On the Koschorke normal bordism sequence (Q1209045)

This paper considers the normal bordism group \(\Omega_ n(X,\varphi)\), \(\varphi\in K\widetilde{O}(X)\), consisting of bordism classes of pairs \((g,G)\), with \(g: M\to X\), \(M\) a closed \(n\)-manifold and \(G\) a trivialization of \(\tau M\oplus g^*\varphi\) (\(\tau M\) the tangent bundle of \(M\)). These groups appear, for example, in the work of Koschorke on the vector field problem and in the classification of immersions. The results in the present paper describe explicit relations between \(\Omega_ n(X,\varphi)\) and \(H_ n(X,Z_ \varphi)\) for \(n=1\) and 2, where \(Z_ \varphi\) is the twisted \(Z\) defined by \(w_ 1(\varphi)\). The precise relations depend on various technical conditions. There are also results on \(\Omega_ n(X\times \text{BO}(2),\varphi\times \Gamma)\), where \(\Gamma\) is the universal bundle (also for \(n=1\) and 2).