Celebratio Mathematica

R. S. Palais and T. E. Stewart: “Deformations of compact differentiable transformation groups,” Am. J. Math. 82 : 4 (October 1960), pp. 935–937. MR 120652 Zbl 0106.16401 article

R. S. Palais and T. E. Stewart: “Torus bundles over a torus,” Proc. Am. Math. Soc. 12 : 1 (1961), pp. 26–29. MR 123638 Zbl 0102.38702 article

If a compact Lie group \( P \) acts on a completely regular topological space \( E \) then \( E \) is said to be a principal \( P \)-bundle if whenever the relation \( px = x \) holds for \( p\in P \), \( x\in E \) it follows that \( p = e \), the identity of \( P \). The orbit space \( X = E/P \) is called the base space and the map \[ \pi:E\to X \] carrying \( y \) into its orbit \( P\cdot y \) is called the projection. Suppose now that \( G \) is a Lie group, \( H_i \) a closed subgroup of \( G \) and \( H \) a closed, normal subgroup of \( H_i \) such that \( P = H_i/H \) is compact. Then \( E = G/H \) becomes a principal \( P \)-bundle with base space \( X = G/H_1 \) and projection \[ gH\mapsto gH_i \] under the action \[ p(gH) = gp^{-1}H .\] Such a principal bundle will be called canonical.

The purpose of this note is to show that if \( X \) is a torus of dimension \( n \) and \( P \) a torus of dimension \( m \) then every principal \( P \)-bundle over \( X \) is canonical and further that the group \( G \) lies in an extremely narrow class. Roughly speaking, our method is to try to lift the action of euclidean \( n \)-space up to the total space of the bundle and observe what obstructs this effort.

R. S. Palais and T. E. Stewart: “The cohomology of differentiable transformation groups,” Am. J. Math. 83 : 4 (1961), pp. 623–644. MR 140613 Zbl 0104.17703 article

If a Lie group \( G \) acts differentiably on a manifold \( \mathscr{M} \) then various spaces of tensor field on \( \mathscr{M} \) become in a natural way modules for the Lie algebra \( \mathscr{G} \) of \( G \) and the cohomology of \( \mathscr{G} \) with these coefficient modules in certain cases carries interesting information about the action of \( G \). In this paper we will discuss this situation, at first in a somewhat more abstract setup, and develop a method for computing these cohomology groups in certain cases. In particular we shall show that if \( G \) is compact and semi-simple then even though these modules are infinite dimensional the conclusions of the First and Second Whitehead Lemmas [1937, 1936] are valid; namely the first and second cohomology groups are trivial. As one consequence we will show that differentiable actions of compact, semi-simple Lie groups admit only trivial infinitesimal deformations, a fact whose global analogue will be found in [Palais and Stewart 1960]. Our second and motivating application of these general cohomology results is to a question initiated by one of the authors in [Stewart 1960]. Namely if a Lie group \( G \) acts differentiably on the base space of a differentiable fiber bundle \( B \) over \( M \) can \( G \) be made to act differentiably on \( B \) so as to be equivariant with respect to the fiber projection and so that each operation of \( G \) on \( B \) is a bundle map. We show here that the answer is yes if \( G \) is compact and simply connected and if the structural group of \( B \) is a solvable Lie group, and moreover that the way of “lifting” the action of \( G \) to \( B \) is essentially unique.

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Richard Sheldon Palais

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