Let $Z$ be a smooth compact $(n+1)$-manifold. We study smooth embeddings and immersions $\beta: M \to Z$ of compact or closed $n$-manifolds $M$ such that the normal line bundle $\nu^\beta$ is trivialized. For a fixed $Z$, we introduce an equivalence relation between such $\beta$'s; it is a crossover between pseudo-isotopies and bordisms. We call this equivalence relation ``{ \sf quasitopy}". It comes in two flavors: $\mathsf{ IMM}(Z)$ and $\mathsf{ EMB}(Z)$, based on immersions and embeddings into $Z$, respectively. We prove that the natural map $\mathsf{ A}:\mathsf{ EMB}(Z) \to \mathsf{ IMM}(Z)$ is injective and admits a right inverse $\mathsf{ R}:\mathsf{ IMM}(Z) \to \mathsf{ EMB}(Z)$, induced by the resolution of self-intersections. As a result, we get a map $$\mathcal B\Sigma:\; \mathsf{ IMM}(Z) \big/ \mathsf{ A}(\mathsf{ EMB}(Z)) \longrightarrow \bigoplus_{ k \in [2, n+1]} \mathbf B_{ n+1-k}(Z)$$ whose target is a collection of smooth bordism groups of the space $Z$ and which differentiate between immersions and embeddings.
Categories: math.GT math.DG