On the homotopy type of (n-1)-connected (3n+1)-dimensional free chain Lie algebra

Abstract

Let R be a subring ring of Q. We reserve the symbol p for the least prime which is not a unit in R; if R ⊒Q, then p=∞. Denote by DGL ^np _n , n≥1, the category of (n-1)-connected np-dimensional differential graded free Lie algebras over R. In [1] D. Anick has shown that there is a reasonable concept of homotopy in the category DGL ^np _n . In this work we intend to answer the following two questions: Given an object (L(V), ϖ) in DGL ³ⁿ⁺² _n and denote by S(L(V), ϖ) the class of objects homotopy equivalent to (L(V), ϖ). How we can characterize a free dgl to belong to S(L(V), ϖ)? Fix an object (L(V), ϖ) in DGL ³ⁿ⁺² _n . How many homotopy equivalence classes of objects (L(W), δ) in DGL ³ⁿ⁺² _n such that H _* (W, d′)≊H _* (V, d) are there? Note that DGL ³ⁿ⁺² _n is a subcategory of DGL ^np _n when p>3. Our tool to address this problem is the exact sequence of Whitehead associated with a free dgl.