# Topological and affine structure of complete flat manifolds

###### Abstract.

The results of the paper concern the topological structure of complete riemannian manifolds with cyclic holonomy groups and low-dimensional orientable complete flat manifolds. We also discuss related results such as the affine classification of orientable complete flat -manifolds, an algebraic criterion of an affine equivalence and the relationship between holonomy homomorphisms and some algebraic and geometric invariants.

###### Key words and phrases:

Complete flat manifold, complete flat manifold with cyclic holonomy group, Bieberbach group, holonomy group, topological and affine classification###### 1991 Mathematics Subject Classification:

Primary: 53C20; Secondary: 5302, 53C10, 57R22## 1. Introduction

The aim of this paper is to collect some results concering topological and affine structure of complete flat manifolds(cf-manifolds). We recall particularly important known results in Section 2. The others seem to be new. Complete flat manifolds play a particular role in geometry. On one hand they are natural generalizations of euclidean spaces, having the same local properties. On the other hand the study of some complicated questions, arising in differential geometry and related fields, often starts with the examination of them in the case of manifolds of constant curvature. A cf-manifold is the orbit space of a properly discontinuous, isometric, and free action of a discrete group on an euclidean space The holonomy homomorphism of is the map carrying onto linear part of and is the holonomy group of The linear isometry can be written as where acts on the universal covering space of the totally geodesic submanifold of , homotopy equivalent to and acts on the orthogonal complement of If is compact, then and In this case we say that is a Bieberbach manifold. If is noncompact, then is determined by the Bieberbach group and the vertical holonomy homomorphism so that the theory of cf-manifolds can be treated as the theory of orthogonal representations of Bieberbach groups. The manifold is the total space of a flat riemannian vector bundle whose structure group can be reduced to The main difference between the noncompact case and the compact one is that is not always finite and is not a topological invariant of (cf. Section 4 below).

The main difficulty in the classification of Bieberbach manifolds is that it is based on the classification of conjugacy classes of finite subgroups of This is a hard problem in integral representaion group theory, solved for cyclic groups of prime order , for cyclic groups of order , and for small values of only. Bieberbach manifolds are described when the holonomy groups of them are cyclic groups of prime order and when the dimensions of them are smaller than The complexity of the problem is also connected with the fact that the number of -dimensional closed flat manifolds increases rapidly with It is known that ([29, Section 3.5]), ([5]), and ([5]).

In this paper we study cf-manifolds in two particular important cases: when they have cyclic holonomy groups and when their dimensions are smaller than These particular cases are starting points in the investigation of more general ones. In the affine classification it is convenient to use a criterion of affine equivalence formulated in Section 4. We also answer some natural questions concerning algebraic and geometric invariants that are used in the paper. Flat manifolds were investigated in many books and papers. Only few of them deal with the noncompact case (see e.g. [10], [29], [16], [22], [25], [26], [28]). Related results can be found in papers dealing with flat vector bundles, for instance in [4], [7], [9], [10], [12], and [15].

Throughout this paper the following notation will be used. The universal covering space of a topological space will be denoted by If is a group, , then will denote the subgroup of generated by . is the group of diffeomorphisms of a smooth manifold and is the group of affine diffeomorphisms of By we denote the holonomy group of a riemannian manifold . The symbol stands for projection of a Bieberbach group onto its abelianization and for the total space of a vector bundle Real and complex -dimensional trivial vector bundles will be denoted by and respectively.

## 2. Holonomy representations and vector bundles

The aim of the first part of this section is to reformulate some known results in a more convenient for our purposes form. The results will be expressed in terms of vector bundles associated with coverings. To describe these bundles consider a covering map and its covering transformation group . Let be the field of reals or the field of complex numbers, and let be an -dimensional representation of Take the diagonal action on and the orbit space Let be the map determined by the projection . Then the triple is a vector bundle associated to the principal bundle with typical fiber In the sequel we identify this bundle with We often use the following.

###### Proposition 2.1.

Let be a connected -dimensional cf-manifold and let be a closed totally geodesic submanifold of , homotopy equivalent to Denote by , by , and by Then there are a riemannian covering and an orthogonal representation such that The projection is affine. The action of on can be written as where and

There are different proofs of Proposition 2.1. For instance it follows from the arguments used in the proof of Theorem 3.3.3 in [29, ch. 3]. The homomorphism is the vertical holonomy homomorphism od

###### Corollary 2.1.

If the holonomy group of is abelian, then the bundle is isomorphic to

where are 1-dimensional complex representations of and are 1-dimensional real representations of

Proposition 2.1 and the following observation show that cf-manifolds correspond to orthogonal representations of Bieberbach groups.

###### Proposition 2.2.

If is a Bieberbach manifold, is an orthogonal representation of and the action of on is given by the formula

then the orbit space is a complete flat manifold homotopy equivalent to

If is another orthogonal representation of then it is natural to ask when is diffeomorphic (affinely diffeomorphic) to . We consider this question in the next sections. In particular, we give an algebraic criterion of an affine equivalence.

The following three simple lemmas are known.

###### Lemma 2.1.

(cf. [13, ch. 16, § 11]). If is a topological space, is a real vector bundle and is a complex vector bundle, then

The maps and carrying onto and onto are called the restriction and the complexification.

###### Lemma 2.2.

(cf. [13, ch. 8, Theorem 2.6]). Let be a topological space homotopy equivalent to a finite -dimensional CW-complex.

a) If and are two real vector bundles over , and , then

b) If and are two complex vector bundles over , and , then

###### Lemma 2.3.

Let be a closed flat manifold. Assume that is an -dimensional real flat vector bundle and is a -dimensional complex flat vector bundle. Then

a) and have the structures of complete flat manifolds,

b) the image of the total Chern class in is equal to

c)

d)

e)

The proof of a) can be found in [20] (see also [22]). Parts b) and c) are consequences of the flatness of , which implies that Chern forms representing Chern classes of are equal to (see e.g. [18, ch. 11] and [21, Appendix C, Corollary 2]). Parts d) and e) follows from the fact that is a monomorphism (cf. [17, ch. 5, Theorem 3.25], [8, ch. 1, Section B, § 4]), from the equality , and from Lemma 2.1.

Now we formulate a particular case of a result of Wilking ([28, Corollaries 6.4 and 6.5]). It is a generalization of the second Bieberbach theorem.

###### Theorem 2.1.

The set of diffeomorphism classes of -dimensional cf-manifolds is finite and each element of contains a cf-manifold with finite holonomy group.

An affine variant of Theorem 2.1 is false because the holonomy groups of affinely diffeomorphic flat manifolds are isomorphic.

## 3. Complete flat manifolds with cyclic holonomy groups

The aim of this section is to give a topological classification of complete flat manifolds whose holonomy groups are cyclic and whose fundamental groups are isomorphic to a fixed group

###### Definition 3.1.

A flat riemannian manifold with cyclic holonomy group will be called a fch-manifold. A nolcyc bundle is a nonorientable line bundle such that is a complete fch-manifold and is a totally geodesic submanifold of .

The main here are the following.

###### Theorem 3.1.

Let , , and be as in Proposition 2.1 and let Assume that is a cyclic group.

a) If is an orientable bundle, then

b) If is a nonorientable bundle, then

###### Theorem 3.2.

Let be a complete,
connected, -dimensional riemannian manifold with
cyclic holonomy group and let
be a closed

-dimensional flat manifold
homotopy equivalent to . Suppose that is a nolcyc-bundle
over and .
Then is diffeomorphic either to
or to the total space of

Theorem 3.2 shows that there are exactly two diffeomorphism classes of complete, noncompact fch-manifolds having the same fundamental group. It reduces the classification of complete fch-manifolds to the classification of compact ones. Recall that the classification of Bieberbach manifolds with a fixed cyclic holonomy group is a difficult problem solved only when has prime order ([2] [3]).

Theorem 3.1 is a consequence of the following algebraic property of the deck groups of fch-manifolds.

###### Lemma 3.1.

If is a finite cyclic group, then there are and a subgroup of such that and

Theorem 3.1 implies that if the bundle is orientable, then is diffeomorphic to . The case when the bundle is nonorientable is more difficult. It follows from the fact that any two nolcyc bundles and , with the same base space , belong to the same orbit of the action of on

Theorem 3.1 is a generalization of a result of Thorpe stating that is parallelizable or is isomorphic to the direct sum of a trivial bundle and a line bundle (cf. [27]). Using Theorem 3.1 it is easy to verify a more general version of the last statement.

###### Theorem 3.3.

Let be a complete fch-manifold.

a) If is orientable, then is parallelizable.

b) If is nonorientable, then for some nolcyc-bundle over .

###### Corollary 3.1.

If is a complete flat manifold and is a cyclic group of odd order, then is parallelizable.

## 4. Affinely equivalent complete flat manifolds

The aim of this section is to describe algebraic invariants corresponding to affine equivalence classes of noncompact cf-manifolds. For details and for the proofs we refer to [23].

###### Theorem 4.1.

Let and be two -dimensional cf-manifolds with isomorphic fundamental groups. Let and be totally geodesic submanifolds of and homotopy equivalent to . Assume that and Let be the vertical holonomy homomorphisms of and Then the following conditions are equivalent.

(a) is affinely diffeomorphic to

(b) there is an isomorphism and a linear isomorphism such that

for .

Let and be as in the formulation of Theorem 4.1. For a fixed discrete group consider the set of all pairs where is an epimorphism and is a representation.

###### Definition 4.1.

Two elements and of are equivalent if there are and a linear isomorphism such that

where

Let be the set of equivalence classes of the elements of let be the set of epimorphisms from to , and let be the set of conjugacy classes of representations of in Applying Theorem 4.1 we have.

###### Theorem 4.2.

If is a Bieberbach group, then there is a bijection

###### Corollary 4.1.

If then

Let be the set of affine diffeomorphism classes of -dimensional complete flat manifolds with the same fundamental group and let be as above.

###### Theorem 4.3.

If and is infinite, then is uncountable.

###### Proposition 4.1.

Let be a Bieberbach group. Then there are infinitely many affine equivalence classes of cf-manifolds whose fundamental groups are isomorphic to and whose holonomy groups are finite.

##
5. Topological and affine classification of low-dimensional

orientable cf-manifolds

The aim of this section is to describe topological and affine equivalence classes of cf--manifolds. For simplicity we deal with the orientable case. A nonorientable case is somewhat more complicated and will be considered elsewhere. For the classification of cf--manifolds we refer to [29]. We shall use the fact that there are affine diffeomorphism classes of closed flat -manifolds (see e.g. [29, Theorems 3.5.5 and 3.5.9]] for the description of them). For the classification of closed -manifolds we refer to [1] or [11].

###### Theorem 5.1.

There are diffeomorphism classes of orientable, noncompact cf--manifolds. They are represented by:

Here is the Klein bottle and is the tangent bundle of To deal with affine classification of cf--manifolds we need some definitions. Consider the action of on , induced by the standard action of on , and the arising orbit space . Let be the equivalence relation in such that

if and only if and for some and some

###### Theorem 5.2.

Affine equivalence classes of orientable noncompact cf--manifolds, not diffeomorphic to , or are represented by and Affine equivalence classes of cf-manifolds, diffeomorphic to , or correspond to the elements of , , and

## 6. Diffeomorphism classes of some cf-manifolds

In this chapter we discuss the problem of the topological classification of cf-manifolds in a more general context than in Chapter 5. We consider cf-manifolds homotopy equivalent to some low-dimensional Bieberbach manifolds. We also deal with stable diffeomorphism classes of some cf-manifolds.

###### Definition 6.1.

Two manifolds and are stably diffeomorphic if there is a positive integer such that is diffeomorphic to

Given a Bieberbach manifold and there is a diffeomorphism such that ([3, ch. 2, Theorem 5.3], [29, ch. 3, Theorem 3.2.2]). This induces an action of on Let be the class of a flat bundle over in The investigation of stable diffeomorphism classes of cf-manifolds is based on the following consequence of a result of Mazur (cf. [19, Theorem 2]).

###### Proposition 6.1.

Let be two cf--manifolds homotopy equivalent to the same Bieberbach manifold and let be the arising flat bundles. Assume that Then the following conditions are equivalent

a) and are diffeomorphic,

b)

As an immediate consequence of Proposition 6.1 we have

###### Corollary 6.1.

Let be two cf--manifolds homotopy equivalent to the same Bieberbach manifold and let be the arising flat bundles. Then the following conditions are equivalent

a) and are stably diffeomorphic,

b)

###### Corollary 6.2.

Diffeomorphism classes of complete flat -manifolds, homotopy equivalent to a fixed Bieberbach -manifold , correspond to the elements of

Let , let be the Möbius bundle:

let , and let be the generators of the deck group of the Klein bottle defined by the formulas

Consider the generators of , dual to the images of in , and line bundles such that and We have

###### Proposition 6.2.

Let be a cf--manifold homotopy equivalent to If , then is diffeomorphic to or

###### Theorem 6.1.

a) If , then the diffeomorphism classes of cf--manifolds homotopy equivalent to are represented by and

b) Diffeomorphism classes of cf--manifolds homotopy equivalent to are represented by and

###### Theorem 6.2.

a) If , then
the diffeomorphism classes of
cf--manifolds homotopy equivalent to the Klein bottle
are represented by
and

b) Diffeomorphism classes of cf--manifolds homotopy equivalent to the Klein bottle are represented by and

The proof of Proposition 6.2 is easy. To describe the idea of the proofs of the other results denote and by The and -dimensional case follows from a direct argument. By Lemma 6.2, the isomorphism classes of flat vector bundles over whose dimension is greater than correspond to their images in Using the Atiyah-Hirzebruch -spectral sequence of the fibration one can check that the map

carrying the class of the bundle onto is a bijection. Now it suffices to find the orbit space of the action of on

## 7. Holonomy homomorphisms and geometric invariants

In this section we express characteristic classes of some flat bundles in terms of their holonomy homomorphisms. We also discuss how to calculate cohomology groups containing some invariants arising in this paper. By Corollary 2.1, any cf-manifold with abelian vertical holonomy group is the total space of the direct sum of complex line bundles and real line bundles These line bundles are determined by their Chern classes and Stiefel-Whitney classes, respectively.

###### Lemma 7.1.

Let be a real flat line bundle over a closed flat manifold Assume that is the holonomy homomorphism of and Let be the projection and let be the isomorphism. Then

To state an analogous description of the Chern classes write the first homology group of a Bieberbach manifold as where Let be the order of the holonomy group of and let be the torsion subgroup of The set , of isomorphism classes of flat complex line bundles over , is a commutative group with tensor product as a group operation. It is known that is a monomorphism (cf. [13, ch. 16, Theorem 3.4]) and ([15, Theorem 6.1]). For every the formula

defines Consider the coboundary homomorphism induced by the short exact sequence

where and is the canonical projection.

###### Lemma 7.2.

Let be a Bieberbach manifold and let , and be as above. Let be a complex flat line bundle over with holonomy homomorphism . Take the factorization

of . Then

a) carrying onto is an isomorphism,

b)

The proof of Lemma 7.1 is an easy exercise. We do not know a reference to the statement and proof of Lemma 7.2.

###### Corollary 7.1.

a)

b) If then all complex flat line bundles over are trivial.

c) If and is a cf-manifold homotopy equivalent to , having abelian vertical holonomy group, then is diffeomorphic to the total space of the direct sum of some real line bundles over

There are different methods allowing to calculate first and second cohomology groups of a Bieberbach -manifold . One can use a general approach based on the Smith normal form of an integer matrix (cf. [14], [24]). We discuss another simple approach that can be applied if the holonomy group of is a cyclic group of order . In this case is affinely diffeomorphic to the mapping torus of an affine diffeomorphism such that

###### Theorem 7.1.

Let be a group isomorphic to acting on and let be a generator of Assume that is an affine diffeomorphism of such that and is the mapping torus of Then

Here The cohomology group is of interest in its own right. To see this denote by the number of connected components of the fixed point set of a smooth action of on a manifold homotopy equivalent to Identifying with we can treat as a -module. If is a -group, then

([6, Theorem A.10]). Recall that a -lattice is a -module that is also a free abelian group of finite rank.

###### Theorem 7.2.

Let be a finite group and let be a -lattice. Assume that and is a positive integer relatively prime to Then

###### Corollary 7.2.

Let and be two different prime numbers and let be a -lattice. Then

Corollary 7.2 is particularly convenient because it reduces the calculation of to the determination of the number of solutions of systems of linear equations in finite fields and Applying Theorem 7.1 and Lemma 7.2 we have

###### Corollary 7.3.

Let be closed flat manifold, whose holonomy group is isomorphic to let be an affine diffeomorphism of such that is diffeomorphic to , and let denote with the induced -action on it. Then

Now we give a convenient criterion of the triviality of

###### Proposition 7.1.

Let be a finite group, let be a -lattice, and let be a prime number such that . Assume that for every prime divisor of we have

Then .

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