Nielsen type numbers and homotopy minimal periods for maps on solvmanifolds with \(\operatorname{Sol}_{1}^{4}\)-geometry
- Jang Hyun Jo^{1} and
- Jong Bum Lee^{1}Email author
https://doi.org/10.1186/s13663-015-0427-x
© Jo and Lee 2015
Received: 4 June 2015
Accepted: 15 September 2015
Published: 26 September 2015
Abstract
For all maps f on the special solvmanifolds with \(\operatorname{Sol}_{1}^{4}\)-geometry, we give explicit formulas for a complete computation of the Nielsen type numbers \(\operatorname{NP}_{n}(f)\) and \(\mathrm{N}\Phi _{n}(f)\). We also give a complete description of the sets of homotopy minimal periods of all such maps.
Keywords
MSC
1 Introduction
For a self-map \(f : X \rightarrow X\), the Lefschetz number \(L(f)\) and the Nielsen number \(N(f)\) give information concerning the fixed points of f. It is known that the Nielsen number gives more precise information about the existence of fixed points than the Lefschetz number, but its computation is in general very difficult. For the periodic points, two Nielsen type numbers \(\operatorname{NP}_{n}(f)\) and \(\mathrm{N}\Phi _{n}(f)\) were introduced by Jiang [1], 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.
It is obvious that these Nielsen numbers are much more powerful than the Lefschetz numbers in describing the periodic point sets of self-maps. Using fiber techniques on nilmanifolds and some solvmanifolds, Heath and Keppelmann [2] (see also [3]) succeeded in showing that the Nielsen numbers and the two Nielsen type numbers are related to each other under certain conditions. However, the computation of the Nielsen type numbers even on low dimensional infra-homogeneous spaces is a hard problem. See [4] for the Klein bottle and [5] for a three-dimensional flat Riemannian manifold.
There are four-dimensional geometries which were classified by Filipkiewicz [11], see also [12]. One of their model spaces is a simply connected four-dimensional unimodular solvable Lie group \(\operatorname{Sol}_{1}^{4}\). This group contains Nil^{3} as a nil-radical and the quotient by its center is Sol^{3}. Recall that Nil^{3} and Sol^{3} are model spaces for three-dimensional geometries.
In this paper, we are concerned with the special solvmanifolds with \(\operatorname{Sol}_{1}^{4}\)-geometry, i.e., the closed manifolds \(\Gamma \backslash \operatorname{Sol}_{1}^{4}\) which are quotient spaces of \(\operatorname{Sol}_{1}^{4}\) by its lattices Γ. For all continuous maps f on any special solvmanifolds with \(\operatorname{Sol}_{1}^{4}\)-geometry, we will give a complete description of the Nielsen type numbers \(\operatorname{NP}_{n}(f)\) and \(\mathrm{N}\Phi _{n}(f)\), and the homotopy minimal periods \(\operatorname{HPer}(f)\).
2 The Lie group \(\operatorname{Sol}_{1}^{4}\) and its Lie algebra
Remark 2.1
3 The lattices of \(\operatorname{Sol}_{1}^{4}\)
In this section we study lattices Γ of \(\operatorname{Sol}_{1}^{4}\) with [12] as our basic reference, and then we study continuous maps on the solvmanifold \(\Gamma \backslash \operatorname{Sol}_{1}^{4}\) up to homotopy.
Theorem 3.1
Notation
We denote such a lattice of \(\operatorname{Sol}_{1}^{4}\) by \(\Gamma_{k,N,\mathbf{p}}\).
Proof
In the theorem below, we study the homomorphisms on any lattice of \(\operatorname{Sol}_{1}^{4}\).
Theorem 3.2
- Type (I)
\(\phi({\gamma}_{0})={\gamma}_{0}{\gamma}_{1}^{r_{1}}{\gamma}_{2}^{r_{2}}\gamma _{3}^{q_{0}}\), \(\phi({\gamma}_{1})={\gamma}_{1}^{\mu}{\gamma}_{2}^{\frac {n_{21}}{n_{12}}\nu}\gamma_{3}^{q_{1}}\), \(\phi({\gamma}_{2})={\gamma}_{1}^{\nu}{\gamma}_{2}^{\mu+\frac {n_{22}-n_{11}}{n_{12}}\nu}\gamma_{3}^{q_{2}}\), \(\phi(\gamma_{3})=\gamma_{3}^{\mu(\mu+\frac {n_{22}-n_{11}}{n_{12}}\nu)-\frac{n_{21}}{n_{12}}\nu^{2}}\);
- Type (II)
\(\phi({\gamma}_{0})={\gamma}_{0}^{-1}{\gamma}_{1}^{r_{1}}{\gamma }_{2}^{r_{2}}\gamma_{3}^{q_{0}}\), \(\phi({\gamma}_{1})={\gamma}_{1}^{-\mu}{\gamma}_{2}^{\nu}\gamma _{3}^{q_{1}}\), \(\phi({\gamma}_{2})={\gamma}_{1}^{\frac{n_{11}-n_{22}}{n_{21}}\mu -\frac{n_{12}}{n_{21}}\nu}{\gamma}_{2}^{\mu}\gamma_{3}^{q_{2}}\), \(\phi(\gamma_{3})=\gamma_{3}^{-\mu^{2}-(\frac {n_{11}-n_{22}}{n_{21}}\mu-\frac{n_{12}}{n_{21}}\nu)\nu}\);
- Type (III)
\(\phi({\gamma}_{0})={\gamma}_{0}^{m}{\gamma}_{1}^{r_{1}}{\gamma }_{2}^{r_{2}}\gamma_{3}^{q_{0}}\), \(\phi({\gamma}_{1})=\gamma_{3}^{q_{1}}\), \(\phi({\gamma}_{2})=\gamma_{3}^{q_{2}}\), \(\phi(\gamma_{3})=1\) with \(m\ne\pm1\).
Proof
Remark 3.3
(Homomorphisms on Γ up to conjugacy)
4 Continuous maps on solvmanifolds \(\Gamma \backslash \operatorname{Sol}_{1}^{4}\)
Since the invariants that we are going to deal with are all homotopy invariants, we will assume in what follows that every continuous map on \(\Gamma \backslash \operatorname{Sol}_{1}^{4}\) is induced by a Lie group homomorphism Φ on \(\operatorname{Sol}_{1}^{4}\) preserving Γ.
From Theorem 3.2, we immediately obtain the following.
Proposition 4.1
Proof
Note that homomorphisms of distinct types are not conjugate to each other. We shall say a map on \(\Gamma \backslash \operatorname{Sol}_{1}^{4}\) is of type (I), (II) or (III) according to its homomorphism on Γ.
Corollary 4.2
Proof
Since \(\operatorname{tr}N=n_{11}+n_{12}>2\), the number \(\sqrt{(n_{11}+n_{22})^{2}-4}\) must be irrational. If \(\alpha=\mu-\tfrac{n_{11}-n_{22}-\sqrt {(n_{11}+n_{22})^{2}-4}}{2n_{12}}\nu=0\), then ν must be zero and hence \(\mu=0\) because μ, ν, \(n_{11}\), \(n_{12}\), \(n_{22}\) are all integers. It follows that \(\beta=\alpha=0\). The converse is the same. Similarly, we can show that \(\gamma=0\) if and only if \(\delta=0\). □
Remark 4.3
If f is of type (II), then \(f^{2}\) is of type (I).
Proposition 4.4
Proof
As it was mentioned earlier, we may assume that f, \(f'\) are induced respectively by Lie group homomorphisms \(\Phi,\Phi':\operatorname{Sol}_{1}^{4}\to \operatorname{Sol}_{1}^{4}\), both of which restrict to homomorphisms \(\varphi,\varphi': \Gamma_{k,N,\mathbf{p}}\to \Gamma_{k,N,\mathbf{p}}\). Because \(f\simeq f'\), φ and \(\varphi'\) differ by the conjugation by an element of \(\Gamma_{k,N,\mathbf{p}}\). Now our assertion follows from Remark 3.3. □
According to this result, \(\det\Phi_{*}\) and \(\varphi_{0}\), \(\varphi_{2}\) are all homotopy invariants. A map becomes of type (I), (II) or (III) according to what the value of \(\varphi_{0}\) is 1, −1 or the others. It should be noticed that the \(\varphi_{1}\) is not a homotopy invariant.
5 Lefschtez numbers and Nielsen type numbers
Recall that \(\operatorname{Sol}_{1}^{4}\) is of type \((\mathrm{R})\). Let f be a map on the special solvmanifold \(\Gamma \backslash \operatorname{Sol}_{1}^{4}\) of type \((\mathrm{R})\) with \(\Gamma=\Gamma_{k,N,\mathbf{p}}\). Then we may assume that f is induced by a Lie group homomorphism Φ. Consider a linearization \(\Phi_{*}=\operatorname{diag}\{\varphi_{0},\varphi_{1},\varphi _{2}\}\) of f. Then the main result, Theorem 3.1, of [19] implies that \(L(f)=\det(I-\Phi_{*})\). By [20], Theorem 2.1, we also have \(N(f)=|L(f)|\). See also the main result, Theorem 3.1, of [3].
Proposition 5.1
Proof
If f is of type (I), then \(\varphi_{0}=\varphi_{0}^{n}=1\) and hence \(L(f^{n})=0\). If f is of type (III), then \(\varphi_{1}=\varphi_{1}^{n}=0\) and \(\varphi _{2}=\varphi_{2}^{n}=0\), and hence \(L(f^{n})=1-\varphi_{0}^{n}\).
A connected solvable Lie group S is called of type \((\mathrm{NR})\) (for ‘no roots’) if the eigenvalues of \(\operatorname{Ad}(x):\mathfrak{S}\to \mathfrak{S}\) are always either equal to 1 or else they are not roots of unity. Solvable Lie groups of type \((\mathrm{NR})\) were considered first in Keppelmann and McCord [3]. Since our solvmanifold \(\Gamma \backslash \operatorname{Sol}_{1}^{4}\) is of type \((\mathrm{NR})\), we have the following.
Theorem 5.2
- (1)If f is of type (I), then$$\operatorname{NP}_{n}(f)=\mathrm{N}\Phi _{n}(f) =0. $$
- (2)Suppose f is of type (II). Then we have:
- (a)
If n is odd and \(\varphi_{2}=\pm1\), then \(\operatorname{NP}_{n}(f)=\mathrm{N}\Phi _{n}(f) =0\).
- (b)If n is odd and \(\varphi_{2}\ne\pm1\), then$$\operatorname{NP}_{n}(f)=2 \sum_{m|n}\mu(n/m)\bigl|1- \varphi_{2}^{2m}\bigr|,\qquad \mathrm{N}\Phi _{n}(f) =2\bigl|1- \varphi_{2}^{2n}\bigr|. $$
- (c)If \(n=2^{r}n_{0}\) with \(n_{0}\) odd, then$$\operatorname{NP}_{n}(f)=0,\qquad \mathrm{N}\Phi _{n}(f) = \textstyle\begin{cases} 0 & \textit{when }\varphi_{2}=\pm1,\\ 2|1-\varphi_{2}^{2n_{0}}| &\textit{when }\varphi_{2}\ne\pm1. \end{cases} $$
- (a)
- (3)If f is of type (III), then$$\operatorname{NP}_{n}(f) =\sum_{m|n}\mu(n/m)\bigl|1- \varphi_{0}^{2m}\bigr|, \qquad \mathrm{N}\Phi _{n}(f) =\bigl|1- \varphi_{0}^{2n}\bigr|. $$
Proof
In case (1), the Nielsen number \(N(f^{m})=0\) for all positive integers m. The map f has no essential periodic orbit classes of any period. It follows that \(\operatorname{NP}_{n}(f)=\mathrm{N}\Phi _{n}(f)=0\).
Since \(\Gamma \backslash \operatorname{Sol}_{1}^{4}\) is a solvmanifold of type \((\mathrm{NR})\), by [2], Theorem 1.2, we have \(\mathrm{N}\Phi _{m}(f)=N(f^{m})\) and \(\operatorname{NP}_{m}(f)= \sum_{q\mid m} \mu(q) N(f^{\frac{m}{q}})\) for all \(m|n\) provided \(N(f^{n})\ne0\). This proves our case (2)(b) and case (3) because of the following reason: When f is of type (III), \(N(f^{n})\ne0\); when f is of type (II), \(N(f^{n})=(1-(-1)^{n})|1-\varphi_{2}^{2n}|\), and \(N(f^{n})\ne0\) if and only if n is odd and \(\varphi_{2}\ne\pm1\). In case (2) when n is odd and \(\varphi_{2}=\pm1\), i.e., in case (2)(a), we note that \(N(f^{m})=0\) for all positive integers m. Hence \(\operatorname{NP}_{n}(f)=\mathrm{N}\Phi _{n}(f)=0\).
Consider the case (2) with n even. Since \(N(f^{n})=0\), it follows that \(\operatorname{NP}_{n}(f)=0\). Let \(n= 2^{r} n_{0}\) for odd \(n_{0}\). By Proposition 5.1, \(f^{q}\) has no essential fixed point class for every even factor q of n. Thus, the set of essential fixed point classes of \(f^{q}\) with \(q\mid n\) is the same as that of \(f^{q}\) with \(q\mid n_{0}\). Thus, \(\mathrm{N}\Phi _{n}(f)=\mathrm{N}\Phi _{n_{0}}(f)\), which is just \(N(f^{n_{0}})=2|1-\varphi_{2}^{2n_{0}}|\) if \(\varphi_{2}\ne\pm1\) and 0 if \(\varphi_{2}=\pm1\). □
6 Homotopy minimal periods
Theorem 6.1
([21], Theorem 6.1)
Let \(f:M\to M\) be a self-map on a compact PL-manifold M of dimension ≥3. Then f is homotopic to a map g with \(P_{n}(g)=\emptyset\) if and only if \(\operatorname{NP}_{n}(f)=0\).
Proposition 6.2
([9], Proposition 3.2)
Let \(f:M\to M\) be a self-map of a compact solvmanifold M of type \((\mathrm{NR})\). Then \(\operatorname{NP}_{n}(f)=0\) if and only if either \(N(f^{n})=0\) or \(N(f^{n})=N(f^{n/q})\) for some prime factor \(q\mid n\).
Now the following is one of our main results.
Theorem 6.3
Proof
(1) If \(\varphi_{0}=0\), then f is of type (III) and \(N(f^{n})=1\) for all n by Proposition 5.1. Hence \(\operatorname{HPer}(f)=\{1\}\).
(2) If \(\varphi_{0}=1\), then f is of type (I) and so \(N(f^{n})=0\) for all n. This implies that \(\operatorname{HPer}(f)=\emptyset\).
- (3-1)
If \(\varphi_{2}=\det\varphi_{1}=\pm1\), by Proposition 5.1, \(N(f^{n}) = 0\) for all n. Thus \(\operatorname{HPer}(f) =\emptyset\).
- (3-2)
If \(\varphi_{2}=\det\varphi_{1}=0\), by Proposition 5.1, \(N(f^{n}) = 2\) for all odd n. Since \(N(f)\ne0\), we have \(1\in \operatorname{HPer}(f)\). By (H), we have \(n\notin \operatorname{HPer}(f)\) for all odd n with \(n>1\). Thus \(\operatorname{HPer}(f) =\{1 \}\).
- (3-3)
If \(\varphi_{2}=\det\varphi_{1}\ne0,\pm1\), by Proposition 5.1, \(N(f^{n})\)’s are all distinct for all odd n. Hence \(\operatorname{HPer}(f) =\mathbb{N}-2\mathbb{N}\).
(4) If \(\varphi_{0}=-2\), by Proposition 5.1, we have \(N(f^{n}) = |1-(-2)^{n}|\). Especially, \(N(f) = N(f^{2})=3\). By (H), \(1\in \operatorname{HPer}(f)\) but \(2\notin \operatorname{HPer}(f)\). Since \(N(f^{n+1})>N(f^{n})\) for all \(n>2\), (H) induces that \(\operatorname{HPer}(f) =\mathbb{N}-\{2\}\).
(5) Consider finally the case where \(\varphi_{0}\ne-2,-1,0,1\). By Proposition 5.1 again, we still have \(N(f^{n}) = |1-\varphi _{0}^{n}|\). In this case, we have that \(N(f^{n+1})>N(f^{n})\) for all n. (H) induces that \(\operatorname{HPer}(f) =\mathbb{N}\). □
Declarations
Acknowledgements
The second-named author was supported by Basic Science Researcher Program through the National Research Foundation of Korea(NRF) funded by the Ministry of Education (No. 2013R1A1A2058693). The authors would like to thank the referees for making careful corrections to a few expressions and computations in the original version of the article.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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