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Benkhalifa, M.

On the homotopy types of 2-connected and 6-dimensional CW-complexes. (English) Zbl 1540.55012

Math. Notes 114, No. 5, 687-703 (2023).
This paper is concerned with the classification of \(2\)-connected \(6\)-dimensional CW-complexes. Under the assumption that \(H_3(X)\) is uniquely \(2\)-divisible (i.e., \(H_3(X)\otimes\mathbb{Z}_2=0\) and \(\mathrm{Tor}(H_3(X),\mathbb{Z}_2)\)=0), the author shows that the Whitehead exact sequence \[ H_6(X)\to \Gamma_5(X) \to \pi_5(X)\twoheadrightarrow H_5(X) \] provides a complete invariant for the homotopy classification of such complexes.
As an application, the author shows that there exist exactly three homotopy types among \(2\)-connected \(6\)-dimensional CW-complexes with \[ H_3(X)\cong \mathbb{Z}_n, H_4(X)\cong\mathbb{Z}_m, H_5(X)\cong\mathbb{Z}_p, H_6(X)\cong\mathbb{Z}, \] where \(n\) is odd and \(m,p\) are even.
Reviewer: Xueqi Wang (Beijing)

MSC:

55P15 Classification of homotopy type
55U99 Applied homological algebra and category theory in algebraic topology
55Qxx Homotopy groups

Keywords:

2-connected 6-dimensional CW-complex; homotopy types; Whitehead’s certain exact sequence; detecting functor
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References:

[1] Baues, H. J., Homotopy Type and Homology, 1996, Oxford: Clarendon Press, Oxford · Zbl 0857.55001 · doi:10.1093/oso/9780198514824.001.0001
[2] Félix, Y.; Halperin, S.; Thomas, J.-C., Rational Homotopy Theory, 2001, New York: Springer, New York · Zbl 0961.55002
[3] Whitehead, J. H. C., A certain exact sequence, Ann. Math., 52, 51-110, 1950 · Zbl 0037.26101 · doi:10.2307/1969511
[4] Benkhalifa, M., The effect of cell-attachment on the group of self-equivalences of an elliptic space, Mich. Math. J., 71, 3, 611-617, 2022 · Zbl 1497.55015
[5] Benkhalifa, M., On the group of self-homotopy equivalences of an elliptic space, Proc. Am. Math. Soc., 148, 6, 2695-2706, 2020 · Zbl 1437.55012 · doi:10.1090/proc/14900
[6] Benkhalifa, M., On the group of self-homotopy equivalences of \((n+1)\)-connected and \((3n+2)\)-dimensional CW-complex, Topology Appl., 233, 1-15, 2018 · Zbl 1378.55004 · doi:10.1016/j.topol.2017.10.018
[7] Benkhalifa, M., Postnikov decomposition and the group of self-equivalences of a rationalized space, Homology Homotopy Appl., 19, 1, 209-224, 2017 · Zbl 1380.55005 · doi:10.4310/HHA.2017.v19.n1.a11
[8] Benkhalifa, M., On the classification problem of the quasi-isomorphism classes of free chain algebras, J. Pure Appl. Algebra, 210, 2, 343-362, 2007 · Zbl 1122.55007 · doi:10.1016/j.jpaa.2006.09.012
[9] Benkhalifa, M., On the homotopy type of a chain algebra, Homology Homotopy Appl., 6, 109-135, 2004 · Zbl 1070.55010 · doi:10.4310/HHA.2004.v6.n1.a8
[10] Benkhalifa, M., Sur le type d’homotopy d’un CW-complexe, Homology Homotopy Appl., 5, 1, 101-120, 2003 · Zbl 1041.55004 · doi:10.4310/HHA.2003.v5.n1.a6
[11] MacLane, S., Homology, 1963, Berlin-Göttingen-Heidelberg: Springer, Berlin-Göttingen-Heidelberg · Zbl 0133.26502 · doi:10.1007/978-3-642-62029-4
[12] Baues, H. J.; Goerss, P., A homotopy operation spectral sequence for the computation of homotopy groups, Topology, 39, 1, 161-192, 2000 · Zbl 0958.55013 · doi:10.1016/S0040-9383(98)00065-2
[13] Miller, C., The second homology group of group; relations among commutators, Proc. Am. Math. Soc., 3, 588-595, 1952 · Zbl 0047.25703 · doi:10.1090/S0002-9939-1952-0049191-5
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