Topological Invariants of Principal G-Bundles with Singularities

Abstract

principal G-bundle with singularities is a principal bundle π: \(\bar P\) → M with structure group \(\bar G\) which reduces to a subgroup G ⊂ \(\bar G\) on the set M \ Σ, where M is an n-dimensional compact manifold and Σ ⊂ M is a k-dimensional submanifold. For example, a vector field on an n-dimensional Riemannian manifold M defines reduction of the orthonormal frame bundle of M to the subgroup O(n − 1) ⊂ O(n) on the set M \ Σ, where Σ is the set of zeros of this vector field. The aim of this paper is to construct topological invariants of principal bundles with singularities. To do this we apply the obstruction theory to the sectionM → \(\bar P\)/Gcorresponding to the reduction and obtain the topological invariant as a class in H^n−k(M,M \ Σ; π_n−k−1(\(\bar G\)/G)). We study the properties of this invariants and, in particular, consider cases k = 0 y k = n − 1.