Real Analytic Fibre Bundles: Some Results on Classification and Homotopies

The classification problem for holomorphic fibre bundles over Stein spaces was solved by H. GRAUERT. Along the same lines, the real coherent analytic case was considered by A. TOGNOLI and V. ANCONA. In this paper we propose a different approach, based on classifying spaces, in order to study the previous problem for real analytic fibre bundles over C -analytic subspaces of R ^m. So, let X be a C -analytic subspace of R ^m and G a compact Lie group. The main result is a characterization of the real analytic G-principal fibre bundles over X for which the analytic and topological equivalence coincide. Moreover, we prove that these bundles can be classified also by means of homotopy classes of analytic maps of X into classifying spaces. Among the others results, are worth recording: a relative approximation theorem of continuous cross sections by analytic ones, a theorem about the equivalence between analytical and topological homotopy between cross sections and a covering homotopy theorem.