.

Theorem 4.4. Suppose forj = 1 or 2 that iFPCA0,Aj

: FPC(f0)→ FPC(fj ) is surjective; thatA0 can be by-passed inAj and thatAj can be by-passed inX. Then

N(f ;A1∪A2)=N(f,P).


Proof. Since

iFPCA0,A1

: FPC(f0)→ FPC(f1)

is surjective, in the proof of Theorem 4.3, everyGi contains an inessential fixed point classof f0. Hence, every one of the pairs {(Gi ,Hj )} is inessentially joined. The assertionfollows from the proof of Theorem 4.3.2Remark 4.5. The surjectivity assumption oniFPC

A0,Ajcan be slightly weakened by simply

assuming that for any essential fixed point classesF1,F2 of f1, f2, respectively, ifiFPCA1,X

(F1) = iFPCA2,X

(F2) then there exists anF0 ∈ FPC(f0) such thatiFPCA0,Aj

(F0) = Fj forj = 1,2.

5. The numberN∗R(f,P)

The objective of this section is to generalize Theorem 4.1 and to extend the definition ofN(f,P;4) (and henceN∗(f,P)) in order to handle the equivariant situation which willbe discussed in Section 6. Recall from Section 2 that〈Ai〉 denotes the principal order idealof Ai ∈P and the rank (〈Ai〉) is well-defined.

Theorem 5.1. Let P = {X,A0, . . . ,Ak} such that (〈A0〉) = 0, `(〈Ai〉) = 1, A0 can beby-passed inAi and

⋃kj=0,j 6=i Aj can be by-passed inX =⋃k

j=0Aj for all i = 1, . . . , k.ThenN(f,P) is well-defined.

Proof. Since `(〈Ai〉) = 1 for i = 1, . . . , k, there are exactlyk! distinct admissibleorderings onP . For eachj , 16 j 6 k, let

Fj,1, . . . ,Fj,mj

be the essential fixed point class offj :Aj → Aj that do not contain any essential fixedpoint class off0 :A0→ A0. For everyFj,r , we associate to it an unorderedm-tupleCm(Fj,r ) consisting ofm6 k fixed point classes of the formFi,s so that

(1) Fj,r ∈ Cm(Fj,r );(2) for i 6= p, Fi,s ,Fp,t ∈Cm(Fj,r ) if, and only ifFi,s andFp,t belong to the same fixed

point class off .Using the same argument as in the proof of Theorem 4.1, under the by-passing

conditions, theCm(Fj,r )’s are well-defined. Thus,Cm(Fj,r ) is the only such unorderedm-tuple, for 16m6 k, that contains the classFj,r . In other words, if bothCm(Fj,r ) andCn(Fi,s ) containFj,r thenm= n andCm(Fj,r )= Cn(Fi,s ) as sets ofm elements.

For any admissible ordering onP ,

N(f,P;4)=N(f0)+k∑i=1

mi −O,


whereO denotes the number of classes that are “overcounted”. Since each setCm(Fj,r )appears exactlym times (once for each member in the set), the classes that are overcountedare precisely all but one of them setsCm(Fj,r ) each of which is of sizem. Thus,

O =k∑j=1

(j − 1) · #(unorderedj -tuples)

and

k∑j=1

mj −O =k∑j=1

j · #(unorderedj -tuples)−O

=k∑j=1

[j · #(unorderedj -tuples)− (j − 1) · #(unorderedj -tuples)

]=

k∑j=1

#(unorderedj -tuples)=k∑j=1

mj∑r=1

#{Cm(Fj,r )

}.

Hence, the numberN(f,P;4) remains constant, independent of the admissible order-ing. 2

Note that the proof of Theorem 4.1 is thek = 2 case of the above proof.In order to incorporate equivariant Nielsen fixed point theory into our unified Nielsen

theory, we need to generalize the Nielsen number as follows.Let R be an equivalence relation on Fixf . Denote by∼R be the smallest equivalence

relation on Fixf containing both∼ andR. Assume thatR has the following property:if xi andyi, i = 1,2, are {ft }-related as fixed points thenx1Rx2⇔ y1Ry2. It followsthat for anyx, y ∈ Fixf , (1) x ∼ y ⇒ x ∼R y; and (2) if xi andyi, i = 1,2, are {ft }-related as fixed points thenx1 ∼R x2⇔ y1 ∼R y2. Thus, every∼R equivalence class isa (finite) disjoint union of ordinary fixed point classes off . Moreover, ifF is an∼Requivalence class, the fixed point indexi(f,F) is well-defined, namely, it is the sum ofthe indices of the ordinary fixed point classes withinF. ThenF is essentialif i(f,F) 6= 0.Note that the definition of∼R essentiality may allow an inessential∼R class to containessential (ordinary) fixed point classes. We define theNielsen numberof f with respect tothe equivalence relation∼R to be the number of essential∼R fixed point classes and isdenoted byNR(f ).

Theorem 5.2. The numberNR(f ) satisfies the basic properties of the ordinary Nielsennumber.

Proof. Since every∼R fixed point class is a disjoint union of ordinary fixed point classesof f , it is clear thatNR(f )6 N(f ). On the other hand, an∼R fixed point class is also adisjoint union ofR equivalence classes soNR(f ) is a lower bound for the number ofRequivalence classes of fixed points off and hence the lower bound property is satisfied.


Let F be an essential∼R fixed point class off consisting ofF1, . . . ,Fr ordinary fixedpoint classes. Since the fixed point index is invariant under a homotopy{ft }, F1, . . . ,Frare{ft }-related toG1, . . . ,Gr , respectively. Note that ifGj is empty theni(f,Fj )= 0. Infact,Fi is essential if and only ifGi is, for eachi = 1, . . . , r. Now, condition (2) of∼Requivalence implies thatG1, . . . ,Gr form an∼R fixed point classG which is{ft }-relatedto F. SinceF is essential, then so isG. Thus, essential∼R classes off0 and off1 arein one-one index preserving correspondence. This shows the homotopy invariance. It isstraightforward to show that commutativity and homotopy type invariance also hold.2

Analogously, we can define the weighted setsFR(f ) andFPCR(f ), where the formerconsists of nonempty classes and the latter is defined in terms of lifts off and may includepossibly empty classes. It is clear thatNR(f ) = N(f ) when∼R coincides with∼ onFixf , i.e., whenR is the trivial equivalence relation.

Extending the definition ofN(f,P;4), we define for any admissible ordering onP theNielsen type number

NR(f,P;4)=k∑i=1

[NR(f (i))−NR(f (i);X6i)

],

where

NR(f (i);X6i) = #{F ∈ FR(f (i)) | i(f (i),F) 6= 0 andF contains

an essentialG ∈ FR(f (j)) for somej < i}.

Similarly, we extend the definition ofN∗(f,P), by taking the maximum value over alladmissible orderings onP , to

N∗R(f,P) := max{NR(f,P;4) |4 is an admissible ordering onP

}.

Remark 5.3. SupposeF ∈ FR(f (i)) is essential andF contains an essentialG ∈FR(f (j)) for somej < i. Sincei(f (i),F) 6= 0 andi(f (j),G) 6= 0, under a homotopy{ft }, the classesF andG must be related to essential classesF′ of f1(i) andG′ off1(j), respectively. Condition (2) of∼R implies thatG′ is contained inF′ and henceNR(f (i);X6i ) is invariant under homotopy. Therefore, the numberNR(f,P;4) (andhenceN∗R(f,P)) is invariant under homotopy. Furthermore, Theorems 3.3, 3.4 and 3.5hold true forNR(f,P;4) using the same arguments as in their proofs. For the lower boundproperty, we will see in Sections 7 and 8 thatNR(f,P;4) will provide lower bounds forthe number of orbits of fixed points.

We generalize Theorem 5.1 to the following

Theorem 5.4. Let f :P → P be the same as in Theorem5.1. ThenNR(f,P;4) isindependent of the choice of the admissible ordering and, in that case, it is denoted byNR(f,P).


6. Fibre maps, equivariant maps and iterated maps

In this section, we study the settings of fibre preserving maps, equivariant maps anditerates of maps. We compare the existing Nielsen type numbers in these settings withNR(f,P;4) for appropriate posetsP .

First, we deduce the Nielsen type number of a fibre preserving mapNF (f,p) [16] fromthe Nielsen number of a posetN(f,P;4) for a suitableP .

Fibre space techniques have been employed to express the Nielsen number of a fibrepreserving map as the product of the Nielsen numbers of maps on the base and the fibre.Necessary and sufficient conditions were given in [43] for such a product formula to hold.Furthermore, these techniques allow us to estimate the Nielsen number in case of non-orientable fibrations [26,29,30]. On the other hand, this product serves as a lower bound(and sometimes a sharp one) for the number of fixed points in the fibrewise homotopyclass of maps and a connection between this product and the relative Nielsen number wasestablished in [14]. For non-orientable fibrations, a similar Nielsen type number for fibrepreserving maps, that does not require orientability of the fibration, was introduced andstudied in [16]. This number is defined as follows.

Let p :E→ B be a Hurewicz fibration andf :E→ E be a fibre preserving map, i.e.,there is mapf :B→ B such thatp ◦ f = f ◦ p. We assume thatE andB are compactconnected ANRs andN(f ) 6= 0. In addition, we assume that a typical fibre is also pathconnected. Arepresentativeof the fixed point classes off is a subsetξ = {b1, . . . , bn} ⊆Fixf such thatn= N(f ) and no twobi ’s belong to the same fixed point class off . Forany representativeξ , we letFξ =⋃bi∈ξ p

−1(bi) ⊆ E. The Nielsen type number off isdefined to be

NF (f,p)=N(f )∑i=1

N(fi )=N(f |Fξ ),

wherefi = f |p−1(bi) :p−1(bi)→ p−1(bi).

Theorem 6.1. Let P be the poset(E,Fξ ) as in Example2.4. For any fibre preservingmapf of p, we have

NF (f,p)=N(f,P).

Proof. First, it is easy to see thatN(f,P) is well-defined. Note thatP = {E,p−1(b1), . . . ,

p−1(bn)} wheren=N(f ). Since the fibres{p−1(bi)} are mutually disjoint, the subspaceXi is a disjoint union

⊔j6i Aj , for eachi = 1, . . . , n. It follows thatN(f (i),X6i ) = 0

for i = 1, . . . , n andN(f )=N(f (n+ 1))=N(f (n+ 1),X6n+1). Thus,

N(f,P) =n+1∑i=1

[N(f (i))−N(f (i);X6i)

]=

n∑i=1

N(f (i)

)= n∑i=1

N(fi )=NF (f,p). 2


Remark 6.2. Theorem 6.1 also follows from [16, 7.1] thatNF (f,p)=N(f ;E,Fξ ), therelative Nielsen number off as a selfmap of the pair(E,Fξ ).

Techniques from relative Nielsen theory have proven to be useful in the Nielsen theoryfor equivariant maps [37,38]. We compare the Nielsen type numberNOG(f ) definedin [38] for equivariant maps to the Nielsen type numberN∗R(f,P) by considering theappropriate posetsP .

LetG be a compact Lie group acting effectively on a compact ANRX and let

Iso(X)= {(H) ∈ C(G) |H =Gx for somex ∈X}be the set of isotropy types ofX. Given aG-map f :X → X, we define for each(H) ∈ Iso(X) an equivalence relation∼R on Fixf (H) as follows. Two fixed pointsx, y ∈ Fixf (H) 6= ∅ are∼R equivalentif, and only if,

(1) y = σx for someσ ∈G, or(2) there exists a pathα : [0,1]→X(H), (H) ∈ F such thatα(0)= x, α(1)= σ ′y for

someσ ′ ∈G andα ∼ h ◦ α (rel endpoints).It is easy to see thatx ∼ y⇒ x ∼R y. Moreover the equivalence relationR is defined by

xRy⇔ y = σx for someσ ∈G.Thus, theR equivalence class of a pointx is theG-orbitGx = {σx | σ ∈G} of x.

Denote byFR(f (H)) the set of non-empty∼R fixed point classes off (H). These are thenon-empty(H)-fixed point classes in [38]. Using lifts, we can define analogously the∼Rfixed point classes, denoted byFPCR(f (H)) (or FPC(H)(f ) in [38]). We should point outthatX(H) is not necessarily connected whileXH is usually assumed to be connected. (IfG is Abelian thenX(H) = XH .) Thus, an element inFPCR(f (H)) consists of conjugacyclasses of lifts off H

′where(H ′)= (H).

Suppose(H), (K) ∈ Iso(X) and (H) 6 (K). Then for any representativeH ′ in (H),there is a representativeK ′ in (K) such thatH ′ is a subgroup ofK ′. Thus,XK

′ ⊂ XH ′ .Any lift f K ′ of f K

′determines a unique liftf H ′ of f H

′as in Section 4. Hence, there is a

(contravariant) functionτ(H)6(,)K : FPCR(f (K))→ FPCR(f (H)).For each(H) ∈ Iso(X),

NOG(fH ) = min

{#C | C ⊂

⋃(H)6(K)

FPCR(f(K)),∀ essentialN ′ ∈ FR(f

(L)),

∃N ∈ C ∩ FPCR(f(K)) such thatτ(L)6(,)K(N)=N ′

}.

WhenH = (1), we have

NOG(f ) = min

{#C | C ⊂

⋃(K)

FPCR(f(K)),∀ essentialN ′ ∈ FR(f

(L)),

∃N ∈ C ∩ FPCR(f(K)) such thatτ(L)6(,)K(N)=N ′

}.

We should point out thatH = (1) may not be an isotropy type ofX; however, theabove definition ofNOG(f ) still makes sense. In [38],NOG(f H ) is only defined for those


isotropy types(H) with finite Weyl groupWH because under the standard hypothesesin [38], fixed points inXH can be removed by an equivariant homotopy whenWH is notfinite.

A minimality theorem was proven in [38, 3.2] forNOG(f H ) under certain codimensionhypotheses.

Theorem 6.3. LetP be the poset{X(H)} as in Example2.6. Then

N∗R(f,P)6NOG(f ).

Proof. Let C ⊂ ⋃(K)FPCR(f (K)) be such that #C = NOG(f ). Choose an admissibleordering onP which gives an admissible ordering(H1), . . . , (Hk). Let Ni be the set ofessential∼R fixed point classes off (i) that do not contain any essential∼R fixed pointclasses off (j) for anyj < i. Thus,

#Ni =[NR(f (i))−NR(f (i);X6i )

]and NR(f,P)=

k∑i=1

#Ni .

SupposeF ∈Ni for somei. Without loss of generality, we may assume, as in the proofof Theorem 4.1, thati(f,F ∩ X(Hj )) = i(f (Hj ),F ∩ X(Hj )) for all j . SinceF does notcontain any essential fixed point class inXj = ⋃r6j X

(Hr ) for j < i, it follows that

i(f,F ∩ X(Hi)) 6= 0. Thus,F must contain an essential fixed point classF of f (Hi).There exists anN ∈ C ∩ FPCR(f (Hr)) such thatτ(Hi)6(,)Hr (N) = F . SupposeG ∈ N`such thatG contains an essential∼R fixed point classG of f (H`) with the property thatτ(H`)6(,)Hr (N)=G.

Case(I). If ` < i, i.e.,X` ⊂Xi , thenF andG belong to the same∼R fixed point class off (i). By definition of[NR(f (i))−NR(f (i);X6i)], F cannot be inNi so such aG cannotexist. The case wherei < ` is similar.

Case(II). If i = `, thenF = G. Thus, there is an injective function from⊔iNi to C.

Hence, the assertion follows since the sets{Ni} and{C} are finite. 2Remark 6.4. While the Nielsen type numberNOG(f ) can be realized under certaincodimension hypotheses as in [38], it is difficult to compute from its definition. Theo-rem 6.3 gives an estimate (in fact, one for each possible admissible ordering) forNOG(f ).

We now turn to periodic points or, equivalently, fixed points of iterates of a map. Letf :X→X be a selfmap of a compact connected ANR (metric) andn be a positive integer.Denote byYn =X×· · ·×X then-fold product ofX. The finite cyclic group ofn elementsZn acts onYn via

ζ · (x1, . . . , xn)= (xn, x1, . . . , xn−1),

whereζ is the generator ofZn.For everyy ∈ Yn, the isotropy subgroup ofy is denoted by

stab(y)= {σ ∈ Zn | σy = y}.


Let Yn(1) = {y ∈ Yn | stab(y)= 1} be the set of points inYn upon which theZn-action isfree. It is known from equivariant topology [2] thatYn(1) is an open subspace ofYn. DefineaZn-equivariant mapgf :Yn→ Yn by

gf (x1, . . . , xn)=(f (xn), f (x1), . . . , f (xn−1)

).

It is easy to see that

Fixgf ={(x, f (x), . . . , f n−1(x)) | x ∈ Fixf n

}.

In fact, if (x, f (x), . . . , f n−1(x)) ∈ Fixgf |Yn(1) thenf i(x) 6= f j (x) unlessi ≡ j modn.Therefore, the fixed points ofgf onYn(1) correspond to the periodic points off with leastperiodn.

It was shown in Theorem 2.1 of [40] that(1) N(f n)=N(gf );(2) there is a bijection betweenFPC(f n) andFPC(gf ).For further connections between fixed points ofgf and periodic points off , see [37,38,

40].The setL= {Yn/m|m|n} as in Example 2.7 is a lattice under inclusion andgf :L→ L

is an order preserving map. We are in the equivariant setting, as in the last section, withG = Zn. Thus, the Nielsen type numberNOZn(gf ) is a lower bound for the number ofn-periodic orbits{x,h(x), . . . , hn−1(x)} wherex ∈ Fixhn (not necessarily of least periodn) for anyh homotopic tof . This follows from the fact thatNOZn (gf ) is aZn-homotopyinvariant and

h∼ f ⇒ gf ∼Zn gh.As a consequence of Theorem 6.3, we obtain

Theorem 6.5. LetX be a compact ANR andL be the poset of Example2.7. Then for anymapf :X→X and for any positive integern,

N∗R(gf ,L)6min{#O |O is ann-periodic orbit ofh, whereh∼ f }.

Proof. Theorem 6.3 implies that

N∗R(gf ,L) 6 NOZn (gf )

6 min{#O |O is ann-periodic orbit ofh, whereh∼ f }. 2

7. Converse Lefschetz theorem

The Minimum Theorem 2.1 states that for selfmaps on many spaces, the classicalNielsen number is a sharp lower bound for the number of fixed points in the homotopyclass of a map, i.e.,N(f )=MF[f ] := {# Fixg | g ∼ f }. Thus, in the case whenL(f )=0⇒N(f )= 0, the mapf is homotopic to a fixed point free map and thus the converse ofthe Lefschetz–Hopf fixed point theorem holds. In this section, we present the Lefschetz–Hopf fixed point theorem in theNP(k) category and we will give conditions onP such


thatN∗(f,P)= 0 is sufficient to deformf to be fixed point free. Its converse will also begiven.

Recall that a spaceX is Weckenif for any selfmapf :X→X, N(f )=MF[f ].

Theorem 7.1. Let {X = Ak,A1, . . . ,Ak−1} = P ∈ NP(k) be a poset of compactpolyhedra. Suppose thatX is connected and thatAi is connected and Wecken for each1 6 i 6 k. Suppose also that for anyAj < Ai , Aj can be by-passed inAi and thatAi −Aj has no local separating points and is not a surface. Letf :P→ P be a selfmap.If N∗(f,P)= 0, then there is a selfmapg :P→ P such thatg ∼P f andFixg = ∅.

Proof. Choose an admissible ordering onP with a filtrationX1 ⊂ · · · ⊂ Xk = X. SinceN∗(f,P)= 0, it follows thatN(f (i))−N(f (i);X6i )= 0 for all i, 16 i 6 k. If k = 2,the assertion is simply Theorem 6.2 of [32]. Using argument similar to that of [32, 6.2]or [12,37], we may assume thatf is fixed point free onXj−1 for somej < k. SinceN(f (j))−N(f (j);X6j )= 0 andf (j − 1) is fixed point free,N(f (j))= 0. Any fixedpoint class off (j) that lies inAj −Xj−1 is inessential and can be removed by a homotopyof f (j) relative toXj−1. Extending the homotopy to all ofX yields a mapf ′ ∼P f whichis fixed point free onXj . Hence the inductive step is proven and the proof is complete.2Corollary 7.2. Let f :P → P be as in Theorem7.1. If N(f ) = 0 = N(fi) for alli = 1, . . . , k − 1, wherefi :Ai→Ai , then there is a selfmapg :P→ P such thatg ∼P fandFixg = ∅. Equivalently,

MF[fi] = 0, ∀i = 1, . . . , k⇒MFP [f ] = 0.

Proof. It suffices to show thatN(f ) = 0 = N(fi ) for all i = 1, . . . , k − 1 impliesN∗(f,P)= 0. This follows from the fact that, given any admissible ordering, an essentialfixed point class off (j) :Xj → Xj must contain an essential fixed point class offr forsomer 6 j . 2

The classical Lefschetz–Hopf fixed point theorem [3, C.2] states that if the Lefschetznumber of a selfmap on a compact connected ANR is nonzero, then any selfmap homotopicto it must have a fixed point. In the categoryNP(k), the classical result extendsimmediately to the following

Theorem 7.3. Let P = {X,A1, . . . ,Ak−1} be a poset of compact connected ANRs andf :P → P a selfmap. IfL(fi) 6= 0 for somei, then for any selfmapg :P → P withg ∼P f , Fixg 6= ∅.

Remark 7.4. Theorem 7.3 holds for general posets so thatP does not necessarily belongtoNP(k).

As an immediate consequence of Corollary 7.2, we obtain the following converseof Theorem 7.3 if we impose conditions on the spaces such that the vanishing of theLefschetz number implies the vanishing of the Nielsen number. We call a spaceX


Lefschetz-likeif for any selfmapf :X→X,L(f )= 0⇒N(f )= 0. Lefschetz-like spacesinclude the classical Jiang spaces (see [24,22,3]), nilmanifolds [1], a certain class ofsolvmanifolds [26], orbit spaces of odd spheres by free action of finite groups [11, 6.38]and orientable homogeneous spacesG/K of compact connected Lie groupsG such thatHn(G)→Hn(G/K) is nonzero wheren= dimG/K [42].

Corollary 7.5. Let f :P→ P be as in Theorem7.1. In addition, we suppose thatX andeachAi are Lefschetz-like spaces. IfL(fi)= 0 for all i, then there is a selfmapg :P→ Psuch thatg ∼P f andFixg = ∅.

Remark 7.6. The connectedness of theAi ’s in Theorem 7.1 can be relaxed but allother conditions must be imposed on the connected components of theAi ’s. With thismodification, it is easy to see that Corollary 7.2 reduces to the Main Theorem of [12] withP = {X(H)}. Since the hypotheses of Theorem 7.1 are satisfied by the standard hypothesesas in [12], it remains to show thatN(f H ) = 0⇒ N(f (H)) = 0. This follows from thefact thatN(f H ) = 0⇒ N(f H

′) = 0 for anyH ′ in the same conjugacy class asH and

N(f (H)) is just the sum of theN(f H′) for H ′ ∈ (H).

Remark 7.7. When P is the poset of Example 2.2, i.e., in the relative setting, thehypotheses of Theorem 7.1 coincide with those in the minimality theorem of Schirmer inTheorem 6.2 of [32]. Thus, Corollary 7.2 states that under these conditions, the vanishingof the relative Nielsen numberN(f ;X,A) is equivalent to the vanishing ofN(f ) andof N(fA). Similarly in the fibre-preserving situation of Example 2.4, the by-passinghypotheses of Theorem 7.1 will automatically be satisfied if the total spaceE, base spaceB and the fibres are compact manifolds with dimB > 2. In this case, Corollary 7.2 reducesto theNF (f,p)= 0 case of the minimality result (Theorem 8.2) of [16].

8. Concluding remarks

In this paper, we have developed, using the combinatorial structure of a poset of spaces,a Nielsen type theory and a Nielsen type numberN(f,P;4) (andNR(f,P;4)) for anygiven admissible ordering. We showed, for appropriate posetsP , the numberN(f,P;4)(andNR(f,P;4)) is related and sometimes equal to some known Nielsen type invariants.In the case thatN(f,P;4) is independent of the choice of the admissible ordering,N(f,P;4) can then be computed in many different ways according to a particularadmissible ordering yielding the same value. For example, in the situation in Theorem 4.4,the Nielsen type numberN(f ;A1 ∪ A2) for the triad can be calculated in two differentways. Thus, this approach gives an alternative method of computing and of estimatingsome of the existing Nielsen type numbers.

Möbius inversion has been employed in relating fixed point type invariants forequivariant maps in [13,28,38]; for iterates of maps in [13,15,17,40]. To make use ofMöbius inversion in the context of partially ordered sets, one needs to make some


modifications as follows. Letf :P→ P be a selfmap of a posetP = {X,A1, . . . ,Ak−1}.Define for eachi,

nR(f,Ai;4) = #{essential∼R fixed point classes offi that do not contain

any ∼R fixed point class inFPCR(fj ), ∀Aj 6Ai}.

Note that when∼R=∼ and k = 2, nR(f,Ai;4) is the Nielsen number on thecomplement introduced by Zhao in [44]. Then we let

nR(f6j )=∑Ai6Aj

nR(f,Ai;4)

which yields, by Möbius inversion,

nR(f,Ai;4)=∑Aj6Ai

µP (Aj ,Ai)nR(f6j ).

One can show that, in the periodic point situation of Section 6, the numbernR(f,Ai;4)coincides with 1

mNPm(f ) (see [22,15,38,40] for definition) wherem = n/d andAi =

YZdn = Yn/d .Applications of Möbius inversion to congruence relations involving Nielsen type

invariants such asnR(f6j ) andnR(f,Ai;4) will be the objective of a future work.

Acknowledgement

I would like to thank the referee for numerous helpful suggestions and for pointing outsome errors in the earlier versions of the paper.

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Nielsen ﬁxed point theory for partially ordered sets · Nielsen ﬁxed point theory for partially ordered sets Peter Wong Department of Mathematics, Bates College, Lewiston, ME