Drozd, Yuriy A. Matrix problems and stable homotopy types of polyhedra. (English) Zbl 1062.55004 Cent. Eur. J. Math. 2, No. 3, 420-447 (2004). Summary: This is a survey of the results on stable homotopy types of polyhedra of small dimensions mainly obtained by H.-J. Baues and the author [Expo. Math. 17, 161–179 (1999; Zbl 0542.55010); Topology 40, 789–821 (2001; Zbl 0984.55006); Contemp. Math. 274, 39–56 (2001; Zbl 0979.55006)]. The proofs are based on the technique of matrix problems (bimodule categories). Cited in 4 Documents MSC: 55P15 Classification of homotopy type 15B36 Matrices of integers 16G60 Representation type (finite, tame, wild, etc.) of associative algebras 55P10 Homotopy equivalences in algebraic topology 55P42 Stable homotopy theory, spectra Keywords:polyhedra; homotopy type; matrix problems; tame and wild problems Citations:Zbl 0542.55010; Zbl 0984.55006; Zbl 0979.55006 PDFBibTeX XMLCite \textit{Y. A. Drozd}, Cent. Eur. J. Math. 2, No. 3, 420--447 (2004; Zbl 1062.55004) Full Text: DOI arXiv References: [1] H.J. Baues:Homotopy Type and Homology, Oxford University Press, 1996.; · Zbl 0857.55001 [2] H.J. Baues: “Atoms of Topology”,Jahresber. Dtsch. Math.-Ver., Vol. 104, (2002), pp. 147-164.; · Zbl 1013.55001 [3] H.J. Baues add Yu. A. Drozd: “The homotopy classification of (n−1)-connected (n+4)-dimensional polyhedra with torsion free homology”,Expo. Math., Vol. 17, (1999), pp. 161-179.; · Zbl 0942.55010 [4] H.J. Baues and Yu. A. Drozd: “Representation theory of homotopy types with at most two non-trivial homotopy groups”,Math. Proc. Cambridge Phil. Soc., Vol. 128, (2000), pp. 283-300. http://dx.doi.org/10.1017/S0305004199004168; · Zbl 0959.55006 [5] H.J. Baues and Yu. A. Drozd: “Indecomposable homotopy types with at most two non-trivial homology groups, in: Groups of Homotopy Self-Equivalences and Related Topics”,Contemporary Mathematics, Vol. 274, (2001), pp. 39-56.; · Zbl 0979.55006 [6] H.J. Baues and Yu. A. Drozd: “Classification of stable homotopy types with torsion-free homology”. Topology,Vol 40, (2001),pp. 789-821. http://dx.doi.org/10.1016/S0040-9383(99)00084-1; · Zbl 0984.55006 [7] H.J. Baues and Hennes: “The homotopy classification of (n−1)-connected (n+3)-dimensional polyhedra,n≥4”,Topology, Vol. 30, (1991), pp. 373-408. http://dx.doi.org/10.1016/0040-9383(91)90020-5; · Zbl 0735.55002 [8] V.M. Bondarenko: “Representations of bundles of semichained sets and their applications”,St. Petersburg Math. J., Vol. 3, (1992), pp. 973-996.; · Zbl 0791.06002 [9] S.C. Chang: “Homotopy invariants and continuous mappings”,Proc. R. Soc. London, Vol. 202, (1950), pp. 253-263. http://dx.doi.org/10.1098/rspa.1950.0098; · Zbl 0041.10201 [10] J.M. Cohen:Stable Homotopy, Lecture Notes in Math., Springer-Verlag, 1970.; · Zbl 0201.55703 [11] Yu.A. Drozd: “Matrix problems and categories of matrices”,Zapiski Nauch. Semin. LOMI, Vol. 28, (1972), pp. 144-153.; [12] Yu.A. Drozd: “Finitely generated quadratic modules”, Manus. Math,Vol 104, (2001),pp. 239-256. http://dx.doi.org/10.1007/s002290170041; · Zbl 1014.16017 [13] Yu.A. Drozd: “Reduction algorithm and representations of boxes and algebras”,Comptes Rendues Math. Acad. Sci. Canada, Vol. 23, (2001), pp. 97-125.; · Zbl 1031.16010 [14] P. Freyd: “Stable homotopy II. Applications of Categorical Algebra”,Proc. Symp. Pure Math., Vol. 17, (1970), pp. 161-191.; [15] I.M. Gelfand and V.A. Ponomarev: “Remarks on the classification of a pair of commuting linear transformations in a finite-dimensional space”,Funk. Anal. Prilozh., Vol. 3:4, (1969), pp. 81-82.; · Zbl 0204.45301 [16] S.I. Gelfand and Yu.I. Manin:Methods of Homological Algebra, Springer-Verlag, 1996.; · Zbl 0855.18001 [17] H.W. Henn: “Classification ofp-local low dimensiona; spectra”,J. Pure and Appl. Algebra, Vol. 45, (1987), pp. 45-71. http://dx.doi.org/10.1016/0022-4049(87)90083-1; [18] Hu Sze-Tsen:Homotopy Theory, Academic Press, 1959.; · Zbl 0088.38803 [19] E. Spanier:Algebraic Topology, McGraw-Hill, 1966.; · Zbl 0145.43303 [20] R.M. Switzer:Algebraic Topology—Homotopy and Homology, Springer-Verlag, 1975.; · Zbl 0305.55001 [21] H. Toda:Composition Methods in the Homotopy Groups of Spheres, Ann. Math. Studies, Vol. 49, Princeton, 1962.; · Zbl 0101.40703 [22] H.M. Unsöld: “A n 4-Polyhedra with free homology”,Manus. Math., Vol. 65, (1989), pp. 123-145. http://dx.doi.org/10.1007/BF01168295; · Zbl 0683.55002 [23] J.H.C. Whitehead: “The homotopy type of a special kind of polyhedron”,Ann. Soc. Polon. Math., Vol. 21, (1948), pp. 176-186.; · Zbl 0041.10103 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.