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 ⊂ N , the inclusion M → N induces an isomorphism in integral cohomology, both M and N have ( n − d − 1 ) -dimensional spines and m ⩾ max { n + 3 , 3 n + 2 − d 2 } . Then the restriction-induced map Emb m ( N ) → Emb m ( M ) is bijective. Here Emb m ( X ) is the set of embeddings X → 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 ≇ D 3 (or for its special 2-spine N) there exists an equivariant map N ˜ → 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 N ˜ → S m − 1 implies embeddability of N into R m ? An answer was known for each pair ( m , n ) except ( 3 , 3 ) and ( 3 , 2 ) .