Homotopy types and Nielsen numbers ofperiodic homotopy idempotents on tori
Yasuzo Nishimura and Naomi Nishimura
Abstract. In this paper, we determine the Nielsen numbers of periodichomotopy idempotents on the n-dimensional torus Tn. In the specialcase n = 2, we classify the homotopy types of periodic homotopy idem-potents on T 2 using an elementary number-theoretic discussion.
Let X be a compact connected polyhedron and let f : X → X be a self-map. The set of all fixed points of f is denoted by Fix(f), and the number ofessential fixed point classes is denoted by N(f) and called the Nielsen numberof f (see Section 2). The Nielsen number is important in the fixed pointtheory. In fact, N(f) is a homotopy-type invariant of f (see Theorem 2.3(iii))and concerns with the least number of fixed points in a homotopy class [f ].A self-map f : X → X is said to be a periodic homotopy idempotent of periodk ≥ 1 if fr is homotopic to fr+k for some integer r ≥ 0. Special cases arehomotopy idempotents (f � f2) and periodic homotopy equivalences of periodk ≥ 1 (idX � fk). Moreover, we say that f is a homotopy-type idempotent iff and f2 have the same homotopy type; i.e., h◦f � f2◦h for some homotopyequivalence h : X → X. For the Nielsen number of a homotopy idempotentwe have the following conjecture (see [6, p. 21]).
Conjecture (Geoghegan 1979). If f is a homotopy idempotent on a compactconnected polyhedron X, then N(f) ≤ 1.
This conjecture is related to the splitting problem of homotopy idem-potents (cf. [3], [4] and [5]). For example, any homotopy idempotent on asurface is splittable (cf. [3, Theorem 1.9]) and N(f) ≤ 1 for any splittablemap f , therefore the above conjecture is true for any surface X.
In this paper, we are interested in the Nielsen numbers of general peri-odic homotopy idempotents on tori. In Section 2 we first recall the definitionsand the basic results in the Nielsen fixed point theory. By virtue of Nielsen–Brouwer’s theorem [6, p. 33, Example 2] we can easily compute the Nielsennumber for a self-map on the torus Tn. It is easily seen that N(f) ≤ 1 for anyhomotopy idempotent f : Tn → Tn. In Section 3 we discuss periodic homo-topy equivalences on the torus. In the special case n = 2 we can classify thehomotopy types of periodic homotopy equivalences on T 2 using the number-theoretic discussion. Moreover, we study the Nielsen numbers of periodichomotopy equivalences on the general n-dimensional torus Tn. In Section 4we discuss periodic homotopy idempotents and homotopy-type idempotentson the torus. In the special case n = 2 we can classify the homotopy typesof periodic homotopy idempotents and homotopy-type idempotents on T 2.Moreover, we study their Nielsen numbers in the general n-dimensional toruscase.
This paper is based on the Master thesis [8] of the second author (hermaiden name is Muragishi).
2. Review of Nielsen fixed point theory
In this section we recall some basic results in the Nielsen fixed point theory(see [2] or [6] for details). Let f : X → X be a self-map of a compact connectedpolyhedron X. Two fixed points x and y are said to be f -equivalent if thereis a path τ from x to y on X such that τ and f ◦ τ are homotopic relative toendpoints. Such an f -equivalence class F is called a fixed point class. Thereare finitely many fixed point classes, each of which is a compact open subsetof the fixed point set Fix(f).
With each fixed point class F we associate an integer index(f, F ) calledthe index of F . Here we explain it briefly (for more details see [2, p. 87]).
Definition 2.1. Let U ⊂ Rn be an open set and let f : U → Rn be a map suchthat its fixed point set Fix(f) is compact. Define a map ϕ : U \ Fix(f) →Sn−1 by
ϕ(x) =x− f(x)
‖x− f(x)‖ .Take an open set V ⊂ U such that Fix(f) ⊂ V ⊂ V ⊂ U and V is a smoothn-dimensional manifold. Then the integer deg(ϕ|∂V ) is independent of thechoice of V , and it is defined to be the fixed point index of f on U , denotedby index(f, U).
Every compact polyhedronX can be embedded in some Euclidean spaceRn as a neighborhood retract, thus there exist an inclusion i : X → Rn
and a neighborhood W of i(X) in Rn and a retraction r : W → X suchthat r ◦ i = idX . Let f : U → X be a map on an open subset U ⊂ X.Using the maps i and r, we can define the fixed point index of f on U byindex(f, U) := index(i ◦ f ◦ r, r−1(U)), the latter being the index in Rn. It
Periodic homotopy idempotents on tori 3
is independent of the choices of n, i, W and r. Moreover, for an isolatedsubset S ⊂ Fix(f) (i.e., compact open subset of Fix(f)), we define the fixedpoint index as index(f, S) := index(f, V ), where V is an open set such thatS = Fix(f) ∩ V . It is also independent of the choice of V .
Definition 2.2. Let f : X → X be a self-map of a compact connected poly-hedron X. A fixed point class F is said to be essential if its fixed point indexindex(f, F ) is nonzero. The number of essential fixed point classes is calledthe Nielsen number of f , denoted by N(f).
Two self-maps f : X → X and g : Y → Y have the same homotopytype if there exists a homotopy equivalence h : X → Y such that h◦f � g◦h.We recall the following basic result for the Nielsen number.
Theorem 2.3.
(i) N(f) is a homotopy invariant of f : X → X.(ii) N(f ◦ g) = N(g ◦ f) for any maps f : X → Y and g : Y → X.(iii) If two self-maps f : X → X and g : Y → Y have the same homotopy
type, then N(f) = N(g).
Let f : X → X be a self-map of a compact connected polyhedron X.The sum of the indices of all fixed point classes is equal to the Lefschetznumber L(f) defined by
L(f) =∑q
(−1)q Trace{f∗ : Hq(X;Q) → Hq(X;Q)
}.
Note that if X is simply connected, then there is only one fixed point classfor a self-map f : X → X, whose index is L(f). Consider the number
MF[f ] := Min{#Fix(g)|g � f},the least number of fixed points in the homotopy class [f ] of f . If X is a nicespace such as a manifold of dimension greater than or equal to 3 or a surfaceof nonnegative Euler characteristic, the equality N(f) = MF[f ] holds for anyself-map f : X → X (cf. [6, Section 1, Theorem 6.3]).
We consider that S1 is a unit circle on C and is a multiplicative groupwith the unit 1. Let f : Tn → Tn be a self-map of the n-dimensional torusTn = S1 × · · · × S1. The endomorphism induced by f on the fundamentalgroup π1(T
n) ∼= Z ⊕ · · · ⊕ Z is represented by an n × n integral matrixA = (ai,j). Conversely, with such an n× n integral matrix A = (ai,j) we canassociate the map fA : Tn → Tn defined by
fA(z1, . . . , zn) = (za1,1
1 · · · za1,nn , . . . , z
an,1
1 · · · zan,nn ).
Evidently its associated map fA is homotopic to the original map f . We caneasily compute the Nielsen number of a self-map f on the torus Tn by use ofthe following theorem (see also [1]).
Theorem 2.4 (Nielsen–Brouwer). Let fA : Tn → Tn be the self-map asso-ciated with an integral matrix A. Then N(fA) = |det(In − A)| = |ΦA(1)|,
4 Y. Nishimura and N. Nishimura JFPTA
where In is the unit matrix of degree n and ΦA is the eigenpolynomial of A.Moreover, N(f) = MF[f ] = |L(f)| for any self-map f on the torus.
By virtue of Theorem 2.4 we can easily verify the conjecture of Geoghe-gan in the case when X is the n-dimensional torus Tn as in the followingproposition.
Proposition 2.5. Let f : Tn → Tn be a homotopy idempotent (f � f2). ThenN(f) ≤ 1, in particular N(f) = 0 if f is homotopically nontrivial.
Proof. Let A be a matrix representing the induced endomorphism f∗ onπ1(T
n). Then A2 must be equal to A because f � f2. Therefore, the eigenval-ues of A are zero or one. If the eigenvalues of A are all zero, then N(f) = 1,else N(f) = 0. When N(f) = 1, A must be zero matrix because A2 = A, andhence f is homotopically trivial. �
An integral n×n matrix P is unimodular if there exists an integral n×nmatrix Q such that PQ = QP = In. The set of n × n unimodular matricesis denoted by GL(n,Z). Note that an integral n× n matrix P is unimodularif and only if detP = ±1. For two integral matrices A and B we say that Ais similar to B if there exists a unimodular matrix P such that AP = PB.Let f, g : Tn → Tn be self-maps whose induced endomorphisms f∗ and g∗ onπ1(T
n) are represented by integral matrices A and B, respectively. Then fand g have the same homotopy type if and only if A is similar to B. We noticethat if A is similar to B, then they have the same eigenvalues and the sameeigenpolynomial. For any integral polynomial Φ(x) = xn+an−1x
n−1+· · ·+a0there exists a matrix A whose eigenpolynomial is Φ(x). For example, we canchoose the matrix A = (ai,j) with entries⎧⎪⎨
⎪⎩an,j = −aj−1 (1 ≤ j ≤ n),
ai,i+1 = 1 (1 ≤ i < n),
ai,j = 0 (otherwise)
(2.1)
so that its eigenpolynomial ΦA(x) is equal to Φ(x).
3. The periodic homotopy equivalences on tori
According to Proposition 2.5 the Nielsen number of any homotopy idem-potent f : Tn → Tn must be zero or one (particularly, N(f) = 0 if f ishomotopically nontrivial). We shall first classify the homotopy types of self-maps on T 2 whose Nielsen number is zero.
Proposition 3.1. Let A =(a bc d
)be a 2 × 2 integral matrix whose eigenval-
ues are integers λ and μ. Then A is similar to the matrix(λ ν0 μ
)for some
integer ν. Here the integer ν is given uniquely such as 0 ≤ ν ≤ [|λ − μ|/2]when λ �= μ, and 0 ≤ ν when λ = μ.
Proof. We may assume that b �= 0 or c �= 0 because the b = c = 0 caseis obvious. Then we can choose an integral eigenvector
( xy
)associated with
the eigenvalue λ where x and y are relatively prime. Choose integers z and w
Periodic homotopy idempotents on tori 5
satisfying xw−yz = 1 and set P =( x zy w
), which is unimodular matrix. Then
there exists an integer ν such that P−1AP =(λ ν0 μ
). It is easy to verify that(
λ ν0 μ
)is similar to
(λ ν′0 μ
)if and only if ν + ν′ ≡ 0 or ν − ν′ ≡ 0 mod |λ− μ|
when λ �= μ, |ν| = |ν′| when λ = μ. Therefore, we can uniquely determine νsuch as 0 ≤ ν ≤ [|λ− μ|/2] when λ �= μ, and 0 ≤ ν when λ = μ. �
Let f : T 2 → T 2 be a self-map whose induced endomorphism f∗ isrepresented by a 2×2 integral matrix A. The Nielsen number of f is zero if andonly if A has the eigenvalue one. Using Proposition 3.1 we can immediatelyobtain the following theorem.
Theorem 3.2. Let f : T 2 → T 2 be a self-map such that N(f) = 0. Then ithas the same homotopy type as one of the following maps:
(i) fμ(z1, z2) = (z1zμ2 , z2), μ ≥ 0;
(ii) fλ,ν(z1, z2) = (z1zν2 , z
λ2 ), λ �= 1, 0 ≤ ν ≤ [|λ− 1|/2].
Corollary 3.3. Let f : T 2 → T 2 be a homotopy idempotent which is neverhomotopically trivial. Then it has the same homotopy type as either idT 2
(μ = 0) or g+ = idS1 ×ε : (z1, z2) �→ (z1, 1) (λ = ν = 0), where ε is aconstant map on S1 such that ε(z) = 1.
Denote the primitive kth root of one by ρk = exp(2π√−1/k) and the
cyclotomic polynomial of ρk by ψk(x) (i.e., ψk(x) is an irreducible polynomialsuch that ψk(ρk) = 0). It is well known that
degψk(x) = ϕ(k) := #{1 ≤ i ≤ k | i and k are relatively prime}is the Euler function. If the factorization into prime factors of k is givenby k =
∏i p
eii , then its Euler function is given by ϕ(k) =
∏i p
ei−1i (pi − 1).
Especially ϕ(pe) = pe−1(p − 1), ψp(x) = xp−1 + · · · + x + 1 and ψpe(x) =
ψp(xpe−1
) for any prime p. Notice that ψk(1) = p if k �= 1 is a power of aprime p, and ψk(1) = 1 otherwise. For example,
ϕ(1) = 1, ψ1(x) = x− 1, A1 = 1,
ϕ(2) = 1, ψ2(x) = x+ 1, A2 = −1,
ϕ(3) = 2, ψ3(x) = x2 + x+ 1, A3 =
(0 1−1 −1
),
ϕ(4) = 2, ψ4(x) = x2 + 1, A4 =
(0 1−1 0
),
ϕ(6) = 2, ψ6(x) = x2 − x+ 1, A6 =
(0 1−1 1
).
(3.1)
Here Ak = Aψkis the matrix with eigenpolynomial ψk(x) given by (2.1).
Since (Ak)k = Iϕ(k), the associated map fAk
on Tϕ(k) is a periodic homo-
topy equivalence of period k (fkAk
� idTϕ(k)) on Tϕ(k). When the integralpolynomial Φ(x) is given as a product of the cyclotomic polynomials such asΦ(x) =
∏j ψkj (x), we usually choose the matrix AΦ =
⊕j Akj so that its
eigenpolynomial is Φ(x).
6 Y. Nishimura and N. Nishimura JFPTA
Next we consider a certain Diophantine equation.
Lemma 3.4. There exists an integral solution (x, y) of the following indeter-minate equations:
cx2 − (2a+ 1)xy − by2 = ±1, (3.2)
cx2 − 2axy − by2 = ±1, (3.3)
where a, b and c are any integers such that a2 + a + 1 = −bc for (3.2) anda2 + 1 = −bc for (3.3).
Proof. (i) For (3.2), we may assume that a �= 0 and c > 0. Since a2 + a+1 isodd, c is also odd and hence 2c is not square. Fix a positive integer e such that(e−1)2 < 2c < e2 and consider the set {(2a+1)Y−V | 0 ≤ Y < e, 0 ≤ V < e}consisting of e2 pairs of integers (Y, V ). Since e2 > 2c, there exist two pairsof integers (Y1, V1) and (Y2, V2) such that (2a + 1)Y1 − V1 ≡ (2a + 1)Y2 −V2 mod 2c. Set y = Y1 − Y2 and v = V1 − V2. Since v ≡ (2a + 1)y mod 2c,it follows that 3y2 + v2 ≡ (3 + (2a + 1)2)y2 = 4(a2 + a + 1)y2 ≡ 0 mod 4c.Since |y|, |v| ≤ e− 1, we obtain that 0 < 3y2 + v2 ≤ 4(e− 1)2 < 8c, that is,3y2 + v2 = 4c. Then x = {(2a+ 1)y − v}/2c is an integer and the pair (x, y)is a solution of (3.2).
(ii) For (3.3), we may assume that a �= 0, b < 0 and c > 0. Since a2 + 1is not square, either −b or c is not square. We can assume that c is not squareby changing −b to c if necessary. Fix an integer e such that (e− 1)2 < c < e2
and consider the set {aY − V | 0 ≤ Y < e, 0 ≤ V < e} consisting of e2 pairsof integers (Y, V ). Since e2 > c, there exist two pairs of integers (Y1, V1) and(Y2, V2) such that aY1−V1 ≡ aY2−V2 mod c. Set y = Y1−Y2 and v = V1−V2.Since v ≡ ay mod c, it follows that v2 + y2 ≡ (a2 + 1)y2 ≡ 0 mod c. Since|y|, |v| ≤ e−1, we obtain that 0 < y2+v2 ≤ 2(e−1)2 < 2c, that is, y2+v2 = c.Then x = (ay−v)/c is an integer and the pair (x, y) is a solution of (3.3). �
Remark 3.5. The Diophantine equations (3.2) and (3.3) of Lemma 3.4 aresimilar to the Pell equation x2−Dy2 = ±1, or more general ax2− by2 = ±1.The latter equation has been studied in [7] and [10].
We shall classify the homotopy types of periodic homotopy equivalencesf (fk � idT 2) on the two-dimensional torus T 2.
Proposition 3.6. A self-map f : T 2 → T 2 is a periodic homotopy equivalenceof period three or four but not one if and only if it has the same homotopytype as one of the following maps:
(i) ω : (z1, z2) �→ (z2, z−11 z−1
2 ), when the period is three;(ii) κ : (z1, z2) �→ (z2, z
−11 ), when the period is four.
Proof. The “if” part is immediate because ω3 = idT 2 and κ4 = idT 2 . We shallprove the “only if” part. Let A =
(a bc d
)be an integral matrix representing
the induced automorphism f∗ on π1(T2).
(i) In case the period of f is three, it is sufficient to show that A is similarto Ω =
(0 1−1 −1
)when A3 = I2 and A �= I2. Since A has the eigenvalues ρ3
and ρ23, we note that a + d = ρ3 + ρ23 = −1 and ad − bc = ρ33 = 1, that is,
Periodic homotopy idempotents on tori 7
a2 + a+1 = −bc. From Lemma 3.4, the indeterminant equation (3.2) has anintegral solution (x, y). Then P =
( x zy w
)is a unimodular matrix such that
AP = PΩ, where z = −ax − by and w = −cx + (1 + a)y, thus A is similarto Ω.
(ii) In case the period of f is four, it is sufficient to show that A is similarto K =
(0 1−1 0
)when A4 = I2 and A2 �= I2. Since A has the eigenvalues ρ4
and ρ34, we note that a + d = ρ4 + ρ34 = 0 and ad − bc = ρ44 = 1, that is,a2 + 1 = −bc. From Lemma 3.4, the indeterminant equation (3.3) has anintegral solution (x, y). Then P =
( x zy w
)is a unimodular matrix such that
AP = PK, where z = −ax−by and w = −cx+ay, thus A is similar to K. �
Theorem 3.7. Let f : T 2 → T 2 be a periodic homotopy equivalence of periodk ≥ 1 (fk � idT 2). Then the period k is 1, 2, 3, 4 or 6 and f has the samehomotopy type as one of the following maps with Nielsen number:
(i) idT 2 : (z1, z2) �→ (z1, z2) with N(idT 2) = 0, when k = 1;(ii) ϑ : (z1, z2) �→ (z1, z
−12 ) with N(ϑ) = 0, ϑ′ : (z1, z2) �→ (z1z2, z
−12 ) with
N(ϑ′) = 0 or κ2 : (z1, z2) �→ (z−11 , z−1
2 ) with N(κ2) = 4, when k = 2;(iii) ω : (z1, z2) �→ (z2, z
−11 z−1
2 ) with N(ω) = 3, when k = 3;(iv) κ : (z1, z2) �→ (z2, z
−11 ) with N(κ) = 2, when k = 4;
(v) κ2ω : (z1, z2) �→ (z2, z−11 z2) with N(κ2ω) = 1, when k = 6.
Proof. Let A be an integral matrix representing the induced automorphismf∗ on π1(T
2). From the assumption on f we see that any eigenvalue λ of Asatisfies λk = 1. Since the Euler function ϕ(k) ≤ deg(A) = 2, the period kmust be 1, 2, 3, 4 or 6. According to Theorem 3.2(i), f is homotopic to theidentity map when the eigenvalues of A are both one, and f is homotopic tothe map κ2 when the eigenvalues of A are both −1, in other words, whenthe eigenvalues of −A are both one. According to Theorem 3.2(ii), f hasthe same homotopy type as either ϑ or ϑ′ when the eigenvalues of A are 1and −1. Hence our result is immediate when k = 1 or 2. When k = 3 or 4 itshomotopy type is uniquely determined by virtue of Proposition 3.6 and itsNielsen number is given as N(ω) = 3 or N(κ) = 2, respectively.
When k = 6 we note that f3 has the same homotopy type as one of thethree maps ϑ, ϑ′ or κ2. If f3 has the same homotopy type as κ2, then κ2f isa periodic homotopy equivalence of period three. Therefore, f has the samehomotopy type as κ2ω. If f3 has the same homotopy type as either ϑ or ϑ′,then P−1A3P =
(1 ν0 −1
)with ν = 0 or 1 for some unimodular matrix P .
Set B = P−1AP . Since B2 = B−1(1 ν0 −1
)and detB = −1, we can easily
verify that B =(1 ν0 −1
), thus f itself has the same homotopy types as either
ϑ or ϑ′. �
Corollary 3.8. For any periodic homotopy equivalence of period greater thanor equal to 3 on the two-dimensional torus T 2 there exists an essential fixedpoint class.
Remark 3.9. Theorem 3.7 can be said in other words as follows. The generatorof any finite cyclic subgroup of GL(2,Z) is similar to one of the matrices I2,
8 Y. Nishimura and N. Nishimura JFPTA
−I2,(1 00 −1
),(1 10 −1
), A3, A4 or A6 (where Ai is defined in (3.1)). In fact it is
known that there are 13 nonconjugate finite subgroups of GL(2,Z) (see [9,Section 14, Chapter 9]). Moreover, we notice that SL(2,Z) may be generatedby A3 and A4 (see [9, Chapter 7]). By using these facts, we can also proveTheorem 3.7.
For a periodic homotopy equivalence f on n-dimensional torus Tn itsNielsen number N(f) is given as follows.
Theorem 3.10. Let f : Tn → Tn be a periodic homotopy equivalence of periodk ≥ 1 (fk � idTn), and let {pi}i=0,1,2,... denote the set of primes, i.e., p0 = 2,p1 = 3, p2 = 5 and so on. Then its Nielsen number N(f) is zero or
∏i≥0 p
eii ,
where ei ≥ 0 are integers such that ei = 0 unless pi is a prime divisor of k,e0+
∑i≥0(pi−1)ei ≤ n and e0− e0 ≡ n mod 2 for some e0 with 0 ≤ e0 ≤ e0.
Proof. Let A be an integral matrix representing the induced automorphismf∗ on π1(T
n). Since f is of period k ≥ 1, any eigenvalue λ of A satisfiesλk = 1. Therefore, its eigenpolynomial is written as ΦA(x) =
∏j ψkj (x),
where kj ’s are divisors of k such that LCM{kj} = k and∑
j ϕ(kj) = n. If
kj = 1 for some j, then N(f) = |ΦA(1)| = 0. Assume that kj ≥ 2 for anyj. Note that if kj is a power of a prime p, then ψkj (1) = p, else ψkj (1) = 1.Then the Nielsen number of f is given by N(f) =
∏j ψkj (1) =
∏i≥0 p
eii ,
where ei = #{kj | kj is a power of pi}. We set e0 = #{kj | kj = 2} ande0 = e0 − e0. Then we obtain the condition that
ϕ(2)e0 + ϕ(4)e0 +∑i≥1
ϕ(pi)ei = e0 +∑i≥0
(pi − 1)ei ≤∑j
ϕ(kj) = n
and n ≡ e0 mod 2 because ϕ(kj) is even except for kj = 2. �Remark 3.11. Conversely, for any integers ei’s satisfying the condition given
in Theorem 3.10 we set Φ = ψe02 ψe0
4 ψm/26
∏i≥1 ψ
eipi, where m = n − e0 −∑
i≥0(pi − 1)ei is even. For the integral polynomial Φ(x) of degree n, we can
choose the n × n matrix AΦ whose eigenpolynomial is Φ(x). The map fAΦ
associated with the matrix AΦ is a periodic homotopy equivalence on Tn andits Nielsen number is given as N(fAΦ) =
∏i≥0 p
eii .
Example. If f : T 4 → T 4 is a periodic homotopy equivalence, then its Nielsennumber N(f) is 0, 1, 2, 3, 4, 5, 6, 8, 9, 12 or 16.
4. The periodic homotopy idempotents on tori
We here consider periodic homotopy idempotents f (fr � fr+k with r ≥ 1)on the torus Tn. First we shall classify their homotopy types in the casen = 2.
Theorem 4.1. Let f : T 2 → T 2 be a periodic homotopy idempotent of periodk ≥ 1 (fr � fr+k), but not a periodic homotopy equivalence (fk �� idT 2).Then the pair (r, k) is (1, 1), (1, 2) or (2, 1) and f has the same homotopytype as one of the following maps with Nielsen number:
Proof. Let λ be an eigenvalue of the matrix A representing the induced endo-morphism f∗ on π1(T
2). The assumption fr � fr+k implies that λr = λr+k,thus λ = 0 or λk = 1. Since f is not a periodic homotopy equivalence,detA = 0 and hence zero must be one of the eigenvalues of A. Another eigen-value is 0 or ±1. According to Proposition 3.1, A is similar to one of thefollowing matrices
(0 ξ0 0
),(1 00 0
)or
(−1 00 0
)with ξ ≥ 0. �
For a periodic homotopy idempotent f on the n-dimensional torus Tn
its Nielsen number N(f) is given as follows.
Theorem 4.2. Let f : Tn → Tn be a periodic homotopy idempotent of periodk ≥ 1 (fr � fr+k), but not a periodic homotopy equivalence (fk �� idTn),and let {pi}i=0,1,2,... denote the set of primes, i.e., p0 = 2, p1 = 3, p2 = 5 andso on. Then its Nielsen number N(f) is zero or
∏i≥0 p
eii , where ei ≥ 0 are
integers such that ei = 0 unless pi is a prime divisor of k and∑
i≥0(pi−1)ei ≤n− 1.
Proof. Let λ be an eigenvalue of the matrix A representing the induced en-domorphism f∗ on π1(T
n). From the assumption on f we see that λ = 0 orλk = 1, and zero must be one of the eigenvalues of A. Therefore, its eigen-polynomial is written as ΦA(x) = xm
∏j ψkj (x) (m ≥ 1). If kj = 1 for
some j, then N(f) = |ΦA(1)| = 1. Assume that kj ≥ 2 for any j. ThenN(f) =
∏i≥0 p
eii , where ei = #{kj | kj is a power of pi}. Obviously we ob-
tain the condition that∑i≥0
ϕ(pi)ei =∑i≥0
(pi − 1)ei ≤ n−m ≤ n− 1. �
Remark 4.3. Conversely, for any integers ei’s satisfying the condition given inTheorem 4.2 we set Φ(x) = xm
∏i≥0 ψ
eipi(x), where m = n−∑
i≥0(pi−1)ei ≥1. For the integral polynomial Φ(x) of degree n, we can choose the n×nmatrixAΦ with detAΦ = 0 whose eigenpolynomial is Φ(x). The map fAΦ associatedwith the matrix AΦ is periodic homotopy idempotent, but not a periodichomotopy equivalence, and its Nielsen number is given asN(fAΦ) =
∏i≥0 p
eii .
On the other hand, g+ = idTn−1 × ε is a periodic homotopy idempotent suchthat N(g+) = 0.
Example. If f : T 5 → T 5 is a periodic homotopy idempotent but not aperiodic equivalence, then its Nielsen number N(f) is 0, 1, 2, 3, 4, 5, 6, 8, 9,12 or 16.
Remark 4.4. In Theorems 4.1 and 4.2, we can relax the condition as: fr andfr+k have the “same homotopy type” (i.e., a self-map f should be calleda periodic “homotopy-type” idempotent) instead of fr � fr+k. Indeed thoseconditions are equivalent as shown in the following lemma.
10 Y. Nishimura and N. Nishimura JFPTA
Lemma 4.5. For any self-map f on Tn, if fr and fr+k have the same ho-motopy type for some integers r ≥ 0 and k ≥ 1, then f must be a periodichomotopy idempotent.
Proof. Let λi (i = 1, . . . , n) be eigenvalues of the matrix A associated withf : Tn → Tn. It is sufficient to show that λi = 0 or λq
i = 1 for some integerq ≥ 1. Since Ar is similar to Ar+k, we have{
λr1, λ
r2, . . . , λ
rn
}=
{λr+k1 , λr+k
2 , . . . , λr+kn
}.
We set λri = λr+k
si �= 0 (1 ≤ i ≤ m) and λj = 0 (m < j ≤ n). Here
a permutation σ =(
1 2 ··· ms1 s2 ··· sm
)can be decomposed into the product of
cyclic permutations ci = (si,1, si,2, . . . , si,ai) (1 ≤ i ≤ l, ai ≥ 1) suchthat
∐1≤i≤l{si,1, si,2, . . . , si,ai
} = {1, 2, . . . ,m}. We can assume that c1 =
(1, 2, . . . , a) by a suitable replacing λi, and so λri = λr+k
i+1 (i ∈ Z/a). Then
λra
1 = (λr1)
ra−1
=(λr+k2
)ra−1
=(λr(r+k)2
)ra−2
=(λ(r+k)2
3
)ra−2
= · · · =(λ(r+k)a−1
a
)r
= λ(r+k)a
1 ,
that is, λq1 = 1, where q = (r + k)a − ra. Similarly when λi �= 0, λq
i = 1 forsome integer q ≥ 1. �
We discuss the special case of r = k = 1 in the following. A self-mapf : X → X is a homotopy-type idempotent if f has the same homotopytype as f2. Here we shall study the homotopy types and Nielsen numbers ofhomotopy-type idempotents on the torus.
Proposition 4.6. If a homotopy equivalence f : T 2 → T 2 is a homotopy-typeidempotent, then it is a periodic homotopy equivalence of period three (orone).
Proof. Let A be an integral matrix representing the induced endomorphismf∗ on π1(T
2). Since A is similar to A2, the eigenvalues {λ1, λ2} of A coincidewith those {λ2
1, λ22} of A2. Since the rank of A is two, the eigenvalues of A
must be {ρ3, ρ23} or both one. In the first case, f3 � idT 2 because A3 = I2.In the latter case, A must be written
(1 μ0 1
)using Proposition 3.1. Since A is
similar to A2, it is immediate that μ = 0, thus A = I2. �Let f : T 2 → T 2 be a homotopy-type idempotent and A an integral
matrix representing the induced endomorphism f∗ on π1(T2). If the rank
of A is one, then the eigenvalues of A must be zero and one. According toTheorem 3.2(ii), f has the same homotopy type as the map g+ = idS1 × ε :(z1, z2) �→ (z1, 1). Using this fact and Proposition 3.6 we obtain the followingtheorem.
Theorem 4.7. Let f : T 2 → T 2 be a homotopy-type idempotent. Then it hasthe same homotopy type as one of the following maps with Nielsen number:
We notice that in the case (iv) f is a periodic homotopy equivalenceof period three and in the other cases f is a homotopy idempotent. For ahomotopy-type idempotent f on the n-dimensional torus its Nielsen numberN(f) is given as follows.
Theorem 4.8. Let f : Tn → Tn be a homotopy-type idempotent, and let{pi}i≥1 denote the set of odd primes. Then its Nielsen number N(f) is zeroor
∏i≥1 p
eii , where ei ≥ 0 are integers such that
∑i≥1(pi − 1)ei ≤ n.
Proof. Let λ1, . . . , λn be the eigenvalues of the matrix A representing theinduced endomorphism f∗ on π1(T
n). If λi = 1 for some i, then N(f) = 0.Assume that λi �= 1 for any i. Since A is similar to A2, the eigenvalues{λ1, . . . , λn} of A coincide with those {λ2
1, . . . , λ2n} of A2. We notice that
λi = 0 if λ2i = λi. We set λi �= λ2
i = λsi (1 ≤ i ≤ m) and λj = 0 (m < j ≤ n).Here a permutation σ =
(1 2 ··· ms1 s2 ··· sm
)can be decomposed into a product of
cyclic permutations ci = (si,1, si,2, . . . , si,ai) (1 ≤ i ≤ l) such that∐1≤i≤l
{si,1, si,2, . . . , si,ai} = {1, 2, . . . ,m}.
From the proof of Lemma 4.5, it follows that λ2ai−1si,1 = 1 (1 ≤ i ≤ l), thus
λsi,1 is a kith root of one for some divisor ki of 2ai − 1. Therefore, theeigenpolynomial of A is given by
ΦA(x) = xn−m∏
1≤i≤l
ψki(x),
where ki is odd and deg(ψki(x)) = ϕ(ki) is even, thus Σiϕ(ki) = m ≤ nis also even. Hence f is a periodic homotopy idempotent of the odd periodk, where k = LCM{ki}. Using Theorems 3.10 and 4.2 we can immediatelyobtain our result. �
Remark 4.9. From the consequence of Theorem 4.8, the Nielsen number ofa homotopy-type idempotent on Tn is zero or an odd number. Moreover,when n is odd, the integers which can be Nielsen numbers of homotopy-typeidempotents on Tn are the same as ones on Tn−1. For example, if f is ahomotopy-type idempotent on T 6 (or T 7), then its Nielsen number N(f) is0, 1, 3, 5, 7, 9, 15 or 27.
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12 Y. Nishimura and N. Nishimura JFPTA
[4] R. Geoghegan, Splitting homotopy idempotents which have essential fixed point.Pacific J. Math. 95 (1981), 95–103.
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Yasuzo NishimuraFaculty of Education and Regional StudiesUniversity of Fukui3-9-1 Bunkyo, Fukui 910-8507Japane-mail: [email protected]
