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Formulas for the Reidemeister, Lefschetz and Nielsen coincidence number of maps between infra-nilmanifolds
Fixed Point Theory and Applications volume 2012, Article number: 39 (2012)
Abstract
We prove practical formulas for the Reidemeister coincidence number, the Lefschetz coincidence number and the Nielsen coincidence number of continuous maps between oriented infra-nilmanifolds of equal dimension. In order to obtain these formulas, we use the averaging formulas for the Lefschetz coincidence number and for the Nielsen coincidence number and we develop an averaging formula for the Reidemeister coincidence number. We also give a simple proof of the averaging formula for the Lefschetz coincidence number.
Mathematics Subject Classification 2000: 55M20; 57S30.
1. Introduction
In order to study the number of fixed points of a continuous selfmap f : M → M on a closed, connected manifold M, three homotopy invariant numbers are associated to f: the Reidemeister number R(f), the Lefschetz number L(f) and the Nielsen number N(f). The non-vanishing of the Lefschetz number of f implies the existence of a fixed point, while the Nielsen number is a lower bound for the number of fixed points. The Nielsen number is of particular interest since a classical result by Wecken [1] states that the Nielsen number coincides with the minimal number of fixed point in the homotopy class of the map when the dimension of M is at least three.
Because of the mixture of geometry and algebra that occur in the definition of the Nielsen number, its computation is very hard in general and it is up to now the subject of a great deal of research. Simple and practical formulas have only been obtained in specific cases (for an overview, see for instance [2, 3]) and one often turns the attention to comparing the Nielsen number to other numbers that are relatively more easy to compute, such as the Lefschetz number and the Reidemeister number (see for instance [4] for an overview).
Closely related to fixed point theory is coincidence theory. A point x ∈ M1 is a coincidence of a pair of continuous maps f, g : M1 → M2 when f(x) = g(x). In the case where M1 and M2 are both oriented and of equal dimension, the Reidemeister coincidence number R(f, g), the Lefschetz coincidence number L(f, g) and the Nielsen coincidence number N(f, g) are defined. The Nielsen coincidence number is particularly interesting since it is, just as in the fixed point case, a strong lower bound for the number of coincidences. Unfortunately, the Nielsen coincidence number is usually at least as hard to compute as the Nielsen number in fixed point theory.
In 1985, Anosov [5] shows that for nilmanifolds, the Nielsen number can easily be computed via the Lefschetz number since N(f) = |L(f)| for any continuous selfmap f on a nilmanifold. The result of Anosov was also proved by Fadell and Husseini [6]. This is the beginning of the fruitful study of fixed point theory and coincidence theory for larger classes of manifolds, such as infra-nilmanifolds [7, 8], solvmanifolds (see for instance [9]) and infra-solvmanifolds (see for example [10]). In [11, 12], formulas for the Lefschetz (fixed point) number and the Nielsen (fixed point) number of continuous selfmaps on infra-nilmanifolds have been proved. In coincidence theory however, only recently a formula has been proved for the Lefschetz coincidence number and the Nielsen coincidence number of a pair of continuous maps between solvmanifolds of type (R) [13]. For infra-nilmanifolds however, formulas for the Lefschetz coincidence number and the Nielsen coincidence number are still open for study.
In this article, we address this problem and we prove in Theorem 6.11 explicit and practical formulas for the Reidemeister coincidence number, the Lefschetz coincidence number and the Nielsen coincidence number of a pair of continuous maps between oriented infra-nilmanifolds of equal dimension, generalizing [12] from fixed point theory to coincidence theory and generalizing [13] from nilmanifolds to infra-nilmanifolds. In order to prove these formulas, we use the averaging formulas for the Nielsen coincidence number [14] and for the Lefschetz coincidence number (see [[9], p. 88] and [10]) and we develop an averaging formula for the Reidemeister coincidence number. We also formulate a simple proof for the averaging formula for the Lefschetz coincidence number.
2. Preliminaries
In this section, we fix notation and we give some definitions that will be needed to prove our results.
Definition 2.1. If is a covering map, then we use or simply to denote the covering transformation group. If and are covering maps, then we say that a continuous map is a homotopy lift of a continuous map f : M1 → M2 when is the lift of a map homotopic to f.
Definition 2.2. Let G be a group and g ∈ G. Then we use τ g : G → G : g' ↦ gg' g-1 to denote the conjugation map λ g : G → G : g' ↦ gg' to denote the left multiplication map. If φ, ψ : G → H are morphisms of groups, then we define coin(φ, ψ) = {g ∈ G | φ(g) = ψ(g)}.
2.1. Coincidence theory
In this section, we introduce basic notions concerning coincidence theory. A reference on coincidence theory is [15].
Let M1 and M2 be oriented, closed, connected manifolds of equal dimension. In order to study the coincidence set Coin(f, g) = {x ∈ M1 | f(x) = g(x)} of a pair of continuous maps f, g : M1 → M2, one splits the coincidence set into so-called coincidence classes. In this article, we define coincidence classes by fixing lifts of f and g, where and are universal covers. Remark that by fixing a lift of f : M1 → M2, the continuous map f : M1 → M2 induces a morphism between the covering transformation groups as follows.
Definition 2.3. Let be the covering transformation group of the cover and the covering transformation group of the cover . For every , let f×(α) be the unique covering transformation in that satisfies .
Fix a lift of f and a lift of g. Then for any , the set is by definition a coincidence class of f, g. Now for any pair of covering transformations , if , then one can show that and that there exists such that β = g× (γ)αf× (γ)-1. This is the motivation for the following definitions.
Definition 2.4. Let G1 and G2 be groups and φ, ψ : G1 → G2 morphisms of groups. Define an equivalence relation ~ on G2 by
The equivalence classes are called coincidence Reidemeister classes or (doubly) twisted conjugacy classes and denotes the set of coincidence Reidemeister classes. For any α ∈ G2, we use [α] to denote the coincidence Reidemeister class containing α. The Reidemeister coincidence number R(φ, ψ) of φ, ψ is defined as the cardinality of .
Definition 2.5. The Reidemeister coincidence number R(f, g) of the continuous maps f and g is defined as the Reidemeister coincidence number R(f×, g×) of the induced morphisms f× and g×. For any coincidence Reidemeister class , the set does not depend on the particular choice of the representative α of the coincidence Reidemeister class and we call the coincidence class of f, g corresponding to [α].
Note that the Reidemeister coincidence number R(f, g) does not depend on the particular choice of the lifts . Also the coincidence classes do not depend on the choice of lifts, although the corresponding coincidence Reidemeister classes may.
To each isolated subset C of Coin(f, g), one associates an integer ind(f, g; C), called the coincidence index, which generalizes the well-known fixed point index to Nielsen coincidence theory in the setting of maps between oriented manifolds of the equal dimension by Schirmer [16] (see also [17]). Each coincidence class is an isolated subset of Coin(f, g). If the coincidence index of a coincidence class is non-zero, then we call the coincidence class essential. One can prove that the number of essential coincidence classes is finite. The Lefschetz coincidence number L(f, g) is by definition the sum of the coincidence indices of the coincidence classes. If L(f, g) ≠ 0, then f and g have a coincidence. The Nielsen coincidence number N(f, g) is defined as the number of essential coincidence classes of f, g. This number plays a central role in coincidence theory since N(f, g) is a lower bound for the cardinality of Coin(f, g). The Nielsen coincidence number, the Lefschetz coincidence number and the Reidemeister coincidence numbers are homotopy invariants: if f' is homotopic to f and g' is homotopic to g, then N(f', g') = N(f, g), L(f', g') = L(f, g) and R(f', g') = R(f, g). Schirmer [16] shows that when the dimension of M1 and M2 is at least three, then for any pair of continuous maps f, g : M1 → M2, there exist maps f' homotopic to f and g' homotopic to g such that the cardinality of Coin(f', g') is precisely N(f, g). In other words: when the dimension is at least three, then the Nielsen number coincides with the minimal number of coincidence points in the homotopy class of the map.
2.2. Infra-nilmanifolds
In this section, we shortly review infra-nilmanifolds. A reference is [18]. Let G be a connected, simply connected, nilpotent Lie group and let C be a maximal compact subgroup of Aut(G). A discrete and cocompact subgroup Π of G ⋊ C ⊂ Aff(G) = G ⋊ Aut(G) is called an almost-crystallographic group. Moreover, if Π is torsion free, then Π is called an almost-Bieberbach group and the quotient space Π\G is a closed manifold that we call an infra-nilmanifold. In particular, if Π ⊂ G, then Π\G is called a nilmanifold. Recall from [19] that Γ = Π ∩ G is the maximal normal nilpotent subgroup of Π with finite quotient group Π/Γ. The quotient Π/Γ is isomorphic to the group {A ∈ Aut(G) | ∃a ∈ G such that (a, A) ∈ Π} which we call the holonomy group of the infra-nilmanifold Π\G or of the almost-Bieberbach group Π.
Let us recall an important result on maps between infra-nilmanifolds.
Theorem 2.6. [[8], Corollary 1.2] Let G1and G2be connected, simply connected, nilpotent Lie groups and M1and M2infra-nilmanifolds modeled on G1and G2respectively. Let f : M1 → M2be a continuous map, then there exist d ∈ G2and a morphism of Lie groups D : G1 → G2such that λ d ○ D : G1 → G2is a homotopy lift of f.
In fact, the result in [8] is only formulated for selfmaps, but it is straight forward to generalize the proof to the setting in Theorem 2.6. There is a similar result for maps between nilmanifolds, cf. [[6], Proposition 3.2] or [[9], Lemma 2.7]:
Theorem 2.7. Let G1and G2be connected, simply connected, nilpotent Lie groups and N1and N2nilmanifolds modeled on G1and G2respectively. Let f : N1 → N2be a continuous map. Then f has a homotopy lift D : G1 → G2that is a morphism of Lie groups.
When D : G1 → G2 is a morphism of Lie groups, then we will use D* to denote the corresponding morphism of Lie algebras.
3. The Reidemeister coincidence number
Suppose we have a commutative diagram of groups:
where the top and bottom sequences are exact and where the quotient groups Π1/Γ1 and Π2/Γ2 are finite. For each and , we have a commutative diagram
Moreover the following sequence of groups
is exact. Remark that i2 : Γ2 → Π2 and u2 : Π2 → Π2/Γ2 induce maps and such that is surjective and . Define and , where Id ∈ Π2 is the identity element.
With this notation:
Lemma 3.1. [[14], Lemma 2.1] Given the commutative diagram (3.1), we have for each α ∈ Π2that
This lemma is stated in [[14], Lemma 2.1] as a straightforward extension of [[11], Lemma 2.1] in which a topological proof of (4) is given. We will translate the topological proof to algebra.
Proof. Choose arbitrary α ∈ Π2.
-
(1)
Choose arbitrary γ ∈ Π2 and define . Then
-
(2)
Because is surjective, equals the disjoint union
Hence the first equality follows. The second equality follows from (1) and the last equality follows from the fact that .
-
(3)
This follows from the fact that equals the disjoint union
-
(4)
Choose arbitrary γ ∈ Γ2. Define
where
and where δ ∈ Π1 is chosen such that .
First we prove that A is well defined:
ψ ( δ ) γ ( τ α φ )( δ )-1 belongs to Γ 2 : This follows from the fact that because .
[ ψ ( δ ) γ ( τ α φ )( δ )-1] does not depend on the choice of δ:
Suppose that . Then there exists β ∈ Γ1 such that δ' = βδ.
Then
so that
Let us now prove that A is surjective. Choose arbitrary . Because , there exists δ ∈ Π1 such that β = ψ(δ)γ(τ α φ)(δ)-1. Because β, γ ∈ Γ2, we have that , where so that . Additionally, .
In order to prove statement (4), it is left to prove that for each , we have that if and only if there exists β ∈ coin(τ γα φ, ψ) such that . Choose arbitrary .
First assume that there exists β ∈ coin(τ γα φ, ψ) such that . Choose δ ∈ Π1 such that and define δ' = δβ, then . Remark that because β ∈ coin(τ γα φ, ψ) we have that ψ(β) = γ(τ α φ)(β)γ-1 so that ψ(β)γ(τ α φ)(β)-1 = γ. Hence
Conversely suppose that . It then suffices to prove that . Choose δ, δ' ∈ Π1 such that and . Then because , there exists η ∈ Γ1 such that
or
Hence and .
-
(5)
This follows from (3) and (4).
Corollary 3.2. [[20], Corollary 2.1] Given the commutative diagram (3.1), for all α ∈ Π2then
Proof. By (2) of Lemma 3.1,
By (4) of Lemma 3.1, is injective and hence the conclusion follows.
Remark 3.3. The group Π1/Γ1 acts on the set by the rule . This action is transitive. The isotropy subgroup is
Hence .
Corollary 3.4. [[21], Proposition 3.8] Suppose we are given the commutative diagram (3.1). Then R(φ, ψ) is finite if and only if R(τ α φ', ψ') is finite for every α ∈ Π2.
Proof. Suppose that R(φ, ψ) is finite. Choose arbitrary α ∈ Π2. By (2) of Lemma 3.1, is finite. Because also is finite, by (5) of Lemma 3.1, R(τ α φ', ψ') is finite. The converse is also easy to check.
Furthermore, the Reidemeister coincidence numbers R(φ, ψ) and R(τ α φ', ψ') are directly related as follows:
Theorem 3.5. Suppose we have a commutative diagram of groups:
where the top and bottom sequences are exact and the quotient groups Π1/Γ1and Π2/Γ2are finite. Then
When either side of the inequality is finite, then equality occurs if and only if coin(τ α φ, ψ)) ⊂ Γ1for each α ∈ Π2.
Remark that R(τ α φ', ψ') depends only on . Indeed, suppose that u2(α) = u2(β), then αβ-1 ∈ Γ2 and βα-1 ∈ Γ2 and we can define
and
A and B are bijections that are each other's inverse, hence R(τ α φ', ψ') = R(τ β φ', ψ').
Proof of Theorem 3.5. This follows from the following observations:
Moreover, when either side of the inequality is finite, then equality holds if and only if for each α ∈ Π2, u1(coin(τ α φ, ψ)) is the trivial group.
4. Reidemeister coincidence numbers for covering spaces
Now we can translate the previous results on algebraic Reidemeister coincidence numbers to results on topological Reidemeister coincidence numbers.
Theorem 4.1. Let M1and M2be closed and connected manifolds with universal coversand. Let (f, g) : M1 → M2be a pair of maps and letbe a pair of lifts of (f, g). Letandbe the covering transformation groups and let f×, g× : Π1 → Π2be the morphisms of groups induced by. Suppose Γ1is a finite index normal subgroup of Π1and Γ2is a finite index normal subgroup of Π2such that f×(Γ1) ⊂ Γ2and g×(Γ1) ⊂ Γ2. Letbe a pair of lifts of (f, g) so that the following diagram commutes
Then:
-
(1)
If for all α ∈ Π2, then
-
(2)
R(f, g) is finite if and only if is finite for every α ∈ Π2.
-
(3)
We have
When either side of the inequality is finite, then equality occurs if and only if coin(τ α f×, g×) ⊂ Γ1for each α ∈ Π2.
Theorem 4.2 (Averaging Formula for the Reidemeister Coincidence Number). Let M1and M2be orientable infra-nilmanifolds of equal dimension modeled on connected, simply connected, nilpotent Lie groups G1and G2respectively. Let (f, g) : M1 → M2be a pair of maps. Let Π1 = A(G1, p1) and Π2 = A(G2, p2) be the covering transformation groups and let f×, g× : Π1 → Π2be the morphisms of groups induced by f, g respectively. Let Γ1and Γ2be finite index normal subgroups of Π1and Π2respectively such that f×(Γ1) ⊂ Γ2and g×(Γ1) ⊂ Γ2and such that N1 = Γ1\G1and N2 = Γ2\G2are nilmanifolds. Ifis a pair of lifts of (f, g), then
Proof. Suppose f and g have an inessential coincidence class. Then from the proof of [[21], Theorem 5.1], it follows that R(f, g) = ∞ and that there exists such that . So without loss of generality, we may assume that all coincidence classes are essential. Choose arbitrary α ∈ Π2. Then by assumption, is an essential coincidence class, where p1 : G1 → M1 is the natural covering projection. By Theorem 4.1 (3), it suffices to show that coin(τ α φ, ψ) ⊂ Γ1. In fact, coin(τ α φ, ψ) = {1} by [[14], Lemma 4.8] together with the proof of [[14], Theorem 4.9].
5. Averaging formula for the Lefschetz coincidence number
The averaging formula for the Lefschetz coincidence number relates the Lefschetz co-incidence number of a pair of continuous maps f, g : M1 → M2 to the Lefschetz coincidence numbers of lifts of f and g to finite sheeted regular covers of M1 and M2. The averaging formula for the Lefschetz coincidence number has been proved in [10] (see also [[9], p. 88]). In this section, we formulate a simple proof.
Remark 5.1 (See [[21], Lemma 3.9], [10] or [[22], p. 37]). Let f, g : M1 → M2 be continuous maps between closed oriented manifolds M1, M2 of equal dimension. Let and be covers of M1 and M2 respectively and suppose that the covering projections and are orientation preserving. Let be a lifting pair of (f, g). Let . Then . Because the covering projections are orientation-preserving local diffeomorphisms,
Theorem 5.2 (Averaging formula for the Lefschetz coincidence number). Letandbe universal covers, letandbe finite sheeted, regular covers and letandbe covers such that the diagrams
commute, where all spaces are connected oriented manifolds, whereandare closed manifolds of equal dimension and where all covering projections are orientation preserving local diffeomorphisms. Suppose that f has a liftand g has a lift. Then
Proof. It is well-known (cf. [[23], Theorem 3.1]) that there exists a pair of maps (f', g') such that (f, g) is homotopic to (f', g') and Coin(f', g') is finite. Since the Lefschetz coincidence number is a homotopy invariant, we may assume that Coin(f, g) is finite.
Because every x ∈ Coin(f, g) has preimages under ,
Now
Indeed, implies that , such that . Hence
because of Remark 5.1. Hence
6. Formulas for the Reidemeister, Lefschetz and Nielsen co-incidence number of maps between infra-nilmanifolds
In this section, we give practical formulas for the Reidemeister coincidence number, the Lefschetz coincidence number and the Nielsen coincidence number of a pair of continuous maps between oriented infra-nilmanifolds of equal dimension. First we give some definitions and recall some useful results.
Definition 6.1. Let G be an n-dimensional oriented, connected, simply connected, nilpotent Lie group and Λ a uniform lattice in G. Let {a1, ..., a n } be a set of generators of Λ and write b i = log a i , then {b1, ..., b n } is a basis for the Lie algebra corresponding to G. Suppose additionally that the basis {b1, ..., b n } is positively oriented. Then we refer to {b1, ..., b n } as a preferred basis of Λ.
Definition 6.2. Let V and W be finite dimensional vector spaces and L : V → W a linear map. Let β V be a basis for V and β W a basis for W, then we use to denote the matrix corresponding to L, where this matrix is expressed with respect to the bases β V and β W .
Let G1 and G2 be oriented, connected, simply connected, nilpotent Lie groups of equal dimension with associated Lie algebras and . Let Λ1 be a uniform lattice in G1 and Λ2 a uniform lattice in G2. Let be a linear map. Let β1 be a preferred basis of Λ1 and β2 a preferred basis of Λ2. Then det does not depend on the particular choice of the preferred bases β1 of Λ1 and β2 of Λ2 (see [[24], p. 253]). Hence we can define
for any choice of preferred bases β1 of Λ1 and β2 of Λ2.
The following theorem gives a formula for the Lefschetz coincidence number of a pair of continuous maps between nilmanifolds of equal dimension.
Theorem 6.3. [[13], Theorem 3.1] Let G1, G2be oriented, connected, simply connected, nilpotent Lie groups of equal dimension. Let Λ1be a uniform lattice in G1and Λ2a uniform lattice in G2. Let N1 = Λ1\G1and N2 = Λ2\G2be the corresponding nilmanifolds. Let f, f' : N1 → N2be continuous maps. Let D, D' : G1 → G2be morphisms of Lie groups such that D is a homotopy lift of f and D' is a homotopy lift of f'. Then
Remark that Theorem 2.7 guarantees the existence of morphisms of Lie groups D, D' : G1 → G2 that are homotopy lifts of f, f' : N1 → N2.
Lemma 6.4. [[24], Lemma 3.2] Let G be an oriented, connected, simply connected, nilpotent Lie group and Λ a uniform lattice in G. Letbe a finite index subgroup of Λ. Then
Proof. Define N = Λ\G and and give N and orientations so that the covering projections G → N and are orientation preserving. Then Id : G → G : g ↦ g induces an orientation preserving map and O : G → G : g ↦ 1 G induces a constant map . By Theorem 6.3,
On the other hand, by [[7], Lemma 3.10], every essential coincidence class is a singleton and has coincidence index . Remark that sign . So it suffices to show that . Now
Hence .
Corollary 6.5. Let G be an oriented, connected, simply connected, nilpotent Lie group and Λ a uniform lattice in G. Letbe a finite index subgroup of Λ. Then
Proof. Let β be a preferred basis of Λ and a preferred basis of . Then
and the corollary follows from the previous lemma.
Corollary 6.6. Let G1and G2be oriented, connected, simply connected, nilpotent Lie groups with associated Lie algebrasand. Let Λ1be a uniform lattice of G1and Λ2a uniform lattice of G2. Letbe a finite index subgroup of Λ1anda finite index subgroup of Λ2. Then for any linear map,
Proof. This follows from a short calculation:
In order to prove the main formulas of this article, we will prove the following technical theorem, which generalizes [[12], Lemma 3.2].
Theorem 6.7. Let G1and G2be connected, simply connected, nilpotent Lie groups of equal dimension with associated Lie algebrasandrespectively. Let D, D' : G1 → G2be morphisms of Lie groups inducing morphisms of Lie algebras D*, . Then for any g ∈ G2,
The proof of this theorem expresses the right hand side of the equality as a polynomial. Then we use the following lemma to prove that either this polynomial is the zero polynomial or it has no roots.
Lemma 6.8. [[21], Lemma 3.5] Let G1and G2be connected, simply connected, nilpotent (complex) Lie groups with associated (complex) Lie algebrasandrespectively. Let D, D' : G1 → G2be morphisms of Lie groups inducing morphisms of Lie algebras D*, . Then for any g ∈ G2, is surjective if and only ifis surjective.
In fact, this lemma was only proved in the real case, but its proof is also valid in the complex case. Now we can prove Theorem 6.7.
Proof of Theorem 6.7. Let and be the complexifications of the real Lie groups G1 and G2 respectively with canonical maps . Since G i is simply connected, α i is injective. Let and denote the respective Lie algebras. Then .
The morphism D : G1 → G2 extends uniquely to a morphism of complex Lie groups so that . Similarly, we define and Ad(g)ℂ for any g ∈ G2.
Let be a basis for and a basis for . Write and , then is a basis for the complex vector space and is a basis for the complex vector space . Now
where on the left hand side, the matrices are expressed with respect to the bases and and on the right hand side, the matrices are expressed with respect to the bases and .
Now we consider the function defined by
By Lemma 6.8, either f is the zero map or f has no roots.
Choose arbitrary . With respect to the complex basis of . ad(Y) is regarded as a matrix [z ij (Y)] where all are complex-valued functions. Now there exist complex numbers such that
Writing , we remark that
So the entries of ad(Y) depend linearly on the λ k , and hence the entries of Ad(exp(Y)) = exp(ad(Y)) depend polynomially on the λ k . Consequently,
is a polynomial in Y.
If f is not the zero map, then f has no roots and from the fundamental theorem of algebra, it follows that f is a constant polynomial. Hence, regardless of whether f is the zero map or not, f is constant and for any g ∈ G2,
does not depend on g. This proves the theorem.
Now we generalize [[12], Lemma 3.1], in which the existence of a fully invariant sub-group of finite index in an almost-Bieberbach group is proved. The proof consists merely of a straightforward adaptation of that of [[12], Lemma 3.1] to this more general, but very analogous situation.
Lemma 6.9. Let Π1and Π2be almost-crystallographic groups and let Γ i be the maximal normal nilpotent subgroup of Π i or, more generally, let Π i ⊂ S i ⋊ Aut(S i ) be a finite extension of the lattice Γ i of a connected, and simply connected solvable Lie group S i . Then there exist fully invariant subgroups Λ i ⊂ Γ i of Π i , which are of finite index, so that any morphism Π1 → Π2maps Λ1into Λ2.
Proof. Let k be the least common multiple of the orders of the holonomy groups Π1/Γ1 and Π2/Γ2. Let Λ1 be the subgroup of Π1 generated by the set
Clearly, the generating set is a subset of Γ1 so that Λ1 is a subgroup of Γ1. Similarly, the subgroup Λ2 of Π2 generated by {yk| y ∈ Π2} is a subgroup of Γ2. Obviously, any morphism θ : Π1 → Π2 sends the generating set {xk| x ∈ Π1} of Λ1 into the generating set {yk| y ∈ Π2} of Λ2. Thus θ maps Λ1 into Λ2.
We claim that Λ1 has finite index in Γ1 (and hence in Π1). Consider the subgroup Γ(k) generated by the set {xk| x ∈ Γ1}. Since Γ1 is a lattice in the connected and simply connected solvable Lie group S1, it is a (strongly) polycyclic group. Then Γ(k) has finite index in Γ1, see [[25], Lemma 4.4]. Since Γ(k) ⊂ Λ1, we find that Λ1 has finite index in Γ1. Similarly, Λ2 has finite index in Γ2 and hence in Π2.
By taking S1 = S2 and Π1 = Π2, we see that any morphism on Π i maps Λ i into Λ i itself. Hence Λ i is a fully invariant subgroup of Π i .
Let us recall the averaging formula for the Nielsen coincidence number in the special case of maps between infra-nilmanifolds.
Theorem 6.10. [[14], Theorem 4.9] Let M1and M2be closed oriented infra-nilmanifolds of equal dimension and f, g : M1 → M2continuous maps. Suppose there exist finite sheeted regular coversand, where N1and N2are nilmanifolds. Suppose thatis a lift of f andis a lift of g. Then
Now we prove practical formulas for the Reidemeister coincidence number, the Lef-schetz coincidence number and the Nielsen coincidence number of a pair of continuous maps between oriented infra-nilmanifolds of equal dimension.
Theorem 6.11. Let G1and G2be connected, simply connected, nilpotent Lie groups of equal dimension. Let Π1and Π2be almost-Bieberbach groups modeled on G1and G2, respectively and suppose that the corresponding infra-nilmanifolds M1 = Π1\G1and M2 = Π2\G2are oriented. Let F1 ⊂ Aut(G1) be the holonomy group of M1and F2 ⊂ Aut(G2) the holonomy group of M2. Let f, g : M1 → M2be continuous maps. Let D, D' : G1 → G2be morphisms of Lie groups and d, d' ∈ G2such that λ d ○D : G1 → G2is a homotopy lift of f and λ d' ○D' : G1 → G2is a homotopy lift of g. (Recall that λ d : G2 → G2 : g' ↦ dg' is the left multiplication map.) Then
and
where the morphisms of Lie groups D*, and A*induced by D, D' and A are expressed with respect to preferred bases of Π1 ∩ G1and Π2 ∩ G2and where σ : ℝ → ℝ ∪ {∞} is defined by σ(0) = ∞ and σ(x) = |x| for all x ≠ 0.
Remark that by Theorem 2.6, there exist d, d' ∈ G2 and morphisms of Lie groups D, D' : G1 → G2 such that λ d ○ D : G1 → G2 is a homotopy lift of f and λ d' ○ D' : G1 → G2 is a homotopy lift of g.
Proof of Theorem 6.11. Without loss of generality, we may assume that λ d ○ D is a lift of f and λ d' ○ D' is a lift of g. Define Γ1 = Π1 ∩ G1 and Γ2 = Π2 ∩ G2. By Lemma 6.9, there exist uniform lattices Λ1 ⊂ G1 and Λ2 ⊂ G2 such that Λ i is a fully invariant subgroup of Π i , Λ i is a finite index subgroup of Γ i and such that θ(Λ1) ⊂ Λ2 ⊂ Γ2 for every morphism of groups θ : Π1 → Π2. Define N1 = Λ1\G1 and N2 = Γ2\G2. Let , , and be natural projections. Then f, g lift to continuous maps so that the following diagram commutes up to homotopy
By Theorem 5.2, the averaging formula for the Lefschetz coincidence number,
By [[9], Theorem 1.1], for every . Hence by Theorem 6.10, the averaging formula for the Nielsen coincidence number,
By the main result of [26], for every . Hence by Theorem 4.2, the averaging formula for the Reidemeister coincidence number,
Choose arbitrary . Then there exist a ∈ G2 and A in the holonomy group F2 ⊂ Aut(G2) such that λ a ○ A : G2 → G2 is a lift of . We now claim that
First remark that induces a morphism defined by
If we apply both sides of this equality to the identity element of G1, then we see that . Hence (τ d ○ D)(Λ1) ⊂ Γ2 and τ d ○ D : G1 → G2 induces a continuous map h : N1 → N2. A similar calculation shows that . Because N1 and N2 are aspherical, and h are homotopic and we see that τ d ○ D is a homotopy lift of .
Similarly, one can show that τ d' ○ D' is a homotopy lift of and that τ a ○ A is a homotopy lift of . Then by Theorem 6.3,
By applying Theorem 6.7 twice, we see that
By Corollary 6.6,
Since was chosen arbitrarily, this equality holds for every , where A ∈ F2 is such that λ a ○ A is a lift of for some a ∈ G2. Hence
Similar calculations show that
and that
The following generalizes [[8], Theorem 2.2] from the fixed point version to the coincidence version.
Corollary 6.12. Let G1and G2be connected, simply connected, nilpotent Lie groups of equal dimension. Let M1and M2be oriented infra-nilmanifolds modeled on G1and G2respectively. Let F2 ⊂ Aut(G2) be the holonomy group of M2. Let f, g : M1 → M2be continuous maps. Let D, D' : G1 → G2be morphisms of Lie groups and d, d' ∈ G2such that λ d ○D : G1 → G2is a homotopy lift of f and λ d '○D' :G1 → G2is a homotopy lift of g. Then N(f, g) = L(f, g) if and only iffor every A ∈ F2and N(f, g) = -L(f, g) if and only iffor every A ∈ F2, where D*, and A*are the morphisms of Lie algebras induced by D, D' and A respectively, expressed with respect to positively oriented bases of the Lie algebras associated to G1and G2.
7. Examples
In this section we illustrate, by some examples, how practical the averaging formulas on infra-nilmanifolds are. For this purpose we will consider maps from a 3-dimensional flat Riemannian manifold to an infra-nilmanifold modeled on the Heisenberg group Nil.
Let Nil be the 3-dimensional Heisenberg group defined by
Then it is a connected and simply connected 2-step nilpotent Lie group. The corresponding Lie algebra is
It is easy to see that (cf. [[27], Proposition 2.2])
and an element
acts on Nil as follows:
where
Example 7.1. Let Π1 be the Bieberbach group generated by the standard basis {e1, e2, e3} of ℝ3. Let M1 = Π1\ℝ3 be the corresponding infra-nilmanifold, the 3-dimensional flat torus, with the trivial holonomy group.
Consider the almost Bieberbach group Π2 given by
This is a 3-dimensional orientable almost Bieberbach group π3 with Seifert bundle type 3 ( [[28], Proposition 6.1], or the list of [[29], p. 800]). We can embed Π2 into Aff(Nil) = Nil ⋊ Aut(Nil) by taking
Then the translation lattice is
and the holonomy group of Π2 is F2 = Π2/Γ2 ≅ ℤ2, which is generated by the image A of α under the natural map Aff(Nil) → Aut(Nil). Thus, A is the automorphism on Nil defined by
Let M2 = Π2\Nil be the corresponding infra-nilmanifold.
Define φ : Π1 → Π2 by
Define the morphism of Lie groups D : ℝ3 → Nil by
Define d = 1Nil, then one can verify that φ(γ) ○ λ d ○ D = λ d ○ D ○ γ for γ = e1, e2, e3 and hence for all γ ∈ Π1. Thus λ d ○ D : ℝ3 → Nil induces a map f : M1 → M2 so that f× = φ.
Define ψ : Π1 → Π2 by
Define d' = 1Nil ∈ Nil and define the morphism of Lie groups D' : ℝ3 → Nil by
Then one can verify that ψ(γ) ○ λd'○ D' = λd'○ D' ○ γ for γ = e1, e2, e3 and hence for all γ ∈ Π1. This implies that λd'○ D' : ℝ3 → Nil induces a map f' : M1 → M2 so that .
Since Γ1 = Π1 ∩ ℝ3 is generated by {e1, e2, e3} and Γ2 = Π2 ∩ Nil is generated by {s1, s2, s3}, the basis {log(e1), log(e2), log(e3)} is a preferred basis for Γ1 and {log(s1), log(s2), log(s3)} is a preferred basis for Γ2. With respect to these preferred bases, the matrices corresponding to the induced morphisms of Lie algebras and are
Hence by Theorem 6.11
Example 7.2. In this example we will consider a 3-dimensional orientable flat Rie-mannian manifold M1 and a 3-dimensional orientable infra-nilmanifold M2.
We consider first a 3-dimensional orientable flat Riemannian manifold M1 = Π1\ℝ3 where Π1 is the 3-dimensional orientable Bieberbach group [[30], Theorem 3.5.5]:
We can embed this group into Aff(ℝ3) by taking {e1, e2, e3} as the standard basis for ℝ3 and
Then the translation lattice is
and the holonomy group is F1 = Π1/Γ1 ≅ ℤ2. The holonomy group F1 ⊂ Aut(ℝ3) is generated by A : ℝ3 → ℝ3 : (x, y, z) ↦ (x, -y, -z). With respect to the preferred basis {e1, e2, e3}, the differential A* : ℝ3 → ℝ3 can be expressed as a matrix as follows:
Next we consider an infra-nilmanifold M2 = Π2\Nil where Π2 is a 3-dimensional orientable almost Bieberbach group π5,3 with Seifert bundle type 5 ( [[28], Proposition 6.1], or the list of [[29], p. 800]):
Now we can embed Π2 into Aff(Nil) = Nil ⋊ Aut(Nil) by taking
Then the translation lattice is
and the holonomy group of Π2 is F2 = Π2/Γ2 ≅ ℤ4. The holonomy group F2 ⊂ Aut(Nil) is generated by the image B of β under the natural map Aff(Nil) → Aut(Nil). Thus, B is the automorphism on Nil defined by
Hence the differential B* of B is given by
With respect to the preferred basis {log s1, log s2, log s3} of nil, the differential B* can be written as a matrix as follows:
Now define a morphism of groups φ : Π1 → Π2 by
Also define d = 1Nil and define the morphism of Lie groups D by
Then φ(γ) ○ λ d ○ D = λ d ○ D ○ γ for γ = t1, t2, t3, α. Hence this equation holds for every γ ∈ Π1 and λ d ○ D : ℝ3 → Nil induces a continuous map f : M1 → M2 so that f× = φ. With respect to the preferred bases chosen above, the differential D* can be written as a matrix as follows:
Similarly, by defining the morphism of groups ψ : Π1 → Π2 by
one can show that λd'○ D' : ℝ3 → Nil induces a continuous map f' : M1 → M2 so that , where
With respect to the same preferred bases, can be written as a matrix as follows:
Hence and . From the formulas in Theorem 6.11, it follows that
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Acknowledgements
The authors would like to thank the referee for pointing out some errors and making careful corrections to a few expressions in the original version of the article. The second-named author is partially supported by the Mid-career Researcher Program through NRF grant funded by the MEST (No. 2010-0008640) and by the Sogang University Research Grant of 2010. The third-named author is supported by a Ph.D. fellowship of the Research Foundation- Flanders (FWO)
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Ha, K., Lee, J. & Penninckx, P. Formulas for the Reidemeister, Lefschetz and Nielsen coincidence number of maps between infra-nilmanifolds. Fixed Point Theory Appl 2012, 39 (2012). https://doi.org/10.1186/1687-1812-2012-39
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DOI: https://doi.org/10.1186/1687-1812-2012-39