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

Open Mathematics (2005)

• Volume: 3, Issue: 1, page 58-75
• ISSN: 2391-5455

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## Abstract

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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.

## How to cite

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Mahmoud Benkhalifa, and Nabilah Abughzalah. "On the homotopy type of (n-1)-connected (3n+1)-dimensional free chain Lie algebra." Open Mathematics 3.1 (2005): 58-75. <http://eudml.org/doc/268709>.

@article{MahmoudBenkhalifa2005,
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 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.},
author = {Mahmoud Benkhalifa, Nabilah Abughzalah},
journal = {Open Mathematics},
keywords = {55Q15; 55U40},
language = {eng},
number = {1},
pages = {58-75},
title = {On the homotopy type of (n-1)-connected (3n+1)-dimensional free chain Lie algebra},
url = {http://eudml.org/doc/268709},
volume = {3},
year = {2005},
}

TY - JOUR
AU - Mahmoud Benkhalifa
AU - Nabilah Abughzalah
TI - On the homotopy type of (n-1)-connected (3n+1)-dimensional free chain Lie algebra
JO - Open Mathematics
PY - 2005
VL - 3
IS - 1
SP - 58
EP - 75
AB - 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.
LA - eng
KW - 55Q15; 55U40
UR - http://eudml.org/doc/268709
ER -

## References

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1. [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 Zbl0681.55006
2. [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 Zbl0684.55010
3. [3] H.J. Baues: Homotopy Type and Homology, Oxford Mathematical Monographs, Oxford University Press, Oxford, 1996.
4. [4] H.J. Baues: “Algebraic homotopy”, Cambridge studies in advanced mathematics, Vol. 15, (1989).
5. [5] M. Benkhalifa: Modèles algebriques et suites exactes de Whitehead, Thesis (PhD), Université de Nice France, 1995.
6. [6] M. Benkhalifa: “Sur le type d’homotopie d’un CW-complexe”, Homology, Homotopy and Applications, Vol. 5(1), (2003), pp. 101–120. Zbl1041.55004
7. [7] M. Benkhalifa: “On the homotopy type of a chain algebra”, Homology, Homotopy and Applications, Vol. 6(1), (2004), pp. 109–135. Zbl1070.55010
8. [8] Y. Felix, S. Halperin and J.C. Thomas: “Rational homotopy theory”, C.M.T., Vol. 205, (2000). Zbl0961.55002
9. [9] S. MacLane: Homology, Springer, 1967.
10. [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 Zbl0163.28202
11. [11] J.C. Moore: Séminaire H. Cartan, Exposé 3, 1954–1955.
12. [12] J.H.C. Whitehead: “A certain exact sequence”, Ann. Math., Vol. 52, (1950), pp. 51–110. http://dx.doi.org/10.2307/1969511 Zbl0037.26101

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