Reports of the Academy of Sciences of the USSR
1969. Volume 188, No. 4

UDC 519.46

MATHEMATICS

B. I. ROSENFELD

ON PRINCIPAL FIBERINGS WHOSE BASES SATISFY THE SECOND AXIOM OF COUNTABILITY

(Presented by Academician P. S. Aleksandrov on 28 II 1969)

In the paper of I. M. Gelfand and D. B. Fuks (\(^1\)) an extension was considered of the classical category of locally trivial principal \(G\)-fiberings, where \(G\) is a connected Lie subgroup of the group \(GL(n,\mathbf R)\). The objects of this category were principal \(G\)-fiberings with Hausdorff bases, locally trivial in the following sense: for each point \(x\) of the base of such a fibering there exists a mapping of the space \(E^\#\) of the associated vector fibering into the space \(\mathbf R^n\), linear on each fiber of the vector fibering and nondegenerate on the fiber over the point \(x\). We emphasize that for fiberings with completely regular bases this notion of local triviality does not differ from the classical one, and that the local triviality of a \(G\)-fibering is defined depending not only on the group \(G\), but also on the embedding \(G \subset GL(n,\mathbf R)\). In (\(^1\)), in this category, a subcategory was singled out that plays a role analogous to that of the universal \(G\)-fibering. It was also indicated there that if the category under consideration is narrowed by requiring the bases of the fiberings to satisfy the second axiom of countability, then there exists a fibering \(\xi_G\) such that for any fibering \(\xi\) of our category there is a mapping \(\xi \to \xi_G\).

The fibering \((\xi_G)_p:\mathcal E_n \to \mathcal S_G\) is constructed as follows. Let \(\overline{\mathcal E}\) be the product of a countable number of real lines in the Tikhonov topology. By \(\mathcal E_n\) we denote the set of all nondegenerate \(n\)-frames in the linear space \(\overline{\mathcal E}\). The group \(GL(n,\mathbf R)\) and its subgroup \(G\) act on the space \(\mathcal E_n\) in the obvious way. We denote the quotient space \(\mathcal E_n/G\) by \(\mathcal S_G\); the projection \(p:\mathcal E_n \to \mathcal S_G\) defines, obviously, a principal \(G\)-fibering, which is denoted by \(\xi_G\).

All the constructions described carry over without change to the case when \(G\) is a Lie subgroup of the group \(GL(n,\mathbf C)\). The corresponding universal fibering will likewise be denoted by \((\xi_G)_p:\mathcal E_n \to \mathcal S_G\).

The aim of the present work is to show that the answers to the basic questions considered in (\(^1\)) change essentially under the above narrowing of the category. For example, it turns out that for a wide class of (noncompact) groups \(G\) the spaces \(\mathcal S_G\) are paracompact, from which the traditional consequences are obtained.

Let us list the main results of the paper.

Denote by \(\mathfrak H\) the class of such subgroups \(G\) of the groups \(GL(n,\mathbf C)\) for which the base \(\mathcal S_G\) of the universal \(G\)-fibering is paracompact. Conditions sufficient for \(G\) to belong to this class are given by the following theorems:

Theorem 1. Lie subgroups of the group \(L(n,\mathbf C)\) of matrices of order \(n\) whose determinant modulus is equal to one belong to the class \(\mathfrak H\).

Theorem 2. If \(G\) is connected, \(G\cap L(n,\mathbf C)\) is a compact group, and in \(G\) there exists a matrix \(C\) with such eigenvalues \(\lambda_1\) and \(\lambda_2\) that \(|\lambda_1|>1,\ |\lambda_2|<1\), then \(G\) belongs to \(\mathfrak H\).

The following theorems follow from the definition of the class \(\mathfrak H\).

Using our narrowed category of \(G\)-bundles, we can, analogously to how this was done in \((^1)\), define the groups \(H^G_{\mathrm{ag}}(G,\mathbf R)\).

Theorem 3. If \(G \in \mathfrak S\), then \(H^q_{\mathrm{alg}}(G,\mathbf R)=0\) for \(q>0\).

Theorem 4. Let \(G \in \mathfrak S\); then for any \(G\)-bundle \(f:X\to Y\), the paracompactness of the space \(X\) implies the paracompactness of the space \(Y\).

On the other hand, for many noncompact groups the base of the universal bundle has poor topological properties.

Put
\[ \|G\|=\inf_{g\in G}\|g\|,\qquad \text{where }\|g\|=\sum_{i,j}|x_{ij}|\quad \text{for }g=(x_{ij}). \]

Theorem 5. If \(\|G\|=0\), then every continuous real-valued function on \(\mathscr P_G\) is constant.

The condition \(\|G\|=0\) is satisfied, for example, for the group \(GL(n,\mathbf C)\). It turns out that for \(GL(n,\mathbf C)\) the groups \(H^q_{\mathrm{alg}}(G,\mathbf R)\) are nontrivial and, moreover, the space \(\mathscr P_{GL(n,\mathbf C)}\) can be the base of a nontrivial Hilbert bundle.

Example. Let \(H=L_2(\mathbf C^n)\) be the space of square-summable functions on \(\mathbf C^n\). Consider the unitary representation of the group \(GL(n,\mathbf C)\) acting in \(H\), which is given by the formula
\[ gf(z)=|\det g|\,f(g^{-1}z), \]
where \(g\in GL(n,\mathbf C)\), \(f(z)\in L_2(\mathbf C^n)\). With the help of Theorem 5 one can show that the bundle induced by this representation,
\[ \mathscr E_n\times H\to \mathscr E_n\times H/G\simeq \mathscr P_{GL,\mathbf C}, \]
with base \(\mathscr P_{GL}\) and fiber \(H\), is nontrivial.

Analogous results can also be obtained for real linear groups.

Proof of Theorem 1.

Case 1. The group \(G=L(n,\mathbf C)\).

Denote by \(A\) the space of all collections \(\{x_{p_1,\ldots,p_n}\}_{p_i=1,2,\ldots}\), where \(x_{p_1,\ldots,p_n}\) are real numbers, endowed with the Tikhonov topology. The continuous mapping \(f^*:\mathscr E_n\to A\), given by the formula
\[ f^*(x)=\{(f^*x)_{p_1,\ldots,p_n}\}, \]
where
\[ (f^*x)_{p_1,\ldots,p_n} = \left| \det \begin{pmatrix} x_{1p_1}&\ldots&x_{1p_n}\\ \ldots&\ldots&\ldots\\ x_{np_1}&\ldots&x_{np_n} \end{pmatrix} \right| \]
for \(x=\{x_{ij}\}\) (recall that the points of the space \(\mathscr E_n\) are \(n\)-frames in the space \(\mathscr E\), and each point \(x\in\mathscr E_n\) can be specified by a collection \(\{x_{ij}\}_{i=1,\ldots,n;\,j=1,2,\ldots}\)), is constant on the orbits of the group \(G\) and therefore defines a continuous mapping \(f:\mathscr P_G\to A\). It is obvious that the mapping \(f\) takes distinct points to distinct points. We shall show that \(f\) is a topological embedding. For this purpose, for every point \(y\in f(\mathscr P_G)\subset A\) we construct a continuous mapping \(h\) of a certain neighborhood of this point (in the space \(A\)) into \(\mathscr E_n\) and show that the mapping \(f\circ p\circ h|_{f(\mathscr P_G)}\) is the identity. Such a mapping can be defined by setting
\[ h\bigl(\{y_{p_1,\ldots,p_n}\}\bigr)=\{y_{ij}\}, \]
where
\[ y_{ij}= \begin{cases} y_{j,p_2^0,\ldots,p_n^0}, & \text{for } i=1,\\[6pt] \dfrac{y_{p_1^0,\ldots,p_{i-1}^0,j,p_{i+1}^0,\ldots,p_n^0}} {y_{p_1^0,\ldots,p_n^0}}, & \text{for } i>1 \end{cases} \]
(here \(p_1^0,\ldots,p_n^0\) is any collection such that
\[ y_{p_1^0,\ldots,p_n^0}\ne 0 \]
).

As is known, \(A\) is a metric space. Consequently, \(\mathscr P_G\) is also a metric space. Since the space \(\mathscr P_G\) satisfies the second axiom of countability, it is paracompact.

Case 2. The group \(G\) is a Lie subgroup of the group \(L(n, C)\). In this case there exists a smooth fibration \(\mathcal S_G \to \mathcal S_{L(n,\,)}\) with fiber \(L(n,C)/G\). The base of this fibration is paracompact, and the fiber is a finite-dimensional manifold. Consequently, \(\mathcal S_G\) is also paracompact.

Proof of Theorem 2. Clearly, the group \(G\) is not contained in \(L(n,C)\). Let \(d:G\to \mathbb R\) be the homomorphism defined by the formula
\[ d(g)=\ln|\det g|. \]
Denote by \(\Gamma\) the closed normal divisor \(G\cap L(n,C)\). By hypothesis \(\Gamma\) is compact. There is an exact sequence
\[ 0\to \Gamma\to G\to \mathbb R\to 0. \]
In an obvious way one constructs a homomorphism \(\bar d:\mathbb R\to G\) such that
\[ d\bar d:\mathbb R\to \mathbb R \]
is the identity mapping. It is clear that \(\operatorname{Im}\bar d\) is a closed one-parameter subgroup of the group \(G\). Its elements are the matrices \(\exp(tA)\), where \(t\in \mathbb R\), and \(A\) is some matrix. We denote this subgroup by \(B\). We may assume that \(\exp(A)=C\), where \(C\) is the matrix from the hypothesis of the theorem. Then the matrix \(A\), in some basis, has the form:
\[ A= \begin{pmatrix} \mu_1&0& \\ 0&\mu_2&0\\ \hline 0&&A' \end{pmatrix}; \qquad \exp(tA)= \begin{pmatrix} \exp(t\mu_1)&0&\\ 0&\exp(t\mu_2)&0\\ \hline 0&&\exp(tA') \end{pmatrix}, \]
where \(A'\) is some matrix and \(\operatorname{Re}\mu_1>0,\ \operatorname{Re}\mu_2<0\). The universal fibration \(\mathcal E_n\to \mathcal S_G\) decomposes into the composition
\[ \mathcal E_n \xrightarrow{\Gamma} \mathcal S_\Gamma \xrightarrow{B} \mathcal S_G, \]
where \(\mathcal S_G\) is obtained after factorizing \(\mathcal S_\Gamma\) by the action of the group \(B\). We prove that the fibration
\[ \mathcal S_\Gamma \xrightarrow{B} \mathcal S_G \]
is trivial.

For this purpose consider real continuous functions \(f_1,f_2\) on \(\mathcal S_\Gamma\), defined by the formulas
\[ f_1(\Gamma\{x_{ij}\})= \sum_{n=1}^{\infty}\frac{1}{2^n} \frac{\displaystyle\int_{\Gamma}|x_{1n}|\,d\Gamma} {\displaystyle 1+\int_{\Gamma}|x_{1n}|\,d\Gamma}; \qquad f_2(\Gamma\{x_{ij}\})= \sum_{n=1}^{\infty}\frac{1}{2^n} \frac{\displaystyle\int_{\Gamma}|x_{2n}|\,d\Gamma} {\displaystyle 1+\int_{\Gamma}|x_{2n}|\,d\Gamma}, \]
where \(\Gamma\{x_{ij}\}\) is the orbit of the element \(\{x_{ij}\}\in \mathcal E_n\).

On each orbit of the group \(B=\{\exp(tA)\}\) the function \(f_1\) increases monotonically, while the function \(f_2\) decreases monotonically (as \(t\) increases). Moreover,
\[ \lim_{t\to+\infty} f_1(\exp(tA)x)=a(x)>0;\qquad \lim_{t\to-\infty} f_2(\exp(tA)x)=b(x)>0; \]
\[ \lim_{t\to-\infty} f_1=\lim_{t\to+\infty} f_2=0. \]
The function \(f=f_1-f_2\) on an orbit of the group \(B\) increases monotonically from \(-b(x)\) to \(a(x)\). Therefore the set \(f^{-1}(0)\subset \mathcal S_\Gamma\) defines a section surface of the fibration
\[ \mathcal S_\Gamma \xrightarrow{B} \mathcal S_G. \]
Consequently, \(\mathcal S_\Gamma\) is homeomorphic to \(\mathcal S_G\times \mathbb R\). The space \(\mathcal S_\Gamma\) is paracompact. Consequently, \(\mathcal S_G\) is also paracompact. Theorem 2 is proved.

Theorem 3 is obvious.

Proof of Theorem 4. According to \((^1)\), there exists a mapping of an arbitrary \(G\)-fibration \((\xi)f:X\to Y\) into the fibration \((\xi_G)p:\mathcal E_n\to \mathcal S_G\). Since \(\mathcal S_G\) is paracompact, there exists a mapping of the fibration \((\xi_G)\) into the classical universal fibration \((\xi_G^*)p_G:EG\to BG\), described in \((^2)\). This latter decomposes into the composition
\[ EG\to B\hat G\to BG, \]
where \(\hat G\subset G\) is a maximal compact subgroup of the group \(G\). The first arrow denotes a principal \(\hat G\)-fibration, and the second is a fibration having a section surface. It follows from this that our fibration \(\xi\) also decomposes into the composition
\[ X\to \hat X\to Y. \]
Here the first arrow denotes a principal \(\hat G\)-fibration, and therefore from the paracompactness of \(X\) there follows the paracompactness

\(\tilde X\); the second arrow denotes a bundle having a cross-section, and therefore \(Y\) is homeomorphic to a subspace of the space \(\tilde X\) and, consequently, is also paracompact.

Proof of Theorem 5. Let \(h\) be a nonconstant continuous function on \(\mathscr P_G\). Denote by \(f\) the continuous function on \(\mathscr E_n\) defined by the formula \(f(z)=h(pz)\) for \(z\in \mathscr E_n\). There exist \(x,y\in \mathscr E_n\) such that \(f(x)\ne f(y)\); put \(|f(x)-f(y)|=\varepsilon\). By the definition of the space \(\mathscr E_n\), its points are \(n\)-tuples in \(\mathscr E\), and each point \(z\in \mathscr E_n\) can be specified by a set \(\{z_{ij}\mid i=1,\ldots,n;\ j=1,2,\ldots\}\). Let, for example, \(x=\{x_{ij}\}\), \(y=\{y_{ij}\}\).

For each natural number \(k\) choose a point \(z^k=\{z_{ij}^k\}\in \mathscr E_n\) such that \(z_{ij}^k=0\) for \(j\le k\). For any element \(g\in G\) put \(g(z^k)=\{(gz)_{ij}^k\}\). Put \(x(g)^k=\{x(g)_{ij}^k\}\), \(y(g)^k=\{y(g)_{ij}^k\}\), where

\[ x(g)_{ij}^k= \begin{cases} x_{ij}, & (j\le k),\\ (gz)_{ij}^k, & (j>k); \end{cases} \qquad y(g)_{ij}^k= \begin{cases} y_{ij}, & (j\le k),\\ (gz)_{ij}^k, & (j>k). \end{cases} \]

It is clear that, for sufficiently large \(k\), the sets \(x(g)^k,\ y(g)^k\) define elements of the space \(\mathscr E_n\) and

\[ |f(x)-f(x(g)^k)|<\varepsilon/4,\qquad |f(y)-f(y(g)^k)|<\varepsilon/4 \tag{1} \]

for every \(g\in G\). Since \(\|G\|=0\), one can find such a \(g_0\in G\) that

\[ |f(z^k)-f(g_0x(g_0^{-1})^k)|<\varepsilon/4,\qquad |f(z^k)-f(g_0y(g_0^{-1})^k)|<\varepsilon/4. \tag{2} \]

Since the function \(f\) is constant on the orbits of the group \(G\), we have

\[ f(g_0x(g_0^{-1})^k)=f(x(g_0^{-1})^k),\qquad f(g_0y(g_0^{-1})^k)=f(y(g_0^{-1})^k), \]

and from inequalities (1) with \(g=g_0^{-1}\) and (2) we obtain \(|f(x)-f(y)|<\varepsilon\), which contradicts the definition of the number \(\varepsilon\).

In conclusion the author expresses deep gratitude to I. M. Gel'fand and D. B. Fuks for posing the problem and for their help in the work.

REFERENCES

1. I. M. Gel'fand, D. B. Fuks, Functional analysis and its applications, 1, no. 4 (1967).
2. N. Steenrod, The Topology of Fibre Bundles, Moscow, 1953.