NIELSEN THEORY ON NILMANIFOLDS OF THE STANDARD FILIFORM LIE GROUP

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1 NIELSEN THEORY ON NILMANIFOLDS OF THE STANDARD FILIFORM LIE GROUP JONG BUM LEE AND WON SOK YOO Abstract Let M be a nilmanifold modeled on the standard filiform Lie group H m+1 and let f : M M be a self-map Using the averaging formulas, we compute the spectra of the Lefschetz, Nielsen and Reidemeister (coincidence) numbers of maps f on M Moreover, we give explicit formulas for a complete computation of the Nielsen type numbers NP n (f) and NΦ n (f) We also give a complete description of the sets of homotopy minimal periods of all such maps on M 1 Introduction Let H m+1 be the Lie group R m σ R where σ = σ m : R GL(m, R) is given by 1 h h2 h 2! m 2 h m 1 (m 2)! (m 1)! h 0 1 h m 3 h m 2 (m 3)! (m 2)! h σ(h) = m 4 h m 3 (m 4)! (m 3)! h There are many different ways of describing the group H m+1, see for example [4] When m = 1, we see that H 2 is the Abelian group R 2 ; when m = 2, we can see that H 3 = R 2 σ R is isomorphic to the classical Heisenberg group Nil 3 of uni-triangular matrices of size 3 The explicit isomorphism is given by (x, h) 1 h x x Hence H m+1 = R m σ R generalizes the classical Heisenberg group We also refer to [2] for another natural generalization of H 3 Date: April 16, Mathematics Subject Classification Primary 55M20; secondary 57S30 Key words and phrases Generalized Heisenberg group, homotopy minimal period, Nielsen number, Nielsen type number, nilmanifold The first-named author was partially supported by Basic Science Research Program through the National Research Foundation of Korea funded by the Ministry of Education (NRF-2016R1D1A1B ) 1

2 2 JONG BUM LEE AND WON SOK YOO It is convenient to embed H m+1 affinely into GL(m + 1, R) as {( ) R m σ(h) x σ R = h R, x R m} 0 1 The Lie algebra of H m+1 is where h m+1 = R m τ R = {( ) τ(h) x h R, x R m} h h τ(h) = h because σ(h) = exp τ(h) Let ( ) ( τ(1) 0 τ(0) em E 1 =, E 2 = ( τ(0) em j+1 E j+1 = 0 0 ),, ),, E m+1 = ( ) τ(0) e1 0 0 Then these E j s from a (fixed) linear basis of h m+1 The only nontrivial Lie brackets between these basis elements are Hence it follows that h (1) m+1 = h m+1, [E 1, E j ] = E j+1, (2 j m) h (2) m+1 = [h m+1, h (1) m+1 ] = E 3,, E m+1, h (m) m+1 = [h m+1, h (m 1) m+1 ] = E m+1 = Z(h m+1 ) This shows that h m+1 is the standard (graded) filiform Lie algebra Hence we can call the Lie group H m+1 the standard filiform Lie group In this paper, we will simply use H and h to denote H m+1 and h m+1 Let Γ be a lattice (ie, a discrete, cocompact subgroup) of H = R m σ R Then it is a torsion-free finitely generated nilpotent group of Hirsch rank m + 1 The subgroup Γ R m of Γ is finitely generated, hence it is a torsionfree finitely generated Abelian group contained in R m and is of Hirsch rank m This implies that Γ R m is a lattice of R m, and Γ/(Γ R m ) is isomorphic to a lattice of R, so that Γ R m = Z m and Γ/(Γ R m ) = Z, and the following

3 NIELSEN THEORY ON NILMANIFOLDS 3 diagram of short exact sequences is commutative 1 R m H R 1 1 Z m Γ Z 1 Choose a basis {x 1,, x m } for Z m and a basis t 0 for Z Then σ(t 0 )(x i ) = l 1i x l mi x m, (i = 1,, m) for some integers l ij With A = [l ij ] GL(m, Z), we have Γ = Z m A Z Letting P = [x 1 x m ] the matrix with columns x 1,, x m, we have (11) P AP 1 = σ(t 0 ) and so A SL(m, Z) is a unipotent matrix with (I A) m 1 0 It is the purpose of this work to study the Nielsen theory for all continuous maps f of any nilmanifold M = Γ\H We will determine the spectra of the fundamental invariants L(f), N(f) and R(f) of the Nielsen theory where L(f), N(f) and R(f) are the Lefschetz, the Nielsen and the Reidemeister numbers of f We will also determine the spectra of the Nielsen coincidence invariants That is, we will determine L(M) = {L(f) f is a self-map of M}, L h (M) = {L(f) f is a self-homeomorphism of M}, LC(M) = {L(f, g) f, g are self-maps of M} Similarly, we will also determine and N(M), N h (M), NC(M), R(M), R h (M), RC(M) These spectral property were studied in [3] for maps on infra-nilmanifolds modeled on the nilpotent Lie group Nil m of uni-triangular matrices of size m For the periodic points, two Nielsen type numbers NP n (f) and NΦ n (f) were introduced by Jiang [17], which are lower bounds for the number of periodic points of least period exactly n and the set of periodic points of period n, respectively One of the natural problems in dynamical systems is the study of the existence of periodic points of least period exactly n The set of homotopy minimal periods of f, HPer(f) = g f {n N P n (g) }, where P n (g) = Fix(g n ) k<n Fix(gk ), was introduced by Alesedà et al [1] A complete description of the Nielsen type numbers NP n (f) and NΦ n (f), and the homotopy minimal periods HPer(f) of all self-maps f was obtained on the Klein bottle [19, 20], on the nilmanifolds with Nil 3 -geometry [16,

4 4 JONG BUM LEE AND WON SOK YOO 24], on a 3-dimensional flat Riemannian manifold [10], and on the special solvmanifolds with Sol 3 -geometry [15, 25] and with Sol 4 1 -geometry [18] In this paper, we will give a complete description of the Nielsen type numbers NP n (f) and NΦ n (f), and the homotopy minimal periods HPer(f) for all self-maps f on any nilmanifold Γ\H 2 Nielsen fixed point theory on Γ\H The Nielsen theory concerns with the following basic invariants: the Lefschetz (coincidence) numbers, the Nielsen (coincidence) numbers and the Reidemeister (coincidence) numbers In the following sections, we shall compute those basis invariants for all maps on any nilmanifold Γ\H Let Γ = Z m A Z be a fixed lattice of H Let f be a continuous self-map of the nilmanifold Γ\H Then f induces an endomorphism ϕ : Γ Γ of the group of covering transformations of p : H Γ\H In fact we first fix a lift f : H H of f Then ϕ is defined as follows: ϕ(γ) f = f γ, γ Γ Due to a result of Mal cev, the endomorphism ϕ extends uniquely to a Lie group endomorphism F of H In particular, the endomorphism F of H restricts to a self-map Φ F of Γ\H which is homotopic to f That is, the Lie group endomorphism F is a homotopy lift of f For the computation of the basic invariants of the Nielsen theory, it suffices to consider all Lie group endomorphisms ϕ of H preserving the lattice Γ Recall that {ϕ Endo(H) ϕ(γ) Γ} = Endo(Γ) With such endomorphisms, we shall use the following formulas: Theorem 21 ([23], [22], [21], [11], [7])] Let f and g be continuous selfmaps on the nilmanifold Γ\H Let F and G be Lie group endomorphisms of H so that they are homotopy lifts of f and g respectively Then we have: and L(f, g) = det(g F ), N(f, g) = det(g F ), R(f, g) = σ (det(g F )), where F and G are the differentials of the Lie group endomorphisms F and G, and where σ : R R { } is defined by σ(0) = and σ(x) = x for x 0 Recall that any self-map f of Γ\H gives a unique endomorphism ϕ of Γ up to conjugation Such an endomorphism ϕ extends uniquely to a Lie group endomorphism F of H and then a Lie algebra endomorphism F of h Conversely any endomorphism of Γ induces a self-map of Γ\H Therefore in applying the above theorem, we may write L(ϕ, ψ) instead of L(f, g) and so on For details, we refer to [8]

5 NIELSEN THEORY ON NILMANIFOLDS 5 Let ϕ be an endomorphism of Γ Then it can be regarded as a Lie group endomorphism of H That is, Endo(Γ) Endo(H) Because of the following commutative diagram ϕ H H log h dϕ h exp we can identify ϕ with its differential dϕ Hence we have Endo(Γ) Endo(H) = Endo(h) We note also that the map exp : h H is given by exp : R m τ R R m σ R, (x, h) (σ(h)x, h), hence the following diagram is commutative First we remark that 1 R m h R 1 = exp = 1 R m H R 1 H (1) = H, H (2) = [H, H (1) ] = R m 1, H (m) = [H, H (m 1) ] = R = Z(H) Here, R j stands for the subgroup of R m = {(x, 0)} H consisting of all elements (x, 0) such that x j+1 = = x m = 0 With the fixed lattice Γ = Z m Z of H, we let a 1 Γ be a generator of the Z-factor Since Z m = Γ R m is a lattice of R m and since R m fits a short exact sequence 1 H (2) R m R 1, Γ H (2) is a lattice of H (2) and Z m /(Γ H (2) ) = Z is a lattice of R We choose a 2 Z m R m so that its projection ā 2 generates Z m /(Γ H (2) ) We can choose a i Γ H (i 1), (i = 3,, m + 1), so that its projection ā i generates Γ H (i 1) /(Γ H (i) ) = Z Recall that Γ H (i) = Γ (i), the isolator of Γ (i) in Γ These form a torsion free central series of Γ It follows that the set {a 1,, a m+1 } is a set of generators of Γ, and the set {log(a 1 ),, log(a m+1 )} form a basis of h Let (21) E 1 = log(a 1 ), E 2 = log(a 2 ), E j+1 = [E 1, E j ] (j = 2,, m)

6 6 JONG BUM LEE AND WON SOK YOO ( ) τ(t0 ) 0 Then E 1 = h by (11), and E i R m h for 2 i m It follows that [E i, E j ] = 0 for i, j 1 and (22) h (1) = h, h (2) = [h, h (1) ] = E 3,, E m+1, h (m) = [h, h (m 1) ] = E m+1 = Z(h) The E i form a basis of h However, it is not necessarily true that E i = log(a i ) for i = 3,, m + 1 We shall determine Endo(h) first Let ϕ : h h be a Lie algebra endomorphism Then ϕ is a linear transformation of the linear space h preserving the nontrivial Lie brackets (21) together with all trivial Lie brackets In particular, ϕ preserves the lower central series (22) of h This implies that with respect to the ordered basis {E 1,, E m+1 }, ϕ is a lower block triangular matrix of the form p 11 p 12 p 21 p 22 p 31 p 32 p 33 ϕ = p 41 p 42 p 43 p 44 p m+1,1 p m+1,2 p m+1,3 p m+1,4 p m+1,m+1 We first consider the trivial Lie brackets [E 2, E j ] = 0 for 3 j m From [E 2, E 3 ] = 0, we have [ϕ(e 2 ), ϕ(e 3 )] = 0, hence Similarly, p 12 p 33 = p 12 p 43 = = p 12 p m,3 = 0 p 12 p 44 = = p 12 p m,4 = 0, p 12 p m,m = 0 The remaining trivial Lie brackets [E i, E j ] = 0 (3 i < j) give no conditions on the entries of ϕ Consequently, (1) if p 12 0 then p ij = 0 for all i, j with 1 i m, 3 j; (2) if p 12 = 0 then p ji (3 i j m) can be any number Now we show that the first column block matrices of ϕ determine the remaining blocks In fact, from [E 1, E 2 ] = E 3, we have [ϕ(e 1 ), ϕ(e 2 )] =

7 NIELSEN THEORY ON NILMANIFOLDS 7 ϕ(e 3 ), hence (23) p 33 = p 11 p 22 p 12 p 21, p 43 = p 11 p 32 p 12 p 31, p m+1,3 = p 11 p m,2 p 12 p m,1 Similarly from [E 1, E 3 ] = E 4,, [E 1, E m ] = E m+1, we have Thus, (24) and (25) p 44 = p 11 p 33, p 54 = p 11 p 43,, p m+1,4 = p 11 p m,3, p m,m = p 11 p m 1,m 1, p m+1,m = p 11 p m,m 1, p m+1,m+1 = p 11 p m,m p j+3,j+3 = p j 11 p 33 (j = 1,, m 2) p j+k,k = p k 3 11 p j+3,3 (k = 4,, m; j = 1,, m (k 1)) Case 1: p 12 0 All p ij = 0 with 3 j i m, and by (23), (24) and (25) we have p 11 p 22 p 12 p 21 = 0, p 11 p 32 p 12 p 31 = 0, (26) p 11 p m 1,2 p 12 p m 1,1 = 0, p m+1,4 = p m+1,5 = = p m+1,m+1 = 0 Therefore, ϕ is of the following form p 11 p 12 p 21 p 22 p 31 p 32 0 p 41 p p m+1,1 p m+1,2 p m+1,3 0 0 where p denotes a nonzero number with the conditions (26) Case 2: p 12 = 0 Then entries p i,1 (1 i m + 1) and p i,2 (2 i m + 1) are arbitrary,

8 8 JONG BUM LEE AND WON SOK YOO and the remaining entries p ij (i j) are determined by the identities (23), (24) and (25) Therefore, ϕ is of the form p 11 0 p 21 p 22 p 31 p 32 p 11 p 12 p 41 p 42 p 11 p 32 p 2 11p 22 p m+1,1 p m+1,2 p 11 p m,2 p 2 11p m 2,2 p m 1 11 p 22 Because of Theorem 21, we need to compute the determinant of ϕ Endo(Γ) Endo(H) = Endo(h) Moreover, the determinant of ϕ is independent of the choice of bases of h Hence, we may assume that ϕ has a matrix presentation which is a diagonal block matrix of one of the following types p 11 p 12 p 21 p 22 (I) 0 0 0, p ij Z, p 11 p 22 p 12p 21 = 0; (II) p 11 0 p 21 p p 11 p 22, p ij Z p m 1 11 p 22 For ϕ Endo(Γ) of type (I), we have 1 p 11 p 12 p 21 1 p 22 det(i m+1 ϕ) = det = 1 (p 11 + p 22 ), and for ϕ Endo(Γ) of type (II), we have 1 p 11 0 p 21 1 p 22 det(i m+1 ϕ) = det p 11 p p m 1 11 p 22 = (1 p 11 )(1 p 22 )(1 p 11 p 22 ) (1 p m 1 11 p 22 )

9 Hence, by Theorem 21 we have NIELSEN THEORY ON NILMANIFOLDS 9 L(M) = Z, N(M) = N {0}, R(M) = N { }, LC(M) = Z, NC(M) = N {0}, RC(M) = N { } In fact, this result is valid for any nilmanifold modeled on any nilpotent Lie group, see [6] If ϕ Aut(Γ) is an automorphism, then ϕ must be of type (II) Therefore, ( ) 1 p11 0 det(i m+1 ϕ) = det (1 p p 21 1 p 33 ) (1 p m+1,m+1 ) 22 where p j+2,j+2 = p j 11 p 22 with j = 1,, m 1 If p 11 = 1 or p 22 = 1 then it is clear that det(i m+1 ϕ) = 0 If p 11 = p 22 = 1 then p 33 = p 11 p 22 = 1 and 1 p 33 = 0, hence det(i m+1 ϕ) = 0 By Theorem 21, L(ϕ) = det(i ϕ) = 0, N(ϕ) = det(i ϕ) = 0, R(ϕ) = σ(det(i ϕ)) = σ(0) = Theorem 22 ([4]) Let M = Γ\H Then L h (M) = {0}, N h (M) = {0}, R h (M) = { } In particular, every lattice Γ of H has the R -property The R -property for free nilpotent groups N r,c was proved when rank r = 2 and class c 9 in [9]; when r = 2 or 3 and c 4r, or r 4 and c 2r or r = 2 and c 4 in [26]; when r > 1 and c 2r in [5] Remark 23 When m = 2, it is easy to see that Endo(h 3 ) = a b 0 c d 0 a, b, c, d, u, v R u v ad bc Unlike the case of m 3, the first block submatrix is the set of all 2 2 real matrices 3 Nielsen periodic point theory on Γ\H Let f be a self-map on Γ\H with linearization ϕ Then the iteration f n of f has ϕ n as its linearization By Theorem 21, we have L(f n ) = L(ϕ n ) = det(i m+1 ϕ n ), N(f n ) = N(ϕ n ) = L(ϕ n ) We consider ϕ Endo(Γ) of type (I) Let ϕ be the first block submatrix of ϕ Then det( ϕ) = 0 and L(ϕ n ) = det(i m+1 ϕ n ) = det(i 2 ϕ n ) = 1 tr ( ϕ n )

10 10 JONG BUM LEE AND WON SOK YOO We denote by λ 1 and λ 2 the eigenvalues of ϕ Since ϕ is an integer matrix, its determinant λ 1 λ 2 = 0, and its trace λ 1 + λ 2 = l is an integer Moreover, we have Let L(ϕ n ) = (1 l n ), N(ϕ n ) = 1 l n p 11 0 p 21 p p ϕ = 11 p p m 1 11 p 22 be an endomorphism of Γ of type (II) The eigenvalues of ϕ are integers p 11 and p 22, and ϕ n is also of type (II) Hence N(ϕ n ) = det(i ϕ n ) = 1 p n 11 1 p n 22 1 p n 11p n 22 1 p n(m 1) 11 p n 22 Heath and Keppelmann in [12, 13] proved that the two Nielsen type numbers NP n (f) and NΦ n (f) can be computed according to the Nielsen numbers for most maps on the nilmanifold Γ\H Proposition 31 ([12, Theorem 1], [13, Theorem 12]) Let f be a continuous map on a nilmanifold Suppose that N(f n ) 0 Then for all k n, NΦ k (f) = N(f k ), NP k (f) = q k µ(q) N(f k q ), where µ is the Möbius function Proposition 32 ([13, Corollary 46]) Let f be a continuous map on a solvmanifold Then NΦ n (f) = k n NP k (f), NP n (f) = k n µ(k) NΦ n k (f) Since there are formulae for the Nielsen numbers of continuous maps on the nilmanifold Γ\H, NP n (f) and NΦ n (f) are computable provided N(f n ) 0 by Proposition 31 Therefore, we shall focus on the case where N(f n ) = 0 Consider ϕ Endo(Γ) of type (I) with eigenvalues 0 and l Z It is clear that N(ϕ n ) = 0 (31) l = 1, or l = 1 with n even (λ 1 λ 2, λ 1 + λ 2 ) = (0, 1), or (0, 1) with n even The following lemma gives us a necessary and sufficient condition for ϕ of type (II) to be N(ϕ n ) = 0

11 NIELSEN THEORY ON NILMANIFOLDS 11 Lemma 33 Let ϕ Endo(Γ) of type (II) Let ϕ be the first block submatrix of ϕ with (integer) eigenvalues λ 1 and λ 2 Then N(ϕ n ) = 0 if and only if one of the following is satisfied: (1) (λ 1 λ 2, λ 1 + λ 2 ) = (1, 2), (2) (λ 1 λ 2, λ 1 + λ 2 ) = (l, l + 1), (3) (λ 1 λ 2, λ 1 + λ 2 ) = (l, l 1) (l ±1), and n is even Proof Suppose N(ϕ n ) = 1 λ n 1 1 λn 2 1 λn 1 λn 2 1 λ(m 1)n 1 λ n 2 = 0 If λ n i = 1 then λ i = 1 or λ i = 1 with n even, hence (λ 1 λ 2, λ 1 + λ 2 ) = (l, l + 1) or (l, l 1) with n even The former case is exactly the same as the condition (2) In the latter case, if l = ±1 then (λ 1 λ 2, λ 1 +λ 2 ) = (1, 2) or ( 1, 0), which is included in (1) or (2) respectively If (λ 2j+1 1 λ 2 ) n = 1 then λ 2j+1 1 λ 2 = 1 or λ 2j+1 1 λ 2 = 1 with n even, hence (λ 1 λ 2, λ 1 + λ 2 ) = (1, ±2) or (λ 1 λ 2, λ 1 + λ 2 ) = ( 1, 0) with n even The former case is included in (2) or exactly the same as (1) respectively The latter case is included in (2) If (λ 2j 1 λ 2) n = 1 then λ 2j 1 λ 2 = 1 or λ 2j 1 λ 2 = 1 with n even, hence (λ 1 λ 2, λ 1 + λ 2 ) = (1, 2) or ( 1, 0), or (λ 1 λ 2, λ 1 + λ 2 ) = ( 1, 0) or (1, 2) and n is even These, except (1, 2), are included in (2) Conversely, it is easy to check that each of the three conditions ensures that N(ϕ n ) = 0 Now we can determine the Nielsen type numbers NP n (ϕ) and NΦ n (ϕ) of ϕ when N(ϕ n ) = 0 Theorem 34 Let ϕ Endo(Γ) Let ϕ be the first block submatrix of ϕ with eigenvalues λ 1 and λ 2 Suppose N(ϕ n ) = 0 and n = 2 r n 0 with r 0 and gcd(2, n 0 ) = 1 Then NP n (ϕ) = 0 and { 2 if n is even; (1) ϕ is of type (I) = NΦ n (ϕ) = 0 if n is odd, (2) ϕ is of type (II) 2 m 1 j=1 1 + ( 1)j l n 0 if n is even and {λ 1, λ 2 } = { 1, l} = NΦ n (ϕ) = with l ±1 0 otherwise Proof Suppose N(ϕ n ) = 0 Then the pair (λ 1 λ 2, λ 1 + λ 2 ) has two or three possibilities listed in (31) or Lemma 33, respectively For ϕ of type (I), if (λ 1 λ 2, λ 1 + λ 2 ) = (0, 1), N(ϕ k ) = 0 for all k, thus NP n (ϕ) = NΦ n (ϕ) = 0; if (λ 1 λ 2, λ 1 + λ 2 ) = (0, 1) with n even, then N(ϕ n 0 ) 0, and the essential periodic orbit classes of period q for all q n are the same as all those of period q for all q n 0 It follows from definition and Proposition 31 that NΦ n (ϕ) = NΦ n0 (ϕ) = N(ϕ n 0 ) = 1 ( 1) n 0 = 2 In this case, using Proposition 32, we compute NP n (ϕ) For k n, if n k is odd

12 12 JONG BUM LEE AND WON SOK YOO then NΦ n (ϕ) = N(ϕ n k ) = 2; if N(ϕ n k ) is even then NΦ n (ϕ) = NΦ k k ( n k ) (ϕ) = 0 N(ϕ ( n k ) 0 ) = 2 Here, we denote by ( ) n k the largest odd factor of the even 0 integer n k Consequently, NP n(ϕ) = k n µ(k)nφ n (ϕ) = 2 k k n µ(k) = 0 We consider ϕ of type (II) If (1) or (2) of Lemma 33 is satisfied, then N(ϕ k ) = 0 for all k Thus NP n (ϕ) = NΦ n (ϕ) = 0 If (3) of Lemma 33 is satisfied, then ϕ has eigenvalues 1 and l with l ±1, n is even and N(ϕ n 0 ) 0 Hence, we have and NP n (ϕ) = 0 m 1 NΦ n (ϕ) = NΦ n0 (ϕ) = N(ϕ n 0 ) = ( 1) j l n 0 j=1 4 Homotopy minimal periods In this section, we shall present the homotopy minimal periods for all maps on the nilmanifold Γ\H Our main tool is { N(f n } ) 0, (41) HPer(f) = n N : N(f n ) N(f n q ) prime q n This formula can be obtained immediately from the following results Theorem 41 ([14, Theorem 61]) Let f : M M be a self-map on a compact PL-manifold M of dimension 3 Then f is homotopic to a map g with P n (g) = if and only if NP n (f) = 0 Theorem 42 ([15, Proposition 32]) Let f : M M be a self-map on a compact solvmanifold M of type (NR) Then NP n (f) = 0 if and only if either N(f n ) = 0 or N(f n ) = N(f n/q ) for some prime factor q n Immediately, we have 1 / HPer(ϕ) N(ϕ) = 0 By (31) and Lemma 33, this implies that all N(ϕ k ) = 0 and hence by the formula (41) all k / HPer(ϕ), which means that HPer(ϕ) = Consequently, Lemma 43 Let ϕ Endo(Γ) of type (I) The following are equivalent: (1) 1 / HPer(ϕ) (2) N(ϕ) = 0 (3) (λ 1 λ 2, λ 1 + λ 2 ) = (0, 1) (4) HPer(ϕ) = Lemma 44 Let ϕ Endo(Γ) of type (II) The following are equivalent: (1) 1 / HPer(ϕ) (2) N(ϕ) = 0 (3) (λ 1 λ 2, λ 1 + λ 2 ) = (1, 2) or (l, l + 1) (4) HPer(ϕ) =

13 NIELSEN THEORY ON NILMANIFOLDS 13 The following lemma characterizes all endomorphisms ϕ of Γ for which 1 HPer(ϕ) and 2 / HPer(ϕ) Lemma 45 Let ϕ Endo(Γ) Let ϕ be the first block submatrix of ϕ with eigenvalues λ 1 and λ 2 Suppose that 1 HPer(ϕ) Then (1) if ϕ is of type (I), then 2 / HPer(ϕ) (λ 1 λ 2, λ 1 + λ 2 ) = (0, 0), (0, 1) or (0, 2); (2) if ϕ is of type (II), then 2 / HPer(ϕ) (λ 1 λ 2, λ 1 + λ 2 ) = (0, 0), (0, 2) or (l, (l + 1)) with l ±1 Proof By the formula (41), 1 HPer(ϕ), and 2 / HPer(ϕ) if and only if N(ϕ) 0, and N(ϕ 2 ) = 0 or N(ϕ 2 ) = N(ϕ) Let ϕ be of type (I) If N(ϕ) 0 and N(ϕ 2 ) = 0 then by (31) (λ 1 λ 2, λ 1 + λ 2 ) = (0, 1) If N(ϕ 2 ) = N(ϕ) 0 then 1 l 2 = 1 l = 0, hence l = 0 or 2 Thus (λ 1 λ 2, λ 1 + λ 2 ) = (0, 0) or (0, 2) Let ϕ be of type (II) Suppose N(ϕ) 0 and N(ϕ 2 ) = 0 By Lemma 33, we have (λ 1 λ 2, λ 1 + λ 2 ) = (l, l 1), l ±1 Suppose next that N(ϕ 2 ) = N(ϕ) 0 Then (1 λ 2 1)(1 λ 2 2)(1 λ 2 1λ 2 2) (1 λ 2(m 1) 1 λ 2 2) = ±(1 λ 1 )(1 λ 2 )(1 λ 1 λ 2 ) (1 λ m 1 1 λ 2 ) 0, so λ 1 1, λ 2 1, λ 1 λ 2 1,, λ m 1 1 λ 2 1, (1 + λ 1 )(1 + λ 2 )(1 + λ 1 λ 2 ) (1 + λ m 1 1 λ 2 ) = ±1 Since the eigenvalues λ i are integers, we must have 1 + λ 1 = ±1, 1 + λ 2 = ±1, 1 + λ 1 λ 2 = ±1,, 1 + λ m 1 1 λ 2 = ±1 From 1 + λ i = ±1, we have λ i = 0 or 2; from 1 + λ 1 λ 2 = ±1, we have λ 1 λ 2 = 0 or 2 Hence (λ 1, λ 2 ) = (0, 0), (0, 2) or ( 2, 0) These satisfy the remaining conditions Hence, we have This proves our assertion (λ 1, λ 2 ) = (0, 0), (0, 2) or ( 2, 0) Lemma 46 Let ϕ Endo(Γ) Let ϕ be the first block submatrix of ϕ with eigenvalues λ 1 and λ 2 Let n be an integer with n 3 such that N(ϕ n ) 0 If n / HPer(ϕ), then (λ 1 λ 2, λ 1 + λ 2 ) = (0, 0) or (0, 1)

14 14 JONG BUM LEE AND WON SOK YOO Proof Since N(ϕ n ) 0 and n / HPer(ϕ), there would be a prime divisor q of n with N(ϕ n ) = N(ϕ n q ) For ϕ of type (I), we have 1 l n = 1 l n q 0 Then l n 1 l n q 1, = ±1 1 l n q Since n 3, l must be 0, or 1 with n and q both odd Thus we have For ϕ of type (II), we have m 1 j=1 (λ 1 λ 2, λ 1 + λ 2 ) = (0, 0) or (0, 1) ( 1 + (λ j 1 λ 2) n q + + (λ j 1 λ 2) (q 1)n q ( ) ( ) n (q 1)n n (q 1)n q 1 + λ1 + + λ q q λ2 + + λ q 2 = ±1 Since λ 1 and λ 2 are integers, each factor of the above identity must be an integer ±1: 1 (λ j 1 λ 2) n 1 (λ j 1 λ 2) n q 1 λ n 1 1 λ ) = ±1 (j = 1,, m 1), n q 1 = ±1, 1 λ n 2 1 λ n q 2 = ±1 Hence we must have λ 1, λ 2, λ j 1 λ 2 = 0, or 1 with n and q both odd The case of λ 1 = λ 2 = 1 is impossible as λ 1 λ 1 = 1 Thus we have (λ 1 λ 2, λ 1 + λ 2 ) = (0, 0) or (0, 1) Now we are ready to prove our main results Theorem 47 Let ϕ Endo(Γ) of type (I) Let ϕ be the first block submatrix of ϕ with eigenvalues λ 1 and λ 2 Then (1) (λ 1 λ 2, λ 1 + λ 2 ) = (0, 1) if and only if HPer(f) = (2) (λ 1 λ 2, λ 1 + λ 2 ) = (0, 0) or (0, 1) if and only if HPer(f) = {1} (3) (λ 1 λ 2, λ 1 + λ 2 ) = (0, 2) if and only if HPer(f) = N {2} (4) Except the above cases, HPer(f) = N Proof Note that the sets HPer(f) in these four cases are distinct It suffices to show the If part Case (1): It follows from Lemma 43 Case (2): If (λ 1 λ 2, λ 1 + λ 2 ) = (0, 0), then λ 1 = λ 1 = 0 and so N(ϕ k ) = 1 for all k By our formula (41), it follows that HPer(ϕ) = {1} If (λ 1 λ 2, λ 1 + λ 2 ) = (0, 1), then two eigenvalues are 0 and 1 Thus, N(ϕ k ) = 1 ( 1) k for all k By (41) again, HPer(ϕ) = {1}

15 NIELSEN THEORY ON NILMANIFOLDS 15 Case (3): If (λ 1 λ 2, λ 1 + λ 2 ) = (0, 2), then N(ϕ k ) = 1 ( 2) k for all k By Lemmas 45 and 46, HPer(ϕ) = N {2} Case (4): In the remaining cases, by Lemma 43, we see that N(ϕ) 0 and 1 HPer(ϕ) Lemma 45 ensures that 2 HPer(f) Since N(ϕ k ) 0 for all k, by Lemma 46, we obtain that k HPer(f) for k 3 Therefore HPer(f) = N Theorem 48 Let ϕ Endo(Γ) of type (II) Let ϕ be the first block submatrix of ϕ with (integer) eigenvalues λ 1 and λ 2 Then (1) (λ 1 λ 2, λ 1 + λ 2 ) = (1, 2) or (l, l + 1) if and only if HPer(ϕ) = (2) (λ 1 λ 2, λ 1 + λ 2 ) = (0, 0) or (0, 1) if and only if HPer(ϕ) = {1} (3) (λ 1 λ 2, λ 1 + λ 2 ) = (0, 2) if and only if HPer(ϕ) = N {2} (4) (λ 1 λ 2, λ 1 +λ 2 ) = (l, (l+1)) with l 0, ±1 if and only if HPer(ϕ) = N 2N (5) Except the above cases, HPer(ϕ) = N Proof The proof is similar to the above proof Case (1): It follows from Lemma 44 Case (2): If (λ 1 λ 2, λ 1 + λ 2 ) = (0, 0), then λ 1 = λ 2 = 0 Thus, N(ϕ k ) = 1 for all k By (41), HPer(ϕ) = {1} If (λ 1 λ 2, λ 1 + λ 2 ) = (0, 1), the two eigenvalues are 0 and 1 Thus, N(ϕ k ) = 1 ( 1) k for all k By (41) again, HPer(ϕ) = {1} Cases (3) and (4): Assume 1 HPer(ϕ) and 2 / HPer(ϕ) By Lemma 45, (λ 1 λ 2, λ 1 + λ 2 ) = (0, 0), (0, 2) or (l, (l + 1)) with l ±1 In the first case, we already observed that HPer(ϕ) = {1} In the second case, by Lemma 33 N(ϕ n ) 0 for all n, and by Lemma 46 all n 3 belong to HPer(ϕ), hence HPer(ϕ) = N {2}, ie, Case (3) In the last case, ϕ n is still of type (II) and has eigenvalues {µ 1, µ 2 } = {( 1) n, ( l) n }, hence by Lemma 44 N(ϕ n ) = 0 if and only if {µ 1, µ 2 } = { 1, 1} or {1, k} for some integer k If {µ 1, µ 2 } = { 1, 1} then µ 1 = µ 2 = 1, hence ( 1) n = ( l) n = 1, which forces l = 1, but l ±1 Hence this case cannot occur If {µ 1, µ 2 } = {1, k} then since l ±1, we must have ( 1) n = 1, hence n is even By (41) all even n 3 cannot belong to HPer(ϕ) By Lemma 46, if l 0 then all odd n 3 belong to HPer(ϕ) Consequently, we obtain Case (4) Case (5): In the remaining cases, from Lemma 44, we see that N(ϕ) 0 Thus, 1 HPer(ϕ) Lemma 45 ensures that 2 HPer(ϕ) Moreover, by Lemma 33 N(ϕ k ) 0 for each k Hence by Lemma 46, we obtain that k HPer(f) for k 3 Therefore HPer(f) = N Acknowledgement The authors would like to thank Karel Dekimpe for thorough reading, pointing out some errors and valuable comments on the original version

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17 NIELSEN THEORY ON NILMANIFOLDS 17 [25] J B Lee and X Zhao, Nielsen type numbers and homotopyminimal periods for maps on the 3-solvmanifolds, Algebr Geom Topol, 8 (2008), [26] V Roman kov, Twisted conjugacy classes in nilpotent groups, J Pure Appl Alg, 215 (2011), Department of Mathematics, Sogang University, Seoul 04107, KOREA address: jlee@sogangackr Department of Applied Mathematics, Kumoh National Institute of Technology, Kumi 39177, KOREA address: wsyoo@kumohackr