On immersions and embeddings with trivial normal line bundles

TL;DR

This paper introduces a new equivalence relation called quasitopy for immersions and embeddings with trivial normal line bundles, and establishes a relationship between these classes and bordism groups of the ambient manifold.

Contribution

It defines quasitopy for immersions and embeddings, proves the injectivity of the natural map between their classes, and constructs a map to bordism groups distinguishing immersions from embeddings.

Findings

The map from embeddings to immersions is injective.

A right inverse for this map is constructed via resolution of self-intersections.

A new invariant map to bordism groups differentiates immersions from embeddings.

Abstract

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.