Embeddings of homology equivalent manifolds with boundary

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).