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Let $Z$ be a smooth compact $(n+1)$-manifold. We study smooth embeddings andimmersions $\beta: M \to Z$ of compact or closed $n$-manifolds $M$ such thatthe normal line bundle $\nu^\beta$ is trivialized. For a fixed $Z$, weintroduce an equivalence relation between such $\beta$'s; it is a crossoverbetween pseudo-isotopies and bordisms. We call this equivalence relation ``{\sfquasitopy}". It comes in two flavors: $\mathsf{IMM}(Z)$ and $\mathsf{EMB}(Z)$,based on immersions and embeddings into $Z$, respectively. We prove that thenatural map $\mathsf{A}:\mathsf{EMB}(Z) \to \mathsf{IMM}(Z)$ is injective andadmits 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 targetis a collection of smooth bordism groups of the space $Z$ and whichdifferentiate between immersions and embeddings.

On immersions and embeddings with trivial normal line bundles