Abstract
In this paper, we classify the congruence classes of \(F_{n(2)}^5\)-polyhedra, i.e., (n − 1)-connected, at most (n + 5)-dimensional polyhedra with 2-torsion free homology. The proof relies on the matrix problem technique which was developed in the classification of representations of algebras and applied to the homotopy theory by Baues and Drozd (1999).
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References
Baues H J, Drozd Y A. The homotopy classification of (n − 1)-connected (n + 4)-dimensional polyhedra with torsion free homology, n ⩾ 5. Expo Math, 1999, 17: 161–180
Baues H J, Drozd Y A. Indecomposable homotopy types with at most two non-trivial homology groups. In: Groups of Homotopy Self-Equivalences and Related Topics. Contemporary Mathematics, vol. 274, Providence: Amer Math Soc, 2001, 39–56
Baues H J, Drozd Y A. Classification of stable homotopy types with torsion-free homology. Topology, 2011, 40: 789–821
Baues H J, Hennes M. The homotopy classification of (n − 1)-connected (n+3)-dimensional polyhedra, n ⩾ 4. Topology, 1991, 30: 373–408
Bondarenko V M. Representations of bundles of semichained sets and their applications. St Petersburg Math J, 1991, 3: 38–61
Chang S C. Homology invariants and continuous mappings. Proc R Soc Lond Ser A Math Phys Eng Sci, 1950, 202: 253–263
Cohen J M. Stable Homotopy. Lecture Notes in Mathematics, vol. 165. Berlin-Heidelberg-New York: Springer-Verlag, 1970
Drozd Y A. Finitely generated quadratic modules. Manuscripta Math, 2001, 104: 239–256
Drozd Y A. Matrix problems and stable homotopy types of polyhedra. Cent Eur J Math, 2004, 2: 420–447
Drozd Y A. On classification of torsion free polyhedra. Https://www.imath.kiev.ua/~drozd/TFpoly.pdf, 2005
Drozd Y A. Matrix problems, triangulated categories and stable homotopy types. São Paulo J Math Sci, 2010, 4: 209–249
Pan J Z, Zhu Z J. The classification of 2 and 3 torsion free polyhedra. Acta Math Sin Engl Ser, 2015, 31: 1659–1682
Pan J Z, Zhu Z J. Stable homotopy classification of \(A_n^4\)-polyhedra with 2-torsion free homology. Sci China Math, 2016, 59: 1141–1162
Switzer R M. Algebraic Topology-Homology and Homotopy. Berlin: Springer-Verlag, 1975
Unsöld H M. \(A_n^4\)-polyhedra with free homology. Manuscripta Math, 1989, 65: 123–146
Acknowledgements
The first author was supported by National Natural Science Foundation of China (Grant No. 11701430). The second author was supported by National Natural Science Foundation of China (Grant Nos. 11661131004 and 11131008).
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Zhu, Z., Pan, J. Classification of the congruence classes of \(A_n^5\) (n ⩾ 6) with 2-torsion free homology. Sci. China Math. 63, 1409–1428 (2020). https://doi.org/10.1007/s11425-018-9413-y
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DOI: https://doi.org/10.1007/s11425-018-9413-y