On embeddings and acyclic maps

TL;DR

This paper investigates the relationship between embeddings of acyclic maps of closed manifolds and their implications for the homotopy types of embedding spaces, using Poincaré duality and surgery theory.

Contribution

It introduces a novel approach to relate embeddings of acyclic maps of manifolds to those of their Poincaré duality space variants and applies surgery theory to derive new results.

Findings

Established a connection between embeddings of acyclic maps and their Poincaré duality space counterparts.

Applied surgery theory to extend results from Poincaré spaces to manifolds.

Derived results on the homotopy type of block embedding spaces for smooth homology spheres.

Abstract

Given an acyclic map $X\to Y$ of closed manifolds dimension $d$, we study the relationship between the embeddings of $Y$ in $S^{n}$ with those of $X$ in $S^{n}$ when $n-d \ge 3$. The approach taken here is to first solve the Poincar\'e duality space variant of the problem. We then apply the surgery machine to obtain the corresponding results for manifolds. Thereafter, we focus on the case when $X$ is a smooth homology sphere and deduce results about the homotopy type of the space of block embeddings of $X$ in a sphere.