Topological invariants of principal \(G\)-bundles with singularities (Q1785040)

A principal \(G\)-bundle with singularities is a principal bundle \(\pi : \overline{P} \rightarrow M\) with structure group \(\overline{G}\) which reduces to a subgroup \(G \subset \overline{G}\) on the set \(M \setminus \Sigma\), where \(M\) is a compact \(n\)-dimensional manifold and \(\Sigma \subset M\) is a \(k\)-dimensional submanifold. This paper constructs topological invariants for principal bundles with singularities. This is done by applying obstruction theory to the section \(M \rightarrow \overline{P}/G\) corresponding to the reduction to obtain a topological invariant in \(H^{n-k}(M, M \setminus \Sigma; \pi_{n-k-1}(\overline{G}/G))\)