Lobachevskii Journal of Mathematics Vol. 10, 2002, 9 – 16 \begin{matrix}c \\ ◯ \end{matrix}S. A. Grigorian, R. N. Gumerov

S. A. Grigorian, R. N. Gumerov

ON A COVERING GROUP THEOREM AND ITS APPLICATIONS

(submitted by M. A. Malakhaltsev)

Abstract. Let p : X → G be an n-fold covering of a compact group G by a connected topological space X. Then there exists a group structure in X turning p into a homomorphism between compact groups. As an application, we describe all n-fold coverings of a compact connected abelian group. Also, a criterion of triviality for n-fold coverings in terms of the dual group and the one-dimensional Čech cohomology group is obtained.

Introduction

The study of algebraic equations with coefficients in function algebras is intimately related to the study of coverings (see, e. g., [F], [GL], [Go2]). In the case in which the coefficients are continuous functions defined on a topological group, in particular, almost periodic functions, we have to do with coverings of groups (see, e. g., [GL]). The questions considered here arose in studying algebraic equations with coefficients in Banach algebras of generalized analytic functions. In this situation one also deals with topological groups ([AS], [G, Ch. VII], [Gr]) and their coverings (see, e. g., [GGK1]).

This note is concerned with finite-sheeted coverings of compact connected groups. Several results on the structure of these coverings will be announced here. We will give a brief outline of the proof of the covering group theorem. The theorem states that an n-fold covering space of a group can be equipped with a multiplication turning a covering map into a homomorphism between topological groups. This result enables us to use a technique of characters for studying finite-sheeted coverings of abelian groups. As an application of the theorem, we describe the structure of such coverings. Namely, all n-fold coverings of a compact connected abelian group G are algebraic. Moreover, it can be shown that if the dual group Ĝ, which coincides with the one-dimensional Čech cohomology H^{1}(G, ℤ), admits taking the n!th root, then all n-fold coverings of G are trivial, and vice versa.

1991 Mathematical Subject Classification. Primary 14E20, 22A10, 43A77, 43A10, 43A40, 46J10; Secondary 20J06, 22C05, 54H11, 57M10.

Key words and phrases. n-fold coverings of compact groups, covering groups, algebraic coverings, criterion of triviality for n-fold coverings, dual group, onedimensional Čech cohomology group, algebraic equations with coefficients in function algebras.

The problem on the existence of a group structure in a covering space is also motivated by the well-known theorem on covering group in the topological groups theory (see, for example, [P, Theorem 79]). In proving our covering group theorem we use this one and the fact that every compact group can be approximated by an inverse system of compact Lie groups ([P, Theorem 68], [We, §25]). It should be noted that we assume neither path connectedness nor local connectedness of considering spaces. In [GGK1] the covering group theorem was proved for the compact solenoidal groups. The method used there does not work in the general case. In [GGK2] the covering group theorem for solenoids is applied for proving the Walter-Bohr-Flanders theorem on solutions of algebraic equations with functional coefficients (see [Wa], [BF]).

The note is organized as follows. It is divided into 4 sections. In Section 1 we introduce a notation needed in the sequel and formulate a series of technical lemmas concerning unramified coverings of compact groups. Section 2 is devoted to the covering group theorem for a compact connected group (Theorem 1). In Section 3 we characterize all finite-sheeted coverings of compact connected abelian groups as algebraic ones (Theorem 2). In concluding Section 4 we formulate the criterion of triviality for unramified finite-sheeted coverings of groups (Theorem 3 and Theorem 4). Also, our results are applied for investigating equations with functional coefficients. More precisely, we prove the criterion of totally solvability for algebraic equations over the Banach algebra of all continuous functions on a compact connected abelian group (see Corollary).

We would like to thank the participants of Alexandrov topological seminar at Moscow State University for the valuable discussions of the results presented here.

1. Notation and Preliminaries

We shall use the terminology of the Riemann surfaces theory for covering maps (see, for example, [F, §4]). Throughout this note p : X → G is an n-fold covering of a compact group G by a Hausdorff topological space X.

As it was remarked in the Introduction, there exists an inverse system {G_{λ}, π_{λ}^{μ}, Λ} of compact Lie groups G_{λ} with an index set Λ upward directed by a relation ≺, such that

G = \underleftarrow{lim}\{G_{λ}, π_{λ}^{μ}, Λ\}.

Recall (see [ES, Chapter VIII]) that a continuous homomorphism π_{λ}^{μ} : G_{μ} → G_{λ} is defined for λ, μ ∈ Λ whenever λ ≺ μ; π_{λ}^{λ} : G_{λ} → G_{λ} is the identity map, and π_{λ}^{ν} = π_{λ}^{μ}π_{μ}^{ν} for all λ, μ, ν ∈ Λ satisfying λ ≺ μ ≺ ν. For every λ ∈ Λ we have the natural projection π_{λ} : G → G_{λ} such that the following diagram

G  G
π_{λ}↓  ↓^{π_{μ}}
G_{λ} ⟵^{π_{λ}^{μ}} G_{μ}

is commutative, i. e., π_{λ} = π_{λ}^{μ}π_{μ} whenever λ ≺ μ. It should be pointed out that in this situation all bonding homomorphisms π_{λ}^{μ} and all projections π_{λ} are open and surjective [We, §5]. We also recall that the family of all sets π_{λ}^{−1}(U), where U is an open subset of G_{λ} and λ runs over a subset cofinal in Λ, is a base for the topology of \underleftarrow{lim}\{G_{λ}, π_{λ}^{μ}, Λ\} (see, e. g., [En, Proposition 2.5.5]). Clearly, there exists an index α ∈ Λ and a finite open cover {W_{1}^{α}, W_{2}^{α}, …, W_{m}^{α}} of a group G_{α} such that a family {W_{k} := π_{α}^{−1}(W_{k}^{α}) : k = 1, 2, …, m} of the inverse images forms a finite open cover of G and every set W_{k} is evenly covered by p.

From now on the index α ∈ Λ and the cover {W_{1}, W_{2}, …, W_{m}} are fixed. For each k = 1, 2, …, m, we denote by {V_{1}^{k}, V_{2}^{k}, …, V_{n}^{k}} a partition of p^{−1}(W_{k}) into non-empty slices. Thus, we have the representation

p^{−1}(W_{k}) = ⨆_{l=1}^{n} V_{l}^{k},

and the restriction p|_{V_{l}^{k}} of p to V_{l}^{k} is a homeomorphism of V_{l}^{k} onto W_{k} for every l ∈ {1, 2, …, n}. We also fix all these partitions {V_{1}^{k}, V_{2}^{k}, …, V_{n}^{k}} and enumerations of their elements by means of the subscripts.

In the sequel we shall use the following notation for maps. Suppose we are given topological spaces Y_{1}, Y_{2}, …, Y_{t} and maps f_{s} : X → Y_{s}, s = 1, 2, …, t. We recall that the diagonal of the maps f_{1}, f_{2}, … f_{t} is defined as follows:

f_{1} △ f_{2} △ … △ f_{t} : X → ∏_{s=1}^{t} Y_{s} : x ↦ (f_{1}(x), f_{2}(x), …, f_{t}(x)).

In order to obtain the main result we need first to formulate three technical lemmas concerning unramified coverings of compact groups. Note, we do not suppose that the groups are connected. The main reason why we present here the first lemma is that its formulation and proof involve the functions and the neighborhoods that we shall work with. Roughly speaking, the other two lemmas show that an n-fold covering of a group can be approximated by unramified finite-sheeted coverings of compact Lie groups.

Using the properties of a topology in G, one can readily see that there exists an open cover {U_{1}, U_{2}, …, U_{m}} of the group G such that

\overline{U}_{k} ⊂ W_{k} and U_{k} = π_{α}^{−1}(U_{k}^{α}),  (1)

where k = 1, 2, …, m, and {U_{1}^{α}, U_{2}^{α}, …, U_{m}^{α}} is an open cover of a group G_{α}.

Let {φ_{1}, φ_{2}, …, φ_{m}} be a partition of unity on G_{α} dominated by the cover {U_{1}^{α}, U_{2}^{α}, …, U_{m}^{α}}. For each k = 1, 2, …, m, we consider the composites

ψ_{k} := φ_{k} ∘ π_{α} : G → [0, 1] and ψ̂_{k} := φ_{k} ∘ π_{α} ∘ p : X → [0, 1].

Clearly, the families {ψ_{1}, ψ_{2}, …, ψ_{m}} and {ψ̂_{1}, ψ̂_{2}, …, ψ̂_{m}} are the partitions of unity on G and X dominated by the covers {U_{1}, U_{2}, …, U_{m}} and {p^{−1}(U_{1}), p^{−1}(U_{2}), …, p^{−1}(U_{m})}, respectively.

Now, for each k = 1, 2, …, m, we define functions f_{k} : X → ℂ and f̂_{k} : X → ℂ by the equations:

f_{k}(x) := \begin{cases}0, & if x ∉ p^{−1}(W_{k}) = ⨆_{l=1}^{n} V_{l}^{k}, \\ exp (i\frac{2π}{n} (l − 1)), & if x ∈ V_{l}^{k}, l = 1, 2, …, n; \end{cases}

f̂_{k}(x) := ψ̂_{k}(x)f_{k}(x) for x ∈ X.

(2)

Of course, we have |f̂_{k}(x)| = |ψ̂_{k}(x)| whenever x ∈ X. It follows from (1) that the functions f̂_{k} are continuous on X.

Let Ψ : X → G × ℂ^{m} denote the diagonal p △ f̂_{1} △ f̂_{2} △ … △ f̂_{m}. Clearly, we have the following statement.

Lemma 1.1. The map Ψ : X → G × ℂ^{m} is an imbedding.

Remark. Certainly, to imbed the space X into the space G × ℂ^{m} it suffices to use the cover {W_{1}, W_{2}, …, W_{m}} without considering a shrinking cover (see [GGK1, Lemma 1]). But the refinement {U_{1}, U_{2}, …, U_{m}} introduced above play an important rôle in proving Lemma 1.3.

We shall next construct a finite-sheeted covering p_{α} : X_{α} → G_{α} of the group G_{α}, i. e., a covering whose fibers p_{α}^{−1}({g}) are finite for all g ∈ G_{α}. This covering will be compatible with p in a certain sense.

Define X_{α} to be the image of the diagonal (π_{α} ∘ p) △ f̂_{1} △ f̂_{2} △ … △ f̂_{m}, that is,

X_{α} := {(π_{α}(p(x)), f̂_{1}(x), f̂_{2}(x), …, f̂_{m}(x)) : x ∈ X} ⊂ G_{α} × ℂ^{m},

and h_{α} : X → X_{α} to be the map given by

x ↦ (π_{α} ∘ p) △ f̂_{1} △ f̂_{2} △ … △ f̂_{m}(x), x ∈ X.

Let p_{α} : X_{α} → G_{α} denote the projection onto the first coordinate.

Lemma 1.2. The map p_{α} is a finite-sheeted covering and the following diagram commutes:

\begin{matrix}X & ⟶^{p} & G \\ ^{h_{α}}↓ & & ↓^{π_{α}} \\ X_{α} & ⟶_{p_{α}} & G_{α} \end{matrix}  (3)

Generally speaking, the finite-sheeted covering p_{α} : X_{α} → G_{α} in the preceding lemma may be ramified. By analogy with the constructions of h_{α} : X → X_{α} and p_{α}, for each successor λ of α we can consider the function h_{λ} : X → X_{λ} and the covering p_{λ} : X_{λ} → G_{λ} with the properties described in Lemma 1.2. It turns out that there exists an unramified covering among these coverings p_{λ}. In other words, the covering p can be approximated by unramified finite-sheeted coverings.

Lemma 1.3. There is an index β ∈ Λ, satisfying α ≺ β, such that p_{β} : X_{β} → G_{β} is an n-fold covering of a group G_{β}.

2. The covering group theorem

Throughout this section X is a connected n-fold covering space of a compact connected group G. In this case G is the inverse limit of an inverse system of connected compact Lie groups G_{λ}, which are abelian provided that G is an abelian group (see [We, §25]).

Using the results of the previous section we can prove the covering group theorem.

Consider the commutative diagram

\begin{matrix}X & ⟶^{p} & G \\ ^{h_{β}} ↓ & & ↓^{π_{β}} \\ X_{β} & ⟶_{p_{β}} & G_{β}. \end{matrix}  (4)

First, the space X_{β} is equipped with a topological group structure turning p_{β} : X_{β} → G_{β} into a homomorphism of topological groups. To do this we use the well-known covering group theorem for path connected and locally path connected groups [P, Theorem 79]. In the case in which G_{β} is abelian we get an abelian covering group X_{β} ([M, Ch.V, Ex.5.4]). Using the multiplication on X_{β} and the diagram (4), we define the desired multiplication on the space X.

Thus we have the covering group theorem.

Theorem 1. Let p : X → G be an n-fold covering of a compact topological group G by a connected topological space X. Then there is a multiplication on X relative to which X is a compact topological group and p is a homomorphism. Moreover, if G is abelian, then X is abelian as well.

3. algebraic coverings

Let us consider a polynomial in one indeterminate w of degree n over the Banach algebra C(G) of all continuous functions on G:

r(w) = w^{n} + f_{1}w^{n−1} + f_{2}w^{n−2} + ⋯ + f_{n},

i.e., the coefficients f_{1}, f_{2}, …, f_{n} are continuous functions on the group G. We set

Y = {(g, z) ∈ G × ℂ : z^{n} + f_{1}(g)z^{n−1} + ⋯ + f_{n}(g) = 0}.

It is well known that if, for each g_{0} ∈ G, the polynomial

w^{n} + f_{1}(g_{0})w^{n−1} + f_{2}(g_{0})w^{n−2} + … + f_{n}(g_{0})

with numerical coefficients does not have multiple roots, then the projection

p̃: Y → G : (g, z) ↦ g

is an n-fold covering of G. In this case, the map p̃ is called the covering generated by the polynomial r(w) of degree n over the algebra C(G).

An n-fold covering p : X → G is said to be algebraic if there is a covering p̃ : Y → G generated by a polynomial of degree n over C(G) and a homeomorphism φ : X → Y such that the following diagram

\begin{matrix}X & ⟶^{φ} & Y \\ ^{p}↓ & & ↓p̃ \\ G & & G \end{matrix}

is commutative.

Theorem 2. All n-fold coverings of a compact connected abelian group G are algebraic.

4. criterion of triviality for n-fold coverings

As usual, Ĝ denotes a multiplicative dual group of a connected compact abelian group G. For each integer m we have a homomorphism

τ_{m} : Ĝ → Ĝ : χ ↦ χ^{m}.

If τ_{m} is an epimorphism, then we say that Ĝ admits taking the mth root. In this case, because Ĝ is a torsion-free group [HR, Theorem 24.25], the homomorphism τ_{m} is an automorphism. We also recall that an n-fold covering p : X → G is said to be trivial if there is a homeomorphism φ : G × {1, 2, …, n} → X making the following diagram

\begin{matrix}G × \{1, 2, …, n\} & ⟶^{φ} & X \\ p̃↓ & & ↓^{p} \\ G & & G \end{matrix}  (5)

commute, where p̃ is the projection onto the first coordinate.

We have the following criterion of triviality for unramified finite-sheeted coverings of abelian groups.

Theorem 3. All n-fold coverings of a compact connected abelian group G are trivial if and only if its dual group Ĝ admits taking the n!th root.

Recall that by the Bohr - van Kampen theorem [Bo],[vK] (see also [Go1]), we have the algebraic isomorphism

Ĝ = C(G)^{−1} /expC(G),

where C(G)^{−1} is the group of invertible elements of the algebra C(G) and expC(G) denotes the subgroup consisting of all the functions of the form expa, where a ∈ C(G). It is well known that this isomorphism along with the Bruschlinsky – Eilenberg theorem ([Br], [Ei]), concerning the one-dimensional Čech cohomology, yield the following isomorphism

H^{1}(G, ℤ) = Ĝ

(see also the Arens - Royden theorem in [R], [G, Ch.III, Theorem 7.2]). Thus we immediately obtain the following criterion.

COVERING GROUP THEOREM 15

Theorem 4. All n-fold coverings of a compact connected abelian group G are trivial if and only if the one - dimensional Čech cohomology group H^{1}(G, ℤ) admits taking the n!th root.

Finally we will apply our criterion to the study of algebraic equations with continuous coefficients. To do this we introduce some terminology and notation from [GL]. Let us consider an algebraic equation

r(w) := w^{n} + f_{1}(g)w^{n−1} + f_{2}(g)w^{n−2} + … + f_{n}(g) = 0  (6)

such that f_{1}, f_{2}, …, f_{n} are elements of the Banach algebra C(G), and for each g_{0} ∈ G the corresponding equation with numerical coefficients does not have multiple roots. A collection of all such equations of degree n is denoted by 𝔄_{n}(G). The equation (6) is said to be totally solvable if there exist n strictly distinct continuous solutions h_{1}, h_{2}, …, h_{n} ∈ C(G) of this equation, i.e., h_{j}(g) ≠ h_{k}(g) whenever j ≠ k and g ∈ G.

It is easy to see that the totally solvability of the equation (6) is equivalent to the triviality of the n-fold covering generated by r(w). In view of Theorems 2 and 3, we get the following criterion of totally solvability for algebraic equations over the Banach algebra of all continuous functions on a compact connected abelian group G.

Corollary. All equations in 𝔄_{n}(G) are totally solvable if and only if the dual group Ĝ admits taking the n!th root.

This statement is the commutative case of the criterion which is due to Gorin and Lin (see [GL, Corollary 1.11]).

References

[AS] R. Arens & I. Singer, Generalized analytic functions, Trans. Amer. Math. Soc., 81(1956), 379–393.

[Bo] H. Bohr, Über fastperiodische ebene Bewegungen, Comm. Math. Helv., 4(1932), 51–64.

[Br] N. Bruschlinsky, Stetige Abbildungen und Bettische Gruppen der Dimensionzahl 1 und 3, Math. Ann., 109(1934), 525–537.

[BF] H. Bohr & D. A. Flanders, Algebraic equations with almost periodic coefficients, Kobenhavn, 1937.

[Ei] S. Eilenberg, Transformations continues en circonférence et la topologie du plan, Fund Math., 26(1936), 61–112.

[En] R. Engelking, General Topology, Warszawa, 1977.

[ES] S. Eilenberg & N. Steenrod, Foundations of Algebraic Topology, Princeton, Univ. Press, New Jersey, 1952.

[F] O. Forster, Riemannsche Flächen, Heidelberger Taschenbücher Band 184, Springer-Verlag, New York, 1977.

[G] T. W. Gamelin, Uniform Algebras, Prentice-Hall Series in Modern Analysis, N.J., 1969.

[Go1] E. A. Gorin, A function-algebra variant of the Bohr - van Kampen theorem, Mat. Sbornik., N. Ser., Vol. 82(124), No. 2 (1970), 260–272 (in Russian). (English translation: Math. USSR, Sb., 11(1970), 233-243).

[Go2] E. A. Gorin, Some examples concerning algebraic equations in Function Algebras, Doklady Akad. Nauk SSSR, Vol. 200, No. 2 (1971), 273–276 (in Russian). (English translation: Sov. Math. Dokl., 12(1971), 1371-1375).

[GL] E. A. Gorin & V. Ya. Lin, Algebraic equations with continuous coefficients and some problems of the algebraic theory of braids, Mat. Sbornik., N. Ser., Vol. 78(120), No. 4

16 S. A. GRIGORIAN, R. N. GUMEROV

(1969), 579-610 (in Russian). (English translation: Math. USSR, Sb., 7(1969), 569-596).

[Gr] S. A. Grigorian, Generalized analytic functions, Uspekhi Mat. Nauk (Russian Mathematical Surveys), Vol. 49, No. 2 (1996), 3–42 (in Russian).

[GGK1] S. A. Grigorian, R. N. Gumerov & A. V. Kazantsev, Group structure in finite coverings of compact solenoidal groups, Lobachevskii Journal of Mathematics, Vol. VI, 2000, 39–46.

[GGK2] S. A. Grigorian, R. N. Gumerov & A. V. Kazantsev, Unramified coverings of solenoidal groups and the Bohr - Walter - Flanders theorem, Mathematical Notes (in Russian) (to appear).

[HR] E. Hewitt & K. A. Ross, Abstract Harmonic Analysis, Vol. 1, Springer, Berlin, 1963.

[vK] E. R. van Kampen, On almost periodic functions of constant absolute value, J. London Math. Soc., XII, No. 1 (1937), 3–6.

[M] W. S. Massey, Algebraic Topology: An Introduction, Harcourt, Brace & World, Inc., New York, 1967.

[P] L. S. Pontrjagin, Continuous groups, Moscow, Nauka, 1984 (in Russian).

[R] H. Royden, Function Algebras, Bull. Amer. Math. Soc., 69, No. 3(1963), 281–298.

[Wa] A. Walter, Algebraishe Funtionen von fastperiodischen Funktionen, Monatsh. Math. und Phys., 40, No 2 (1933), 444–457.

[We] A. Weil, L’intégration dans les groupes topologiques et ses applications. Paris, 1940.

Suren A. Grigorian, Faculty of Mechanics and Mathematics, Zelenodolsk Filiation of Kazan State University, Saidashev street 4, Zelenodolsk, 420520, Tatarstan, Russia E-mail address: sugrigor@green.kcn.ru Renat N. Gumerov, Mathematics and Mechanics Institute, Kazan State University, Kremlevskaya street 17, Kazan, 420008, Tatarstan, Russia E-mail address: Renat.Gumerov@ksu.ru

Received February, 20, 2002