Vector bundles over three-dimensional spherical space forms
The authors consider the problem of enumerating the \(G\)-bundles over closed manifolds \(M\) of dimension \(d \leq 3\) where \(G\) is a compact Lie group. In particular their interest is focused on real vector bundles over the 3-dimensional spherical space forms (= the compact connected 3-dimensional manifolds with positive constant curvature). Clearly the set \({\mathcal{B}}(M, G)\) of equivalence classes of principal \(G\)-bundles over \(M\) coincides with \([M, BG]\). When \(G\) is connected, since \(\pi_2(G)=0\), it therefore follows that \({\mathcal{B}}(M, G)=\{ 0 \}\) for \(d=1\) and \({\mathcal{B}}(M, G)=H^2(M; \pi_1(G))\) for \(d=2, 3\). The authors first generalize this result for the group \(G\) such that the projection \(G \to G/G_0\) to the quotient by the connected component of the identity has a section \(G/G_0 \to G\) (Theorem 2.2). It is known that all the spherical space forms of dimension 3 can be written as \(S_\Gamma=S^3/\Gamma\) where \(\Gamma\) is an arbitrary finite subgroup of \(SO(4)\) operating freely on \(S^3\), and a complete list of these finite subgroups has been known. Using the formula obtained the authors compute \({\mathcal{B}}(S_\Gamma, O(n))\) for every \(\Gamma\) and \(n=1, 2, 3\) and give tables for all the possible real vector bundles of ranks 1, 2, 3 over \(S_\Gamma\) (Tables 5.1 and 5.2). To quote an example from the authors' calculations, the case \(\Gamma = C(2n+1)\), the cyclic subgroup of order \(2n+1\), is perhaps the most simple one. In this case one finds that \([S_\Gamma, BO(1)]=\{0\}\), \([S_\Gamma, BO(2)]={\mathbb Z}/(2n+1)\) and \([S_\Gamma, BO(3)]=\{0\}\). This means that \({\text{Vect}}_1(S_\Gamma)=\{1\}\) and \({\text{Vect}}_3(S_\Gamma)=\{1\}\), where 1 denotes the trivial real vector bundle of rank 1 and 3, respectively, and that \({\text{Vect}}_2(S_\Gamma)\) consists of \(2n+1\) real vector bundles of rank 2, which are in fact described in terms of the Stiefel-Whitney classes.
scientific article; zbMATH DE number 5164005
scientific article