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• # Artykuł - szczegóły

## Open Mathematics

2005 | 3 | 1 | 58-75

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

EN

### Abstrakty

EN
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 nnp, 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 nnp. In this work we intend to answer the following two questions: Given an object (L(V), ϖ) in DGL n3n+2 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 n3n+2. How many homotopy equivalence classes of objects (L(W), δ) in DGL n3n+2 such that H * (W, d′)≊H * (V, d) are there? Note that DGL n3n+2 is a subcategory of DGL nnp when p>3. Our tool to address this problem is the exact sequence of Whitehead associated with a free dgl.

EN

58-75

wydano
2005-03-01
online
2005-03-01

### Twórcy

autor
• King Khalid University
autor
• Girls College of Education

### Bibliografia

• [1] D.J. Anick: “Hopf algebras up to homotopy”, J. Amer. Math. Soc., Vol. 2(3), (1989). pp. 417–452. http://dx.doi.org/10.2307/1990938
• [2] D.J. Anick: “An R-local Milnor-Moore Theorem”, Advances in Math, Vol. 77, (1989), pp. 116–136. http://dx.doi.org/10.1016/0001-8708(89)90016-9
• [3] H.J. Baues: Homotopy Type and Homology, Oxford Mathematical Monographs, Oxford University Press, Oxford, 1996.
• [4] H.J. Baues: “Algebraic homotopy”, Cambridge studies in advanced mathematics, Vol. 15, (1989).
• [5] M. Benkhalifa: Modèles algebriques et suites exactes de Whitehead, Thesis (PhD), Université de Nice France, 1995.
• [6] M. Benkhalifa: “Sur le type d’homotopie d’un CW-complexe”, Homology, Homotopy and Applications, Vol. 5(1), (2003), pp. 101–120.
• [7] M. Benkhalifa: “On the homotopy type of a chain algebra”, Homology, Homotopy and Applications, Vol. 6(1), (2004), pp. 109–135.
• [8] Y. Felix, S. Halperin and J.C. Thomas: “Rational homotopy theory”, C.M.T., Vol. 205, (2000).
• [9] S. MacLane: Homology, Springer, 1967.
• [10] J. Milnor and J.C. Moore: “On the structure of Hopf algebras”, Ann. Math., Vol. 81, (1965), pp. 211–264. http://dx.doi.org/10.2307/1970615
• [11] J.C. Moore: Séminaire H. Cartan, Exposé 3, 1954–1955.
• [12] J.H.C. Whitehead: “A certain exact sequence”, Ann. Math., Vol. 52, (1950), pp. 51–110. http://dx.doi.org/10.2307/1969511