The Topology of Rotation Groups

Abstract

The Betti numbers of the rotation group, R(n), in n real variables were first determined by R. Brauer [5].2 Later Pontrjagin [12] extended slightly these results by computing the integral homology groups of R(n). The Betti numbers combined with Hopf's Theorem on group-manifolds [9], which states that the rational cohomology ring of a group-manifold is isomorphic to that of a product of odd dimensional spheres, can be used to determine abstractly the rational cohomology ring of R(n). Pontrjagin's results can be used to find the additive structure of the cohomology groups of R(n) over an arbitrary coefficient domain, but there are no general theorems such as Hopf's Theorem available for the determination of the cup product over coefficient rings other than the rational numbers. This paper gives a treatment of the cohomology structure of R(n) which links that structure more intimately with the space R(n) than does Pontrjagin's treatment and extends the results mentioned above to other coefficient domains. The basic tool used is the notion of a cell complex. The particular cell complex used on R(n) gives rise to a cell complex on the Stiefel manifold V(n, r) of r-frames in euclidian n-space, which J. H. C. Whitehead used [17] to compute some of the higher homotopy groups of V(n, r). In Section 2 we give a description of this cell complex, rephrasing Whitehead's definitions slightly in order to facilitate later computations. Sections 3, 5, and 7 are devoted to the computation of the homology and cohomology groups and the calculation of the cup product. In Section 6 the Steenrod squaring homomorphisms are determined for V(n, r). These are then used in Section 9 to give a proof of the Steenrod-Whitehead Theorem that 2k independent vector fields do not exist on S'-', where 2k is the largest power of 2 dividing n [16], and are also used in Section 11 to show that, unless 2n = 2-k 2, S2n has no quasi-complex structure.3 In each case the desired result is obtained by employing the squaring homomorphism to show the nonexistence of cross sections in certain fibre bundles. The appendix gives a brief account of a technique for finding the integral cup