Double Covers of Pseudo-orthogonal Groups

Abstract

For every pair (m, n) of non-negative integers one defines E _m,n to be the group of equivalence classes of central extensions of the pseudo-orthogonal group O _m,n by ℤ₂. The isomorphism k : E _m,n → H²(BO _m,n ℤ₂) is used to show that E _m,n is isomorphic to the group ℤ ^l(m,n)₂ where l(0,0) = 0, l(l,0) = 1, l(m,0) = 2, l(1.1) = 3, l(1,n) = 4 and l(m,n) = 5 for m,n > 1. If M is a manifold with a metric tensor g of signature (m, n) and f is a smooth map from M to the classifying space BO _m,n inducing the principal O _m,n -bundle P of orthonormal frames defined by g, then the bundle P can be reduced to an element H of E _m,n —i.e. to a double cover of O _m,n —if, and only if, the element f*k(H) of H²(M, ℤ₂) vanishes. This generalizes the classical topological condition for the existence of a pin structure on a pseudo-Riemannian manifold. The set of all 32 = 2⁵ inequivalent double covers of O _m × O _n , the maximal compact subgroup of O _m,n , m,n > 1, is described explicitly.