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Formulas for the Reidemeister, Lefschetz and Nielsen coincidence number of maps between infra-nilmanifolds

Fixed Point Theory and Applications20122012:39

https://doi.org/10.1186/1687-1812-2012-39

  • Received: 3 December 2011
  • Accepted: 14 March 2012
  • Published:

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.

Keywords

  • infra-nilmanifold
  • Lefschetz coincidence number
  • Nielsen coincidence number
  • Reidemeister coincidence number.

1. Introduction

In order to study the number of fixed points of a continuous selfmap f : MM 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 : M1M2 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 M ̃ M is a covering map, then we use A M ̃ , p or simply A M ̃ to denote the covering transformation group. If p 1 : M ̃ 1 M 1 and p 2 : M ̃ 2 M 2 are covering maps, then we say that a continuous map f ̃ : M ̃ 1 M ̃ 2 is a homotopy lift of a continuous map f : M1M2 when f ̃ 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 : GG : g' gg' g-1 to denote the conjugation map λ g : GG : g' gg' to denote the left multiplication map. If φ, ψ : GH 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 : M1M2, one splits the coincidence set into so-called coincidence classes. In this article, we define coincidence classes by fixing lifts f ̃ , g ̃ : M ̃ 1 M ̃ 2 of f and g, where p 1 : M ̃ 1 M 1 and p 2 : M ̃ 2 M 2 are universal covers. Remark that by fixing a lift f ̃ : M ̃ 1 M ̃ 2 of f : M1M2, the continuous map f : M1M2 induces a morphism f × : A M ̃ 1 , p 1 A M ̃ 2 , p 2 between the covering transformation groups as follows.

Definition 2.3. Let A M 1 ̃ , p 1 be the covering transformation group of the cover p 1 : M ̃ 1 M 1 and A M 2 ̃ , p 2 the covering transformation group of the cover p 2 : M ̃ 2 M 2 . For every α A M ̃ 1 , p 1 , let f×(α) be the unique covering transformation in A M 2 ̃ , p 2 that satisfies f ̃ α = f × ( α ) f ̃ .

Fix a lift f ̃ : M ̃ 1 M ̃ 2 of f and a lift g ̃ : M ̃ 1 M ̃ 2 of g. Then for any α A M ̃ 2 , p 2 , the set p 1 Coin α f ̃ , g ̃ is by definition a coincidence class of f, g. Now for any pair of covering transformations α , β A M ̃ 2 , p 2 , if p 1 Coin α f ̃ , g ̃ p 1 Coin β f ̃ , g ̃ , then one can show that p 1 Coin α f ̃ , g ̃ = p 1 Coin β f ̃ , g ̃ and that there exists γ A M ̃ 1 , p 1 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 : G1G2 morphisms of groups. Define an equivalence relation ~ on G2 by
α ~ β γ G 1 : β = ψ ( γ ) α φ ( γ ) - 1 .

The equivalence classes are called coincidence Reidemeister classes or (doubly) twisted conjugacy classes and R [ φ , ψ ] 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 R [ φ , ψ ] .

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 [ α ] R f × , g × , the set p 1 Coin α f ̃ , g ̃ does not depend on the particular choice of the representative α of the coincidence Reidemeister class and we call p 1 Coin α f ̃ , g ̃ 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 f ̃ , g ̃ . 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 : M1M2, 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 : M1M2be a continuous map, then there exist d G2and a morphism of Lie groups D : G1G2such that λ d D : G1G2is 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 : N1N2be a continuous map. Then f has a homotopy lift D : G1G2that is a morphism of Lie groups.

When D : G1G2 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:
(3.1)
where the top and bottom sequences are exact and where the quotient groups Π11 and Π22 are finite. For each α ̄ Π 2 / Γ 2 and α u 2 - 1 ( α ̄ ) , we have a commutative diagram
(3.2)
Moreover the following sequence of groups
1 coin τ α φ , ψ i 1 coin τ α φ , ψ u 1 coin τ α ̄ φ ̄ , ψ ̄

is exact. Remark that i2 : Γ2 → Π2 and u2 : Π2 → Π22 induce maps i ^ 2 α : R τ α φ , ψ R τ α φ , ψ and u ^ 2 α : R τ α φ , ψ R τ α ̄ φ ̄ , ψ ̄ such that u ^ 2 α is surjective and u ^ 2 α - 1 [ 1 ̄ ] = im i ^ 2 α . Define i ^ 2 = i ^ 2 Id and u ^ 2 = u ^ 2 Id , 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
# u ^ 2 - 1 α ̄ = # u ^ 2 α - 1 1 ̄ ,
(1)
R φ , ψ = [ α ̄ ] R [ φ ̄ , ψ ̄ ] # u ^ 2 - 1 α ̄ = [ α ̄ ] R [ φ ̄ , ψ ̄ ] # u ^ 2 α - 1 1 ̄ = [ α ̄ ] R [ φ ̄ , ψ ̄ ] # im i ^ 2 α ,
(2)
R τ α φ , ψ = [ γ ] im i ^ 2 α # i ^ 2 α - 1 γ ,
(3)
# i ^ 2 α - 1 γ = coin τ α ̄ φ ̄ , ψ ̄ : u 1 coin τ γ α φ , ψ f o r e a c h γ Γ 2 ,
(4)
R τ α φ , ψ = [ γ ] im i ^ 2 α coin τ α ̄ φ ̄ , ψ ̄ : u 1 coin τ γ α φ , ψ .
(5)

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. (1)
    Choose arbitrary γ Π2 and define γ ̄ = u 2 ( γ ) . Then
    [ γ ] u ^ 2 - 1 [ α ̄ ] [ γ ̄ ] = [ α ̄ ] in R [ φ ̄ , ψ ̄ ] δ ̄ Π 1 / Γ 1 such that γ ̄ α ̄ - 1 = ψ ̄ ( δ ̄ ) α ̄ φ ̄ ( δ ̄ ) - 1 α ̄ - 1 = φ ̄ ( δ ̄ ) 1 ̄ τ α ̄ φ ̄ ( δ ̄ ) - 1 u ^ 2 α γ α - 1 = γ ̄ α ̄ - 1 = [ 1 ̄ ] in R τ α ̄ φ ̄ , ψ ̄ γ α - 1 u ^ 2 α - 1 1 ̄ .
     
  2. (2)
    Because u ^ 2 is surjective, R [ φ , ψ ] equals the disjoint union
    R [ φ , ψ ] = [ α ̄ ] R [ φ ̄ , ψ ̄ ] u ^ 2 - 1 α ̄ .
     
Hence the first equality follows. The second equality follows from (1) and the last equality follows from the fact that u ^ 2 α - 1 1 ̄ = im i ^ 2 α .
  1. (3)
    This follows from the fact that R τ α φ , ψ equals the disjoint union
    R [ τ α φ , ψ ] = [ γ ] im i ^ 3 α i ^ 2 α - 1 γ .
     
  2. (4)
    Choose arbitrary γ Γ2. Define
    A : coin τ α ̄ φ ̄ , ψ ̄ i ^ 2 α - 1 γ : δ ̄ A δ ̄ ,
     
where
A ( δ ̄ ) = ψ ( δ ) γ τ α φ ( δ ) - 1 i ^ 2 α - 1 γ R τ α φ , ψ ,

and where δ Π1 is chosen such that u 1 ( δ ) = δ ̄ .

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 u 2 ψ ( δ ) γ ( τ α φ ) ( δ ) - 1 = ψ ̄ ( δ ̄ ) τ α ̄ φ ̄ ( δ ̄ ) - 1 = 1 because δ ̄ coin τ α ̄ φ ̄ , ψ ̄ .

[ ψ ( δ ) γ ( τ α φ )( δ ) -1 ] does not depend on the choice of δ:

Suppose that u 1 ( δ ) = u 1 ( δ ) = δ ̄ . Then there exists β Γ1 such that δ' = βδ.

Then
ψ ( δ ) γ ( τ α φ ) ( δ ) 1 = ψ ( β ) ( ψ ( δ ) γ ( τ α φ ) ( δ ) 1 ) ( τ α φ ) ( β ) 1
so that
ψ ( δ ) γ τ α φ δ - 1 = ψ ( δ ) γ τ α φ ( δ ) - 1 .

Let us now prove that A is surjective. Choose arbitrary [ β ] i ^ 2 α - 1 γ . Because i ^ 2 α β = [ γ ] , there exists δ Π1 such that β = ψ(δ)γ(τ α φ)(δ)-1. Because β, γ Γ2, we have that 1 = u 2 ( β ) = ψ ̄ ( δ ̄ ) τ α ̄ φ ̄ ( δ ̄ ) - 1 , where δ ̄ = u 2 ( δ ) so that δ ̄ coin τ α ̄ φ ̄ , ψ ̄ . Additionally, A ( δ ̄ ) = [ β ] .

In order to prove statement (4), it is left to prove that for each δ ̄ , δ ̄ coin τ α ̄ φ ̄ , ψ ̄ , we have that A ( δ ̄ ) = A ( δ ̄ ) if and only if there exists β coin(τ γα φ, ψ) such that δ ̄ = δ ̄ u 1 ( β ) . Choose arbitrary δ ̄ , δ ̄ coin τ α ̄ φ ̄ , ψ ̄ .

First assume that there exists β coin(τ γα φ, ψ) such that δ ̄ = δ ̄ u 1 ( β ) . Choose δ Π1 such that u 1 ( δ ) = δ ̄ and define δ' = δβ, then u 1 ( δ ) = δ ̄ . 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
A ( δ ̄ ) = ψ ( δ ) γ ( τ α φ ) ( δ ) - 1 = ψ δ β γ ( τ α φ ) ( δ β ) - 1 = ψ ( δ ) ψ ( β ) γ ( τ α φ ) ( β ) - 1 ( τ α φ ) ( δ ) - 1 = ψ ( δ ) γ ( τ α φ ) ( δ ) - 1 = A ( δ ̄ ) .
Conversely suppose that A ( δ ̄ ) = A ( δ ̄ ) . It then suffices to prove that δ ̄ - 1 δ ̄ u 1 coin τ γ α φ , ψ . Choose δ, δ' Π1 such that u 1 ( δ ) = δ ̄ and δ ̄ = δ ̄ u 1 ( β ) . Then because A ( δ ̄ ) = A ( δ ̄ ) , there exists η Γ1 such that
ψ ( δ ) γ ( τ α φ ) ( δ ) - 1 = ψ ( η ) ψ ( δ ) γ ( τ α φ ) ( δ ) - 1 ( τ α φ ) ( η ) - 1 , = ψ ( η δ ) γ ( τ α φ ) ( η δ ) - 1
or
ψ δ - 1 η - 1 δ γ = γ τ α φ δ - 1 η - 1 δ .
Hence δ - 1 η - 1 δ coin τ γ α φ , ψ and δ ̄ - 1 δ ̄ = u 1 δ - 1 η - 1 δ coin τ γ α φ , ψ .
  1. (5)

    This follows from (3) and (4).

     
Corollary 3.2. [[20], Corollary 2.1] Given the commutative diagram (3.1), coin τ α ̄ φ ̄ , ψ ̄ = { 1 ̄ } for all α Π2then
R ( φ , ψ ) = [ α ̄ ] R [ φ ̄ , ψ ̄ ] R τ α φ , ψ .
Proof. By (2) of Lemma 3.1,
R ( φ , ψ ) = [ α ̄ ] R [ φ ̄ , ψ ̄ ] # im i ^ 2 α = [ α ̄ ] R [ φ ̄ , ψ ̄ ] # i ^ 2 α R τ α φ , ψ .

By (4) of Lemma 3.1, i ^ 2 α is injective and hence the conclusion follows.

Remark 3.3. The group Π11 acts on the set [ α ̄ ] R [ φ ̄ , ψ ̄ ] by the rule α ̄ ψ ̄ β ̄ α ̄ φ ̄ β ̄ - 1 . This action is transitive. The isotropy subgroup is
β ̄ | ψ ̄ β ̄ α ̄ φ ̄ β ̄ - 1 = α ̄ = coin τ α ̄ φ ̄ , ψ ̄ .

Hence Π 1 : Γ 1 = # [ α ̄ ] # coin τ α ̄ φ ̄ , ψ ̄ .

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, im i ^ 2 α is finite. Because also τ α ̄ φ ̄ , ψ ̄ 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 Π11and Π22are finite. Then
R ( φ , ψ ) 1 Π 1 : Γ 1 α ̄ Π 2 / Γ 2 R τ α φ , ψ .

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 α ̄ = u 2 ( α ) . Indeed, suppose that u2(α) = u2(β), then αβ-1 Γ2 and βα-1 Γ2 and we can define
A : R τ α φ , ψ R τ β φ , ψ : [ γ ] [ γ α β - 1 ]
and
B : R τ β φ , ψ R τ α φ , ψ : [ γ ] [ γ β α - 1 ] .

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:
α ̄ Π 2 / Γ 2 R τ α φ , ψ = α ̄ Π 2 / Γ 2 [ γ ] im i ^ 2 α coin τ α ̄ φ ̄ , ψ ̄ : u 1 coin τ γ α φ , ψ by ( 5 ) of Lemma 3 . 1 = [ Π 1 : Γ 1 ] α ̄ Π 2 / Γ 2 [ γ ] im i ^ 2 α coin τ α ̄ φ ̄ , ψ ̄ : u 1 coin τ γ α φ , ψ # [ α ̄ ] # coin τ α ̄ φ ̄ , ψ ̄ by Remark 3 . 3 = [ Π 1 : Γ 1 ] α ̄ Π 2 / Γ 2 1 # [ α ̄ ] [ γ ] i m i 2 α 1 # u 1 coin τ γ α φ , ψ [ Π 1 : Γ 1 ] α ̄ Π 2 / Γ 2 1 # [ α ̄ ] [ γ ] i m i ^ 2 α 1 = [ Π 1 : Γ 1 ] α ̄ R [ φ ̄ , ψ ̄ ] # im i ^ 2 α = [ Π 1 : Γ 1 ] R ( φ , ψ ) by (2) of Lemma 3 . 1 .

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 covers p 1 : M ̃ 1 M 1 and p 2 : M ̃ 2 M 2 . Let (f, g) : M1M2be a pair of maps and let f ̃ , g ̃ : M ̃ 1 M ̃ 2 be a pair of lifts of (f, g). Let Π 1 = A M ̃ 1 , p 1 and Π 2 = A M ̃ 2 , p 2 be the covering transformation groups and let f×, g× : Π1 → Π2be the morphisms of groups induced by f ̃ , g ̃ . Suppose Γ1is a finite index normal subgroup of Π1and Γ2is a finite index normal subgroup of Π2such that f×1) Γ2and g×1) Γ2. Let f ̄ , : Γ 1 \ M ̃ 1 Γ 2 \ M ̃ 2 be a pair of lifts of (f, g) so that the following diagram commutes
Then:
  1. (1)
    If coin τ α ̄ f ̄ × , × = { 1 ̄ } for all α Π2, then
    R ( f , g ) = [ α ̄ ] R [ f ̄ × , × ] R α ̄ f ̄ , .
     
  2. (2)

    R(f, g) is finite if and only if R α ̄ f ̄ , is finite for every α Π2.

     
  3. (3)
    We have
    R ( f , g ) 1 Π 1 : Γ 1 α ̄ A Γ 2 \ M ̃ 2 , p ̄ 2 R α ̄ f ̄ , .
     

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) : M1M2be 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×1) Γ2and g×1) Γ2and such that N1 = Γ1\G1and N2 = Γ2\G2are nilmanifolds. If f ̄ , : N 1 N 2 is a pair of lifts of (f, g), then
R ( f , g ) = 1 Π 1 : Γ 1 α ̄ A N 2 , p ̄ 2 R α ̄ f ̄ , .

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 α ̄ A N 2 , p ̄ 2 such that R α ̄ f ̄ , = . So without loss of generality, we may assume that all coincidence classes are essential. Choose arbitrary α Π2. Then by assumption, p 1 Coin α f ̃ , g ̃ is an essential coincidence class, where p1 : G1M1 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 : M1M2 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 : M1M2 be continuous maps between closed oriented manifolds M1, M2 of equal dimension. Let M ¯ 1 and M ¯ 2 be covers of M1 and M2 respectively and suppose that the covering projections p ̄ 1 : M ¯ 1 M 1 and p ̄ 2 : M ¯ 2 M 2 are orientation preserving. Let ( f ̄ , ) : M ¯ 1 M ¯ 2 be a lifting pair of (f, g). Let x ̄ Coin ( f ̄ , ) . Then x = p 1 ( x ̄ ) p 1 Coin ( f ̄ , ) Coin ( f , g ) . Because the covering projections are orientation-preserving local diffeomorphisms,
ind f , g ; { x } = ind f ̄ , ; { x ̄ } .
Theorem 5.2 (Averaging formula for the Lefschetz coincidence number). Let p 1 : M ̃ 1 M 1 and p 2 : M ̃ 2 M 2 be universal covers, let p ̄ 2 : M ¯ 2 M 2 and ( f ̄ , ) : M ¯ 1 M ¯ 2 be finite sheeted, regular covers and let p 1 : M ̃ 1 M ¯ 1 and p 2 : M ̃ 2 M ¯ 2 be covers such that the diagrams
commute, where all spaces are connected oriented manifolds, where M ¯ 1 and M ¯ 2 are closed manifolds of equal dimension and where all covering projections are orientation preserving local diffeomorphisms. Suppose that f has a lift f ̄ : M ¯ 1 M ¯ 2 and g has a lift : M ¯ 1 M ¯ 2 . Then
L ( f , g ) = 1 # A ( M ¯ 1 , p ̄ 1 ) α ̄ A M ¯ 2 , p ̄ 2 L α ̄ , f ̄ , .

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 # A M ¯ 1 , p ̄ 1 preimages under p ̄ 1 ,
L ( f , g ) = x Coin ( f , g ) ind ( f , g ; { x } ) = 1 # A M ¯ 1 , p ̄ 1 x ̄ p 1 - 1 Coin ( f , g ) ind ( f , g ; { p ̄ 1 ( x ̄ ) } ) .
Now
p ̄ 1 - 1 Coin ( f , g ) = α ̄ A M ¯ 2 , p ̄ 2 Coin α ̄ f ̄ , .
Indeed, x ̄ Coin ( α ̄ f ̄ , ) Coin ( β ̄ f ̄ , ) implies that α ̄ ( f ̄ ( x ̄ ) ) = ( x ̄ ) = β ̄ ( f ̄ ( x ̄ ) ) , such that α ̄ = β ̄ . Hence
L ( f , g ) = 1 # A M ¯ 1 , p ̄ 1 α ̄ A M ¯ 2 , p ̄ 2 x ̄ Coin ( α ̄ f ̄ , ) ind f , g ; { p ̄ 1 ( x ̄ ) } = 1 # A M ¯ 1 , p ̄ 1 α ̄ A M ¯ 2 , p ̄ 2 x ̄ Coin ( α ̄ f ̄ , ) ind α ̄ f ̄ , ; { x ̄ }
because of Remark 5.1. Hence
L ( f , g ) = 1 # A M ¯ 1 , p ̄ 1 α ̄ A M ¯ 2 , p ̄ 2 ind α ̄ f ̄ , ; Coin α ̄ f ̄ , = 1 # A M ¯ 1 , p ̄ 1 α ̄ A M ¯ 2 , p ̄ 2 L ( α ̄ f ̄ , ) .

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 g 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 : VW a linear map. Let β V be a basis for V and β W a basis for W, then we use L β W β V 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 g 1 and g 2 . Let Λ1 be a uniform lattice in G1 and Λ2 a uniform lattice in G2. Let L : g 1 g 2 be a linear map. Let β1 be a preferred basis of Λ1 and β2 a preferred basis of Λ2. Then det L β 2 β 1 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
det Λ 2 Λ 1 ( L ) = det L β 2 β 1

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' : N1N2be continuous maps. Let D, D' : G1G2be morphisms of Lie groups such that D is a homotopy lift of f and D' is a homotopy lift of f'. Then
L ( f , f ) = det Λ 2 Λ 1 D * - D * .

Remark that Theorem 2.7 guarantees the existence of morphisms of Lie groups D, D' : G1G2 that are homotopy lifts of f, f' : N1N2.

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. Let Λ ^ be a finite index subgroup of Λ. Then
det Λ Λ ^ ( Id ) = Λ : Λ ^ .
Proof. Define N = Λ\G and N ^ = Λ ^ \ G and give N and N ^ orientations so that the covering projections GN and G N ^ are orientation preserving. Then Id : GG : g g induces an orientation preserving map f : N ^ N and O : GG : g 1 G induces a constant map f : N ^ N . By Theorem 6.3,
L ( f , f ) = det Λ Λ ^ ( Id ) .
On the other hand, by [[7], Lemma 3.10], every essential coincidence class is a singleton and has coincidence index sign ( det Λ Λ ^ ( Id ) ) . Remark that sign sign ( det Λ Λ ^ ( Id ) ) = 1 . So it suffices to show that # Coin ( f , f ) = Λ : Λ ^ . Now
Coin ( f , f ) = Λ ^ g N ^ | f ( Λ ^ g ) = f ( Λ ^ g ) = Λ ^ g N ^ | Λ g = Λ = Λ ^ g N ^ | g = Λ .

Hence # Coin ( f , f ) = Λ : Λ ^ .

Corollary 6.5. Let G be an oriented, connected, simply connected, nilpotent Lie group and Λ a uniform lattice in G. Let Λ ^ be a finite index subgroup of Λ. Then
det Λ Λ ^ ( Id ) = 1 Λ : Λ ^ .
Proof. Let β be a preferred basis of Λ and β ^ a preferred basis of Λ ^ . Then
det Λ Λ ^ ( Id ) det Λ ^ Λ ( Id ) = det ( Id Λ Λ ^ Id Λ ^ Λ ) = det ( Id Λ Λ ) = 1

and the corollary follows from the previous lemma.

Corollary 6.6. Let G1and G2be oriented, connected, simply connected, nilpotent Lie groups with associated Lie algebras g 1 and g 2 . Let Λ1be a uniform lattice of G1and Λ2a uniform lattice of G2. Let Λ ^ 1 be a finite index subgroup of Λ1and Λ ^ 2 a finite index subgroup of Λ2. Then for any linear map L : g 1 g 2 ,
det Λ ^ 2 Λ ^ 1 ( L ) = Λ 1 : Λ ^ 1 Λ 2 : Λ ^ 2 det Λ 2 Λ 1 ( L ) .
Proof. This follows from a short calculation:
det Λ ^ 2 Λ ^ 1 ( L ) = det Λ ^ 2 Λ 2 ( Id ) det Λ 2 Λ 1 ( L ) det Λ 1 Λ ^ 1 ( Id ) = Λ 1 : Λ ^ 1 Λ 2 : Λ ^ 2 det Λ 2 Λ 1 ( L ) .

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 algebras g 1 and g 2 respectively. Let D, D' : G1G2be morphisms of Lie groups inducing morphisms of Lie algebras D*, D * : g 1 g 2 . Then for any g G2,
det D * - D * = det D * - Ad ( g ) D * .

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 algebras g 1 and g 2 respectively. Let D, D' : G1G2be morphisms of Lie groups inducing morphisms of Lie algebras D*, D * : g 1 g 2 . Then for any g G2, D * - D * is surjective if and only if D * - Ad ( g ) D * is 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 G 1 and G 2 be the complexifications of the real Lie groups G1 and G2 respectively with canonical maps α i : G i G i . Since G i is simply connected, α i is injective. Let g i and g i denote the respective Lie algebras. Then g i g i .

The morphism D : G1G2 extends uniquely to a morphism D : G 1 G 2 of complex Lie groups so that D * = D * . Similarly, we define D : G 1 G 2 , D * : g 1 g 2 and Ad(g) for any g G2.

Let X 1 1 , . . . , X 1 n be a basis for g 1 and X 2 1 , . . . , X 2 n a basis for g 2 . Write Y 1 i = ( α 1 ) * X 1 i and Y 2 i = ( α 2 ) * X 2 i , then Y 1 1 . . . , Y 1 n is a basis for the complex vector space g 1 and Y 2 1 . . . , Y 2 n is a basis for the complex vector space g 2 . Now
det D * - Ad ( g ) D * = det D * - A d α 2 ( g ) D * ,

where on the left hand side, the matrices are expressed with respect to the bases X 1 1 , . . . , X 1 n and X 2 1 , . . . , X 2 n and on the right hand side, the matrices are expressed with respect to the bases Y 1 1 . . . , Y 1 n and Y 2 1 . . . , Y 2 n .

Now we consider the function f : g 2 defined by
f ( Y ) = det D * - Ad exp ( Y ) D * .

By Lemma 6.8, either f is the zero map or f has no roots.

Choose arbitrary Y g 2 . With respect to the complex basis Y 2 1 . . . , Y 2 n of g 2 . ad(Y) is regarded as a matrix [z ij (Y)] where all z i j : g 2 are complex-valued functions. Now there exist complex numbers c i k j such that
Y 2 k , Y 2 j = i c i k j Y 2 i .
Writing Y = k λ k Y 2 k , we remark that
i z i j ( Y ) Y 2 i = ad ( Y ) Y 2 i = Y , Y 2 j = k λ k Y 2 k , Y 2 j = k i λ k c i k j Y 2 i = i k λ k c i k j Y 2 i .
So the entries z i j ( Y ) = k c i k j λ k 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,
f ( Y ) = det D * - exp ad ( Y ) D *

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,
det ( D * Ad ( g ) D * ) = det ( D * C Ad ( α 2 ( g ) ) D * C ) = f ( log ( α 2 ( g ) ) )

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 Π11 and Π22. Let Λ1 be the subgroup of Π1 generated by the set
x k | x 1 .

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 {y k | y Π2} is a subgroup of Γ2. Obviously, any morphism θ : Π1 → Π2 sends the generating set {x k | x Π1} of Λ1 into the generating set {y k | 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 {x k | 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 : M1M2continuous maps. Suppose there exist finite sheeted regular covers p ̄ 1 : N 1 M 1 and p ̄ 2 : N 2 M 2 , where N1and N2are nilmanifolds. Suppose that f ̄ : N 1 N 2 is a lift of f and : N 1 N 2 is a lift of g. Then
N ( f , g ) = 1 # A N 1 , p ̄ 1 α ̄ A N 2 , p ̄ 2 N α ̄ f ̄ , .

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 : M1M2be continuous maps. Let D, D' : G1G2be morphisms of Lie groups and d, d' G2such that λ d D : G1G2is a homotopy lift of f and λ d' D' : G1G2is a homotopy lift of g. (Recall that λ d : G2G2 : g' dg' is the left multiplication map.) Then
L ( f , g ) = 1 # F 1 A F 2 det D * - A * D * , N ( f , g ) = 1 # F 1 A F 2 det D * - A * D * ,
and
R ( f , g ) = 1 # F 1 A F 2 σ det D * - A * D * ,

where the morphisms of Lie groups D*, D * and A*induced by D, D' and A are expressed with respect to preferred bases of Π1G1and Π2G2and where σ : {∞} 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' : G1G2 such that λ d D : G1G2 is a homotopy lift of f and λ d' D' : G1G2 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 = Π1G1 and Γ2 = Π2G2. 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 θ1) Λ2 Γ2 for every morphism of groups θ : Π1 → Π2. Define N1 = Λ1\G1 and N2 = Γ2\G2. Let p ̄ 1 : N 1 M 1 , p ̄ 2 : N 2 M 2 , p 1 : G 1 N 1 and p 2 : G 2 N 2 be natural projections. Then f, g lift to continuous maps f ̄ , : N 1 N 2 so that the following diagram commutes up to homotopy
By Theorem 5.2, the averaging formula for the Lefschetz coincidence number,
L ( f , g ) = 1 1 : Λ 1 α ̄ A ( N 2 , p ̄ 2 ) L α ̄ f ̄ , .
By [[9], Theorem 1.1], N α ̄ f ̄ , = L α ̄ f ̄ , for every α ̄ A N 2 , p ̄ 2 . Hence by Theorem 6.10, the averaging formula for the Nielsen coincidence number,
N ( f , g ) = 1 1 : Λ 1 α ̄ A N 2 , p ̄ 2 N α ̄ f ̄ , = 1 1 : Λ 1 α ̄ A N 2 , p ̄ 2 L α ̄ f ̄ , .
By the main result of [26], R α ̄ f ̄ , = σ L ( α ̄ f ̄ , ) for every α ̄ A N 2 , p ̄ 2 . Hence by Theorem 4.2, the averaging formula for the Reidemeister coincidence number,
R ( f , g ) = 1 1 : Λ 1 α ̄ A N 2 , p ̄ 2 R α ̄ f ̄ , = 1 1 : Λ 1 α ̄ A N 2 , p ̄ 2 σ L α ̄ f ̄ , .
Choose arbitrary α ̄ A N 2 , p ̄ 2 . Then there exist a G2 and A in the holonomy group F2 Aut(G2) such that λ a A : G2G2 is a lift of α ̄ . We now claim that
L α ̄ f ̄ , = Γ 1 : Λ 1 det Γ 2 Γ 1 D * - A * D * .
First remark that f ̄ : N 1 N 2 induces a morphism f ̄ × : A G 1 , p i A G 2 , p 2 defined by
f ̄ × ( λ ) λ d D = λ d D λ for every λ A G 1 , p 1 .

If we apply both sides of this equality to the identity element 1 G 1 of G1, then we see that f ̄ × ( λ ) = τ d D ( λ ) . Hence (τ d D)(Λ1) Γ2 and τ d D : G1G2 induces a continuous map h : N1N2. A similar calculation shows that h × = τ d D = f ̄ × . Because N1 and N2 are aspherical, f ̄ and h are homotopic and we see that τ d D is a homotopy lift of f ̄ .

Similarly, one can show that τ d' D' is a homotopy lift of and that τ a A is a homotopy lift of α ̄ . Then by Theorem 6.3,
L ( α ̄ f ̄ , ) = det Γ 2 Λ 1 τ d D * - τ a A τ d D * = det Γ 2 Λ 1 τ d * D * - τ a A ( d ) * A * D * .
By applying Theorem 6.7 twice, we see that
L α ̄ f ̄ , = det Γ 2 Λ 1 τ d * D * - τ a A ( d ) * A * D * = det Γ 2 Λ 1 D * - A * D * .
By Corollary 6.6,
L α ̄ f ̄ , = det Γ 2 Λ 1 D * - A * D * = Γ 1 : Λ 1 det Γ 2 Γ 1 D * - A * D * .
Since α ̄ was chosen arbitrarily, this equality holds for every α ̄ , where A F2 is such that λ a A is a lift of α ̄ for some a G2. Hence
L ( f , g ) = 1 Π 1 : Λ 1 α ̄ A N 2 , p ̄ 2 L α ̄ , f ̄ , = 1 Π 1 : Λ 1 A F 2 Γ 1 : Λ 1 det Γ 2 Γ 1 D * - A * D * = 1 Π 1 : Γ 1 A F 2 det Γ 2 Γ 1 D * - A * D * = 1 # F 1 A F 2 det Γ 2 Γ 1 D * - A * D * .
Similar calculations show that
N ( f , g ) = 1 # F 1 A F 2 det Γ 2 Γ 1 D * - A * D *
and that
R ( f , g ) = 1 # F 1 A F 2 σ det Γ 2 Γ 1 D * - A * D * .

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 : M1M2be continuous maps. Let D, D' : G1G2be morphisms of Lie groups and d, d' G2such that λ d D : G1G2is a homotopy lift of f and λ d 'D' :G1G2is a homotopy lift of g. Then N(f, g) = L(f, g) if and only if det D * - A * D * 0 for every A F2and N(f, g) = -L(f, g) if and only if det D * - A * D * 0 for every A F2, where D*, 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
Nil = 1 x z 0 1 y 0 0 1 | x , y , z .
Then it is a connected and simply connected 2-step nilpotent Lie group. The corresponding Lie algebra is
n i l = 0 x z 0 0 y 0 0 0 | x , y , z .
It is easy to see that (cf. [[27], Proposition 2.2])
Aut ( Nil ) = det ( A ) p 0 A | A GL ( 2 , ) , p 2 is a row vector
and an element
det ( A ) p 0 A Aut ( Nil ) with p = u v , A = p q r s
acts on Nil as follows:
1 x z 0 1 y 0 0 1 1 p x + q y z 0 1 r x + s y 0 0 1 ,
where
z = ( p s - q r ) z + 1 2 p r x 2 + 2 q r x y + q s y 2 + u x + v y .

Example 7.1. Let Π1 be the Bieberbach group generated by the standard basis {e1, e2, e3} of 3. Let M1 = Π1\3 be the corresponding infra-nilmanifold, the 3-dimensional flat torus, with the trivial holonomy group.

Consider the almost Bieberbach group Π2 given by
Π 2 = s 1 , s 2 , s 3 , α [ s 2 , s 1 ] = s 3 2 , [ s 3 , s 1 ] = [ s 3 , s 2 ] = 1 , α 2 = s 1 α s 1 α - 1 = s 1 , α s 2 = s 2 - 1 α s 3 - 1 .
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
s 1 = 1 1 0 0 1 0 0 0 1 , I , s 2 = 1 0 0 0 1 1 0 0 1 , I , s 3 = 1 0 - 1 2 0 1 0 0 0 1 , I , α = 1 1 2 0 0 1 0 0 0 1 , - 1 0 0 0 1 0 0 0 - 1 .
Then the translation lattice is
Γ 2 = Π 2 Nil = s 1 , s 2 , s 3 = 1 p r 2 0 1 q 0 0 1 | p , q , r Z
and the holonomy group of Π2 is F2 = Π22 2, 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
A = - 1 0 0 0 1 0 0 0 - 1 : 1 x z 0 1 y 0 0 1 1 x - z 0 1 - y 0 0 1 .

Let M2 = Π2\Nil be the corresponding infra-nilmanifold.

Define φ : Π1 → Π2 by
φ ( e 1 ) = α and φ ( e 2 ) = φ ( e 3 ) = 1 Π 2 .
Define the morphism of Lie groups D : 3 → Nil by
D ( x , y , z ) = 1 x 2 0 0 1 0 0 0 1 .

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 : 3 → Nil induces a map f : M1M2 so that f× = φ.

Define ψ : Π1 → Π2 by
ψ ( e 1 ) = 1 Nil , ψ ( e 2 ) = s 2 and ψ ( e 3 ) = s 3 - 1 .
Define d' = 1Nil Nil and define the morphism of Lie groups D' : 3 → Nil by
D ( x , y , z ) = 1 0 - z 2 0 1 y 0 0 1 .

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' : 3 → Nil induces a map f' : M1M2 so that f × = ψ .

Since Γ1 = Π13 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 D * , D * : 3 n i l and A * : n i l n i l are
D * = 1 2 0 0 0 0 0 0 0 0 , D * = 0 0 0 0 1 0 0 0 1 , A * = 1 0 0 0 - 1 0 0 0 - 1 .
Hence by Theorem 6.11
L ( f , f ) = det ( D * - D * ) + det D * - A * D * = - 1 2 - 1 2 = - 1 , N ( f , f ) = det D * - D * + det D * - A * D * = 1 , R ( f , f ) = σ det D * - D * + σ det D * - A * D * = 1 .

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\3 where Π1 is the 3-dimensional orientable Bieberbach group G 2 [[30], Theorem 3.5.5]:
Π 1 = t 1 , t 2 , t 3 , α | [ t i , t j ] = 1 , α 2 = t 1 , α t 2 α - 1 = t 2 - 1 , α t 3 α - 1 = t 3 - 1 .
We can embed this group into Aff(3) by taking {e1, e2, e3} as the standard basis for 3 and
t i = ( e i , I ) ( i = 1 , 2 , 3 ) , α = 1 2 0 0 , 1 0 0 0 - 1 0 0 0 - 1 .
Then the translation lattice is
Γ 1 = Π 1 3 = t 1 , t 2 , t 3 = e 1 , e 2 , e 3 = 3
and the holonomy group is F1 = Π11 2. The holonomy group F1 Aut(3) is generated by A : 33 : (x, y, z) (x, -y, -z). With respect to the preferred basis {e1, e2, e3}, the differential A* : 33 can be expressed as a matrix as follows:
A * = 1 0 0 0 - 1 0 0 0 - 1 .
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]):
Π 2 = s 1 , s 2 , s 3 , β [ s 2 , s 1 ] = s 3 4 , [ s 3 , s 1 ] = [ s 3 , s 2 ] = 1 , β 4 = s 3 β s 1 β - 1 = s 2 , β s 2 β - 1 = s 1 - 1 .
Now we can embed Π2 into Aff(Nil) = Nil Aut(Nil) by taking
s 1 = 1 1 0 0 1 0 0 0 1 , I , s 2 = 1 0 0 0 1 1 0 0 1 , I , s 3 = 1 0 - 1 4 0 1 0 0 0 1 , I , β = 1 0 - 1 16 0 1 0 0 0 1 , 1 0 0 0 0 - 1 0 1 0 .
Then the translation lattice is
Γ 2 = Π 2 Nil = s 1 , s 2 , s 3 = 1 p r 4 0 1 q 0 0 1 | p , q , r Z
and the holonomy group of Π2 is F2 = Π22 4. 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
B = 1 0 0 0 0 - 1 0 1 0 : 1 x z 0