ON INFINITE-DIMENSIONAL HOMOGENEOUS SPACES

E. G. SKLYARENKO

(Presented by Academician P. S. Aleksandrov, 3 VII 1961)

The purpose of this note is to show how various homogeneity conditions affect the dimensional properties of infinite-dimensional spaces. It turns out that if a locally bicompact group of transformations (whose space is finally compact) acts transitively on an infinite-dimensional locally bicompact space, then such a space is strongly infinite-dimensional*. Apparently, this result, in some form, remains valid under weaker homogeneity assumptions as well. The following proposition (which, however, has a rather special character) also speaks in favor of this hypothesis:

If an infinite-dimensional metrizable locally bicompact space \(X\) is doubly homogeneous (i.e., for any two points there exists a homeomorphism carrying these points into some fixed pair of points), then it cannot have transfinite dimension \(\omega\)**.

Indeed, suppose that \(X\) has transfinite dimension \(\omega\). Let \(x\) be an arbitrary point of the space \(X\), and let \(U\) be a neighborhood of this point such that \(X[U]\) is bicompact, is not equal to \(X\), and the boundary \(\operatorname{Fr}_X U\) of the set \(U\) is finite-dimensional. Let \(n = \operatorname{ind} \operatorname{Fr}_X U\); we shall show that then \(\operatorname{ind} X \le n+1\). In view of the homogeneity of the space \(X\), it is enough to show that every neighborhood \(Ox\) of the point \(x\) contains a neighborhood \(V\) such that \(\operatorname{ind} \operatorname{Fr}_X V \le n\). Let \(y \in X \setminus X[U]\). For every point \(z \in \operatorname{Fr}_X Ox\), consider a homeomorphism \(f_z\) of the space \(X\) onto itself such that \(f_z x = x\), \(f_z y = z\). Let \(f_1,\ldots,f_s\) be such a finite collection of homeomorphisms for which the set

\[ \bigcup_{i=1}^{s} f_i (X \setminus X[U]) \]

contains \(\operatorname{Fr}_X Ox\). Then, as \(V\), one can obviously take the set

\[ \left\{\bigcap_{i=1}^{s} f_i U\right\} \cap Ox . \]

We pass to the proof of the proposition mentioned at the beginning of the paper.

Theorem. Let \(G\) be an infinite-dimensional locally bicompact group, \(H\) a closed subgroup in \(G\), and let the quotient space \(B = G/H\) be infinite-dimensional. Then \(B\) is strongly infinite-dimensional; moreover, it contains a Hilbert parallelepiped.

Proof. First of all we note that if \(G_1\) is an open subgroup in \(G\) and \(H_1 = H \cap G_1\), then the quotient space \(B_1 = G_1/H_1\) is also infinite-dimensional. Indeed, the natural mapping \(f : B_1 \to B\)

* According to the definition proposed by P. S. Aleksandrov (see (3), p. 14), a space is called strongly infinite-dimensional if there exists in it a countable system of pairs of closed sets \(\{A_i, B_i\}\), \(A_i \cap B_i = \Lambda\), such that for every system of closed sets \(C_i\) separating \(A_i\) and \(B_i\), the set \(\bigcap_i C_i\) is nonempty.

** Transfinite dimension is defined by induction: a space has dimension \(\le \alpha\) if each of its points has arbitrarily small neighborhoods with boundaries of dimension \(<\alpha\); in particular, a space has transfinite dimension \(\omega\) if each of its points has arbitrarily small neighborhoods with finite-dimensional boundary.

is an embedding of the set \(B_1\) into an open subset of the space \(B\); moreover, \(B\) turns out to be the union of pairwise disjoint locally homeomorphic open sets, namely the sets that are images, under the canonical projection of \(G\) onto \(B\), of the double adjacent classes of the group \(G\) with respect to the subgroups \(H\) and \(G_1\).

As is known (see \((^2)\), p. 39), every locally bicompact group contains an open projective-Lie subgroup\(*\); this subgroup, in turn, contains an open subgroup of bicompact origin\(**\), which is also projective-Lie (\((^2)\), p. 5). Therefore, in view of the remark made above, without loss of generality one may assume that the group \(G\) itself is projective-Lie and of bicompact origin.

Let \(\tau\) be the local weight of the topological group \(G\); let \(\theta\) be the least ordinal number of cardinality \(\tau\). Represent the group \(G\) as the limit of a completely ordered inverse spectrum of locally bicompact groups \(\{G_\alpha,\pi_\beta^\alpha\}\) of length \(\theta\), satisfying the following conditions: 1) for every limit ordinal \(\alpha<\theta\) we have \(G_\alpha=\lim_{\beta<\alpha}G_\beta\); 2) the kernel of the homomorphism
\[ \pi_1:G\to G_1 \]
is a bicompact normal divisor of the group \(G\) (whence it follows that for every \(\alpha<\theta\) the kernel of the homomorphism \(\pi_\alpha:G\to G_\alpha\) is bicompact); 3) the group \(G_1\) is a Lie group; for every \(\alpha<\theta\) the kernel of the homomorphism \(\pi_\alpha^{\alpha+1}:G_{\alpha+1}\to G_\alpha\) is a Lie group. Such an inverse spectrum is called a Lie series of the group \(G\). In the book of L. S. Pontryagin \((^6)\) a Lie series is constructed for bicompact groups; the construction given there remains fully valid also for projective-Lie locally bicompact groups. It remains only to verify that the limit group \(G'\) of the spectrum \(\{G_\alpha,\pi_\beta^\alpha\}\) is isomorphic to the group \(G\). The natural homomorphism \(\pi:G\to G'\), generated by the homomorphisms \(\pi_\alpha:G\to G_\alpha\), is, by the construction of the spectrum, a monomorphism. We shall show that it is an epimorphism. Let \(g'=\{g_\alpha\}\), \(g_\alpha\in G_\alpha\), be an arbitrary element of the group \(G'\). In accordance with condition 2), the sets \(\pi_\alpha^{-1}g_\alpha\) are bicompact and, moreover, if \(\alpha<\beta\), then \(\pi_\alpha^{-1}g_\alpha\supset\pi_\beta^{-1}g_\beta\); hence
\[ \bigcap_{\alpha<\theta}\pi_\alpha^{-1}g_\alpha\ne\Lambda . \]
If \(g\in\bigcap_{\alpha<\theta}\pi_\alpha^{-1}g_\alpha\), then \(\pi_\alpha g=g_\alpha\), i.e. \(\pi g=g'\). Thus \(\pi G=G'\). From condition 2) it follows that the group \(G'\) is locally bicompact\(***)\), and therefore, in accordance with Theorem 12 of \((^6)\), the homomorphism \(\pi\) is a topological isomorphism.

We now proceed to the construction of the spectrum for the quotient space \(B\). From condition 2) it follows that the homomorphisms \(\pi_\alpha\) are closed mappings, and therefore for each \(\alpha\) the subgroup \(H_\alpha=\pi_\alpha H\) is closed in \(G_\alpha\); moreover
\[ H=\lim_{\alpha<\theta}H_\alpha . \]
Let \(B_\alpha=G_\alpha/H_\alpha\). The spaces \(B_\alpha\) also, obviously, form an inverse spectrum. We shall denote by \(\varphi_\beta^\alpha\) the mapping \(B_\alpha\to B_\beta\) induced by the homomorphism \(\pi_\beta^\alpha\). By one theorem of Mostert (\((^4)\), theorem 4),
\[ B=\lim_{\alpha<\theta}B_\alpha . \]

We shall show that for each \(\alpha<\theta\) the mapping \(\varphi_\alpha^{\alpha+1}\) is a locally trivial fibration whose fiber is a manifold. Let
\[ N_\alpha=\operatorname{Ker}\pi_\alpha^{\alpha+1}. \]
In accordance with conditions 2) and 3), \(N_\alpha\) is a compact Lie group. The group \(N_\alpha\) acts on the space \(B_{\alpha+1}\), and the fibration
\[ \varphi_\alpha^{\alpha+1}: B_{\alpha+1}\to B_\alpha \]
is a fibration onto the orbits of the group \(N_\alpha\). These orbits are homeomorphic to the quotient space
\[ M_\alpha=N_\alpha/N_\alpha\cap H_{\alpha+1}, \]
which is

\(*\) A locally bicompact group is called projective-Lie if every neighborhood of its identity contains a normal divisor such that the quotient group by it is a Lie group.

\(**\) A topological group is called a group of bicompact origin if it is generated by some bicompact neighborhood of the identity.

\(***)\) Since \(G_1\) is locally bicompact, and the mapping \(\pi_1\) is closed and bicompact (\((^5)\), theorem 4).

manifold. The local triviality of the fibration \(\varphi_\alpha^{\alpha+1}\) follows from Gleason’s theorem \((({}^1),\) Theorem 3.6), in which the local triviality of the fibration onto the orbits is established under the assumption that the stationary subgroups of different points are conjugate to one another. In our case this condition is evidently satisfied, since the stationary subgroup of the element \(aH_{\alpha+1}\in B_{\alpha+1}\) \((a\in G_{\alpha+1})\) is \(aH_{\alpha+1}a^{-1}\cap N_\alpha\).

Thus, let us prove that the space \(B\) contains a Hilbert parallelepiped. To this end we prove by induction that each \(B_\alpha\) contains a certain cube \(I_\alpha\) in such a way that the mapping \(\varphi_\alpha^{\alpha+1}\) on the cube \(I_{\alpha+1}\) coincides with the projection of \(I_{\alpha+1}\) onto \(I_\alpha\) parallel to a finite-dimensional face of this cube whose dimension is equal to the dimension of the manifold \(M_\alpha\). As \(I_1\) we choose a closed neighborhood in the manifold \(B_1\) homeomorphic to a cube. Suppose that, for all \(\alpha<\beta\), the cubes \(I_\alpha\) have already been constructed. Then, if \(\beta\) is not a limit transfinite number, then, since the set \(J_{\beta-1}\) is contractible,

\[ (\varphi_{\beta-1}^{\beta})^{-1} I_{\beta-1} = I_{\beta-1}\times M_{\beta-1}, \]

and as \(I_\beta\) we take the cube \(I_{\beta+1}\times I'\), where \(I'\) is some closed neighborhood in the manifold \(M_{\beta-1}\) homeomorphic to a cube. If, however, \(\beta\) is a limit number, then we set

\[ I_\beta=\lim_{\alpha<\beta} I_\alpha . \]

The cube \(I_\beta\) is canonically embedded in \(B_\beta\), since, by the Moestert theorem cited above and condition 1),

\[ B_\beta=\lim_{\alpha<\beta} B_\alpha . \]

Then

\[ I=\lim_{\alpha<\theta} I_\alpha \]

will be the desired parallelepiped in the space \(B\). The theorem is proved.

Moscow State University
named after M. V. Lomonosov

Received
24 VI 1961

REFERENCES

\({}^1\) A. M. Gleason, Proc. Am. Math. Soc., 1, 35 (1950).
\({}^2\) V. M. Glushkov, UMN, 12, No. 2, 3 (1957).
\({}^3\) V. Hurewicz, H. Wallman, Dimension Theory, Moscow, 1948.
\({}^4\) P. S. Mostert, Duke Math. J., 23, 57 (1956).
\({}^5\) V. I. Ponomarev, 14, No. 4, 203 (1959).
\({}^6\) L. S. Pontryagin, Continuous Groups, Moscow, 1954.