Embeddings of homology equivalent manifolds with boundary

TL;DR

This paper establishes new conditions under which embeddings of homology equivalent manifolds with boundary are unique and explores the relationship between equivariant maps and embeddability into Euclidean spaces.

Contribution

It proves a theorem on equivariant maps that leads to new results on embeddings of manifolds with boundary and addresses a key question in topological embedding theory.

Findings

Bijective restriction-induced maps for certain manifolds and dimensions.

Existence of equivariant maps without embeddings for specific 3-manifolds.

Completes the classification of (m,n) pairs relating equivariant maps to embeddability.

Abstract

We prove a theorem on equivariant maps implying the following two corollaries: (1) Let N and M be compact orientable n-manifolds with boundaries such that M\subset N, the inclusion M\to N induces an isomorphism in integral cohomology, both M and N have (n-d-1)-dimensional spines and m > max {n+2, (3n+1-d)/2} . Then the restriction-induced map E^m(N)\to E^m(M) is bijective. Here E^m(X) is the set of embeddings X\to R^m up to isotopy (in the PL or smooth category). (2) For a 3-manifold N with boundary whose integral homology groups are trivial and such that N\not\cong D^3 (or for its special 2-spine N) there exists an equivariant map from the deleted product of N to S^2, although N does not embed into R^3. The second corollary completes the answer to the following question: for which pairs (m,n) for each n-polyhedron N the existence of an equivariant map from the deleted product of N to S^{m-1} implies the embeddability of N into R^m? An answer was known for each pair (m,n) except (3,3) and (3,2).