On topological classification of singular fiber bundles with 2-D oriented base spaces (Q1961994)

Let \(M\) be a smooth manifold with boundary; from the tangent bundle \(TM\), one passes to the associated unit sphere bundle. We cite the author: ''By compressing the unit sphere fiber over every point of the boundary of \(M\) into a point, we get a new manifold; this manifold is called a singular fiber bundle and is denoted by \(\beta(TM)\).'' The aim of the paper is to prove that if \(M\) is an oriented \(2\)-dimensional manifold with boundary, then the topology of \(\beta(TM)\) is uniquely determined by the Euler characteristic \(\chi(M)\), \(\beta(TM)\) is homeomorphic to the sphere \(S^3\) when \(\chi(M)=1\), and \(\beta(TM)\) is homeomorphic to the connected sum of \(1-\chi(M)\) copies of \(S^1 \times S^2\) when \(\chi(M) < 1\).