Cobordism of manifolds with strong almost tangent structures (Q801567)

This paper studies the possible cobordism classes for a closed manifold \(M^ n\) whose tangent bundle has the form \(r\xi\) \(\oplus \eta\). In differential geometry this is called an almost tangent structure of order r-1. Typical results say that \(M^{4k+3}\) with \(\tau =4k\xi^ 1\oplus \eta^ 3\) bounds and that there are manifolds of dimension \(2^ p(2q+1)-1\), \(p\geq 2\), \(q\geq 1\) which are indecomposable in unoriented cobordism for which \(\tau =(2^{p+1}q-1)\xi^ 1+\eta\).