  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.
