Suspension Theorems for Links and Link Maps

We present a new short proof of the explicit formula for the group of links (and also link maps) in the ’quadruple point free’ dimension. Denote by Lp,q (respectively, C m−p p ) the group of smooth embeddings S p ⊔S → S (respectively, S → S) up to smooth isotopy. Denote by LM p,q the group of link maps S ⊔ S → S up to link homotopy. Theorem 1. If p ≤ q ≤ m− 3 and 2p+ 2q ≤ 3m− 6 then Lmp,q ∼= πp(S )⊕ πp+q+2−m(SO/SOm−p−1)⊕ C m−p p ⊕ C m−q q . Theorem 2. If p, q ≤ m−3 and 2p+2q ≤ 3m−5 then LM p,q ∼= π p+q+1−m. Our approach is based on the use of the suspension operation for links and link maps, and suspension theorems for them.