On immersions and embeddings with trivial normal line bundles

2023/08/11 by Katz, Gabriel
#57R42 #57R90 #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Topology (math.GT)

paper · doi:10.48550/arxiv.2308.06150

Abstract

Let Z be a smooth compact (n+1)-manifold. We study smooth embeddings and immersions β: M → Z of compact or closed n-manifolds M such that the normal line bundle ν^β is trivialized. For a fixed Z, we introduce an equivalence relation between such β's; it is a crossover between pseudo-isotopies and bordisms. We call this equivalence relation ``\sf quasitopy". It comes in two flavors: IMM(Z) and EMB(Z), based on immersions and embeddings into Z, respectively. We prove that the natural map A:EMB(Z) → IMM(Z) is injective and admits a right inverse R:IMM(Z) → EMB(Z), induced by the resolution of self-intersections. As a result, we get a map \mathcal BΣ: IMM(Z) / A(EMB(Z)) \longrightarrow \bigoplus_{k ∈ [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.

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