On projected embeddings and desuspending the α-invariant

A map f :K→L is called a projected embedding from L×Bs if there is an embedding F :K→L×Bs such that f=π∘F, where π :L×Bs→L is the projection. A map f :Sp⊔Sq→Sm is a link map if fSp∩fSq=∅. We apply projected embeddings to desuspending the α-invariant of link maps and to embeddings of double covers into Euclidean space.