V. V. Gorbatsevich, “On the fibre structure of compact homogeneous spaces”, Izv. Math., 87:6 (2023), 1161

The article studies the topological structure of arbitrary compact homogeneous spaces. The first works of the author in this direction were published about forty years ago. Several fibrations have been described that allow one to understand how an arbitrary compact manifold is topologically structured if there exists a transitive action of some finite-dimensional Lie group on it (a partial exposition of results of this kind was given in the survey [1]). This description (which, as it turned out, can naturally be carried out up to a finite covering) was quite detailed, but it also contained many unsolved problems. This article contains results that continue these studies and contain solutions to some of these problems. This was done both with the use of new topological results of various authors, and with the help of the development of some methods previously used by the author himself. In fact, almost all the results obtained below are also valid for a wider class of homogeneous spaces, viz. for plesiocompact homogeneous spaces (see [2], [3]). Plesiocompact homogeneous spaces can be considered as some generalizations of compact homogeneous spaces and a homogeneous spaces with a finite invariant measure. Below, we will occasionally mention the possibility of extending the results obtained to the plesiocompact case, but we will not dwell on this in more detail here.

Sometimes it will be convenient for us not to specify a Lie group that is transitive on some manifold. In this case, we will speak of a homogeneous manifold, meaning by it a manifold on which there exists a transitive action of some finite-dimensional Lie group.

In this first section of the article, we will give (or recall) the main definitions and constructions related to the concepts of natural, structure, and Borel fibrations introduced by the author for arbitrary compact homogeneous spaces. In § 2, some properties of the base of the natural fibration are considered. In § 3, we study properties of the structural fibration of this base, and in §§ 4 and 5, those of the structural group and fibre of the natural fibration, respectively. In § 6, we consider a fibre of a Borel fibration, and in § 7, the principal fibration corresponding to the natural fibration and study the concept of principal homogeneous space. Here we also give a description of the topological structure of compact homogeneous spaces close to principal ones. The concluding § 8 discusses the relationship between the properties of the fibre and the base of the natural fibration.

In the present paper, we will give a number of examples illustrating some of the concepts introduced and the assertions made, as well as several counterexamples to some natural assumptions about the elements of the fibre structure of compact homogeneous spaces.

Lie groups will be denoted by uppercase Latin letters, and their Lie algebras, by the corresponding lowercase Latin letters. Lie groups that are transitive on manifolds will usually be assumed to be connected and simply connected, and their transitive actions themselves to be locally effective. Note that the local efficiency of the action of a Lie group is equivalent to the fact that the set of its elements, whose actions on the manifold are trivial, is discrete. In particular, the action of any subgroup of such a Lie group will also be locally effective. For an arbitrary Lie group $F$, we denote by $F_0$ the connected component of the identity of this Lie group, and by $\pi_0(F)$, the group of connected components. In general, $\pi_k$ will denote homotopy groups. A smooth locally trivial fibration of $M$ over base $B$ and with fibre $F$ will be denoted by $F\to M \to B$.

We begin by presenting some basic properties of compact homogeneous spaces related to their fibre structure (for more details, see, for example, [1], [3]).

Let $M = G/H$ be a compact homogeneous space of some connected Lie group $G$ (the Lie subgroup $H$ is always assumed to be closed). We can (and will) consider this Lie group to be simply connected and locally effective on $M$ (however, it will not always be effective). Let $K$ be some maximal compact subgroup of $G$. It is simply connected (since so is the Lie group $G$) and uniquely defined up to conjugation in $G$. There is a natural action of the Lie group $K$ on the manifold $M$. In the general case, the orbits of this action are not isomorphic to each other, although all of them here will be of the same dimension. They may have different orbit types (that is, the stationary subgroups corresponding to different points of $M$ will not always be conjugate in $K$).

An action of a compact group $K$ on a manifold $M$ is called equiorbital if all its orbits are of the same orbital type, that is, if for any points $m, m' \in M$ the corresponding stationary subgroups $K_m$, $K_{m'}$ are conjugate in $K$. It turns out that for an arbitrary compact homogeneous space $G/H$ there exists some compact homogeneous space $M' = G/H'$ (of the same Lie group $G$) covering it finitely (here $H'$ is some subgroup of finite index in $H$) and which is equiorbital under the natural action of the maximal compact subgroup $K$. In this case, the space of orbits $K \setminus M' $ of this action is a smooth manifold, and the natural mapping $M' \to K \setminus M' $ is a smooth locally trivial fibration. This fibration is called the natural fibration for $M$ (although in fact it is often not the manifold $M$ itself that is fibrationed, but an appropriate manifold $M'$ that covers it finitely). We denote the base of this fibration by $M_a$ and its fibre by $M_c$.

In the general case, when constructing a natural fibration, passing to $M'$ is necessary if we want to obtain a smooth manifold structure on $K \setminus M'$. Here, there is no canonical choice of a finite-sheeted covering. However the original manifold $M$ also has a kind of fibration – this will be the Seifert fibration over the base $K \setminus M$ which is an orbifold (and therefore, generally speaking, a manifold with singularities; for more details, see [4]). We will not consider this generalization of the natural fibration here.

So, the natural fibration $M_c \to M' \to M_a$ is a smooth locally trivial fibration for an appropriate manifold $M'$ finite-sheeted covering the original compact homogeneous space $M$. This fibration over the manifold $M'$ is defined uniquely up to a fibrewise homotopic equivalence.

The base of the natural fibration $M_a$ has the form $K \setminus M' $ and it is an aspherical smooth manifold (asphericity means that the homotopy groups $\pi_i(M_a)$ are trivial for all $i \geqslant 2$). Moreover, the universal covering manifold of $M_a$ is diffeomorphic to Euclidean space (note that not every aspherical manifold has this property, although the universal covering manifold will always be contractible for them). Moreover, up to diffeomorphism, the base of the natural fibration can be written (perhaps again by passing to some finite covering) in the form $\Gamma \setminus F/C$, where $F$ is some connected, although not necessarily simply connected, group Lie group (in fact, it is closely related to the original transitive Lie group; see for example [5], [2]), $C$ is the maximal compact Lie subgroup of $F$, and $\Gamma$ is a uniform lattice (that is, a discrete subgroup with a compact quotient space) in $F$ that is torsion-free. Moreover, passing if necessary to a subgroup of finite index in $\Gamma$ (which is equivalent to passing from $M_a$ to some manifold covering it finitely), we can assume that $\Gamma \cap C$ is contained in the centre $Z(F)$ of the Lie group $F$ (the necessary argument from [5] will be recalled below in § 7). In addition, we can assume that this intersection is trivial by factorizing the Lie group $F$ with respect to it.

Representation of the base $M_a$ of the natural fibration as $\Gamma \setminus F/C$ (where $C$ is the maximal compact subgroup in the connected Lie group $F$ and $\Gamma$ is the uniform lattice in $F$ torsion-free) will be called the standard representation of the base. The base of a natural fibration can have several different (and essentially different) standard representations.

The fibre of the natural bundle has the form $M_c = K/L$, where $L = K \cap H'$ is a stationary subgroup for some point $m_0 \in M'$ (it is a closed subgroup in the compact group Lie $K$). If we assume that the Lie group $G$ is simply connected (and we will do so), then, as is well known, the maximal compact subgroup $K$ will also be simply connected, and therefore, it will necessarily be semisimple. Therefore $M_c$ is a homogeneous space of a compact semisimple Lie group. The fibre $M_c$ of the natural bundle is almost simply connected, that is, its fundamental group $\pi_1(M_c)$ is finite. An important particular case is when this fibre is simply connected. A necessary and sufficient condition for the fibre $M_c$ (or, equivalently, for the subgroup $L = K \cap H'$ to be connected) is the condition that the fundamental group $\pi_1(M)$ of the manifold $M$ has no torsion (under the assumption that the action of $K$ on $M$ is equiorbital).

The compact Lie group $Q = N_K (L)/L$ is the structure group of the natural bundle, where $N_K(L)$ is the normalizer of the Lie subgroup $L$ of $K$. This Lie group $Q$ (disconnected, in general) can be considered as the group of all automorphisms of the homogeneous spaces $K/L$ (that is, diffeomorphisms commuting with the transitive action of the Lie group $K$). The natural action of the group $Q$ on the fibre of the natural bundle is free.

Passing to an appropriate manifold $M'$, the structural Lie group can be considered connected (by reducing the natural bundle to it). If it is possible to reduce the structure group to $\{ e \}$, then the natural bundle will be trivial (which is by no means always the case; conditions for such triviality in some special cases are given in [6]). The structure group $Q$ of the above natural bundle can, in some special cases, be reduced to some of its subgroups or extended (which also sometimes turns out to be useful).

Consider now the natural action of the fundamental group $\pi_1(M_a)$ (which is the only non-trivial homotopy group of the aspherical manifold $M_a$) of the base of the natural bundle on its fibre $M_c$. With an appropriate choice of $M'$, which covers finitely the manifold $M$, this action will be homotopy trivial (and therefore, the natural bundle will be homotopy simple). In particular, then the action of $\pi_1(M_a)$ on $\pi_1(M_c)$ will also be trivial.

There is a bundle, which we call the Borel bundle, of the compact homogeneous space $M=G/H$, which is in a certain sense dual to the natural bundle. It has the form $M^L \to M' \to K/N_K(L)$, where $K$ is the maximal compact subgroup of the simply connected Lie group $G$, $L$ is the stationary subgroup of the natural action of $K$ on the manifold $M$, $M^L$ is the set (in fact, a subvariety) of fixed points of the action of $L$ on $M'$. The structure group of this bundle is the group $Q=N_K(L)/L$, which is the same as for the natural bundle. In contrast to the case of a natural bundle, the base and fibre of a Borel bundle, even their dimensions, in the general case will not be defined topologically uniquely (including up to a finite covering). But the Borel bundle allows one, in some special cases, to describe in detail the structure of compact homogeneous spaces, and therefore, its study often turns out to be useful.

For the base $M_a$ of the natural fibration (or for some manifold $M'_a$ corresponding to an appropriate homogeneous space $M''$ finitely covering $M'$) one can, in turn, construct another useful fibration, viz. the structural fibration $M_r \to M_a \to M_s$. This is a smooth locally trivial fibration whose fibre has the form $M_r=R/D$ (where $D$ is a uniform lattice in some simply connected solvable Lie group $R$) and whose base $M_s$ has the form $U\setminus S/\Pi$, where $S$ is some semisimple connected Lie group with finite centre, $U$is the maximal compact Lie subgroup in $S$, and $\Pi$ is a uniform lattice in $S$. Moreover, $U \setminus S$ is a symmetric space of negative curvature, and $U \setminus S/\Pi$ is a locally symmetric space, which is a compact geometric form for the symmetric space $U\setminus S$. A structure fibration can be considered as some analogue of the Levy decomposition for Lie groups, constructed for compact homogeneous spaces (more precisely, for the bases of their natural fibrations).

Note that the cases, where the fibration components $M_c$, $M_r$, $M_s$ degenerate to a point, were previously studied in detail by the author.

Consider two examples of natural fibrations for compact three-dimensional homogeneous spaces. Let $M_1= (\mathrm{SU}(2)/\mathrm{SO}(2)) \times \mathrm{SO}(2)$ be the homogeneous space of the compact Lie group $\mathrm{SU}(2)\times \mathrm{SO}(2)$. It is diffeomorphic to $S^2\times S^1$. The natural fibration is trivial for it, the base is diffeomorphic to the circle $S^1$, and the fibre, to the two-dimensional sphere $S^2$.

Now consider another homogeneous space $M_2= \mathrm{U}(2)/\mathrm{O}(2)$. For it, the base of the natural fibration is also diffeomorphic to $S^1$, and the fibre, to the manifold $S^2$ (for details, see below). In other words, the two homogeneous manifolds under consideration have the same fibres and bases of natural fibrations. Note that the homogeneous manifold $M_2$ was omitted in [7] (due to incomplete use of the result from [8]). It is easy to check that the fundamental groups of both homogeneous spaces are isomorphic to the group $\mathbf{Z}$. The manifold $M_1$ covers the manifold $M_2$ (this covering corresponds to the subgroup $2\mathbf{Z} \subset \mathbf{Z}$ in the fundamental group $\pi_1(M_2)$). In [7] (and later in [1]) it was stated that a three-dimensional compact homogeneous manifold is determined up to diffeomorphism by its fundamental group. However, this general statement has one exception, namely, the above homogeneous spaces $M_1$, $M_2$ have isomorphic fundamental groups; however, it will be proved below that they are not diffeomorphic. Moreover, all their homotopy groups are pairwise isomorphic.

Let us prove that two manifolds $M_1$, $M_2$ are not diffeomorphic. To this end, we will examine $M_2$ in more detail. Following the general construction of the natural fibration, we need to pass from the Lie group $\mathrm{U}(2)$ to its universal covering, isomorphic in this case to the Lie group $\mathrm{SU}(2)\times \mathbf R$, and consider in it the stationary subgroup corresponding to the subgroup $\mathrm{O}(2)$. But the action of the group $\mathrm{SU}(2)$ on $M_2$ does not change and the space of orbits does not change either. Therefore, one can simply consider the natural action of the subgroup (maximal compact semisimple) $\mathrm{SU}(2) \subset \mathrm{U}(2)$ on the manifold $M_2=\mathrm{U}(2)/\mathrm{O}(2)$. Note that $\mathrm{SU}(2)\cap \mathrm{O}(2) =\mathrm{SO}(2)$. It is clear that the space of orbits of this action is diffeomorphic to the circle $S^1$, which will be the base of the natural fibration. The fibre of the natural fibration for this homogeneous space $M_2$ has the form $\mathrm{SU}(2)/(\mathrm{SU}(2)\cap \mathrm{O}(2))$ and is diffeomorphic to the two-dimensional sphere $S^2$. The structure group of this fibration is the group $Q=N_{\mathrm{SU}(2)} (\mathrm{SU}(2) \cap \mathrm{O}(2)) / (\mathrm{SU}(2) \cap \mathrm{O}(2))$. Since $N_{\mathrm{SU}(2)} (\mathrm{SU}(2) \cap \mathrm{O}(2))$ is exactly $\mathrm{O}(2)$, it is clear that the group $Q$ is isomorphic to $\mathbf{Z}_2$. The action of the group $Q$ on the fibre of the natural fibration $S^2$ is free, and the quotient by it in this case is the two-dimensional projective plane $\mathbf RP^2$.

As a result, we obtain a natural fibration $S^2\to \mathrm{U}(2)/\mathrm{O}(2) \to S^1$. The natural action of the group $\pi_1(S^1)$ on the fibre of this fibration is non-trivial (it factorizes through the free action of the group $Q=\mathbf{Z}_2$), which implies that the action of the group $\pi_1(M_2)$ (isomorphic, as is easy to check, to the group $\pi_1(S^1) = \mathbf{Z}$) on the second homotopy group $\pi_2(M_2)$ (isomorphic to $\pi_2(S^2) = \mathbf{Z}$) is non-trivial. Since the action of $\pi_1$ on $\pi_2$ for the manifold $M_1$ is obviously trivial, we conclude that the manifolds $M_1$ and $M_2$ are not homeomorphic (they will not even be homotopically equivalent). Thus, we have obtained an example of two compact three-dimensional homogeneous manifolds in which all homotopy groups are pairwise isomorphic, but these manifolds themselves are not homotopically equivalent (and, moreover, homeomorphic). For three-dimensional compact homogeneous manifolds, this is the only example of this kind.

If the homogeneous space $M=G/H$ is plesiocompact, then it also has a natural fibration, but the base of this fibration is not necessarily compact, although it has a finite measure (with respect to the measure naturally introduced on it). For such a base $M_a$ a structural fibration is also constructed, the fibre of which is compact and the base has a finite measure. Everything said above about the structure group is also valid for the plesiocompact case. It is also generalized to the plesiocompact case the construction of the Borel fibration.

We now turn to the presentation of new results. We start with the base of the natural fibration.

§ 2. Base of the natural fibration

The base $M_a$ of a natural fibration, due to its asphericity, is uniquely determined up to homotopy equivalence by its fundamental group $\pi_1(M_a)$. And this fundamental group itself is determined uniquely by the original compact homogeneous space $M=G/H$ up to weak commensurability. Let us explain this.

Recall that two groups $D_1$ and $D_2$ are said to be weakly commensurable if they contain some subgroups of finite indices $D_i' \subset D_i$, and in those groups there are finite normal subgroups $\Phi_i\subset D_i'$ ($i=1,2$) such that the factor groups $D_1'/\Phi_1$ and $D_2'/\Phi_2$ are isomorphic to each other. Groups are said to be commensurable when the appropriate subgroups of $D_i'$ themselves are isomorphic. Similarly, two manifolds are said to be commensurable if they are both finitely covered by some third manifold. It is easy to see that two manifolds are commensurable if and only if their fundamental groups are commensurable.

From the above description of the construction of a natural fibration it follows that the fundamental groups of all bases of natural fibrations of some compact homogeneous space are commensurable. Therefore, these bases or some manifolds that cover them finitely are homotopy equivalent.

A natural question here is whether there a closer connection for a compact homogeneous space between different bases of natural fibrations (considered up to a finite covering) than the homotopy equivalence. It turns out that such a connection exists – they are diffeomorphic up to a finite covering. This was proved by the author (an improved version of the proof can be found in [2]). But another question here is whether two compact aspherical manifolds with isomorphic (and not only commensurable) fundamental groups (and not necessarily both are bases of natural fibrations) are necessarily homeomorphic. The assumption of diffeomorphism is no longer valid here, as can be seen from the following example.

Note that ordinary (non-false) tori are themselves homogeneous spaces (even compact Lie groups).

Further, an interesting question is how necessary is the transition to finite coverings (inevitable in the proof from [2]) for the homeomorphism (let alone diffeomorphism) of the bases of natural fibrations? The answers to these questions will be given below.

There is one very general conjecture due to A. Borel. It assumes that if two aspherical compact manifolds are homotopically equivalent (which is equivalent to their fundamental groups being isomorphic), then these manifolds are homeomorphic (but they are not necessarily diffeomorphic; see Example 1 above). This conjecture follows from the Farrell–Jones (FJC) conjectures, formulated in terms of $K$- and $L$-theories (there are also a number of generalizations of these conjectures). In [10] it was proved that these FJC conjectures are valid for arbitrary uniform lattices in almost connected Lie groups (in [10], such Lie groups are called virtually connected; they are characterized by having only a finite number of connected components). Another proof was given for the same class of groups in [11], even in a more general situation. In particular, the following assertion follows from these results.

For dimension $4$, in the case of bases of natural fibrations, a separate consideration is necessary. If the fundamental group of the base of a natural fibration is polycyclic (and it will always be so if the fundamental group of a compact homogeneous space is solvable), then such groups and the corresponding solvmanifolds are easy to describe. For them, in fact, the FJC has already been tested. If the group $\pi_1(M_a)$ for a four-dimensional $M_a$ is not polycyclic, then it is isomorphic to the direct product of $\mathbf{Z}$, and the lattice in the three-dimensional simple Lie group $\mathcal A$, to the universal covering for $\mathrm{SL}(2,\mathbf{R})$. Here everything reduces to some extent to the three-dimensional case, for which the above results about FJC apply. Thus, for compact homogeneous spaces (more precisely, for four-dimensional bases of natural fibrations), the assertion of Theorem 1 and Corollary 1 below is apparently also true.

Note that the fundamental group of an aspherical manifold is always torsion-free, and therefore, the lattices appearing in Theorem 1 are torsion-free. The following result follows from Theorem 1 in view of the above facts about the bases of natural fibrations.

Proof. As mentioned above, the base of a natural fibration can (if necessary, passing to a finite covering) be written in the standard form $\Gamma \setminus F /L$, where $F$ is some connected Lie group (in fact, it is closely related to the original transitive Lie group), $L$ is the maximal compact Lie subgroup of $F$, and $\Gamma$ is a uniform lattice (that is, a discrete subgroup with a compact quotient space) in $F$. Moreover, the quotient space $F/L$ is contractible, and the action of the subgroup $\Gamma$ on it can be considered free (which follows from the property $\Gamma \cap L = \{e\}$; see above).

As a result, we find that for the base $M_a$ of the natural fibration, considered up to a finite covering, the fundamental group $\pi_1(M_a)$ is isomorphic to the uniform lattice $\Gamma$ in the connected Lie group $F$. Therefore, Theorem 1 applies (that is, the Borel conjecture holds for the manifold $M_a$). Corollary 1 is proved.

There is reason to hope that in the second part of Corollary 1 it will be possible to replace the assertion that the bases are homeomorphic by their being diffeomorphic (so far this is true only when passing to suitable finite-sheeted coverings).

As for the generalization of the results presented in this section to the case of plesiocompact homogeneous spaces, the question remains open (because it is related to lattices in Lie groups that are not uniform, for which FJC has not yet been verified).

§ 3. Structural fibration of the base of a natural fibration

Here, based on the results of the previous section, we consider the questions on uniqueness of the objects of the structural fibration, viz., its fibre and base.

For the base $M_a$ of the natural fibration of the compact homogeneous space $M=G/H$, the structure fibration has the form $M_r \to M_a\to M_s$. The selection of the manifolds $M_r$ and $M_s$ is analogous to that of the radical and semisimple part in a Lie group.

Let $\pi_r$ and $\pi_s$ be, respectively, the fundamental groups of aspherical manifolds $M_r$ and $M_s$. These fundamental groups are uniquely determined by the fundamental group $\pi_1(M)$ up to commensurability. And this fibration itself, due to the asphericity of the manifold $M_a$, is uniquely determined up to homotopy equivalence of fibrations (taking into account the possibility of passing to a finite-sheeted covering). Just as in the case of the manifold $M_a$, the manifolds $M_r$, $M_s$ are defined up to a diffeomorphism and a finite covering. The Borel conjecture (see above) also holds for these two manifolds, and therefore, if the bases (or fibres) of a structure fibration of dimension $\ne 4$ have isomorphic fundamental groups, then they are homeomorphic (presumably, they will also be diffeomorphic).

The structure fibration generates the following exact sequence of groups (this is the only non-trivial segment of the exact homotopy sequence of this fibration): $\{e\} \to \pi_r \to \pi_1(M_a) \to \pi_s \to \{ e \} $.

All three groups here are torsion-free here. By virtue of this exact sequence, the group $\pi_1(M_a)$ is an extension (generally speaking, non-abelian) of the solvable group $\pi_r$ with the aid of the group $\pi_s$ (note that sometimes a different terminology is used – one speaks of an extension of the group $\pi_s$ using the $\pi_r$ group). As is known, such a (non-Abelian) extension is given by the cohomology class $c \in H^2(\pi_s, Z(\pi_r))$, which is the characteristic class of the extension lying in the two-dimensional cohomology group of the group $\pi_s$ with coefficients in the centre $Z(\pi_r)$ of the group $\pi_r$ (see, for example, [12]). In particular, if this cohomology class is $0$ (this happens, for example, if the group $H^2(\pi_s,Z(\pi_r))$ is trivial, which is quite possible – this will always be the case if the centre $Z(\pi_r)$ is trivial), then the extension splits and the group $\pi_1(M_a)$ can be represented as a semidirect product of a subgroup isomorphic to $\pi_s$ and a solvable normal subgroup isomorphic to $\pi_r$. In this case, the structural fibration has a section. In the general case, however, this cohomology class $c$ constructed for the base of the natural fibration will be non-trivial. Here is the simplest possible example.

Example 2. Consider a compact three-dimensional homogeneous space of the form $M = \mathrm{SL}(2, \mathbf{R})/\Gamma$, where $\Gamma$ is some torsion-free uniform lattice in $\mathrm{SL}(2,\mathbf{R})$. There are a lot of such lattices (and even arithmetic ones) in the group $\mathrm{SL}(2,\mathbf{R})$. The group $\mathrm{SL}(2,\mathbf{R})$ itself is aspherical, and therefore, the fibre of the natural fibration $M_c$ here degenerates to a point, so $M_a=M$. The structure fibration here has the form $S^1\to M_a\to F_g$. This is the fibration on the circle $S^1$ over a compact orientable surface $F_g$ of genus $g \geqslant 2$ (the genus of $g$ depends on the choice of the lattice $\Gamma$). The cohomology group $H^2(F_g, \mathbf{Z})$ of interest to us here is isomorphic to $\mathbf{Z}$, and the characteristic class of this fibration $k \in \mathbf{Z}$ (depending on $\Gamma)$, as is well known, is not zero.

For plesiocompact homogeneous spaces the uniqueness (up to a finite covering) of the bases of the natural fibration, of the base, and of the fibre of the structural fibration follow from Mostov’s rigidity theorem (see [1]). Note that here the manifold $M_r$ is always compact.

§ 4. Structure group of the natural fibration

We now turn to the consideration of the second element of the natural fibration – its structure group. As already mentioned, for a compact homogeneous space $M=G/H$, considered up to a finite covering, the structure group of the natural fibration can be taken to be the compact Lie group $N_K L/L$, where $K$ is the maximal connected compact Lie subgroup in the transitive Lie group $G$, which is assumed to be simply connected (hence the subgroup $K$ is semisimple and also simply connected), and $L=K\cap H$ is a stationary subgroup, which is also compact (but not necessarily connected or simply connected). In fact, sometimes it is possible to reduce the structural group $Q$ to some of its subgroups. For example, it is always possible to reduce the group $Q$ to its connected unity component $Q_0$ (thereby obtaining a connected structural group). In fact, in this case, it may be necessary to replace the homogeneous space $G/H$ by some other one (of the form $G/H'$) its finite covering (where $H'$ is some subgroup of finite index in $H$). In particular, sometimes even a reduction to the identity subgroup is possible, which gives us the triviality of the natural fibration (for some possibilities for such a reduction, see [6]). On the other hand, sometimes it is useful to extend the structure group $Q$ in order to reduce it later, going beyond $Q$ (for an example of this kind, see [13]).

Speaking of reductions or extensions of a structure group, we mean transitions to fibrations of a compact homogeneous space for which two structure groups (old and new) have different dimensions. But the following question also naturally arises – can there exist on some compact manifold transitively and locally effectively acting connected Lie groups for which the structural groups have the same dimensions, but are not isomorphic? It turns out that such a phenomenon is possible. We obtain a classical example of this kind from the fact that the Lie groups $\mathrm{SO}(4)$ and $\mathrm{SU}(2)\times \mathrm{SO}(3)$ are diffeomorphic but not isomorphic (they are both two-sheeted covered by the spinor Lie group $\operatorname{Spin}(4)$). A direct proof that these Lie groups are diffeomorphic can be found, for example, in [14]. As for their non-isomorphism, this follows from the fact that subgroups in $\operatorname{Spin}(4)$ isomorphic to $\mathbf{Z}_2$ whose factor groups give us $\mathrm{SO}(4)$ and $\mathrm{SU}(2)\times \mathrm{SO}(3)$, will not be conjugate in the automorphism group of the Lie group $\operatorname{Spin}(4)$ by any automorphism (neither inner nor outer). These two Lie groups, which are diffeomorphic with each other, considered as homogeneous spaces, have different structure groups of natural fibrations (the bases of these fibrations degenerate to a point, and the fibres and structure groups coincide with these same Lie groups).

But it turns out that there are many other examples of this kind. Here is a description of a series of such examples.

Example 3. Let $K$ be some semisimple simply connected compact Lie group, and $Z$ some non-trivial subgroup of its centre (which is always finite).

Consider the compact semisimple group $G = K \times K$ and define the mapping $f\colon K \,{\times}\, K \,{\to}\, K \,{\times}\, K$ which takes $(k_1,k_2) \,{\in}\, K \,{\times}\, K$ to $(k_1, k_1^s k_2)$ (here $s$ is some nonzero integer coprime with the order of the subgroup $Z$). It is easy to see that this mapping is a diffeomorphism. Under its action, the subgroup $Z \times \{ e\} \subset K \times K$ maps to the central subgroup of the group $K \times K$, which consists of elements of the form $(z_1, z_1^s z_2)$, where $z_1,z_2 \in Z$.

Consider now two Lie groups $G_1= G/Z\times \{e\}$ and $G_2= G/f(Z)$. These two groups are diffeomorphic; the required diffeomorphism is induced by the diffeomorphism $f$ due to the fact that the mapping $f$ permutes with the action of the isomorphic groups $Z$ and $f(Z)$. But Lie groups $G_1$, $G_2$ will not always be isomorphic.

For $K=\mathrm{SU}(2)$ and $s=1$, our general construction generates the aforementioned example of diffeomorphic but not isomorphic groups $\mathrm{SO}(3) \times \mathrm{SU}(2)$ and $\mathrm{SO}(4)$. In general, if the group $K$ is simple and $s=1$, then it is clear that the diffeomorphic compact semisimple Lie groups $G_1$, $G_2$ constructed by us will never be isomorphic with each other.

Note that values of $s$ other than $\pm 1$ can be useful to us only if the group $\mathrm{SU}(n)$ for $n >2$ is involved in the considerations, since only then the centre of this group contains elements of order $>2$ (we allow ourselves to ignore two groups $\operatorname{Spin}(8)$ and the exceptional group $E_6$, whose centers are isomorphic to $S_3$ and $\mathbf{Z}_3$, respectively). For more details on the connection between isomorphism and diffeomorphism of compact Lie groups, see [15], as well as [16], where this connection is studied in detail and Example 3 is considered in a more general situation.

When considering structural groups $Q_0$, the following question naturally arises: what compact connected Lie groups can be realized as structural groups of the form $(N_K (L)/L)_0$ for compact homogeneous spaces? The answer to this question is the following result.

Proof. First, we note that as a Lie group of the form $N_K(L)/L$ (for some $K,L$), we can realize any torus (that is, an Abelian compact Lie group). Any torus $T$ can be represented as a direct product of several one-dimensional connected Lie groups (isomorphic to $\mathrm{SO}(2)$): $T=\times_i\, \mathrm{SO}(2)$. Let us consider a simply connected simple Lie group $K=\mathrm{SU}(3)$, some of its maximal (two-dimensional) torus $A$, and some generic one-dimensional torus $A'$ in it. It is clear that for such $K$ and $L=A'$ the group $(N_K(L)/L)_0$ is isomorphic to $\mathrm{SO}(2)$. This implies that for a compact homogeneous space of the Lie group $\times_i\, \mathrm{SU}(3)$ with some stationary subgroup isomorphic to $\times_i\, \mathrm{SO}(2)$ the connected component of the automorphism group is isomorphic to the Lie group $\times_i\, \mathrm{SO}(2)$, that is, to the original torus $T$. Thus, we have obtained a realization of the given torus as a connected component of the identity of the automorphism group of the compact homogeneous space $\times_i \, \mathrm{SU}(3)/\times_i\, \mathrm{SO}(2)$ of a compact semisimple Lie group.

We now turn to the consideration of the general case.

An arbitrary connected compact Lie group $U$, as is well known, can be represented as an almost direct product of a central torus (the connected component of the identity of the centre of this Lie group) and a semisimple compact Lie group (coinciding with the commutator subgroup of the original compact Lie group). An almost direct product is a decomposition of a group into a product of two subgroups whose intersection is discrete. In the case of a compact Lie group $U$, its centre and semisimple part are compact and their intersection is finite. Therefore, for each such Lie group $U$, there exists a finite covering by the Lie group $U'$ which is a compact Lie group decomposing into a direct product of a torus (=compact Abelian Lie group) and a semisimple compact Lie group. Replacing the original Lie group $U$ with a locally isomorphic compact Lie group $U'$, we can assume that the original Lie group $U$ decomposes into a direct product of a semisimple compact Lie group $C$ and a torus $T$: $U=C\times T$.

Let $K= C \times (\times _i\, \mathrm{SU}(3))$ (where the Lie group $\times_i\, \mathrm{SU}(3)$ is described above and is constructed from the torus $T$). The Lie group $K$ is compact connected and semisimple. Let $L=\times\, A_i'$ (where the one-dimensional tori $A'$ are described above). It is easy to understand that we have an isomorphism $(N_K(L)/L)_0 \simeq Q$. Thus, for an arbitrary compact homogeneous space $K/L$, the connected identity component of the automorphism group is locally isomorphic to the given compact connected Lie group $Q$.

But Theorem 2 also states that the Lie group $K$ can be chosen to be simply connected. We have already proved that $(N_K(L)/L_0) \simeq Q$. Let $p\colon \widetilde C \to C$ be the universal cover of the Lie group $C$. Let $\widetilde K=\widetilde C \times (\times _i\,\mathrm{SU}(3))$; this is a simply connected Lie group that universally covers the group $K$. Denoting the mapping of this universal covering by $p$, let us consider the homogeneous space $\widetilde K/p^{-1}(L)$. It is easy to see that the automorphism groups of the homogeneous spaces $K/L$ and $\widetilde K/p^{-1}(L)$ are isomorphic. Thus, we have obtained the required representation for $Q$, in which the transitive Lie group is simply connected. Theorem 2 is proved.

As a corollary to Theorem 2, we obtain the possibility of realizing (up to a local isomorphism) of any connected compact Lie group in the form of a connected structural group of the natural fibration of an appropriate compact homogeneous space. Moreover, as such a homogeneous space, one can, for example, take homogeneous spaces of the form $K/L$, as well as spaces of the form $M=K/L\times R/\Gamma$, which are direct products of the compact homogeneous space $K/L$ of a compact semisimple Lie group and a homogeneous space $R/\Gamma$ (a solvmanifold), factor spaces of a solvable Lie group $R$ with respect to some of its uniform lattice $\Gamma$. For such a homogeneous space $M$, the fibre of the natural fibration is $K/L$, and the base is the solvmanifold $R/\Gamma$ (it is always aspherical due to solvability of the Lie group $R$). For the indicated homogeneous space, the structure group of its natural fibration is exactly the group $Q$, although it is clear from the very construction of this homogeneous space that, in this case, this structure group can be reduced to $\{ e\}$ (since this homogeneous space is the direct product of the fibre $M_c=K/L$ and the base $M_a=R/\Gamma$ of the natural fibration). This just confirms the above remark to the effect that the “standard” structure group $Q$ can sometimes be reduced (to some subgroup). But sometimes it can be reduced in a different way: first expand, and then reduce the extended structural group (even sometimes to a trivial subgroup). It was in this way that the triviality of the natural fibration for the manifold $\mathrm{SL}(3,\mathbf{R})/H$ (described in detail, for example, in [13]) was proved, which, as it turned out, is diffeomorphic to the manifold $F_g\times \mathbf{R}P^3$ (the direct product of a compact orientable surface $F_g$ of genus $g \geqslant 2$ and the three-dimensional projective space).

The above remarks about reductions of the structure group $Q$ to its subgroups suggest the natural question of whether there exists a universal reduction of the structure group $Q=N_K(L)/L$ for arbitrary compact homogeneous spaces. It turns out that, in some cases, the group $Q$ cannot be reduced to any of its proper subgroups, and therefore, only the specified group $Q$ is a universally valid structural group of the natural fibration (which does not exclude the possibility of its reduction in some special cases). To illustrate this assertion, consider the following example.

Thus, we have refined the properties of the base $M_a$ and the structure group $Q$ of natural bundles for arbitrary compact homogeneous spaces $M=G/H$. The natural bundle itself is classified (like any bundle over a given base with a given structure group) by the homotopy class of the corresponding (classifying) continuous mapping $f\colon M_a \to BQ$ of the aspherical base $M_a$ into the classifying space $BQ$ of the structure group $Q$. Here, of course, the natural question is for which homotopy classes of mappings $f\colon M_a \to BQ$ (for given $M_a$ and $Q$) the space of the corresponding bundle over $M_a$ with fibre $K/L$ and structure group $Q = N_K(L)/L$ is a homogeneous space of some finite-dimensional Lie group? But the author believes that it is very difficult to get a meaningful answer to this question, because in terms of homotopy topology (in particular, classifying mappings) it is very difficult to talk about the homogeneity of manifolds with respect to finite-dimensional Lie groups. Another reason for this skepticism is also the variety of bases of natural bundles. Let us give an example illustrating possible surprises that take place here.

Example 5. Let $F_g$ be a compact orientable surface of genus $g \geqslant 2$. This two-dimensional manifold is not homogeneous, but it can be the base of a natural bundle for homogeneous spaces. Such homogeneous spaces are discussed in detail, for example, in [18]. In particular, the direct product $F_g \times S^3$ of any such surface and a three-dimensional sphere is homogeneous (for example, with respect to the Lie group $\mathrm{SL}(3,\mathbf{R})$). However, it turns out that the structure groups of natural bundles with base $F_g$ cannot be compact simple Lie groups $Sp_n$ (and the only Lie groups locally isomorphic to them are quotient groups with centre isomorphic to $\mathbf{Z}_2$) for $n\geqslant 7$ (see [18]). Although other simple classical simple compact Lie groups can be as structural groups of natural bundles! It is hard to understand why the base $F_g$ “did not please” the series of simple Lie groups $Sp_n$. Let us take a closer look at the situation with these groups.

Bundles over $F_g$ with structure group $Sp_n$ are classified by homotopy classes of mappings $f\colon F_g\to BSp_n$ into classifying spaces. Such a mapping is, in particular, the trivial one (the mapping to a point). But in view of the above, this mapping does not correspond to a homogeneous space. In fact, any continuous mapping $f$ in this case is homotopic to the trivial one (this follows from the fact that the first and second homotopy groups of the space $BSp_n$ are trivial, and the surface $F_g$ is two-dimensional).

§ 5. Fibre of the natural fibration

We pass to the third element of the natural fibration, its fibre $K/L$. Here $K$ is a connected simply connected compact Lie group, and $L$ is some of its closed (but, in general, not necessarily connected or simply connected) Lie subgroup. We note right away that even when considering compact homogeneous spaces up to a finite covering, we cannot always ensure that the fibre of the natural fibration becomes simply connected (for which it would be sufficient to replace the subgroup $L$ with a connected one). Here is an attempt to build an example of this kind. First, we note that a fibre of a natural fibration can be simply connected if and only if the fundamental group of the homogeneous space $G/H$ is torsion-free (see [19]).

Example 6. In fact, this is not a real, but a hypothetical example.

Let $\Gamma$ be some group. It is called virtually torsion-free if it contains a torsion-free subgroup of finite index. An example (not hypothetical!) of such a group is, for example, any finitely generated linear group (Selberg’s lemma; see, for example, [20]). An example (and a very striking one) of a group that is not virtually torsion-free is any Tarski Monster corresponding to an arbitrarily given prime number $p$. Tarski’s monsters are such finitely generated (in fact, generated by two elements) infinite groups, each non-trivial subgroup of which is a cyclic group (finite!) of a fixed prime order. It is clear that any Tarski Monster is not a virtual torsion-free group.

The existence of Tarski’s monsters (and even simple groups among them) was proved by Ol’shansky [21] in 1979. There are many other examples of groups that are not virtually torsion-free. Among such examples are some lattices in semisimple classical Lie groups (see [22], for uniform lattices, see [23]). But all known examples of such lattices have one feature, which for us is a significant drawback – these are lattices in (simple) Lie groups that are not simply connected (and, moreover, have an infinite fundamental group). But uniform lattices in simply connected simple (or semisimple) Lie groups that are not virtually torsion-free have not yet been found. But we will assume that such lattices exist and build our hypothetical example on this basis.

Let $\Gamma$ be a lattice in some simply connected semisimple Lie group $S$ that is not virtually torsion-free (recall that such groups have not yet been discovered). Consider a compact homogeneous space $M = S/\Gamma$ and a natural fibration for it (or for some compact homogeneous space $S/\Gamma'$ covering it finitely). This natural fibration has the form $K/L \to M \to M_a$, where the fibre $K/L$ is the homogeneous space of the maximal compact Lie subgroup $K$ with the stationary subgroup $L\subset K$, which in our case will obviously be discrete. But since the lattice $\Gamma$ is not virtually free of torsion, so is its subgroup $\Gamma'$ of finite index. Therefore, the subgroup $L$ in our hypothetical example will be a non-trivial finite subgroup, and therefore, the fibre $K/L$ of the natural fibration will not be simply connected. Moreover, replacing $\Gamma$ by any of its subgroups of finite index $\Gamma'$ cannot result in a trivial subgroup $L$. In other words, any natural fibration of our compact homogeneous space $M=S/\Gamma$ will always have a fibre that is not simply connected.

Let us continue the consideration of properties of the fibre of the natural fibration.

The fibre of a natural bundle is uniquely defined up to tangential homotopy equivalence (see [13]). Let us explain this. To begin with, we recall that a homotopy equivalence of manifolds is called tangential if it induces a stable (that is, up to the addition of trivial bundles) isomorphism of tangent bundles. For a natural bundle $M_c\to M\to M_a $, consider the covering over $M$ which corresponds to the universal covering over $M_a$. Since the manifold $\widetilde M_a$ universally covering $M_a$ is diffeomorphic to some Euclidean space $\mathbf{R}^k$, the bundle induced by this covering is trivial, and the space of this covering over $M$ is diffeomorphic to the direct product $M_c\times \mathbf{R}^k$. This implies that if, for a given homogeneous space $M$, there exists another natural bundle with some other fibre, then both of these fibres are tangentially homotopy equivalent (because some of their direct products are diffeomorphic to parallelizable manifolds, for example, to $\mathbf{R}^k$). However, it turns out that there are compact homogeneous spaces with a finite fundamental group that are not homeomorphic, although they are simply (we are talking about the concept of simple homotopy equivalence) and tangentially homotopy equivalent. Manifolds of dimension $5$ of this kind are given in [24]. They are spaces of bundles over a two-dimensional sphere in which the fibre is a lens space. The present paper asserts that all the varieties constructed there are homogeneous spaces of the compact Lie group $\mathrm{SU}(2)\times \mathrm{SU}(2) \times \mathrm{U}(1)$. In fact, the semisimple Lie group $\mathrm{SU}(2)\times \mathrm{SU}(2)$ is also transitive on these manifolds. The point is that the varieties under consideration are compact, have a finite fundamental group, and therefore, by virtue of the very general Montgomery theorem (see, for example, [1]) the maximal semisimple subgroup of a transitive Lie group is also transitive on this variety (in the examples from [24] this will be the group $\mathrm{SU}(2)\times \mathrm{SU}(2)$ – it is easy to find out even by a detailed study of the transitive action described there).

It is interesting to note that no tangentially homotopy equivalent compact Lie groups that are not homeomorphic have yet been found. Although if compact Lie groups are simple and simply connected, then even their isomorphism follows from their usual homotopy equivalence.

We now note that tangentially homotopy equivalent manifolds become diffeomorphic [25] when directly multiplied by a parallelizable manifold of sufficiently large dimension. We use this fact to prove the possibility of non-uniqueness of the fibre of the natural bundle for compact homogeneous spaces. To do this, take some two (denoted $N_1^5$, $N_2^5$) from five-dimensional manifolds constructed in [24] which are not homeomorphic, but are tangentially homotopy equivalent. Consider the direct products of these two manifolds by the torus $T$ (although here they are usually multiplied by an open disc, but in fact only the parallelizability of this factor is important) of a sufficiently large dimension (it is enough to take this dimension by $2$ greater than the dimension of the multiplied manifolds). We get two manifolds $M_1=N_1\times T$, $M_2=N_2 \times T$. They will be homeomorphic and even diffeomorphic, and they are obviously homogeneous (as direct products of homogeneous spaces $N_i$ and $T$). Thus, we obtain here one compact homogeneous manifold $M$ (with different transitive actions of Lie groups). Here $N_i$ will be fibres of natural bundles for $M$, and the torus $T$ will be the base of these bundles (both natural bundles are trivial).

As a result, we arrive at the following conclusion: the fibres of a natural bundle are always tangentially homotopy equivalent, but not always homeomorphic. Therefore, the author’s earlier hope that the fibre of the natural bundle would be unique was not justified.

But even if the fibres of the natural bundle turn out to be homeomorphic, they will not necessarily be diffeomorphic. For example, [26] contains examples of simply connected compact homogeneous spaces of dimension $7$ that are homeomorphic but not are diffeomorphic. The natural bundles for these manifolds degenerate – the base is a point, and the layers are the manifold itself. Therefore, the non-diffeomorphism of these homeomorphic manifolds gives us the necessary example for fibres of natural bundles.

Note that the transitive reductive (with one-dimensional centre) compact group $\mathrm{SU}(3)\times\mathrm{SU}(2)\times\mathrm{SU}(1)$ indicated in [26] (following E. Witten) can be replaced (which the authors did not notice) by its transitive proper semisimple subgroup $\mathrm{SU}(3)\times \mathrm{SU}(2)$ (using Montgomery’s theorem already used above).

Everything said in this section is automatically carried over to the case of plesiocompact homogeneous spaces, since there the fibre has the form $K/L$ as well.

§ 6. Borel fibration

A Borel fibration is another “natural” fibration that can be constructed for an arbitrary (considered up to a finite covering) compact homogeneous space. The base of this fibration has the form $K/N_K(L)$ (in the above notation) and is of no particular interest to us now.

Here we will consider in detail the fibre $M^L$ of the Borel fibration for compact homogeneous spaces $M=G/H$. In [27] it was proved that this fibre is a compact homogeneous space of some Lie group (closely related to the original transitive Lie group $G$) provided that the fundamental group $\pi_1(M)$ is torsion-free. However, this condition is rather restrictive when considering arbitrary compact homogeneous spaces. And passing to an appropriate finite-sheeted covering does not always improve the situation here (the problem of passing to a torsion-free subgroup of finite index has already been discussed above).

But it turns out that the assumption made about the virtual absence of torsion of the fundamental group can be discarded. This section is devoted to the proof of this assertion.

For an arbitrary group $\Gamma$ (we are only interested in discrete groups), we denote by $\mathrm{Tors}(\Gamma)$ the set of all elements of finite order in $\Gamma$. This set is not always a subgroup. It may even turn out that it generates the entire group $\Gamma$, which also has elements of infinite order.

We will call a group $\Gamma$ having weakly torsion if the set $\mathrm{Tors}(\Gamma)$ is a central finite subgroup of $\Gamma$.

Note that if the set $\mathrm{Tors}(\Gamma)$ is contained in some Abelian subgroup of the group $\Gamma$, then it is necessarily a subgroup of $\Gamma$, and moreover, it is normal. Further, if $\mathrm{Tors}(\Gamma)$ is a finite subgroup (automatically normal), then there exists a subgroup of finite index in $\Gamma$ whose torsion is weak. This is easy to see if we consider the action of conjugations of the group $\Gamma$ on $\mathrm{Tors}(\Gamma)$ and take the kernel of this action as the desired subgroup of finite index.

Proof. We set $\Gamma = \pi_1(M)$. Since the Lie group $G$, which is transitive on $M$, is assumed to be simply connected (as usual), the group $\Gamma$ is isomorphic to the group of connected components $H/H_0$ of the Lie group $H$. We need to find in $\Gamma$ a subgroup of finite index such that the set of its elements of finite order forms a finite central subgroup.

Denote by $c \colon H\to H/H_0$ the natural epimorphism and set $H_{\mathrm{tors}} = c^{-1}(\mathrm{Tors}(\Gamma))$. The subset $H_{\mathrm{tors}}$ is generally not a subgroup. But if $\mathrm{Tors}(\Gamma)$ is a subgroup of $\Gamma$, then $H_{\mathrm{tors}}$ is a subgroup of $H$.

Let us take the first step in choosing the subgroup of finite index in $\Gamma$ that we need. At this step, we will partially use the same reasoning that was given in [13] when proving Lemma 2.3 there.

Consider the subgroup $P = \{(N_G(H_0)_0)), H)\}$ generated by the Lie subgroup $H$ and the connected component of the normalizer in $G$ of its connected unit component $H_0$. For this Lie subgroup $P$, the quotient space $P/H$ is connected (it is the presence of this property that causes the transition from $N_G(H_0)$ to its subgroup $P$ of finite index). Let us show that, for an appropriate subgroup of finite index $H' \subset H$, the corresponding subset $\mathrm{Tors}(H'/H_0 )$ is contained in $P_0/H_0$.

In [13], in the proof of Lemma 2.3, it was shown that the group $\pi_0(P)$ of connected components of the Lie group $P$ contains a finitely generated torsion-free Abelian subgroup $C$ of finite index. Denote by $\alpha \colon P \to \pi_0(P)=P/P_0$ the natural epimorphism and set $P' = \alpha ^{-1} (C)$, $H' = H \cap P'$. The Lie subgroup $H'$ is of finite index in $H$ and the group $\Gamma'= H'/H_0'$ is a subgroup of finite index in $\Gamma$. Moreover, $H'/ (H' \cap P_0)$ is a subgroup of finite index in $C$, and therefore, the subset $H'_{\mathrm{tors}}$ is contained in $P_0$. Let us replace $H$ by a subgroup of finite index $H'$.

We pass to the next step of constructing the subgroup we need in $\Gamma=\pi_1(M)$.

The adjoint representation $\operatorname{Ad} \colon G \to \mathrm{GL}(g)$ of the Lie group $G$ acting on the Lie algebra $g$ of the Lie group $G$ induces the representation $\operatorname{Ad}|_H \colon H\to \mathrm{GL}(g)$ of the Lie subgroup $H$. The action of $H$ on $g$ preserves the Lie subalgebras $h$ and $p$. Consider the Lie algebra $a=p/h$. The action of $H$ on $g$ generates a representation of $H$ on $a$. Moreover, the representation $H \to \mathrm{GL}(a)$ is factorized via the natural epimorphism $c \colon H\to H/H_0$ onto the group $H/H_0 =\pi_0(H)$ of connected components of the Lie subgroup $H$. As a result, we obtain the representation $\rho \colon \Gamma \to \mathrm{GL}(a)$. The manifold $M$ is compact, so its fundamental group is finitely generated. Therefore, the group $\rho(\Gamma)$ is finitely generated. By virtue of Selberg’s lemma (stating the existence of a torsion-free subgroup of finite index in a finitely generated linear group), in the image $\rho (\Gamma)$ there exists a torsion-free subgroup $\pi$ (even, as it is easy to show, a normal subgroup) of finite index. Let $\widehat H= c^{-1}\rho^{-1}(\pi)$ – this will be a Lie subgroup of finite index in $H$. We replace $H$ by its subgroup $\widehat H$ of finite index. In this case, of course, we will have the inclusion $\widehat H_{\mathrm{tors}} \subset P_0$.

Let us move on to the third and final step. It follows from the construction of the subgroup $\pi$ that $\mathrm{Tors}(\pi)$ lies at the centre of the Lie group $P_0/H_0$, and therefore, as noted before the proof of this Theorem), is an Abelian central subgroup. But then the subset $\widehat H_{\mathrm{tors}}$ will also be a subgroup (even closed) in $H$. This completes the proof of Theorem 3.

The assertion of Corollary 2 allows one to study the topology of the Borel fibration for arbitrary compact homogeneous spaces (considered up to a finite covering) using for its fibre all the results on compact homogeneous spaces.

§ 7. Principal fibrations and those close to them

Here we consider one type of homogeneous spaces of a very special form, in which the base $M_a$ of a natural fibration for a compact homogeneous space $M=G/H$ and the structure group $Q$ of this fibration are in a sense joined together.

In § 3, it was already noted that it is difficult to find out by purely topological methods for a given compact aspherical manifold as the base of a natural fibration what structural groups of such a fibration will be possible. However, in one particular case, it is still possible to specify at least one compact homogeneous space with a given base $M_a$ and an arbitrary structural Lie group $Q$ (considered up to local isomorphism). This will be the case if the base $M_a$ is a compact homogeneous space. A Lie group transitive on such a manifold is necessarily aspherical (for more details on the structure of such Lie groups, see [1] and a little more below). For example, such are solvmanifolds (homogeneous spaces of solvable Lie groups).

So, let us be given a connected compact Lie group $Q$ (a candidate for the title of the structural group of the natural fibration) and an aspherical compact homogeneous space $F/U$ (candidate for the title of the base of the natural fibration). By virtue of Theorem 2, a Lie group $Q$ or a compact Lie group locally isomorphic to it can always be represented as $N_K(L)/L$ for some compact semisimple Lie group $K$ and its closed Lie subgroup $L$ (more precisely, we are talking about the connected component of the identity of the group $N_K(L)/L$, but we will not pay attention to this difference). We set $G=K\times F$, $H=L\times U$ and consider a compact homogeneous space $M=G/H$. It is easy to understand that the base $M_a$ of the natural fibration for it is the manifold $F/U$ given to us, and the structure group is the given Lie group $Q$ or locally isomorphic to it. Thus, at least one compact homogeneous space with given base and structure group exists. However, if the base is not homogeneous (such cases are quite possible – for example, this property is satisfied by the surface $F_g$ and, in general, by any locally symmetric manifold of non-compact type; see [1]), then it is not always possible to indicate even a single compact homogeneous a space with given structure group of natural fibration (see above). Therefore, to describe possible pairs (aspherical compact manifold, compact Lie group) realized as pairs $(M_a,Q)$ for compact homogeneous spaces, due to the insufficiency of purely topological methods, one should also use some algebraic and geometric ones. One such approach will be described in this section.

Let us now consider the structure of that principal fibration over the manifold $M_a$, which is generated by the natural fibration with structure group $Q = (N_K(L)/L)_0$. Let $M^L$ be the space of fixed points of the natural action of the subgroup $L \subset G$ on $M$. If the action of a compact Lie group $K$ on $M$ is equiorbital (which, as mentioned above, can always be achieved by passing from a compact homogeneous space $M=G/H$ to some $M'= G/H'$ covering it finitely), then $M^L$is a smooth manifold (having no singular points). Moreover, the group $Q=N_K(L)/L$ acts naturally on $M^L$, and this action is free and its orbit space $Q\setminus M^L$ is naturally diffeomorphic to the base of the natural fibration (for details see [27]). We get the principal fibration $Q=N_K(L)/L \to M^L \to M_a$ – it is the principal fibration $Q$ corresponding to the natural fibration provided that the group $Q$ (although this is optional, see above). It turns out that the space $M^L$ of this principal fibration is a compact homogeneous space with respect to the transitive action of some Lie group on it (see Corollary 2 above and [27]). We also add that the compact homogeneous space $M=G/H$ under consideration can be represented as a fibre product $M=K/L\times _Q M^L$.

Of particular interest are compact homogeneous spaces for which the natural fibration is the principal one. This is the case if and only if $M=M^L$. Such homogeneous spaces will be called principal. We immediately note that a manifold, being principal with respect to one transitive action of a Lie group, may not be principal with respect to another.

Proof. 1) Assertion 1) is obvious on noting that, in the case under consideration, the set $\mathrm{Tors}(\Gamma)$ is a central subgroup in the maximal compact subgroup $K$ of the Lie group $G$, and therefore, the natural fibration for $G/\Gamma$ is principal with respect to the action of the compact group Lie $K/\mathrm{Tors}(\Gamma)$.

2) Consider the adjoint representation of the Lie group $G$. Its image is linear (that is, it is a linear Lie group), and the kernel is the centre of the Lie group $G$. Therefore, the image of the lattice $\Gamma$ is also linear and, therefore, by virtue of Selberg’s lemma, it has a torsion-free subgroup of finite index. Its preimage $\Gamma'$ under the adjoint representation is a subgroup of finite index in $\Gamma$. But the subgroup $\mathrm{Tors}(\Gamma')$ is central in $G$ by the construction of the group $\Gamma'$, and therefore, assertion 1) applies to the homogeneous space $G/\Gamma'$.

This proves Proposition 2.

In connection with these Example 10 and Proposition 2, the question arises as to what can be said in the case when the lattice $\Gamma$ is not supposed to be torsion-free. This again raises the question of the existence of lattices in simply connected Lie groups that are not virtually torsion-free. This issue, as noted above, is currently open.

The study of the structure of principal homogeneous spaces of the form $G/\Gamma$ and their fibred structure is, in the author’s opinion, of considerable interest.

Note that in [27] one can find descriptions of some principal homogeneous spaces.

The following assertion describes compact homogeneous spaces, which in a certain sense slightly different from principal ones.

Theorem 5. Let $M=G/H$ be some compact homogeneous space of a simply connected Lie group $G$, and let $K$ be a maximal compact (simply connected and semisimple) subgroup of $G$. Then

1) $\dim M^L= \dim M$ or $\dim M^L \leqslant \dim M -2$;

2) for $\dim M^L= \dim M$ the homogeneous space $G/H$ is principal;

3) if $\dim M^L= \dim M-2$, then the compact semisimple Lie group $K$ decomposes into a direct product $K=\mathrm{SU}(2)\times K_1$, where $K_1$ is some semisimple (and simply connected) compact Lie group, $N_K(L)_0 = \mathrm{SO}(2) \times K_1$, and $L_0= \mathrm{SO}(2)$;

4) if $\dim M^L= \dim M-3$, then the compact semisimple Lie group $K$ decomposes into one of the following direct products: $K=\mathrm{SU}(2) \times K_1$ or $K=\operatorname{Spin}(4) \times K_1$, where $K_1$ is some semisimple (and simply connected) compact Lie group, $N_K(L)_0 = K_1$ or $\operatorname{Spin}(3) \times K_1$, and $L_0 = \{e\} $ or $\operatorname{Spin}(3)$, respectively.

Proof. Our proof is based on the classification of compact semisimple Lie groups acting transitively and locally effectively on homogeneous manifolds of low dimension.

The codimension of the submanifold $M^L$ in $M$ is equal to that of the normalizer $N_K(L)$ in $K$. Moreover, $K/N_K(L)$ is a homogeneous space of a compact semisimple Lie group $K$, and therefore, $K/N_K(L)$ has a finite fundamental group.

1) The only thing that needs to be proved in this assertion is that, for $\dim M^L \ne \dim M$, it will be $\dim M^L \leqslant \dim M -2$. But this follows from the fact that there are no one-dimensional compact manifolds with a finite fundamental group.

2) This result is obvious.

3) Let $\dim M^L= \dim M-2$. This means that $K/N_K(L)$ is a two-dimensional homogeneous surface with a finite fundamental group. There are only two such surfaces: $S^2$ and $\mathbf{R}P^2$ (for this and other facts on homogeneous spaces of compact Lie groups, see, for example, [1].

The classification of compact semisimple Lie groups that are transitive and locally effective on two-dimensional manifolds is well known. Namely, among the simply connected compact Lie groups, only the Lie group $\mathrm{SU}(2)$ is possible here, and the connected component of the identity of the stationary subgroup is $\mathrm{U}(1)$ (or, which is actually the same, $\mathrm{SO}(2)$). The action of the Lie group $K$ on $K/N_K(L)$ does not have to be effective (that is, it can have a non-trivial connected normal subgroup that acts trivially on $K/N_K(L)$), and therefore, we get that in the case we are considering here the Lie group has a decomposition $K=\mathrm{SU}(2) \times K_1$, where $K_1$ is some compact simply connected semisimple Lie group (possibly trivial) whose action on $K/N_K(L)$ is trivial. The connected component of the identity of the subgroup $N_K(L)$ in this case obviously has the form $\mathrm{SO}(2)\times K_1$. Consider the Lie subgroup $L_0 \cap K_1$; this subgroup is normal in $N_K(L)$, and therefore, due to local efficiency of the action of $K$ on $G/H$, is discrete. And due to its normality in $N_K(L)$, we can assume that $L_0$ coincides with the direct factor $\mathrm{SU}(2)$ (if it is chosen properly). This proves assertion 3).

The proof of assertion 4) is similar to that of assertion 3). But here $K/N_K(L)$ is no longer a two-dimensional, but a three-dimensional homogeneous space with a finite fundamental group. There are quite a few such homogeneous manifolds (these are all homogeneous lens spaces), but there is only one manifold universally covering them – the three-dimensional sphere $S^3$ (see [1] or [29]). And compact simply connected Lie groups that act transitively and locally effectively on these manifolds are only $\operatorname{Spin}(3)$ and $\operatorname{Spin}(4)$ (the stationary subgroups here are $\mathrm{SO}(2)$ and $\operatorname{Spin}(3)$). Note that $\operatorname{Spin}(3)$ is isomorphic to $\mathrm{SU}(2)$ and diffeomorphic to $S^3$, and the Lie group $\operatorname{Spin}(4)$ is isomorphic to the Lie group $\mathrm{SU}(2)\times \mathrm{SU}(2)$ and is diffeomorphic to $S^3\times S^3$.

Further, we argue in the same way as in the proof of assertion 3) to verify that the Lie group $K$ must be isomorphic to $K=\mathrm{SU}(2) \times K_1$ or $K=\operatorname{Spin}(3) \times K_1$, where $K_1$ is some compact simply connected semisimple Lie group (possibly trivial), whose action on $K/N_K(L)$ is trivial. The assertion about the form of the Lie groups $L_0$ follows from what was said above using an argument similar to that given in the proof of assertion 3). This proves assertion 4), and therefore, Theorem 5.

Note that $\dim K/L \geqslant \dim K/N_K(L)$ (moreover, if these dimensions are equal, the structure group of the natural fibration can be reduced to the trivial one). The possibility of reduction of the structure group of the natural fibration and the topological structure of compact homogeneous spaces for $\dim K/L \leqslant 4$ were studied in [13], [27], and the algebraic structure of the structure groups and stationary subgroups of transitive actions is also considered there.

From Theorem 5 we obtain the following corollary on fibres of natural fibrations.

It can be seen from the assertions of Corollary 3 that, in the case of small $\operatorname{codim}_M M^L$, the fibres of Borel fibrations are indeed close to Lie groups, and thus the corresponding homogeneous spaces are close to principal ones. A similar consideration could be carried out for the case of codimension $4$. Let us denote the main one here: the four-dimensional simply connected homogeneous space of a compact Lie group is diffeomorphic to $S^4$, $S^2\times S^2$ or to the projective plane $\mathbf{C}P^2$. The transitive Lie groups here are, respectively, $\operatorname{Spin}(5)$, $\operatorname{Spin}(3)\,{\times}\,\operatorname{Spin} (3)$ (which is $\mathrm{SU}(2)\times \mathrm{SU}(2)$), and $\mathrm{SU}(3)$. Stationary subgroups are well known here, and therefore, the description of fibres of Borel fibrations in this case is obtained according to the same scheme as in Theorem 5 and Corollary 3. The author sees no need to give further details here.

In connection with the concept of a principal homogeneous space, it is of particular interest to consider the simplest principal compact homogeneous spaces, the compact Lie groups (which, by the way, are also fibres of natural fibrations of arbitrary principal homogeneous spaces). It turns out that compact Lie groups will not be uniquely determined by their topological characteristics. This has already been discussed above.

Note also that for any connected Lie groups, their homeomorphisms and diffeomorphisms are equivalent. For more details on the relationship between homotopy equivalence, diffeomorphism, and isomorphism of compact Lie groups, see [15]. In particular, there are numerous examples of diffeomorphic but not isomorphic compact Lie groups.

§ 8. Base and fibre of the natural fibration

The properties of the base and the fibre of the natural fibration have some kinds of mutual connections. Let us describe one of them. In this case, the concept of the Euler characteristic of the fundamental group is used, which is defined by the usual formula for the Euler characteristic in terms of the ranks of the cohomology groups of this discrete group.

Proof. As is known, the Euler characteristic is multiplicative. Therefore $\chi(M) = \chi(M_a) \cdot \chi (M_c)$. If the base $M_a$ does not degenerate to a point, then the Euler characteristic of a homogeneous space $M$ with an infinite fundamental group is equal to zero (see [1]). Therefore, for $\chi(M_a) \ne 0$ it should be $\chi(M_c)=0$. But the Euler characteristic of the homogeneous space $K/L$ of a compact semisimple Lie group (which is the Lie group $K$) vanishes if and only if the rank of its closed subgroup $L$ is strictly less than that of the Lie group $K$ itself.

The condition for the component $M_r$ of the structure fibration to degenerate to a point also follows from the multiplicativity of the Euler characteristic and the fact that the Euler characteristic of an infinite polycyclic group (which is $\pi_1(M_r)$ if $\dim M_r > 0$) is always zero. Since the manifold $M_r$ is aspherical, the Euler characteristic of this manifold is equal to the Euler characteristic of its fundamental group. Therefore, for $\chi(M_a) \ne 0$ the component $M_r$ must degenerate into a point. This proves Proposition 3.

Note that the condition of infinity of the fundamental group $\pi_1(M)$ in this proposition is equivalent to the non-compactness of the manifold $\widetilde M$ universally covering $M$. An example of the application of Proposition 3 is the case when the base $M_a$ of the natural fibration is some compact orientable surface $F_g$ of genus $g \geqslant 2$. We have $\chi(F_g) = 2-2g$ and therefore, $\chi (F_g) < 0$ for $g \geqslant 2$. Therefore, by Proposition 3, for a compact homogeneous space with base $M_a=F_g$ for the fibre $M_c=K/L$ of the natural fibration, the rank of the subgroup $L$ will be strictly less than that of the group $K$, and will also be $\chi(M_c) = 0$. In particular, even-dimensional spheres and complex projective spaces (whose Euler characteristics are known to be positive) cannot be fibres of natural fibrations for $M_a = F_g$. There are many other applications of Proposition 3.

§ 1. Introduction