On bordism invariance of an obstruction to topological embeddings (Q1356394)

In previous papers, the authors have defined an obstruction class \(\theta(f)\) for a map \(f:M \to N\) between smooth manifolds with \(k=\dim N- \dim M>0\) to be homotopic to an embedding. The class is defined using Stiefel-Whitney classes and Gysin homomorphisms. Here they consider two maps \(f,g:M\to N\) which are bordant and ask whether \(\theta(f)= \theta(g)\). This is shown to be true if \(H^{m-k} (M;Z_2)\) \((m= \dim M)\) consists entirely of characteristic classes. This unpleasant condition is used to force \(f^*=g^*\) on \(k\)-dimensional cohomology and is not really what one wishes.