On the homotopy type of (<Emphasis Type="Italic">n</Emphasis>-1)-connected (3<Emphasis Type="Italic">n</Emphasis>+1)-dimensional free chain Lie algebra |
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Authors: | Mahmoud Benkhalifa Nabilah Abughzalah |
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Institution: | (1) Department of Mathematics of College of Sciences, King Khalid University, P.O. Box 9004, Abha, Saudi Arabia;(2) Department of Mathematics, Girls College of Education, Abha, Saudi Arabia |
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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 n np , 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 n np . In this work we intend to answer the following two questions: Given an object (L(V), ?) in DGL n 3n+2 and denote by S(L(V), ?) the class of objects homotopy equivalent to (L(V), ?). How we can characterize a free dgl to belong to S(L(V), ?)? Fix an object (L(V), ?) in DGL n 3n+2 . How many homotopy equivalence classes of objects (L(W), δ) in DGL n 3n+2 such that H * (W, d′)?H * (V, d) are there? Note that DGL n 3n+2 is a subcategory of DGL n np when p>3. Our tool to address this problem is the exact sequence of Whitehead associated with a free dgl. |
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Keywords: | Anick model differential graded free Lie algebra whitehead exact sequence homotopy type |
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