On the classification problem of the quasi-isomorphism classes of free chain algebras. (English) Zbl 1122.55007
The paper under review is devoted to the study of some properties of the Whitehead exact sequence associated to a free differential graded algebra (free dga). More precisely, the author discusses the following questions:
1) Can one define a notion of homomorphism between two Whitehead exact sequences associated with two free dgas generalizing the notion given by Baues?
2) For which class of free dgas does the Whitehead exact sequence determine the quasi-isomorphism class?
3) Suppose the Whitehead exact sequences of two free dgas are isomorphic (in the sense of 1)). What conditions should we add to have these two dgas quasi-isomorphic?
The author gives answers to these questions by introducing new classes of free dgas (perfect and quasi-perfect) and a notion of strong isomorphism between Whitehead exact sequences.
1) Can one define a notion of homomorphism between two Whitehead exact sequences associated with two free dgas generalizing the notion given by Baues?
2) For which class of free dgas does the Whitehead exact sequence determine the quasi-isomorphism class?
3) Suppose the Whitehead exact sequences of two free dgas are isomorphic (in the sense of 1)). What conditions should we add to have these two dgas quasi-isomorphic?
The author gives answers to these questions by introducing new classes of free dgas (perfect and quasi-perfect) and a notion of strong isomorphism between Whitehead exact sequences.
Reviewer: Saïd Zarati (Tunis)
MSC:
| 55P15 | Classification of homotopy type |
| 55Q15 | Whitehead products and generalizations |
| 55U35 | Abstract and axiomatic homotopy theory in algebraic topology |
| 55U40 | Topological categories, foundations of homotopy theory |
References:
| [1] | Adams, J. F.; Hilton, P. J., On the chain algebra of a loop space, Comm. Math. Helv., 30, 305-330 (1955) · Zbl 0071.16403 |
| [2] | H.J. Baues, Algebraic homotopy, Cambridge Stud. Adv. Math. 15, 1989; H.J. Baues, Algebraic homotopy, Cambridge Stud. Adv. Math. 15, 1989 · Zbl 0688.55001 |
| [3] | Baues, H. J.; Lemaire, J. M., Minimal models in homotopy theory, Math. Ann., 225, 219-242 (1977) · Zbl 0322.55019 |
| [4] | Baues, H. J.; Hennes, M., The homotopy classification of \((n - 1)\)-connected \((n + 3)\)-dimensional polydra, \(n \leq 4\), Topology, 30, 3, 373-408 (1991) · Zbl 0735.55002 |
| [5] | Benkhalifa, M., On the homotopy type of a chain algebra, Homology Homotopy Appl., 6, 1, 109-137 (2004) · Zbl 1070.55010 |
| [6] | Mac lane, S., Homology (1967), Springer · Zbl 0818.18001 |
| [7] | J.C. Moore, Séminaire H. Cartan, 1954-1955, Exposé 3; J.C. Moore, Séminaire H. Cartan, 1954-1955, Exposé 3 |
| [8] | Whitehead, J. H.C., A certain exact sequence, Ann. of Math, 52, 51-110 (1950) · Zbl 0037.26101 |
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